Verification in POMDPs

Transkrypt

1 Verification in OMDs rivacy, machine teaching and other belief-related problems Ufuk Topcu The University of Texas at Austin Slides originally prepared by Bo Wu. Ufuk Topcu

2 rotecting mission-critical information Ufuk Topcu

3 rotecting mission-critical information System model - discrete evolution - actuation uncertainties - secret states Ufuk Topcu

4 rotecting mission-critical information System model - discrete evolution - actuation uncertainties - secret states Intruder model - Knows the system model - Observes the actions - Observes the states partially - Wants to infer the secret states Ufuk Topcu

5 A formulation based on OMDs The system is modeled by a Markov decision process (MD) M = (Q,, A, T ) <latexit sha1_base64="8bes3ynijlhuqfc+ckvkdcvmjxo=">aaacahicbvdlssnafj3uv6vqasxbgalukgupapi1tcubfa6auaucbtattmgkze6gebwvny4ucetnunvnlrzaoubc4dz7uxee7yquaks69vira1vbg7ltws7u3v7b+bhuucgkcckjqmwij6hjgguk7aiipfekajyua63vqu9burega8jaahct1zjtecviawlgnjg+uhomwyq3jsazsek5dty6jgfqknqdcjxzgiibdyb+ocmarz7hcjmkzd+quxgscikgukktirjiaujulfu458it14/kacz7uyhkna6oikztxfezhypzz5nu5mz5xlxir+5/ujnaq5mevhpajhiwjieevwdqnokscymvmmiasql4v4gksccudwughyc+/veo61yp9wakr4r1whzhhpycm1acnrggdxagqanmejam3gfb8at8wk8gx+l1pyrzrydza+fwdnwjuz</latexit> q <latexit sha1_base64="rayxyqhzeh85qbmemfdudy4ns8=">aaacaxicbzc9sgnbfixvxr8y/6kwnonbsaq7imozslfmwxasotzyd1kyozsojmrhcwvpa+hj3y+iq+gy/hjnlcyw4mfjxzl9w5qsk4nq775rtw1jct4rbpz3dvfd8ufrs8epythksyhvj6aabzfynnwi7cqkarqibafj1nefksleszvzsrb6jdyuoqlfw46ffrrhvdy7yh7wckpcr3i9/9wyxsyouhgmqdddze+nnvbnobe5lvvrjqtmydrfrudiitz/nd5sm+smsbgr+6qhc/f3rkyjrsdryccjakz6ozuzk7ewqucbmrcgz/jmkknsra4iewfmtgz1uegxcezymkbmsxtjwgbuuwzsawvbdvechf/oxvr9dyq17is1k7ynopwaqdwdh5cqw3uoa5nyidwdc/w6jw5b86787eyltj5zjh8kf5a1fylwo=</latexit> Decision Maker (Controller) <latexit sha1_base64="vpwb4gmcckr1nizfbx6wlwnzzk=">aaacdnicbzc7sgnbfibxlumt6ilzwaqrmjucrumagejrdaxsnywozmbdjm9mdmrhcxvygmrdentr6cz+blojtsoul+gj4/3gzo/fgitt99wyw19y3oruf3ad3bygfhrvuleigtrajshy8qldwejuaa4gdwciniftb3yd5enliph4yoexoggdbhynzoqjfv4g4xngbmjy5t9cswujorzxbyqecurr/8xtelakw1exqpbqohws3pvjzjnba6iuky8rgdihdgyenulnp7oopotogirnc/uzob+3uhponqk8mxkqvilwazutkljbuq6cbav3jthsajxpdnl/atqxresm7igetkwkwmuca5+qrhiyop6bbkmnhwexigvq1qmnxnftapx6r91seezifc3dgeupwcw1oagmjl/akb9az9w59wj/zykv7xzd1lfvxmpnfc=</latexit> A : a finite set of actions Ufuk Topcu a <latexit sha1_base64="mjbtb4cqg5ya+nby/bscvzccg=">aaacbnicbzc9sgnbfixvxr8y/6kwnonbsaq7kdqyygmzfxasotzywwyznzmbkrhcw9pa+hj3y+ho+gy/hjnlcyw4mfjxzl9w5qskfqdf9cgpr6xubw8xts7u3v5b+fcozejum95ksyx1j6cgs6f4ewvk3kkp1egetsyx8/y9ixrstqhicj9ym6vciujkk1wndikzn+uejw3bnif/byqecurr/83rveli4qiapmv3tddqebbjj+weqnhcwvjourdi4pg3jz/nopobogisxtk8hmbu/nziagtojajszuryz5wxmrswsa6kuimgv34mvjiivxxqzhkgjgzduigqnogcmkbmi3sjwgbuuzuzkth1vuyv/kpvbfq3dyq9yu8pykcwcmcgwexuicbaeatgdzam7zaq/kvdnvzsditodko8fwr87nd885moo=</latexit> <latexit sha1_base64="h9y5illi1bbvoion+kzfkdt8c=">aaacaxicbzc9sgnbfixvxr8y/6kwnonbsaq7imozslfmwxasotzyd1kyozsmjmrhcwvpa+hj3y+iq+gy/hjnlckxwy+djnxrhzgkrwbvz3ylsbg5t7xr3s3v7b4dh5eotlo5txbdjyhgrtka1ci6xabgrekuigqa7g97o8/yrk81g+mkmcfkshkoecuwotbuxk7vnyusgpddbxlv++wf3ibmaytsmeg17npuyvymksozwgmplpmkbvtixytshqh9r5ovnyyzbcwnlnzrk7v7dygik9sqk7grezugvzznzbzzya13qtu145dcjqlbyryxhkkgjiazosiakrgtcxqprj9bgejqigztrssbcdb7mivwldvz616jetk7sbvqqhnca6x4met1oab6taebggv8apvzrz7nw4n4vrgpvnmi/ov+/giw+g==</latexit> <latexit : Q! [, 1] initial distribution <latexit sha1_base64="v6a/vzdqgd1b1xjgn3af5wfb4=">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</latexit> State <latexit sha1_base64="r6dqgmgs9wnksjet1vcfltohgh=">aaacbnicbvc7tsnaefyhvwivacxniqijkrjtaguqdwwqyenkroh8usrhzmfrbouwekpaeej6batv8e38bocexcqmtjqrzo7u4esrqgxfflkaytbxufbdlo7t7+wflw6owirlnejnfmtkdgbouhejnfch5j9achohk7wbyk/ntr66ninq9tmuh3skxfawilzqxbos9csvt+roqf4tlycvynhol797g4glivfijdwm67kx+invkjjks1ivmtymbejhvgupoie3fjq/dkbordigwjbukjm6u+nlibgtmatoyuxbzy8svxmylvuy3wegvnwovj8gvw1wwtctbigszkihqnkgcwkkzfvyjwszuu4yuzjnx1vo4j9p1aqew/xuapx6rz5teu7gfm7bguowyoakmhuazxudvexlenhfnyzfackdy/gd5/mhmuczjw==</latexit> Q : a finite set of states T : T (q, a, q ) = (qt = q qt Action <latexit sha1_base64="6wmdksz8p/kfvihjb+dizdbth4=">aaacdhicbzdntgixfiu7+if4h7phmcenmwkhlejyumzgfbcakuzrqgmb9o4jibycs7f6eo6mw9/bz/alldalbu7s5ms59ya3j9kcw/d9by+3tbzu5fflxwchhffe/wlalhrimvukztkqse1yyjnaqrkmnikkwdsa1+d5+4kzy5v8hilmyukgksecenbwr1xxeowsgpnrfrhkv/yf8doegzrqpka/+nmbkjomtaivxnpu4gsip8qap4lncr3umk3omaxz16ekcbhdhhzdf85z4bjzdytgbfu34psaydjjgbtaim7gonzdmlmbgm4k8v45vknwcrdxhcnaoc8bwgbtgquwcegq4+wsmiiibddfwbutrhaxdq1qjfarwuo1vlvjesqjc3sjyihat6ig7lednrffgrgv/tmxvv3of3urznednoofon7+sxgvgbeg==</latexit> 1, at 1 ) 3

6 A formulation based on OMDs The system is modeled by a Markov decision process (MD) M = (Q,, A, T ) <latexit sha1_base64="8bes3ynijlhuqfc+ckvkdcvmjxo=">aaacahicbvdlssnafj3uv6vqasxbgalukgupapi1tcubfa6auaucbtattmgkze6gebwvny4ucetnunvnlrzaoubc4dz7uxee7yquaks69vira1vbg7ltws7u3v7b+bhuucgkcckjqmwij6hjgguk7aiipfekajyua63vqu9burega8jaahct1zjtecviawlgnjg+uhomwyq3jsazsek5dty6jgfqknqdcjxzgiibdyb+ocmarz7hcjmkzd+quxgscikgukktirjiaujulfu458it14/kacz7uyhkna6oikztxfezhypzz5nu5mz5xlxir+5/ujnaq5mevhpajhiwjieevwdqnokscymvmmiasql4v4gksccudwughyc+/veo61yp9wakr4r1whzhhpycm1acnrggdxagqanmejam3gfb8at8wk8gx+l1pyrzrydza+fwdnwjuz</latexit> Action State a q <latexit sha1_base64="mjbtb4cqg5ya+nby/bscvzccg=">aaacbnicbzc9sgnbfixvxr8y/6kwnonbsaq7kdqyygmzfxasotzywwyznzmbkrhcw9pa+hj3y+ho+gy/hjnlcyw4mfjxzl9w5qskfqdf9cgpr6xubw8xts7u3v5b+fcozejum95ksyx1j6cgs6f4ewvk3kkp1egetsyx8/y9ixrstqhicj9ym6vciujkk1wndikzn+uejw3bnif/byqecurr/83rveli4qiapmv3tddqebbjj+weqnhcwvjourdi4pg3jz/nopobogisxtk8hmbu/nziagtojajszuryz5wxmrswsa6kuimgv34mvjiivxxqzhkgjgzduigqnogcmkbmi3sjwgbuuzuzkth1vuyv/kpvbfq3dyq9yu8pykcwcmcgwexuicbaeatgdzam7zaq/kvdnvzsditodko8fwr87nd885moo=</latexit> <latexit sha1_base64="r6dqgmgs9wnksjet1vcfltohgh=">aaacbnicbvc7tsnaefyhvwivacxniqijkrjtaguqdwwqyenkroh8usrhzmfrbouwekpaeej6batv8e38bocexcqmtjqrzo7u4esrqgxfflkaytbxufbdlo7t7+wflw6owirlnejnfmtkdgbouhejnfch5j9achohk7wbyk/ntr66ninq9tmuh3skxfawilzqxbos9csvt+roqf4tlycvynhol797g4glivfijdwm67kx+invkjjks1ivmtymbejhvgupoie3fjq/dkbordigwjbukjm6u+nlibgtmatoyuxbzy8svxmylvuy3wegvnwovj8gvw1wwtctbigszkihqnkgcwkkzfvyjwszuu4yuzjnx1vo4j9p1aqew/xuapx6rz5teu7gfm7bguowyoakmhuazxudvexlenhfnyzfackdy/gd5/mhmuczjw==</latexit> Q : a finite set of states <latexit sha1_base64="h9y5illi1bbvoion+kzfkdt8c=">aaacaxicbzc9sgnbfixvxr8y/6kwnonbsaq7imozslfmwxasotzyd1kyozsmjmrhcwvpa+hj3y+iq+gy/hjnlckxwy+djnxrhzgkrwbvz3ylsbg5t7xr3s3v7b4dh5eotlo5txbdjyhgrtka1ci6xabgrekuigqa7g97o8/yrk81g+mkmcfkshkoecuwotbuxk7vnyusgpddbxlv++wf3ibmaytsmeg17npuyvymksozwgmplpmkbvtixytshqh9r5ovnyyzbcwnlnzrk7v7dygik9sqk7grezugvzznzbzzya13qtu145dcjqlbyryxhkkgjiazosiakrgtcxqprj9bgejqigztrssbcdb7mivwldvz616jetk7sbvqqhnca6x4met1oab6taebggv8apvzrz7nw4n4vrgpvnmi/ov+/giw+g==</latexit> <latexit <latexit sha1_base64="rayxyqhzeh85qbmemfdudy4ns8=">aaacaxicbzc9sgnbfixvxr8y/6kwnonbsaq7imozslfmwxasotzyd1kyozsojmrhcwvpa+hj3y+iq+gy/hjnlcyw4mfjxzl9w5qsk4nq775rtw1jct4rbpz3dvfd8ufrs8epythksyhvj6aabzfynnwi7cqkarqibafj1nefksleszvzsrb6jdyuoqlfw46ffrrhvdy7yh7wckpcr3i9/9wyxsyouhgmqdddze+nnvbnobe5lvvrjqtmydrfrudiitz/nd5sm+smsbgr+6qhc/f3rkyjrsdryccjakz6ozuzk7ewqucbmrcgz/jmkknsra4iewfmtgz1uegxcezymkbmsxtjwgbuuwzsawvbdvechf/oxvr9dyq17is1k7ynopwaqdwdh5cqw3uoa5nyidwdc/w6jw5b86787eyltj5zjh8kf5a1fylwo=</latexit> : Q! [, 1] initial distribution Decision Maker (Controller) <latexit sha1_base64="vpwb4gmcckr1nizfbx6wlwnzzk=">aaacdnicbzc7sgnbfibxlumt6ilzwaqrmjucrumagejrdaxsnywozmbdjm9mdmrhcxvygmrdentr6cz+blojtsoul+gj4/3gzo/fgitt99wyw19y3oruf3ad3bygfhrvuleigtrajshy8qldwejuaa4gdwciniftb3yd5enliph4yoexoggdbhynzoqjfv4g4xngbmjy5t9cswujorzxbyqecurr/8xtelakw1exqpbqohws3pvjzjnba6iuky8rgdihdgyenulnp7oopotogirnc/uzob+3uhponqk8mxkqvilwazutkljbuq6cbav3jthsajxpdnl/atqxresm7igetkwkwmuca5+qrhiyop6bbkmnhwexigvq1qmnxnftapx6r91seezifc3dgeupwcw1oagmjl/akb9az9w59wj/zykv7xzd1lfvxmpnfc=</latexit> A : a finite set of actions <latexit sha1_base64="v6a/vzdqgd1b1xjgn3af5wfb4=">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</latexit> T : T (q, a, q ) = (qt = q qt <latexit sha1_base64="6wmdksz8p/kfvihjb+dizdbth4=">aaacdhicbzdntgixfiu7+if4h7phmcenmwkhlejyumzgfbcakuzrqgmb9o4jibycs7f6eo6mw9/bz/alldalbu7s5ms59ya3j9kcw/d9by+3tbzu5fflxwchhffe/wlalhrimvukztkqse1yyjnaqrkmnikkwdsa1+d5+4kzy5v8hilmyukgksecenbwr1xxeowsgpnrfrhkv/yf8doegzrqpka/+nmbkjomtaivxnpu4gsip8qap4lncr3umk3omaxz16ekcbhdhhzdf85z4bjzdytgbfu34psaydjjgbtaim7gonzdmlmbgm4k8v45vknwcrdxhcnaoc8bwgbtgquwcegq4+wsmiiibddfwbutrhaxdq1qjfarwuo1vlvjesqjc3sjyihat6ig7lednrffgrgv/tmxvv3of3urznednoofon7+sxgvgbeg==</latexit> 1, at 1 ) Noisy Sensor <latexit sha1_base64="iideejbtov97s6unmqd1urgg=">aaacdhicbzdnsgmxfiuz/tb6v3xpjlgevwmc3vzconkktofaiesse+ozkkjbmhdhfl71idyjw9/bz/altntzansdgy9z7owbeynojx9b9tfwnza7uwu9zdz84lbdn41mnyuglvzqdkqmccagyznlfyasbjxaewjmneegjtmbsdqwgtmhasjhryp3vvzmjedccn1r1tk/5mebmchmoov71x+unjutejzyykwn8junm6itoxwmxw5qqbe6igoobqkarnms5sn+nw5frxl7z6weob+3chiysw4idxkquzqlgztcwmnluq6kqvg4zjlrqqdd5bxhkszv4gzumw3u8redqjvzn8bsdsh1vvxdoei1sq7nacfxkcf8t1y7zngrofjhcxsgk1rdt6iogogihv7qk3rznr13787ni+uefnocfon7+sxwabsq==</latexit> Observation Action a <latexit sha1_base64="ni3ck5qthsmhuhzl/i1oifjzy=">aaacc3icbzdnsgmxfixv+fvrx9wlmarxjwzltrlwy7k9gfma4lkbaewyjjlcgfoilt3qq7gttz6ez+blme5nowbd7ourdutphwpo3rfjsbm1vbo7ulvfl+wehrcexktk1lqghtecml6ozyu84ebrlmoomiui45lqtju/mewdclwzsjlpqomydwwlgmhgwv5dqkma5nyvvnamwutgldafqo1+5wf3kcsnkbcei619j3mugglwge1m5lqaydlgq+pbfdimosjykfojodfellnzaod/9uzdjwehqhdjlgzqsxs7m5nkusts7wuxdbhktswqoiislopqji9g8gdrgihldpxywucx+apervpgyw1/ztumtd7ek7xrncvey73auc56kse5xmavehaddbihjrsagiqxeiu359l5dz6cz8xohlsnme/ov+/f3mbhw==</latexit> <latexit sha1_base64="h9y5illi1bbvoion+kzfkdt8c=">aaacaxicbzc9sgnbfixvxr8y/6kwnonbsaq7imozslfmwxasotzyd1kyozsmjmrhcwvpa+hj3y+iq+gy/hjnlckxwy+djnxrhzgkrwbvz3ylsbg5t7xr3s3v7b4dh5eotlo5txbdjyhgrtka1ci6xabgrekuigqa7g97o8/yrk81g+mkmcfkshkoecuwotbuxk7vnyusgpddbxlv++wf3ibmaytsmeg17npuyvymksozwgmplpmkbvtixytshqh9r5ovnyyzbcwnlnzrk7v7dygik9sqk7grezugvzznzbzzya13qtu145dcjqlbyryxhkkgjiazosiakrgtcxqprj9bgejqigztrssbcdb7mivwldvz616jetk7sbvqqhnca6x4met1oab6taebggv8apvzrz7nw4n4vrgpvnmi/ov+/giw+g==</latexit> <latexit <latexit sha1_base64="r6dqgmgs9wnksjet1vcfltohgh=">aaacbnicbvc7tsnaefyhvwivacxniqijkrjtaguqdwwqyenkroh8usrhzmfrbouwekpaeej6batv8e38bocexcqmtjqrzo7u4esrqgxfflkaytbxufbdlo7t7+wflw6owirlnejnfmtkdgbouhejnfch5j9achohk7wbyk/ntr66ninq9tmuh3skxfawilzqxbos9csvt+roqf4tlycvynhol797g4glivfijdwm67kx+invkjjks1ivmtymbejhvgupoie3fjq/dkbordigwjbukjm6u+nlibgtmatoyuxbzy8svxmylvuy3wegvnwovj8gvw1wwtctbigszkihqnkgcwkkzfvyjwszuu4yuzjnx1vo4j9p1aqew/xuapx6rz5teu7gfm7bguowyoakmhuazxudvexlenhfnyzfackdy/gd5/mhmuczjw==</latexit> z <latexit sha1_base64="dwk1vkanlbct3la/uyovdtawnuu=">aaab6hicbvdlsgnbeoynrxhfuy9ebogkexgqbfvhhmwdwgwclspdczmzu7zmwkmeqlvhhqxkuf5m/czlsqrmlgoqqbrq7gkrwbvz38mtrw9sbuw3czu7e/shxcojpo5txbdbyhgrdka1ci6xybgre4uigqapgtzo/9yhk81jem3gcfkqhkoecuwol+lovwhll7hxklxgzkuggwq/41e3hli1qgiao1h3tyw/ocpwjnba6kyae8pgdiadsywnuut+aftcmavgljzusamld/txopu4cmxnrm1ql3sz8t+vk5rwp9wmaqgjvssclnbtexmx5m+v8imgftcmelvskgvffmbdyfg4k3/iqavbk3kw5ur8svw+yojwaqdwdh5cqrxuoaynyidwdk/w5jw4l86787foztnzzdh8gf5a+fajk=</latexit> Intruder: partially observe (O) the system Belief Update <latexit sha1_base64="anur1anuq89qzmk/ckzmoxfmza=">aaab9hicbvc7tgjbfj3ff+iltbszseysyc4wwhjtldfxgqqzhblkyyfthzl4rs+a4bc4x9ws/bshelbkxycs49uxeonqhba/rdlg5tbtnm3srd/chhut5p6yrthfyeyer1fazbihhcfcihmypgks+h44/v5n5nakqljh7eaqpexiaxcavnactv1gqhpg4amirbtwbx7qxoonekuimfwoqvz9iebzbjfwyrxuonakxm4wcs5hv+pmglexgljhfol18cfsmxhgloggiziurlttfizxfwk8j3xgded61zul/3m9dmmblxdxmiheflkozctfhm4boifqwffodwfccxmr5somgeftu8wu4kx+ezg3xnqt54ansatlfhmzyrc3jjhhjnmusetihlohkiz+svvfkt68v6tz6woywryjys7a+fwbfopg3</latexit> = (Q,, A, T, Z, O) (Belief) b <latexit sha1_base64="xmfnnadevmr1hwpzhgw5igkqvc=">aaacaxicbzc9sgnbfixvxr8y/6kwnonbsaq7imozslfmwxasotzyd1kyozsmjmrhcwvpa+hj3y+iq+gy/hjnlckxwy+djnxrhzgkrwbvz3ylsbg5t7xr3s3v7b4dh5eotlo5txbdjyhgrtka1ci6xabgrekuigqa7g97o8/yrk81g+mkmcfkshkoecuwotrtavv9yqoxdzbs+hcusq98s/vuhmgilyyjq3fxcxgzvyyzgdnsl9wyudamq+xaldrc7wfzq6fkwjodesbkmni37kdfi6kummimpfezmbmiyx1rqgm5rwzs+4tfkdkiucfnbtexmdzabv8immfigthh7ccjgvffmbgkl4633muqtk6qnlv1gtevk3euxho4bwuwynbqmed1kejdbbe4bxengfn3flwhejbsffoyv/cr5+at+llvs=</latexit> <latexit <latexit sha1_base64="jjwlmfqjw+olwwi6zgj3l65rckm=">aaacbnicbvdlssnafj3uv6vqesrbotqizskcnyjvny4s4w+sallmpqyczizccv58vfcufderd/gzr9xmahrqcuhm65l3vvcsnghttnby3srqvphflgxt7+zu6fshbrhghjmwdlniuy4shngatcsvjhqjtpdvmtjxxzep33kgxnawampjrbwfdqquyykkvr6sejockijfupviklhmfh1lgmsa9cxfw14tmzwblhmriwqod7xv+xbigofbbizjetmipjihlihmzfuxykajhmrqsnqib8olwktkbu3iqlahqq4qkhcm/p5ikc/exhdvz3qwrs8t+vfuv6iqigjjajxf5mumyhcmmcab5qrlnleeyu7vrrceedyquqkkgrr8evlq6urfnypxfrrfwzolgcjyaerdajaibw1ahlydbi3ggr+bne9jethfty96a7kzq/ahucxgkwsw==</latexit> <latexit sha1_base64="uohbydgybd8w/khqreym1ncbbs=">aaaccxicbvdltgjbeozff+il9ehlijhbc9nlob6jxjxiiqcbdzkdemhczo5mztaeel7ao1f9cg/gq1/hn/gtdrahbsrppflvlxrxkaiujet+o7m19y3nrfxywd3b/+gehju1hgqgdzylgl1efcngkfymnwifeguuhkibaxdm6nfekklerzdm1gcvqt9iiecuwolx/k1twj4trrfkltxzydlxmtictluu8wfti9mqctimeg1bntuyvwxvyyzgzncj9wyudakfwxbglgjh/dp6qm6vsbgro5ehm/vvykyl1imze1jzuavelnxpzdyazxrtk145y95lkqgiza/iewfmtgz1kj6xcezymqjzyrbjwgbuewzseuvbdveyhflpfmteg7fu6uwahdzt3k4gvmogwexuinbqemdgeh4gvd4c56dd+fd+zyv5pwscwz/4hz9ajd+mz8=</latexit> M = (Q,, A, T ) <latexit sha1_base64="8bes3ynijlhuqfc+ckvkdcvmjxo=">aaacahicbvdlssnafj3uv6vqasxbgalukgupapi1tcubfa6auaucbtattmgkze6gebwvny4ucetnunvnlrzaoubc4dz7uxee7yquaks69vira1vbg7ltws7u3v7b+bhuucgkcckjqmwij6hjgguk7aiipfekajyua63vqu9burega8jaahct1zjtecviawlgnjg+uhomwyq3jsazsek5dty6jgfqknqdcjxzgiibdyb+ocmarz7hcjmkzd+quxgscikgukktirjiaujulfu458it14/kacz7uyhkna6oikztxfezhypzz5nu5mz5xlxir+5/ujnaq5mevhpajhiwjieevwdqnokscymvmmiasql4v4gksccudwughyc+/veo61yp9wakr4r1whzhhpycm1acnrggdxagqanmejam3gfb8at8wk8gx+l1pyrzrydza+fwdnwjuz</latexit> Z : a finite set of observations O : observation function where O(q, a, z) = (zt = z qt = q, at <latexit sha1_base64="yhhz4q7fskply79qqghqqbulli=">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</latexit> Ufuk Topcu 1 = a) 3

7 A formulation based on OMDs The system is modeled by a Markov decision process (MD) M = (Q,, A, T ) <latexit sha1_base64="8bes3ynijlhuqfc+ckvkdcvmjxo=">aaacahicbvdlssnafj3uv6vqasxbgalukgupapi1tcubfa6auaucbtattmgkze6gebwvny4ucetnunvnlrzaoubc4dz7uxee7yquaks69vira1vbg7ltws7u3v7b+bhuucgkcckjqmwij6hjgguk7aiipfekajyua63vqu9burega8jaahct1zjtecviawlgnjg+uhomwyq3jsazsek5dty6jgfqknqdcjxzgiibdyb+ocmarz7hcjmkzd+quxgscikgukktirjiaujulfu458it14/kacz7uyhkna6oikztxfezhypzz5nu5mz5xlxir+5/ujnaq5mevhpajhiwjieevwdqnokscymvmmiasql4v4gksccudwughyc+/veo61yp9wakr4r1whzhhpycm1acnrggdxagqanmejam3gfb8at8wk8gx+l1pyrzrydza+fwdnwjuz</latexit> Action State a q <latexit sha1_base64="mjbtb4cqg5ya+nby/bscvzccg=">aaacbnicbzc9sgnbfixvxr8y/6kwnonbsaq7kdqyygmzfxasotzywwyznzmbkrhcw9pa+hj3y+ho+gy/hjnlcyw4mfjxzl9w5qskfqdf9cgpr6xubw8xts7u3v5b+fcozejum95ksyx1j6cgs6f4ewvk3kkp1egetsyx8/y9ixrstqhicj9ym6vciujkk1wndikzn+uejw3bnif/byqecurr/83rveli4qiapmv3tddqebbjj+weqnhcwvjourdi4pg3jz/nopobogisxtk8hmbu/nziagtojajszuryz5wxmrswsa6kuimgv34mvjiivxxqzhkgjgzduigqnogcmkbmi3sjwgbuuzuzkth1vuyv/kpvbfq3dyq9yu8pykcwcmcgwexuicbaeatgdzam7zaq/kvdnvzsditodko8fwr87nd885moo=</latexit> <latexit sha1_base64="r6dqgmgs9wnksjet1vcfltohgh=">aaacbnicbvc7tsnaefyhvwivacxniqijkrjtaguqdwwqyenkroh8usrhzmfrbouwekpaeej6batv8e38bocexcqmtjqrzo7u4esrqgxfflkaytbxufbdlo7t7+wflw6owirlnejnfmtkdgbouhejnfch5j9achohk7wbyk/ntr66ninq9tmuh3skxfawilzqxbos9csvt+roqf4tlycvynhol797g4glivfijdwm67kx+invkjjks1ivmtymbejhvgupoie3fjq/dkbordigwjbukjm6u+nlibgtmatoyuxbzy8svxmylvuy3wegvnwovj8gvw1wwtctbigszkihqnkgcwkkzfvyjwszuu4yuzjnx1vo4j9p1aqew/xuapx6rz5teu7gfm7bguowyoakmhuazxudvexlenhfnyzfackdy/gd5/mhmuczjw==</latexit> Q : a finite set of states <latexit sha1_base64="h9y5illi1bbvoion+kzfkdt8c=">aaacaxicbzc9sgnbfixvxr8y/6kwnonbsaq7imozslfmwxasotzyd1kyozsmjmrhcwvpa+hj3y+iq+gy/hjnlckxwy+djnxrhzgkrwbvz3ylsbg5t7xr3s3v7b4dh5eotlo5txbdjyhgrtka1ci6xabgrekuigqa7g97o8/yrk81g+mkmcfkshkoecuwotbuxk7vnyusgpddbxlv++wf3ibmaytsmeg17npuyvymksozwgmplpmkbvtixytshqh9r5ovnyyzbcwnlnzrk7v7dygik9sqk7grezugvzznzbzzya13qtu145dcjqlbyryxhkkgjiazosiakrgtcxqprj9bgejqigztrssbcdb7mivwldvz616jetk7sbvqqhnca6x4met1oab6taebggv8apvzrz7nw4n4vrgpvnmi/ov+/giw+g==</latexit> <latexit <latexit sha1_base64="rayxyqhzeh85qbmemfdudy4ns8=">aaacaxicbzc9sgnbfixvxr8y/6kwnonbsaq7imozslfmwxasotzyd1kyozsojmrhcwvpa+hj3y+iq+gy/hjnlcyw4mfjxzl9w5qsk4nq775rtw1jct4rbpz3dvfd8ufrs8epythksyhvj6aabzfynnwi7cqkarqibafj1nefksleszvzsrb6jdyuoqlfw46ffrrhvdy7yh7wckpcr3i9/9wyxsyouhgmqdddze+nnvbnobe5lvvrjqtmydrfrudiitz/nd5sm+smsbgr+6qhc/f3rkyjrsdryccjakz6ozuzk7ewqucbmrcgz/jmkknsra4iewfmtgz1uegxcezymkbmsxtjwgbuuwzsawvbdvechf/oxvr9dyq17is1k7ynopwaqdwdh5cqw3uoa5nyidwdc/w6jw5b86787eyltj5zjh8kf5a1fylwo=</latexit> : Q! [, 1] initial distribution Decision Maker (Controller) <latexit sha1_base64="vpwb4gmcckr1nizfbx6wlwnzzk=">aaacdnicbzc7sgnbfibxlumt6ilzwaqrmjucrumagejrdaxsnywozmbdjm9mdmrhcxvygmrdentr6cz+blojtsoul+gj4/3gzo/fgitt99wyw19y3oruf3ad3bygfhrvuleigtrajshy8qldwejuaa4gdwciniftb3yd5enliph4yoexoggdbhynzoqjfv4g4xngbmjy5t9cswujorzxbyqecurr/8xtelakw1exqpbqohws3pvjzjnba6iuky8rgdihdgyenulnp7oopotogirnc/uzob+3uhponqk8mxkqvilwazutkljbuq6cbav3jthsajxpdnl/atqxresm7igetkwkwmuca5+qrhiyop6bbkmnhwexigvq1qmnxnftapx6r91seezifc3dgeupwcw1oagmjl/akb9az9w59wj/zykv7xzd1lfvxmpnfc=</latexit> A : a finite set of actions <latexit sha1_base64="v6a/vzdqgd1b1xjgn3af5wfb4=">aaacj3icdvfnb9naefbrzyumskxlxurasqfya5iruklgrjalzgstljswenoll1/bu7bilxf4cfxi1/wzr1avoyabvb+bn7l4jmykos4vy37w8nhjjzu7rad7z57vtw9exoiuizwsptdrwc5likfi4cjb/kfic4lwyv5u5y+/cqvfmsxwk3e/hluiiseadrwfy7i+oh/44lbrqjrccnminni6orkovk81pejrqjqafeaogvqnlpy+6ze6wiraqv5rhjhuxeuvr+7o/9blbxbfvuqnzl+9dz86p/8gd86obcjxjvv1wdybwxh7y4zclzb7wo3ar3sxcro//swkstinictoxcdtls1aomorvyysz4bdw4ovdewg5tovt35w9lvhljrkltkji37p6kewotnhjrkghct7+zq8l+5ryhrqv+kjcuqj+xufriiimtlmcvjyh3bgatblxgwvrumyus8kwmcg9++x74gi4cn8ohtn3nffpy8coossvsi+45ismywcyixcrd1ral3zuwd+8t+yi9vsrbwkf4x95teq48q+</latexit> T : T (q, a, q ) = (qt = q qt <latexit sha1_base64="6wmdksz8p/kfvihjb+dizdbth4=">aaacdhicbzdntgixfiu7+if4h7phmcenmwkhlejyumzgfbcakuzrqgmb9o4jibycs7f6eo6mw9/bz/alldalbu7s5ms59ya3j9kcw/d9by+3tbzu5fflxwchhffe/wlalhrimvukztkqse1yyjnaqrkmnikkwdsa1+d5+4kzy5v8hilmyukgksecenbwr1xxeowsgpnrfrhkv/yf8doegzrqpka/+nmbkjomtaivxnpu4gsip8qap4lncr3umk3omaxz16ekcbhdhhzdf85z4bjzdytgbfu34psaydjjgbtaim7gonzdmlmbgm4k8v45vknwcrdxhcnaoc8bwgbtgquwcegq4+wsmiiibddfwbutrhaxdq1qjfarwuo1vlvjesqjc3sjyihat6ig7lednrffgrgv/tmxvv3of3urznednoofon7+sxgvgbeg==</latexit> 1, at 1 ) Noisy Sensor <latexit sha1_base64="iideejbtov97s6unmqd1urgg=">aaacdhicbzdnsgmxfiuz/tb6v3xpjlgevwmc3vzconkktofaiesse+ozkkjbmhdhfl71idyjw9/bz/altntzansdgy9z7owbeynojx9b9tfwnza7uwu9zdz84lbdn41mnyuglvzqdkqmccagyznlfyasbjxaewjmneegjtmbsdqwgtmhasjhryp3vvzmjedccn1r1tk/5mebmchmoov71x+unjutejzyykwn8junm6itoxwmxw5qqbe6igoobqkarnms5sn+nw5frxl7z6weob+3chiysw4idxkquzqlgztcwmnluq6kqvg4zjlrqqdd5bxhkszv4gzumw3u8redqjvzn8bsdsh1vvxdoei1sq7nacfxkcf8t1y7zngrofjhcxsgk1rdt6iogogihv7qk3rznr13787ni+uefnocfon7+sxwabsq==</latexit> Observation Action a <latexit sha1_base64="ni3ck5qthsmhuhzl/i1oifjzy=">aaacc3icbzdnsgmxfixv+fvrx9wlmarxjwzltrlwy7k9gfma4lkbaewyjjlcgfoilt3qq7gttz6ez+blme5nowbd7ourdutphwpo3rfjsbm1vbo7ulvfl+wehrcexktk1lqghtecml6ozyu84ebrlmoomiui45lqtju/mewdclwzsjlpqomydwwlgmhgwv5dqkma5nyvvnamwutgldafqo1+5wf3kcsnkbcei619j3mugglwge1m5lqaydlgq+pbfdimosjykfojodfellnzaod/9uzdjwehqhdjlgzqsxs7m5nkusts7wuxdbhktswqoiislopqji9g8gdrgihldpxywucx+apervpgyw1/ztumtd7ek7xrncvey73auc56kse5xmavehaddbihjrsagiqxeiu359l5dz6cz8xohlsnme/ov+/f3mbhw==</latexit> <latexit sha1_base64="h9y5illi1bbvoion+kzfkdt8c=">aaacaxicbzc9sgnbfixvxr8y/6kwnonbsaq7imozslfmwxasotzyd1kyozsmjmrhcwvpa+hj3y+iq+gy/hjnlckxwy+djnxrhzgkrwbvz3ylsbg5t7xr3s3v7b4dh5eotlo5txbdjyhgrtka1ci6xabgrekuigqa7g97o8/yrk81g+mkmcfkshkoecuwotbuxk7vnyusgpddbxlv++wf3ibmaytsmeg17npuyvymksozwgmplpmkbvtixytshqh9r5ovnyyzbcwnlnzrk7v7dygik9sqk7grezugvzznzbzzya13qtu145dcjqlbyryxhkkgjiazosiakrgtcxqprj9bgejqigztrssbcdb7mivwldvz616jetk7sbvqqhnca6x4met1oab6taebggv8apvzrz7nw4n4vrgpvnmi/ov+/giw+g==</latexit> <latexit <latexit sha1_base64="r6dqgmgs9wnksjet1vcfltohgh=">aaacbnicbvc7tsnaefyhvwivacxniqijkrjtaguqdwwqyenkroh8usrhzmfrbouwekpaeej6batv8e38bocexcqmtjqrzo7u4esrqgxfflkaytbxufbdlo7t7+wflw6owirlnejnfmtkdgbouhejnfch5j9achohk7wbyk/ntr66ninq9tmuh3skxfawilzqxbos9csvt+roqf4tlycvynhol797g4glivfijdwm67kx+invkjjks1ivmtymbejhvgupoie3fjq/dkbordigwjbukjm6u+nlibgtmatoyuxbzy8svxmylvuy3wegvnwovj8gvw1wwtctbigszkihqnkgcwkkzfvyjwszuu4yuzjnx1vo4j9p1aqew/xuapx6rz5teu7gfm7bguowyoakmhuazxudvexlenhfnyzfackdy/gd5/mhmuczjw==</latexit> z <latexit sha1_base64="dwk1vkanlbct3la/uyovdtawnuu=">aaab6hicbvdlsgnbeoynrxhfuy9ebogkexgqbfvhhmwdwgwclspdczmzu7zmwkmeqlvhhqxkuf5m/czlsqrmlgoqqbrq7gkrwbvz38mtrw9sbuw3czu7e/shxcojpo5txbdbyhgrdka1ci6xybgre4uigqapgtzo/9yhk81jem3gcfkqhkoecuwol+lovwhll7hxklxgzkuggwq/41e3hli1qgiao1h3tyw/ocpwjnba6kyae8pgdiadsywnuut+aftcmavgljzusamld/txopu4cmxnrm1ql3sz8t+vk5rwp9wmaqgjvssclnbtexmx5m+v8imgftcmelvskgvffmbdyfg4k3/iqavbk3kw5ur8svw+yojwaqdwdh5cqrxuoaynyidwdk/w5jw4l86787foztnzzdh8gf5a+fajk=</latexit> Intruder: partially observe (O) the system Belief Update <latexit sha1_base64="anur1anuq89qzmk/ckzmoxfmza=">aaab9hicbvc7tgjbfj3ff+iltbszseysyc4wwhjtldfxgqqzhblkyyfthzl4rs+a4bc4x9ws/bshelbkxycs49uxeonqhba/rdlg5tbtnm3srd/chhut5p6yrthfyeyer1fazbihhcfcihmypgks+h44/v5n5nakqljh7eaqpexiaxcavnactv1gqhpg4amirbtwbx7qxoonekuimfwoqvz9iebzbjfwyrxuonakxm4wcs5hv+pmglexgljhfol18cfsmxhgloggiziurlttfizxfwk8j3xgded61zul/3m9dmmblxdxmiheflkozctfhm4boifqwffodwfccxmr5somgeftu8wu4kx+ezg3xnqt54ansatlfhmzyrc3jjhhjnmusetihlohkiz+svvfkt68v6tz6woywryjys7a+fwbfopg3</latexit> = (Q,, A, T, Z, O) (Belief) b <latexit sha1_base64="xmfnnadevmr1hwpzhgw5igkqvc=">aaacaxicbzc9sgnbfixvxr8y/6kwnonbsaq7imozslfmwxasotzyd1kyozsmjmrhcwvpa+hj3y+iq+gy/hjnlckxwy+djnxrhzgkrwbvz3ylsbg5t7xr3s3v7b4dh5eotlo5txbdjyhgrtka1ci6xabgrekuigqa7g97o8/yrk81g+mkmcfkshkoecuwotrtavv9yqoxdzbs+hcusq98s/vuhmgilyyjq3fxcxgzvyyzgdnsl9wyudamq+xaldrc7wfzq6fkwjodesbkmni37kdfi6kummimpfezmbmiyx1rqgm5rwzs+4tfkdkiucfnbtexmdzabv8immfigthh7ccjgvffmbgkl4633muqtk6qnlv1gtevk3euxho4bwuwynbqmed1kejdbbe4bxengfn3flwhejbsffoyv/cr5+at+llvs=</latexit> <latexit <latexit sha1_base64="jjwlmfqjw+olwwi6zgj3l65rckm=">aaacbnicbvdlssnafj3uv6vqesrbotqizskcnyjvny4s4w+sallmpqyczizccv58vfcufderd/gzr9xmahrqcuhm65l3vvcsnghttnby3srqvphflgxt7+zu6fshbrhghjmwdlniuy4shngatcsvjhqjtpdvmtjxxzep33kgxnawampjrbwfdqquyykkvr6sejockijfupviklhmfh1lgmsa9cxfw14tmzwblhmriwqod7xv+xbigofbbizjetmipjihlihmzfuxykajhmrqsnqib8olwktkbu3iqlahqq4qkhcm/p5ikc/exhdvz3qwrs8t+vfuv6iqigjjajxf5mumyhcmmcab5qrlnleeyu7vrrceedyquqkkgrr8evlq6urfnypxfrrfwzolgcjyaerdajaibw1ahlydbi3ggr+bne9jethfty96a7kzq/ahucxgkwsw==</latexit> <latexit sha1_base64="uohbydgybd8w/khqreym1ncbbs=">aaaccxicbvdltgjbeozff+il9ehlijhbc9nlob6jxjxiiqcbdzkdemhczo5mztaeel7ao1f9cg/gq1/hn/gtdrahbsrppflvlxrxkaiujet+o7m19y3nrfxywd3b/+gehju1hgqgdzylgl1efcngkfymnwifeguuhkibaxdm6nfekklerzdm1gcvqt9iiecuwolx/k1twj4trrfkltxzydlxmtictluu8wfti9mqctimeg1bntuyvwxvyyzgzncj9wyudakfwxbglgjh/dp6qm6vsbgro5ehm/vvykyl1imze1jzuavelnxpzdyazxrtk145y95lkqgiza/iewfmtgz1kj6xcezymqjzyrbjwgbuewzseuvbdveyhflpfmteg7fu6uwahdzt3k4gvmogwexuinbqemdgeh4gvd4c56dd+fd+zyv5pwscwz/4hz9ajd+mz8=</latexit> M = (Q,, A, T ) <latexit sha1_base64="8bes3ynijlhuqfc+ckvkdcvmjxo=">aaacahicbvdlssnafj3uv6vqasxbgalukgupapi1tcubfa6auaucbtattmgkze6gebwvny4ucetnunvnlrzaoubc4dz7uxee7yquaks69vira1vbg7ltws7u3v7b+bhuucgkcckjqmwij6hjgguk7aiipfekajyua63vqu9burega8jaahct1zjtecviawlgnjg+uhomwyq3jsazsek5dty6jgfqknqdcjxzgiibdyb+ocmarz7hcjmkzd+quxgscikgukktirjiaujulfu458it14/kacz7uyhkna6oikztxfezhypzz5nu5mz5xlxir+5/ujnaq5mevhpajhiwjieevwdqnokscymvmmiasql4v4gksccudwughyc+/veo61yp9wakr4r1whzhhpycm1acnrggdxagqanmejam3gfb8at8wk8gx+l1pyrzrydza+fwdnwjuz</latexit> Z : a finite set of observations Intruder s belief O : observation function where O(q, a, z) = (zt = z qt = q, at <latexit sha1_base64="yhhz4q7fskply79qqghqqbulli=">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</latexit> 1 = a) b : Q! [, 1], <latexit sha1_base64="fauym4m3wmdjkwc8klgwre4jh6k=">aaacenicbvdlssnafj3uv6vqes3gvoozskchzbklhxyj9qbczdpth4m6cxekahf4mzfcencebeu3k3ttsstxahcm593lvx7eqfsw9w1k1ty3nrey7md3b39a/wqcxdwgdsxcelrcdhkjdksvnrxugnegqfintf3qz89v3rega8js1iygboagnfyqrpjnfvrhioykiqeoed7foly1br8abl4wdymfj6hfgxwvbm/nwzodrhi7jxmqou6zx4vxhfaumimsdm1rui5crkkykamosewjej4haakqylhazfumn9pcs+o9uojics7v3xmjcqscbl7udjaaymvvjv7ndwvr7oj5vgscmelrfyqrxcwt6wrwxbikqvhqfsveqyqqvjrfna7bxn55lbqqzfu8xglc5gvvni4soagnoabscalq4bbuqrng8aiewst4m56mf+d+fizox5hj8gfh5a7krnde=</latexit> Ufuk Topcu 3 X qq b(q) = 1

8 <latexit sha1_base64="vqoleg+axbwbfseewac8xere=">aaab7nicbvdlsgnbeoz1gemr6thlybdijexgwvyegbeecwdkixmtibjknnzdazxces+woshrbz6d78gyfjhjsxokgo6qa7k4ilmoi6387a+sbm1nzuj7+7t39wwdg6bpooyw3wcqj3q6o4vio3kcbkrdjzwkysn4kxrczv/xetrgresbjzqdpuycebrsqgh6xhi5teoeiw3tnikveyuoqm9v7hq9uwbjyhuxsyzqeg6ofuocst7ndxdy8rgdmg7lioacuon83on5nwqftkitcfzk7+nkhpamwkdgxnshfklrz+j/xsxbq9voh4gs5yotfgqsjmjsd9ixmjoueso8lestiiasrqjps3ixjll6+szqxsxzyr91ffwjwliwencayl8oaaanahdwgagze8wyu8obhz4rw7h4vwnseboye/cd5/agqhju4=</latexit> Belief evolution as a switched dynamical system b t (q) = f q a(b t 1,z t )= (q z t,a t 1,b t 1 ) Ufuk Topcu 4

9 <latexit sha1_base64="vqoleg+axbwbfseewac8xere=">aaab7nicbvdlsgnbeoz1gemr6thlybdijexgwvyegbeecwdkixmtibjknnzdazxces+woshrbz6d78gyfjhjsxokgo6qa7k4ilmoi6387a+sbm1nzuj7+7t39wwdg6bpooyw3wcqj3q6o4vio3kcbkrdjzwkysn4kxrczv/xetrgresbjzqdpuycebrsqgh6xhi5teoeiw3tnikveyuoqm9v7hq9uwbjyhuxsyzqeg6ofuocst7ndxdy8rgdmg7lioacuon83on5nwqftkitcfzk7+nkhpamwkdgxnshfklrz+j/xsxbq9voh4gs5yotfgqsjmjsd9ixmjoueso8lestiiasrqjps3ixjll6+szqxsxzyr91ffwjwliwencayl8oaaanahdwgagze8wyu8obhz4rw7h4vwnseboye/cd5/agqhju4=</latexit> B<latexit sha1_base64="escsizef6nwru8bx/jzbyg6wwt8=">aaab73icbvbns8naej3ur1q/qh69lbbbulegopf48v7aeowyk3bpzpuboqs+ie8efdeq3/hm//gbzudtj4yelw3w8y8ibfcg9f9dgobm1vbo8xdt7+wefr+fikpenumwyywmsqe1cngktsgm4edhkfnaoetox3dxv6hsjazpqgh9gh5cfn1fipk/uyfaq+65crbtvdgkwtlycvynhol796g5ileurdbnw667mj8toqdgccz6veqjghbeyhlvugi1ny3unzelqwxigctbpcf+nsio5hwyiwnre1i73qzcx/vg5qwls/4zjjduqxbsmgpiyzj8na66qgtg1hdlf7aejaiizniisjyeb/xlddk6qnpu1xu4rttqerxfoinzuaqbqag99cajjaq8ayv8ozmnbfn3flythacfoyu/sd5/agiyi+w</latexit> <latexit sha1_base64="escsizef6nwru8bx/jzbyg6wwt8=">aaab73icbvbns8naej3ur1q/qh69lbbbulegopf48v7aeowyk3bpzpuboqs+ie8efdeq3/hm//gbzudtj4yelw3w8y8ibfcg9f9dgobm1vbo8xdt7+wefr+fikpenumwyywmsqe1cngktsgm4edhkfnaoetox3dxv6hsjazpqgh9gh5cfn1fipk/uyfaq+65crbtvdgkwtlycvynhol796g5ileurdbnw667mj8toqdgccz6veqjghbeyhlvugi1ny3unzelqwxigctbpcf+nsio5hwyiwnre1i73qzcx/vg5qwls/4zjjduqxbsmgpiyzj8na66qgtg1hdlf7aejaiizniisjyeb/xlddk6qnpu1xu4rttqerxfoinzuaqbqag99cajjaq8ayv8ozmnbfn3flythacfoyu/sd5/agiyi+w</latexit> <latexit sha1_base64="escsizef6nwru8bx/jzbyg6wwt8=">aaab73icbvbns8naej3ur1q/qh69lbbbulegopf48v7aeowyk3bpzpuboqs+ie8efdeq3/hm//gbzudtj4yelw3w8y8ibfcg9f9dgobm1vbo8xdt7+wefr+fikpenumwyywmsqe1cngktsgm4edhkfnaoetox3dxv6hsjazpqgh9gh5cfn1fipk/uyfaq+65crbtvdgkwtlycvynhol796g5ileurdbnw667mj8toqdgccz6veqjghbeyhlvugi1ny3unzelqwxigctbpcf+nsio5hwyiwnre1i73qzcx/vg5qwls/4zjjduqxbsmgpiyzj8na66qgtg1hdlf7aejaiizniisjyeb/xlddk6qnpu1xu4rttqerxfoinzuaqbqag99cajjaq8ayv8ozmnbfn3flythacfoyu/sd5/agiyi+w</latexit> <latexit sha1_base64="escsizef6nwru8bx/jzbyg6wwt8=">aaab73icbvbns8naej3ur1q/qh69lbbbulegopf48v7aeowyk3bpzpuboqs+ie8efdeq3/hm//gbzudtj4yelw3w8y8ibfcg9f9dgobm1vbo8xdt7+wefr+fikpenumwyywmsqe1cngktsgm4edhkfnaoetox3dxv6hsjazpqgh9gh5cfn1fipk/uyfaq+65crbtvdgkwtlycvynhol796g5ileurdbnw667mj8toqdgccz6veqjghbeyhlvugi1ny3unzelqwxigctbpcf+nsio5hwyiwnre1i73qzcx/vg5qwls/4zjjduqxbsmgpiyzj8na66qgtg1hdlf7aejaiizniisjyeb/xlddk6qnpu1xu4rttqerxfoinzuaqbqag99cajjaq8ayv8ozmnbfn3flythacfoyu/sd5/agiyi+w</latexit> Belief evolution as a switched dynamical system new belief old belief b t (q) = action f q a(b t 1,z t )= (q z t,a t 1,b t 1 ) observation b<latexit sha1_base64="xjd5v7unvnn+uddndprnub1vhrs=">aaab6nicbvbns8naej3ur1q/qh69lbbbumqomeif48v7qeowyk3bpzhnnij/qlecji1v/kzx/jts1bwx8mn6bywzekaiujet+o4w19y3nrejawd3b/+gfhjungqgdzzlglvcahgwsudtcco4lcgguc8h4dua3n1bphsthmnqj+hq8pazaqzetdfrnivt5ycrxclkbhi1++as3ifkaotrmuk7npsy6kcczwwuqlghkxnsixusljvd7fzuktmzyocesbildzmrvycygmk9iqlbgvezsvetz66ymvyzlpuogslrweqiinj7g8y4aqzernlkfc3kryicrkjenzewll9eja1a1buo1u4vk/wbi4inmapnimhv1chohaexgm4rle4crzovz7nwswgtonmmf+b8/gdp54n</latexit> Ufuk Topcu 4

10 <latexit sha1_base64="vqoleg+axbwbfseewac8xere=">aaab7nicbvdlsgnbeoz1gemr6thlybdijexgwvyegbeecwdkixmtibjknnzdazxces+woshrbz6d78gyfjhjsxokgo6qa7k4ilmoi6387a+sbm1nzuj7+7t39wwdg6bpooyw3wcqj3q6o4vio3kcbkrdjzwkysn4kxrczv/xetrgresbjzqdpuycebrsqgh6xhi5teoeiw3tnikveyuoqm9v7hq9uwbjyhuxsyzqeg6ofuocst7ndxdy8rgdmg7lioacuon83on5nwqftkitcfzk7+nkhpamwkdgxnshfklrz+j/xsxbq9voh4gs5yotfgqsjmjsd9ixmjoueso8lestiiasrqjps3ixjll6+szqxsxzyr91ffwjwliwencayl8oaaanahdwgagze8wyu8obhz4rw7h4vwnseboye/cd5/agqhju4=</latexit> B<latexit sha1_base64="escsizef6nwru8bx/jzbyg6wwt8=">aaab73icbvbns8naej3ur1q/qh69lbbbulegopf48v7aeowyk3bpzpuboqs+ie8efdeq3/hm//gbzudtj4yelw3w8y8ibfcg9f9dgobm1vbo8xdt7+wefr+fikpenumwyywmsqe1cngktsgm4edhkfnaoetox3dxv6hsjazpqgh9gh5cfn1fipk/uyfaq+65crbtvdgkwtlycvynhol796g5ileurdbnw667mj8toqdgccz6veqjghbeyhlvugi1ny3unzelqwxigctbpcf+nsio5hwyiwnre1i73qzcx/vg5qwls/4zjjduqxbsmgpiyzj8na66qgtg1hdlf7aejaiizniisjyeb/xlddk6qnpu1xu4rttqerxfoinzuaqbqag99cajjaq8ayv8ozmnbfn3flythacfoyu/sd5/agiyi+w</latexit> <latexit sha1_base64="escsizef6nwru8bx/jzbyg6wwt8=">aaab73icbvbns8naej3ur1q/qh69lbbbulegopf48v7aeowyk3bpzpuboqs+ie8efdeq3/hm//gbzudtj4yelw3w8y8ibfcg9f9dgobm1vbo8xdt7+wefr+fikpenumwyywmsqe1cngktsgm4edhkfnaoetox3dxv6hsjazpqgh9gh5cfn1fipk/uyfaq+65crbtvdgkwtlycvynhol796g5ileurdbnw667mj8toqdgccz6veqjghbeyhlvugi1ny3unzelqwxigctbpcf+nsio5hwyiwnre1i73qzcx/vg5qwls/4zjjduqxbsmgpiyzj8na66qgtg1hdlf7aejaiizniisjyeb/xlddk6qnpu1xu4rttqerxfoinzuaqbqag99cajjaq8ayv8ozmnbfn3flythacfoyu/sd5/agiyi+w</latexit> <latexit sha1_base64="escsizef6nwru8bx/jzbyg6wwt8=">aaab73icbvbns8naej3ur1q/qh69lbbbulegopf48v7aeowyk3bpzpuboqs+ie8efdeq3/hm//gbzudtj4yelw3w8y8ibfcg9f9dgobm1vbo8xdt7+wefr+fikpenumwyywmsqe1cngktsgm4edhkfnaoetox3dxv6hsjazpqgh9gh5cfn1fipk/uyfaq+65crbtvdgkwtlycvynhol796g5ileurdbnw667mj8toqdgccz6veqjghbeyhlvugi1ny3unzelqwxigctbpcf+nsio5hwyiwnre1i73qzcx/vg5qwls/4zjjduqxbsmgpiyzj8na66qgtg1hdlf7aejaiizniisjyeb/xlddk6qnpu1xu4rttqerxfoinzuaqbqag99cajjaq8ayv8ozmnbfn3flythacfoyu/sd5/agiyi+w</latexit> <latexit sha1_base64="escsizef6nwru8bx/jzbyg6wwt8=">aaab73icbvbns8naej3ur1q/qh69lbbbulegopf48v7aeowyk3bpzpuboqs+ie8efdeq3/hm//gbzudtj4yelw3w8y8ibfcg9f9dgobm1vbo8xdt7+wefr+fikpenumwyywmsqe1cngktsgm4edhkfnaoetox3dxv6hsjazpqgh9gh5cfn1fipk/uyfaq+65crbtvdgkwtlycvynhol796g5ileurdbnw667mj8toqdgccz6veqjghbeyhlvugi1ny3unzelqwxigctbpcf+nsio5hwyiwnre1i73qzcx/vg5qwls/4zjjduqxbsmgpiyzj8na66qgtg1hdlf7aejaiizniisjyeb/xlddk6qnpu1xu4rttqerxfoinzuaqbqag99cajjaq8ayv8ozmnbfn3flythacfoyu/sd5/agiyi+w</latexit> Belief evolution as a switched dynamical system new belief old belief b t (q) = action f q a(b t 1,z t )= (q z t,a t 1,b t 1 ) observation b<latexit sha1_base64="xjd5v7unvnn+uddndprnub1vhrs=">aaab6nicbvbns8naej3ur1q/qh69lbbbumqomeif48v7qeowyk3bpzhnnij/qlecji1v/kzx/jts1bwx8mn6bywzekaiujet+o4w19y3nrejawd3b/+gfhjungqgdzzlglvcahgwsudtcco4lcgguc8h4dua3n1bphsthmnqj+hq8pazaqzetdfrnivt5ycrxclkbhi1++as3ifkaotrmuk7npsy6kcczwwuqlghkxnsixusljvd7fzuktmzyocesbildzmrvycygmk9iqlbgvezsvetz66ymvyzlpuogslrweqiinj7g8y4aqzernlkfc3kryicrkjenzewll9eja1a1buo1u4vk/wbi4inmapnimhv1chohaexgm4rle4crzovz7nwswgtonmmf+b8/gdp54n</latexit> b<latexit sha1_base64="ib8zac5wbd9/ovfzlt7xpersce=">aaab6nicbvbns8naej3ur1q/qh69lbbbumqomeif48v7qeowyk3bpzhnnij/qlecji1v/kzx/jts1bwx8mn6bywzekaiujet+o4w19y3nrejawd3b/+gfhjungqgdzzlglvcahgwsudtcco4lcgguc8h4dua3n1bphsthmnqj+hq8pazaqzes9frnivt5ycrxclkbhi1++as3ifkaotrmuk7npsy6kcczwwuqlghkxnsixusljvd7fzuktmzyocesbildzmrvycygmk9iqlbgvezsvetz66ymvyzlpuogslrweqiinj7g8y4aqzernlkfc3kryicrkjenzewll9eja1a1buo1u4vk/wbi4inmapnimhv1chohaexgm4rle4crzovz7nwswgtonmmf+b8/gdra4o</latexit> 1 a<latexit sha1_base64="gl1urywrwjdcnj6f9achk1cng4m=">aaab6nicbvbns8naej3ur1q/qh69lbbbumqomeif48v7qeowyk3bpzhnnij/qlecji1v/kzx/jts1bwx8mn6bywzekaiujet+o4w19y3nrejawd3b/+gfhjungqgdzzlglvcahgwsudtcco4lcgguc8h4dua3n1bphsthmnqj+hq8pazaqzqtev1xxq+4czjv4oalajka//nubxcynubomqnzdzmn1flobm4lfvsjqllyzrerqwsrqj9bh7qljxzzudcwnmshszv3xmzjbserihtjkgz6wvvjv7ndvmtxvszllqullfojavxmrk9jczcixmiikllclubyvsrbvlxqztsif4yy+vklat6l1ua/exlfpnhkcrtuauzsgdk6jdhtsgcqyg8ayv8oyi58v5dz4wrqunnzmg3a+fwdp5yn</latexit> 1 z<latexit sha1_base64="ao7yovg3txfgnebi4fv/eoii=">aaab6nicbvbns8naej34wetx1aoxxsj4kkkv9fj4rgi/yalm1i7dbmlurqih8glbw8+ou8+w/ctjlo64obx3szzmwlesg1cd1vzv1bx1js7bv3n7z3dsvhrwdzwqhgwi1i1a6prciknw43adqkqrohavjc6mfqtr1sax/lbjbizqqosmgivd/w8xqnsvtwzydlxclkghve6avbj1kaotrmuk7npsy6kcczwuuymghkrnsahusljvd7ezuctm1sp+esbildzmpvycygmk9jglbgvez1ivevz66qmvizlpuogtzrweqiinj9g/s5wqzewnlkfc3krykcrkjenaewfl9ejs1qxtuvvo8uyrxri4chmmjniehl1cdw6hdaxgm4ble4crzovz7nzmw1ecfoyi/sd5/aeqcom</latexit> 1 Ufuk Topcu 4

11 <latexit sha1_base64="vqoleg+axbwbfseewac8xere=">aaab7nicbvdlsgnbeoz1gemr6thlybdijexgwvyegbeecwdkixmtibjknnzdazxces+woshrbz6d78gyfjhjsxokgo6qa7k4ilmoi6387a+sbm1nzuj7+7t39wwdg6bpooyw3wcqj3q6o4vio3kcbkrdjzwkysn4kxrczv/xetrgresbjzqdpuycebrsqgh6xhi5teoeiw3tnikveyuoqm9v7hq9uwbjyhuxsyzqeg6ofuocst7ndxdy8rgdmg7lioacuon83on5nwqftkitcfzk7+nkhpamwkdgxnshfklrz+j/xsxbq9voh4gs5yotfgqsjmjsd9ixmjoueso8lestiiasrqjps3ixjll6+szqxsxzyr91ffwjwliwencayl8oaaanahdwgagze8wyu8obhz4rw7h4vwnseboye/cd5/agqhju4=</latexit> B<latexit sha1_base64="escsizef6nwru8bx/jzbyg6wwt8=">aaab73icbvbns8naej3ur1q/qh69lbbbulegopf48v7aeowyk3bpzpuboqs+ie8efdeq3/hm//gbzudtj4yelw3w8y8ibfcg9f9dgobm1vbo8xdt7+wefr+fikpenumwyywmsqe1cngktsgm4edhkfnaoetox3dxv6hsjazpqgh9gh5cfn1fipk/uyfaq+65crbtvdgkwtlycvynhol796g5ileurdbnw667mj8toqdgccz6veqjghbeyhlvugi1ny3unzelqwxigctbpcf+nsio5hwyiwnre1i73qzcx/vg5qwls/4zjjduqxbsmgpiyzj8na66qgtg1hdlf7aejaiizniisjyeb/xlddk6qnpu1xu4rttqerxfoinzuaqbqag99cajjaq8ayv8ozmnbfn3flythacfoyu/sd5/agiyi+w</latexit> <latexit sha1_base64="escsizef6nwru8bx/jzbyg6wwt8=">aaab73icbvbns8naej3ur1q/qh69lbbbulegopf48v7aeowyk3bpzpuboqs+ie8efdeq3/hm//gbzudtj4yelw3w8y8ibfcg9f9dgobm1vbo8xdt7+wefr+fikpenumwyywmsqe1cngktsgm4edhkfnaoetox3dxv6hsjazpqgh9gh5cfn1fipk/uyfaq+65crbtvdgkwtlycvynhol796g5ileurdbnw667mj8toqdgccz6veqjghbeyhlvugi1ny3unzelqwxigctbpcf+nsio5hwyiwnre1i73qzcx/vg5qwls/4zjjduqxbsmgpiyzj8na66qgtg1hdlf7aejaiizniisjyeb/xlddk6qnpu1xu4rttqerxfoinzuaqbqag99cajjaq8ayv8ozmnbfn3flythacfoyu/sd5/agiyi+w</latexit> <latexit sha1_base64="escsizef6nwru8bx/jzbyg6wwt8=">aaab73icbvbns8naej3ur1q/qh69lbbbulegopf48v7aeowyk3bpzpuboqs+ie8efdeq3/hm//gbzudtj4yelw3w8y8ibfcg9f9dgobm1vbo8xdt7+wefr+fikpenumwyywmsqe1cngktsgm4edhkfnaoetox3dxv6hsjazpqgh9gh5cfn1fipk/uyfaq+65crbtvdgkwtlycvynhol796g5ileurdbnw667mj8toqdgccz6veqjghbeyhlvugi1ny3unzelqwxigctbpcf+nsio5hwyiwnre1i73qzcx/vg5qwls/4zjjduqxbsmgpiyzj8na66qgtg1hdlf7aejaiizniisjyeb/xlddk6qnpu1xu4rttqerxfoinzuaqbqag99cajjaq8ayv8ozmnbfn3flythacfoyu/sd5/agiyi+w</latexit> <latexit sha1_base64="escsizef6nwru8bx/jzbyg6wwt8=">aaab73icbvbns8naej3ur1q/qh69lbbbulegopf48v7aeowyk3bpzpuboqs+ie8efdeq3/hm//gbzudtj4yelw3w8y8ibfcg9f9dgobm1vbo8xdt7+wefr+fikpenumwyywmsqe1cngktsgm4edhkfnaoetox3dxv6hsjazpqgh9gh5cfn1fipk/uyfaq+65crbtvdgkwtlycvynhol796g5ileurdbnw667mj8toqdgccz6veqjghbeyhlvugi1ny3unzelqwxigctbpcf+nsio5hwyiwnre1i73qzcx/vg5qwls/4zjjduqxbsmgpiyzj8na66qgtg1hdlf7aejaiizniisjyeb/xlddk6qnpu1xu4rttqerxfoinzuaqbqag99cajjaq8ayv8ozmnbfn3flythacfoyu/sd5/agiyi+w</latexit> Belief evolution as a switched dynamical system new belief old belief b t (q) = action f q a(b t 1,z t )= (q z t,a t 1,b t 1 ) observation b<latexit sha1_base64="xjd5v7unvnn+uddndprnub1vhrs=">aaab6nicbvbns8naej3ur1q/qh69lbbbumqomeif48v7qeowyk3bpzhnnij/qlecji1v/kzx/jts1bwx8mn6bywzekaiujet+o4w19y3nrejawd3b/+gfhjungqgdzzlglvcahgwsudtcco4lcgguc8h4dua3n1bphsthmnqj+hq8pazaqzetdfrnivt5ycrxclkbhi1++as3ifkaotrmuk7npsy6kcczwwuqlghkxnsixusljvd7fzuktmzyocesbildzmrvycygmk9iqlbgvezsvetz66ymvyzlpuogslrweqiinj7g8y4aqzernlkfc3kryicrkjenzewll9eja1a1buo1u4vk/wbi4inmapnimhv1chohaexgm4rle4crzovz7nwswgtonmmf+b8/gdp54n</latexit> b<latexit sha1_base64="ib8zac5wbd9/ovfzlt7xpersce=">aaab6nicbvbns8naej3ur1q/qh69lbbbumqomeif48v7qeowyk3bpzhnnij/qlecji1v/kzx/jts1bwx8mn6bywzekaiujet+o4w19y3nrejawd3b/+gfhjungqgdzzlglvcahgwsudtcco4lcgguc8h4dua3n1bphsthmnqj+hq8pazaqzes9frnivt5ycrxclkbhi1++as3ifkaotrmuk7npsy6kcczwwuqlghkxnsixusljvd7fzuktmzyocesbildzmrvycygmk9iqlbgvezsvetz66ymvyzlpuogslrweqiinj7g8y4aqzernlkfc3kryicrkjenzewll9eja1a1buo1u4vk/wbi4inmapnimhv1chohaexgm4rle4crzovz7nwswgtonmmf+b8/gdra4o</latexit> 1 a<latexit sha1_base64="gl1urywrwjdcnj6f9achk1cng4m=">aaab6nicbvbns8naej3ur1q/qh69lbbbumqomeif48v7qeowyk3bpzhnnij/qlecji1v/kzx/jts1bwx8mn6bywzekaiujet+o4w19y3nrejawd3b/+gfhjungqgdzzlglvcahgwsudtcco4lcgguc8h4dua3n1bphsthmnqj+hq8pazaqzqtev1xxq+4czjv4oalajka//nubxcynubomqnzdzmn1flobm4lfvsjqllyzrerqwsrqj9bh7qljxzzudcwnmshszv3xmzjbserihtjkgz6wvvjv7ndvmtxvszllqullfojavxmrk9jczcixmiikllclubyvsrbvlxqztsif4yy+vklat6l1ua/exlfpnhkcrtuauzsgdk6jdhtsgcqyg8ayv8oyi58v5dz4wrqunnzmg3a+fwdp5yn</latexit> 1 z<latexit sha1_base64="ao7yovg3txfgnebi4fv/eoii=">aaab6nicbvbns8naej34wetx1aoxxsj4kkkv9fj4rgi/yalm1i7dbmlurqih8glbw8+ou8+w/ctjlo64obx3szzmwlesg1cd1vzv1bx1js7bv3n7z3dsvhrwdzwqhgwi1i1a6prciknw43adqkqrohavjc6mfqtr1sax/lbjbizqqosmgivd/w8xqnsvtwzydlxclkghve6avbj1kaotrmuk7npsy6kcczwuuymghkrnsahusljvd7ezuctm1sp+esbildzmpvycygmk9jglbgvez1ivevz66qmvizlpuogtzrweqiinj9g/s5wqzewnlkfc3krykcrkjenaewfl9ejs1qxtuvvo8uyrxri4chmmjniehl1cdw6hdaxgm4ble4crzovz7nzmw1ecfoyi/sd5/aeqcom</latexit> 1 a<latexit sha1_base64="niruwwwhuyejnuz/tbo71yq8=">aaab6nicbvbns8naej3ur1q/qh69lbbbumqomeif48v7qeowyk3bpzhnnij/qlecji1v/kzx/jts1bwx8mn6bywzekaiujet+o4w19y3nrejawd3b/+gfhjungqgdzzlglvcahgwsudtcco4lcgguc8h4dua3n1bphsthmnqj+hq8pazaqzqu1frnivt5ycrxclkbhi1++as3ifkaotrmuk7npsy6kcczwwuqlghkxnsixusljvd7fzuktmzyocesbildzmrvycygmk9iqlbgvezsvetz66ymvyzlpuogslrweqiinj7g8y4aqzernlkfc3kryicrkjenzewll9eja1a1buo1u4vk/wbi4inmapnimhv1chohaexgm4rle4crzovz7nwswgtonmmf+b8/gdrayo</latexit> z<latexit sha1_base64="zzkjtcjfguoajipvjcjutanly=">aaab6nicbvbns8naej34wetx1aoxxsj4kkkv9fj4rgi/yalm1y7dbmlurkih8glbw8+ou8+w/ctjlo64obx3szzmwleikmuu63s7k6tr6xwdgqbu/s7uxdg6bjk414wwy1i3aq4fio3ukdk7urzggwst4lrzdrvxjtrkwecjxw6idjulbkfrp/qlx7zxkbswdgswtlydlyfhvlb66/zilevfijdwm47kj+hnvkjjkki3ntyhbeqhvgopohe3fjy7dujordinyaxtksqz9fderinjxlfgoyokq7otcx/ve6k4zwfczwkybwblwptstam79jxjoui4touwleythq6opq5toybglb68tjrvindeqd5dlgvxerwfoiytoamlqegt1chbjaywdo8wpsjnrfn3fmyt644+cwr/ihz+qmrjon</latexit> b<latexit sha1_base64="ambb7fjcw54wngon41/vohqvm=">aaab6nicbvbns8naej3ur1q/qh69lbbbumqomeif48v7qeowyk3bpzhnnij/qlecji1v/kzx/jts1bwx8mn6bywzekaiujet+o4w19y3nrejawd3b/+gfhjungqgdzzlglvcahgwsudtcco4lcgguc8h4dua3n1bphsthmnqj+hq8pazaqzerr/xlfrbpzkfxi5aqcorr98ldvelmqmmyofp3tcxfkav4uzgtnrlnsaujekqu5zkgqhs/mpu3jmlqejyvlgjjxf9knnj6egwm6jmpje9mfif11neo1nxcapqckwi8juebotd9kwbuyiyawuka4vzwwevwugztoyybglb+8slq1qndrrd1fvuo3erxfoiftoacrqaod9cajjaywjo8wpsjnbfn3flytbacfoyy/sd5/ahs74</latexit> Ufuk Topcu 4

12 <latexit sha1_base64="vqoleg+axbwbfseewac8xere=">aaab7nicbvdlsgnbeoz1gemr6thlybdijexgwvyegbeecwdkixmtibjknnzdazxces+woshrbz6d78gyfjhjsxokgo6qa7k4ilmoi6387a+sbm1nzuj7+7t39wwdg6bpooyw3wcqj3q6o4vio3kcbkrdjzwkysn4kxrczv/xetrgresbjzqdpuycebrsqgh6xhi5teoeiw3tnikveyuoqm9v7hq9uwbjyhuxsyzqeg6ofuocst7ndxdy8rgdmg7lioacuon83on5nwqftkitcfzk7+nkhpamwkdgxnshfklrz+j/xsxbq9voh4gs5yotfgqsjmjsd9ixmjoueso8lestiiasrqjps3ixjll6+szqxsxzyr91ffwjwliwencayl8oaaanahdwgagze8wyu8obhz4rw7h4vwnseboye/cd5/agqhju4=</latexit> B<latexit sha1_base64="escsizef6nwru8bx/jzbyg6wwt8=">aaab73icbvbns8naej3ur1q/qh69lbbbulegopf48v7aeowyk3bpzpuboqs+ie8efdeq3/hm//gbzudtj4yelw3w8y8ibfcg9f9dgobm1vbo8xdt7+wefr+fikpenumwyywmsqe1cngktsgm4edhkfnaoetox3dxv6hsjazpqgh9gh5cfn1fipk/uyfaq+65crbtvdgkwtlycvynhol796g5ileurdbnw667mj8toqdgccz6veqjghbeyhlvugi1ny3unzelqwxigctbpcf+nsio5hwyiwnre1i73qzcx/vg5qwls/4zjjduqxbsmgpiyzj8na66qgtg1hdlf7aejaiizniisjyeb/xlddk6qnpu1xu4rttqerxfoinzuaqbqag99cajjaq8ayv8ozmnbfn3flythacfoyu/sd5/agiyi+w</latexit> <latexit sha1_base64="escsizef6nwru8bx/jzbyg6wwt8=">aaab73icbvbns8naej3ur1q/qh69lbbbulegopf48v7aeowyk3bpzpuboqs+ie8efdeq3/hm//gbzudtj4yelw3w8y8ibfcg9f9dgobm1vbo8xdt7+wefr+fikpenumwyywmsqe1cngktsgm4edhkfnaoetox3dxv6hsjazpqgh9gh5cfn1fipk/uyfaq+65crbtvdgkwtlycvynhol796g5ileurdbnw667mj8toqdgccz6veqjghbeyhlvugi1ny3unzelqwxigctbpcf+nsio5hwyiwnre1i73qzcx/vg5qwls/4zjjduqxbsmgpiyzj8na66qgtg1hdlf7aejaiizniisjyeb/xlddk6qnpu1xu4rttqerxfoinzuaqbqag99cajjaq8ayv8ozmnbfn3flythacfoyu/sd5/agiyi+w</latexit> <latexit sha1_base64="escsizef6nwru8bx/jzbyg6wwt8=">aaab73icbvbns8naej3ur1q/qh69lbbbulegopf48v7aeowyk3bpzpuboqs+ie8efdeq3/hm//gbzudtj4yelw3w8y8ibfcg9f9dgobm1vbo8xdt7+wefr+fikpenumwyywmsqe1cngktsgm4edhkfnaoetox3dxv6hsjazpqgh9gh5cfn1fipk/uyfaq+65crbtvdgkwtlycvynhol796g5ileurdbnw667mj8toqdgccz6veqjghbeyhlvugi1ny3unzelqwxigctbpcf+nsio5hwyiwnre1i73qzcx/vg5qwls/4zjjduqxbsmgpiyzj8na66qgtg1hdlf7aejaiizniisjyeb/xlddk6qnpu1xu4rttqerxfoinzuaqbqag99cajjaq8ayv8ozmnbfn3flythacfoyu/sd5/agiyi+w</latexit> <latexit sha1_base64="escsizef6nwru8bx/jzbyg6wwt8=">aaab73icbvbns8naej3ur1q/qh69lbbbulegopf48v7aeowyk3bpzpuboqs+ie8efdeq3/hm//gbzudtj4yelw3w8y8ibfcg9f9dgobm1vbo8xdt7+wefr+fikpenumwyywmsqe1cngktsgm4edhkfnaoetox3dxv6hsjazpqgh9gh5cfn1fipk/uyfaq+65crbtvdgkwtlycvynhol796g5ileurdbnw667mj8toqdgccz6veqjghbeyhlvugi1ny3unzelqwxigctbpcf+nsio5hwyiwnre1i73qzcx/vg5qwls/4zjjduqxbsmgpiyzj8na66qgtg1hdlf7aejaiizniisjyeb/xlddk6qnpu1xu4rttqerxfoinzuaqbqag99cajjaq8ayv8ozmnbfn3flythacfoyu/sd5/agiyi+w</latexit> Belief evolution as a switched dynamical system new belief old belief b t (q) = action f q a(b t 1,z t )= (q z t,a t 1,b t 1 ) observation b<latexit sha1_base64="xjd5v7unvnn+uddndprnub1vhrs=">aaab6nicbvbns8naej3ur1q/qh69lbbbumqomeif48v7qeowyk3bpzhnnij/qlecji1v/kzx/jts1bwx8mn6bywzekaiujet+o4w19y3nrejawd3b/+gfhjungqgdzzlglvcahgwsudtcco4lcgguc8h4dua3n1bphsthmnqj+hq8pazaqzetdfrnivt5ycrxclkbhi1++as3ifkaotrmuk7npsy6kcczwwuqlghkxnsixusljvd7fzuktmzyocesbildzmrvycygmk9iqlbgvezsvetz66ymvyzlpuogslrweqiinj7g8y4aqzernlkfc3kryicrkjenzewll9eja1a1buo1u4vk/wbi4inmapnimhv1chohaexgm4rle4crzovz7nwswgtonmmf+b8/gdp54n</latexit> b<latexit sha1_base64="ib8zac5wbd9/ovfzlt7xpersce=">aaab6nicbvbns8naej3ur1q/qh69lbbbumqomeif48v7qeowyk3bpzhnnij/qlecji1v/kzx/jts1bwx8mn6bywzekaiujet+o4w19y3nrejawd3b/+gfhjungqgdzzlglvcahgwsudtcco4lcgguc8h4dua3n1bphsthmnqj+hq8pazaqzes9frnivt5ycrxclkbhi1++as3ifkaotrmuk7npsy6kcczwwuqlghkxnsixusljvd7fzuktmzyocesbildzmrvycygmk9iqlbgvezsvetz66ymvyzlpuogslrweqiinj7g8y4aqzernlkfc3kryicrkjenzewll9eja1a1buo1u4vk/wbi4inmapnimhv1chohaexgm4rle4crzovz7nwswgtonmmf+b8/gdra4o</latexit> 1 a<latexit sha1_base64="gl1urywrwjdcnj6f9achk1cng4m=">aaab6nicbvbns8naej3ur1q/qh69lbbbumqomeif48v7qeowyk3bpzhnnij/qlecji1v/kzx/jts1bwx8mn6bywzekaiujet+o4w19y3nrejawd3b/+gfhjungqgdzzlglvcahgwsudtcco4lcgguc8h4dua3n1bphsthmnqj+hq8pazaqzqtev1xxq+4czjv4oalajka//nubxcynubomqnzdzmn1flobm4lfvsjqllyzrerqwsrqj9bh7qljxzzudcwnmshszv3xmzjbserihtjkgz6wvvjv7ndvmtxvszllqullfojavxmrk9jczcixmiikllclubyvsrbvlxqztsif4yy+vklat6l1ua/exlfpnhkcrtuauzsgdk6jdhtsgcqyg8ayv8oyi58v5dz4wrqunnzmg3a+fwdp5yn</latexit> 1 z<latexit sha1_base64="ao7yovg3txfgnebi4fv/eoii=">aaab6nicbvbns8naej34wetx1aoxxsj4kkkv9fj4rgi/yalm1i7dbmlurqih8glbw8+ou8+w/ctjlo64obx3szzmwlesg1cd1vzv1bx1js7bv3n7z3dsvhrwdzwqhgwi1i1a6prciknw43adqkqrohavjc6mfqtr1sax/lbjbizqqosmgivd/w8xqnsvtwzydlxclkghve6avbj1kaotrmuk7npsy6kcczwuuymghkrnsahusljvd7ezuctm1sp+esbildzmpvycygmk9jglbgvez1ivevz66qmvizlpuogtzrweqiinj9g/s5wqzewnlkfc3krykcrkjenaewfl9ejs1qxtuvvo8uyrxri4chmmjniehl1cdw6hdaxgm4ble4crzovz7nzmw1ecfoyi/sd5/aeqcom</latexit> 1 a<latexit sha1_base64="niruwwwhuyejnuz/tbo71yq8=">aaab6nicbvbns8naej3ur1q/qh69lbbbumqomeif48v7qeowyk3bpzhnnij/qlecji1v/kzx/jts1bwx8mn6bywzekaiujet+o4w19y3nrejawd3b/+gfhjungqgdzzlglvcahgwsudtcco4lcgguc8h4dua3n1bphsthmnqj+hq8pazaqzqu1frnivt5ycrxclkbhi1++as3ifkaotrmuk7npsy6kcczwwuqlghkxnsixusljvd7fzuktmzyocesbildzmrvycygmk9iqlbgvezsvetz66ymvyzlpuogslrweqiinj7g8y4aqzernlkfc3kryicrkjenzewll9eja1a1buo1u4vk/wbi4inmapnimhv1chohaexgm4rle4crzovz7nwswgtonmmf+b8/gdrayo</latexit> z<latexit sha1_base64="zzkjtcjfguoajipvjcjutanly=">aaab6nicbvbns8naej34wetx1aoxxsj4kkkv9fj4rgi/yalm1y7dbmlurkih8glbw8+ou8+w/ctjlo64obx3szzmwleikmuu63s7k6tr6xwdgqbu/s7uxdg6bjk414wwy1i3aq4fio3ukdk7urzggwst4lrzdrvxjtrkwecjxw6idjulbkfrp/qlx7zxkbswdgswtlydlyfhvlb66/zilevfijdwm47kj+hnvkjjkki3ntyhbeqhvgopohe3fjy7dujordinyaxtksqz9fderinjxlfgoyokq7otcx/ve6k4zwfczwkybwblwptstam79jxjoui4touwleythq6opq5toybglb68tjrvindeqd5dlgvxerwfoiytoamlqegt1chbjaywdo8wpsjnrfn3fmyt644+cwr/ihz+qmrjon</latexit> b<latexit sha1_base64="ambb7fjcw54wngon41/vohqvm=">aaab6nicbvbns8naej3ur1q/qh69lbbbumqomeif48v7qeowyk3bpzhnnij/qlecji1v/kzx/jts1bwx8mn6bywzekaiujet+o4w19y3nrejawd3b/+gfhjungqgdzzlglvcahgwsudtcco4lcgguc8h4dua3n1bphsthmnqj+hq8pazaqzerr/xlfrbpzkfxi5aqcorr98ldvelmqmmyofp3tcxfkav4uzgtnrlnsaujekqu5zkgqhs/mpu3jmlqejyvlgjjxf9knnj6egwm6jmpje9mfif11neo1nxcapqckwi8juebotd9kwbuyiyawuka4vzwwevwugztoyybglb+8slq1qndrrd1fvuo3erxfoiftoacrqaod9cajjaywjo8wpsjnbfn3flytbacfoyy/sd5/ahs74</latexit> b<latexit sha1_base64="zvwt5nu+mj/gcgd9djcmxqz+y=">aaab6nicbvbns8naej3ur1q/qh69lbbbulaqy9flx4rlpoq9lsj+3szsbsboqs+ho8efdeq7/im//gbzudtj4yelw3w8y8ibfcg9f9dgpr6xubw8xts7u3v5b+fcorenumwyxwmsqe1cngktsgw4edhkfnaoegbjm5n/+irk81g+memcfkshkoecuwol+6bf75crbtwdg6wslycvynhsl796g5ileurdbnw667mj8toqdgccp6veqjghbeyhlvugi1n81nzizqwxigctbpc5+nsio5hwkyiwnre1i73szct/vg5qwis/4zjjduqwbsmgpiyz4ma66qgtgxhdlf7aejaiiznhsjyeb/nlvdkuvb16txz3uwlc53eu4qro4rw8uiqg3eitwsbgcm/wcm+ocf6cd+djvpw8plj+an8wfuc4q</latexit> 3 a<latexit sha1_base64="kmvxvtnrvdvyhfr6amxaab5vho=">aaab6nicbvbns8naej3ur1q/qh69lbbbulaqy9flx4rlpoq9lsj+3szsbsboqs+ho8efdeq7/im//gbzudtj4yelw3w8y8ibfcg9f9dgpr6xubw8xts7u3v5b+fcorenumwyxwmsqe1cngktsgw4edhkfnaoegbjm5n/+irk81g+memcfkshkoecuwole9qv98svt+roqvajl5mk5gjy1+9qczsckvhgmrd9dze+blvhjob1iv1zhqnqzd7foqaytaz+antsmzvqykjjutachc/trujrsrtyzoiakv7zuj/xjc14zwfczmkbivblaptquxmzn+tavfijjhyqpni9lbcrlrrzmw6jruct/zykmnxql69wru7qdsu8zikcaknca4exeidbqejlwawhgd4htdhoc/ou/oxac4+cwx/ihz+qs7y</latexit> 3 z<latexit sha1_base64="xauhsnrws5733kvfckyzinjbile=">aaab6nicbvdltgjbeozff+il9ehlijhxrhbbri9elx4xyiobdzkdemhc7oxmztyecz/gxygewlvk3drahbsvppflvne6uibfcg9f9dnjr6xubw/ntws7u3v5b8fcoqenumwywwmsqhvcngktsgg4ethofnaoetolrzcxvalsjyzpygh9gb5cfn1fjp/qlx7rvlbtmdg6wslymlyfdvfb+6/zileurdbnw647mj8sdugc4etgvdvgncygosgoppbfqfzi/durorninyaxssum6u+jcyhkeb7yyogeplbyb+53vse175ey6t1kbkivhkoijyexvuckmrfjsyht3n5kjaqyoxnpbd8jzfxixnstmrlit3f6xadrzhhk7gfm7bguows3uoqembvamr/dmcoffexc+fq5j5s5hj9wn8aexknqa==</latexit> 3 b<latexit sha1_base64="az+vsf6gsgih53caje9izzjol8=">aaab6nicbvbns8naej34wetx1aoxxsj4kkkt6lhoxwnf+wftkjvtpl6ytdivbcf4ixd4p49rd589+4bxq1gcdj/dmmjkxjfiydn1vz19y3nru7bt3n3bzgshr3tjxqxpsslrhubnrwkrrvokdjo4nmnaokbwfj5nffulaifg94ithfkshsoscubtsq9cv9utlt+loqvajl5my5gjs1+9qczsictkkhrt9dwe/yxqfezyabgxgp5qnqzd3rvuygb5ufoixnvhmqmnafjk5+nsio5exkyiwnrhfkvnzuj/xjff8nrheps5iotfowpjbitd9kidrnkcewukafvzwwedwuounaewll9eja1qxbusvo9r5fpnhkcbtuemlscdk6jdhtsgcqyg8ayv8ozi58v5dz4wrwtonmcf+b8/gdv94r</latexit> 4 Ufuk Topcu 4

13 <latexit sha1_base64="vqoleg+axbwbfseewac8xere=">aaab7nicbvdlsgnbeoz1gemr6thlybdijexgwvyegbeecwdkixmtibjknnzdazxces+woshrbz6d78gyfjhjsxokgo6qa7k4ilmoi6387a+sbm1nzuj7+7t39wwdg6bpooyw3wcqj3q6o4vio3kcbkrdjzwkysn4kxrczv/xetrgresbjzqdpuycebrsqgh6xhi5teoeiw3tnikveyuoqm9v7hq9uwbjyhuxsyzqeg6ofuocst7ndxdy8rgdmg7lioacuon83on5nwqftkitcfzk7+nkhpamwkdgxnshfklrz+j/xsxbq9voh4gs5yotfgqsjmjsd9ixmjoueso8lestiiasrqjps3ixjll6+szqxsxzyr91ffwjwliwencayl8oaaanahdwgagze8wyu8obhz4rw7h4vwnseboye/cd5/agqhju4=</latexit> B<latexit sha1_base64="escsizef6nwru8bx/jzbyg6wwt8=">aaab73icbvbns8naej3ur1q/qh69lbbbulegopf48v7aeowyk3bpzpuboqs+ie8efdeq3/hm//gbzudtj4yelw3w8y8ibfcg9f9dgobm1vbo8xdt7+wefr+fikpenumwyywmsqe1cngktsgm4edhkfnaoetox3dxv6hsjazpqgh9gh5cfn1fipk/uyfaq+65crbtvdgkwtlycvynhol796g5ileurdbnw667mj8toqdgccz6veqjghbeyhlvugi1ny3unzelqwxigctbpcf+nsio5hwyiwnre1i73qzcx/vg5qwls/4zjjduqxbsmgpiyzj8na66qgtg1hdlf7aejaiizniisjyeb/xlddk6qnpu1xu4rttqerxfoinzuaqbqag99cajjaq8ayv8ozmnbfn3flythacfoyu/sd5/agiyi+w</latexit> <latexit sha1_base64="escsizef6nwru8bx/jzbyg6wwt8=">aaab73icbvbns8naej3ur1q/qh69lbbbulegopf48v7aeowyk3bpzpuboqs+ie8efdeq3/hm//gbzudtj4yelw3w8y8ibfcg9f9dgobm1vbo8xdt7+wefr+fikpenumwyywmsqe1cngktsgm4edhkfnaoetox3dxv6hsjazpqgh9gh5cfn1fipk/uyfaq+65crbtvdgkwtlycvynhol796g5ileurdbnw667mj8toqdgccz6veqjghbeyhlvugi1ny3unzelqwxigctbpcf+nsio5hwyiwnre1i73qzcx/vg5qwls/4zjjduqxbsmgpiyzj8na66qgtg1hdlf7aejaiizniisjyeb/xlddk6qnpu1xu4rttqerxfoinzuaqbqag99cajjaq8ayv8ozmnbfn3flythacfoyu/sd5/agiyi+w</latexit> <latexit sha1_base64="escsizef6nwru8bx/jzbyg6wwt8=">aaab73icbvbns8naej3ur1q/qh69lbbbulegopf48v7aeowyk3bpzpuboqs+ie8efdeq3/hm//gbzudtj4yelw3w8y8ibfcg9f9dgobm1vbo8xdt7+wefr+fikpenumwyywmsqe1cngktsgm4edhkfnaoetox3dxv6hsjazpqgh9gh5cfn1fipk/uyfaq+65crbtvdgkwtlycvynhol796g5ileurdbnw667mj8toqdgccz6veqjghbeyhlvugi1ny3unzelqwxigctbpcf+nsio5hwyiwnre1i73qzcx/vg5qwls/4zjjduqxbsmgpiyzj8na66qgtg1hdlf7aejaiizniisjyeb/xlddk6qnpu1xu4rttqerxfoinzuaqbqag99cajjaq8ayv8ozmnbfn3flythacfoyu/sd5/agiyi+w</latexit> <latexit sha1_base64="escsizef6nwru8bx/jzbyg6wwt8=">aaab73icbvbns8naej3ur1q/qh69lbbbulegopf48v7aeowyk3bpzpuboqs+ie8efdeq3/hm//gbzudtj4yelw3w8y8ibfcg9f9dgobm1vbo8xdt7+wefr+fikpenumwyywmsqe1cngktsgm4edhkfnaoetox3dxv6hsjazpqgh9gh5cfn1fipk/uyfaq+65crbtvdgkwtlycvynhol796g5ileurdbnw667mj8toqdgccz6veqjghbeyhlvugi1ny3unzelqwxigctbpcf+nsio5hwyiwnre1i73qzcx/vg5qwls/4zjjduqxbsmgpiyzj8na66qgtg1hdlf7aejaiizniisjyeb/xlddk6qnpu1xu4rttqerxfoinzuaqbqag99cajjaq8ayv8ozmnbfn3flythacfoyu/sd5/agiyi+w</latexit> Belief evolution as a switched dynamical system new belief old belief b t (q) = action f q a(b t 1,z t )= (q z t,a t 1,b t 1 ) observation b<latexit sha1_base64="xjd5v7unvnn+uddndprnub1vhrs=">aaab6nicbvbns8naej3ur1q/qh69lbbbumqomeif48v7qeowyk3bpzhnnij/qlecji1v/kzx/jts1bwx8mn6bywzekaiujet+o4w19y3nrejawd3b/+gfhjungqgdzzlglvcahgwsudtcco4lcgguc8h4dua3n1bphsthmnqj+hq8pazaqzetdfrnivt5ycrxclkbhi1++as3ifkaotrmuk7npsy6kcczwwuqlghkxnsixusljvd7fzuktmzyocesbildzmrvycygmk9iqlbgvezsvetz66ymvyzlpuogslrweqiinj7g8y4aqzernlkfc3kryicrkjenzewll9eja1a1buo1u4vk/wbi4inmapnimhv1chohaexgm4rle4crzovz7nwswgtonmmf+b8/gdp54n</latexit> b<latexit sha1_base64="ib8zac5wbd9/ovfzlt7xpersce=">aaab6nicbvbns8naej3ur1q/qh69lbbbumqomeif48v7qeowyk3bpzhnnij/qlecji1v/kzx/jts1bwx8mn6bywzekaiujet+o4w19y3nrejawd3b/+gfhjungqgdzzlglvcahgwsudtcco4lcgguc8h4dua3n1bphsthmnqj+hq8pazaqzes9frnivt5ycrxclkbhi1++as3ifkaotrmuk7npsy6kcczwwuqlghkxnsixusljvd7fzuktmzyocesbildzmrvycygmk9iqlbgvezsvetz66ymvyzlpuogslrweqiinj7g8y4aqzernlkfc3kryicrkjenzewll9eja1a1buo1u4vk/wbi4inmapnimhv1chohaexgm4rle4crzovz7nwswgtonmmf+b8/gdra4o</latexit> 1 a<latexit sha1_base64="gl1urywrwjdcnj6f9achk1cng4m=">aaab6nicbvbns8naej3ur1q/qh69lbbbumqomeif48v7qeowyk3bpzhnnij/qlecji1v/kzx/jts1bwx8mn6bywzekaiujet+o4w19y3nrejawd3b/+gfhjungqgdzzlglvcahgwsudtcco4lcgguc8h4dua3n1bphsthmnqj+hq8pazaqzqtev1xxq+4czjv4oalajka//nubxcynubomqnzdzmn1flobm4lfvsjqllyzrerqwsrqj9bh7qljxzzudcwnmshszv3xmzjbserihtjkgz6wvvjv7ndvmtxvszllqullfojavxmrk9jczcixmiikllclubyvsrbvlxqztsif4yy+vklat6l1ua/exlfpnhkcrtuauzsgdk6jdhtsgcqyg8ayv8oyi58v5dz4wrqunnzmg3a+fwdp5yn</latexit> 1 z<latexit sha1_base64="ao7yovg3txfgnebi4fv/eoii=">aaab6nicbvbns8naej34wetx1aoxxsj4kkkv9fj4rgi/yalm1i7dbmlurqih8glbw8+ou8+w/ctjlo64obx3szzmwlesg1cd1vzv1bx1js7bv3n7z3dsvhrwdzwqhgwi1i1a6prciknw43adqkqrohavjc6mfqtr1sax/lbjbizqqosmgivd/w8xqnsvtwzydlxclkghve6avbj1kaotrmuk7npsy6kcczwuuymghkrnsahusljvd7ezuctm1sp+esbildzmpvycygmk9jglbgvez1ivevz66qmvizlpuogtzrweqiinj9g/s5wqzewnlkfc3krykcrkjenaewfl9ejs1qxtuvvo8uyrxri4chmmjniehl1cdw6hdaxgm4ble4crzovz7nzmw1ecfoyi/sd5/aeqcom</latexit> 1 a<latexit sha1_base64="niruwwwhuyejnuz/tbo71yq8=">aaab6nicbvbns8naej3ur1q/qh69lbbbumqomeif48v7qeowyk3bpzhnnij/qlecji1v/kzx/jts1bwx8mn6bywzekaiujet+o4w19y3nrejawd3b/+gfhjungqgdzzlglvcahgwsudtcco4lcgguc8h4dua3n1bphsthmnqj+hq8pazaqzqu1frnivt5ycrxclkbhi1++as3ifkaotrmuk7npsy6kcczwwuqlghkxnsixusljvd7fzuktmzyocesbildzmrvycygmk9iqlbgvezsvetz66ymvyzlpuogslrweqiinj7g8y4aqzernlkfc3kryicrkjenzewll9eja1a1buo1u4vk/wbi4inmapnimhv1chohaexgm4rle4crzovz7nwswgtonmmf+b8/gdrayo</latexit> z<latexit sha1_base64="zzkjtcjfguoajipvjcjutanly=">aaab6nicbvbns8naej34wetx1aoxxsj4kkkv9fj4rgi/yalm1y7dbmlurkih8glbw8+ou8+w/ctjlo64obx3szzmwleikmuu63s7k6tr6xwdgqbu/s7uxdg6bjk414wwy1i3aq4fio3ukdk7urzggwst4lrzdrvxjtrkwecjxw6idjulbkfrp/qlx7zxkbswdgswtlydlyfhvlb66/zilevfijdwm47kj+hnvkjjkki3ntyhbeqhvgopohe3fjy7dujordinyaxtksqz9fderinjxlfgoyokq7otcx/ve6k4zwfczwkybwblwptstam79jxjoui4touwleythq6opq5toybglb68tjrvindeqd5dlgvxerwfoiytoamlqegt1chbjaywdo8wpsjnrfn3fmyt644+cwr/ihz+qmrjon</latexit> b<latexit sha1_base64="ambb7fjcw54wngon41/vohqvm=">aaab6nicbvbns8naej3ur1q/qh69lbbbumqomeif48v7qeowyk3bpzhnnij/qlecji1v/kzx/jts1bwx8mn6bywzekaiujet+o4w19y3nrejawd3b/+gfhjungqgdzzlglvcahgwsudtcco4lcgguc8h4dua3n1bphsthmnqj+hq8pazaqzerr/xlfrbpzkfxi5aqcorr98ldvelmqmmyofp3tcxfkav4uzgtnrlnsaujekqu5zkgqhs/mpu3jmlqejyvlgjjxf9knnj6egwm6jmpje9mfif11neo1nxcapqckwi8juebotd9kwbuyiyawuka4vzwwevwugztoyybglb+8slq1qndrrd1fvuo3erxfoiftoacrqaod9cajjaywjo8wpsjnbfn3flytbacfoyy/sd5/ahs74</latexit> b<latexit sha1_base64="zvwt5nu+mj/gcgd9djcmxqz+y=">aaab6nicbvbns8naej3ur1q/qh69lbbbulaqy9flx4rlpoq9lsj+3szsbsboqs+ho8efdeq7/im//gbzudtj4yelw3w8y8ibfcg9f9dgpr6xubw8xts7u3v5b+fcorenumwyxwmsqe1cngktsgw4edhkfnaoegbjm5n/+irk81g+memcfkshkoecuwol+6bf75crbtwdg6wslycvynhsl796g5ileurdbnw667mj8toqdgccp6veqjghbeyhlvugi1n81nzizqwxigctbpc5+nsio5hwkyiwnre1i73szct/vg5qwis/4zjjduqwbsmgpiyz4ma66qgtgxhdlf7aejaiiznhsjyeb/nlvdkuvb16txz3uwlc53eu4qro4rw8uiqg3eitwsbgcm/wcm+ocf6cd+djvpw8plj+an8wfuc4q</latexit> 3 a<latexit sha1_base64="kmvxvtnrvdvyhfr6amxaab5vho=">aaab6nicbvbns8naej3ur1q/qh69lbbbulaqy9flx4rlpoq9lsj+3szsbsboqs+ho8efdeq7/im//gbzudtj4yelw3w8y8ibfcg9f9dgpr6xubw8xts7u3v5b+fcorenumwyxwmsqe1cngktsgw4edhkfnaoegbjm5n/+irk81g+memcfkshkoecuwole9qv98svt+roqvajl5mk5gjy1+9qczsckvhgmrd9dze+blvhjob1iv1zhqnqzd7foqaytaz+antsmzvqykjjutachc/trujrsrtyzoiakv7zuj/xjc14zwfczmkbivblaptquxmzn+tavfijjhyqpni9lbcrlrrzmw6jruct/zykmnxql69wru7qdsu8zikcaknca4exeidbqejlwawhgd4htdhoc/ou/oxac4+cwx/ihz+qs7y</latexit> 3 z<latexit sha1_base64="xauhsnrws5733kvfckyzinjbile=">aaab6nicbvdltgjbeozff+il9ehlijhxrhbbri9elx4xyiobdzkdemhc7oxmztyecz/gxygewlvk3drahbsvppflvne6uibfcg9f9dnjr6xubw/ntws7u3v5b8fcoqenumwywwmsqhvcngktsgg4ethofnaoetolrzcxvalsjyzpygh9gb5cfn1fjp/qlx7rvlbtmdg6wslymlyfdvfb+6/zileurdbnw647mj8sdugc4etgvdvgncygosgoppbfqfzi/durorninyaxssum6u+jcyhkeb7yyogeplbyb+53vse175ey6t1kbkivhkoijyexvuckmrfjsyht3n5kjaqyoxnpbd8jzfxixnstmrlit3f6xadrzhhk7gfm7bguows3uoqembvamr/dmcoffexc+fq5j5s5hj9wn8aexknqa==</latexit> 3 = O(q,a,z) qq T (q, a, q )b t 1 (q) q Q O(q,a,z) qq,t(q, a, q )b t 1 (q) b<latexit sha1_base64="az+vsf6gsgih53caje9izzjol8=">aaab6nicbvbns8naej34wetx1aoxxsj4kkkt6lhoxwnf+wftkjvtpl6ytdivbcf4ixd4p49rd589+4bxq1gcdj/dmmjkxjfiydn1vz19y3nru7bt3n3bzgshr3tjxqxpsslrhubnrwkrrvokdjo4nmnaokbwfj5nffulaifg94ithfkshsoscubtsq9cv9utlt+loqvajl5my5gjs1+9qczsictkkhrt9dwe/yxqfezyabgxgp5qnqzd3rvuygb5ufoixnvhmqmnafjk5+nsio5exkyiwnrhfkvnzuj/xjff8nrheps5iotfowpjbitd9kidrnkcewukafvzwwedwuounaewll9eja1qxbusvo9r5fpnhkcbtuemlscdk6jdhtsgcqyg8ayv8ozi58v5dz4wrwtonmcf+b8/gdv94r</latexit> 4 Ufuk Topcu 4

14 <latexit sha1_base64="vqoleg+axbwbfseewac8xere=">aaab7nicbvdlsgnbeoz1gemr6thlybdijexgwvyegbeecwdkixmtibjknnzdazxces+woshrbz6d78gyfjhjsxokgo6qa7k4ilmoi6387a+sbm1nzuj7+7t39wwdg6bpooyw3wcqj3q6o4vio3kcbkrdjzwkysn4kxrczv/xetrgresbjzqdpuycebrsqgh6xhi5teoeiw3tnikveyuoqm9v7hq9uwbjyhuxsyzqeg6ofuocst7ndxdy8rgdmg7lioacuon83on5nwqftkitcfzk7+nkhpamwkdgxnshfklrz+j/xsxbq9voh4gs5yotfgqsjmjsd9ixmjoueso8lestiiasrqjps3ixjll6+szqxsxzyr91ffwjwliwencayl8oaaanahdwgagze8wyu8obhz4rw7h4vwnseboye/cd5/agqhju4=</latexit> B<latexit sha1_base64="escsizef6nwru8bx/jzbyg6wwt8=">aaab73icbvbns8naej3ur1q/qh69lbbbulegopf48v7aeowyk3bpzpuboqs+ie8efdeq3/hm//gbzudtj4yelw3w8y8ibfcg9f9dgobm1vbo8xdt7+wefr+fikpenumwyywmsqe1cngktsgm4edhkfnaoetox3dxv6hsjazpqgh9gh5cfn1fipk/uyfaq+65crbtvdgkwtlycvynhol796g5ileurdbnw667mj8toqdgccz6veqjghbeyhlvugi1ny3unzelqwxigctbpcf+nsio5hwyiwnre1i73qzcx/vg5qwls/4zjjduqxbsmgpiyzj8na66qgtg1hdlf7aejaiizniisjyeb/xlddk6qnpu1xu4rttqerxfoinzuaqbqag99cajjaq8ayv8ozmnbfn3flythacfoyu/sd5/agiyi+w</latexit> <latexit sha1_base64="escsizef6nwru8bx/jzbyg6wwt8=">aaab73icbvbns8naej3ur1q/qh69lbbbulegopf48v7aeowyk3bpzpuboqs+ie8efdeq3/hm//gbzudtj4yelw3w8y8ibfcg9f9dgobm1vbo8xdt7+wefr+fikpenumwyywmsqe1cngktsgm4edhkfnaoetox3dxv6hsjazpqgh9gh5cfn1fipk/uyfaq+65crbtvdgkwtlycvynhol796g5ileurdbnw667mj8toqdgccz6veqjghbeyhlvugi1ny3unzelqwxigctbpcf+nsio5hwyiwnre1i73qzcx/vg5qwls/4zjjduqxbsmgpiyzj8na66qgtg1hdlf7aejaiizniisjyeb/xlddk6qnpu1xu4rttqerxfoinzuaqbqag99cajjaq8ayv8ozmnbfn3flythacfoyu/sd5/agiyi+w</latexit> <latexit sha1_base64="escsizef6nwru8bx/jzbyg6wwt8=">aaab73icbvbns8naej3ur1q/qh69lbbbulegopf48v7aeowyk3bpzpuboqs+ie8efdeq3/hm//gbzudtj4yelw3w8y8ibfcg9f9dgobm1vbo8xdt7+wefr+fikpenumwyywmsqe1cngktsgm4edhkfnaoetox3dxv6hsjazpqgh9gh5cfn1fipk/uyfaq+65crbtvdgkwtlycvynhol796g5ileurdbnw667mj8toqdgccz6veqjghbeyhlvugi1ny3unzelqwxigctbpcf+nsio5hwyiwnre1i73qzcx/vg5qwls/4zjjduqxbsmgpiyzj8na66qgtg1hdlf7aejaiizniisjyeb/xlddk6qnpu1xu4rttqerxfoinzuaqbqag99cajjaq8ayv8ozmnbfn3flythacfoyu/sd5/agiyi+w</latexit> <latexit sha1_base64="escsizef6nwru8bx/jzbyg6wwt8=">aaab73icbvbns8naej3ur1q/qh69lbbbulegopf48v7aeowyk3bpzpuboqs+ie8efdeq3/hm//gbzudtj4yelw3w8y8ibfcg9f9dgobm1vbo8xdt7+wefr+fikpenumwyywmsqe1cngktsgm4edhkfnaoetox3dxv6hsjazpqgh9gh5cfn1fipk/uyfaq+65crbtvdgkwtlycvynhol796g5ileurdbnw667mj8toqdgccz6veqjghbeyhlvugi1ny3unzelqwxigctbpcf+nsio5hwyiwnre1i73qzcx/vg5qwls/4zjjduqxbsmgpiyzj8na66qgtg1hdlf7aejaiizniisjyeb/xlddk6qnpu1xu4rttqerxfoinzuaqbqag99cajjaq8ayv8ozmnbfn3flythacfoyu/sd5/agiyi+w</latexit> Belief evolution as a switched dynamical system new belief old belief b t (q) = action f q a(b t 1,z t )= (q z t,a t 1,b t 1 ) observation b<latexit sha1_base64="xjd5v7unvnn+uddndprnub1vhrs=">aaab6nicbvbns8naej3ur1q/qh69lbbbumqomeif48v7qeowyk3bpzhnnij/qlecji1v/kzx/jts1bwx8mn6bywzekaiujet+o4w19y3nrejawd3b/+gfhjungqgdzzlglvcahgwsudtcco4lcgguc8h4dua3n1bphsthmnqj+hq8pazaqzetdfrnivt5ycrxclkbhi1++as3ifkaotrmuk7npsy6kcczwwuqlghkxnsixusljvd7fzuktmzyocesbildzmrvycygmk9iqlbgvezsvetz66ymvyzlpuogslrweqiinj7g8y4aqzernlkfc3kryicrkjenzewll9eja1a1buo1u4vk/wbi4inmapnimhv1chohaexgm4rle4crzovz7nwswgtonmmf+b8/gdp54n</latexit> b<latexit sha1_base64="ib8zac5wbd9/ovfzlt7xpersce=">aaab6nicbvbns8naej3ur1q/qh69lbbbumqomeif48v7qeowyk3bpzhnnij/qlecji1v/kzx/jts1bwx8mn6bywzekaiujet+o4w19y3nrejawd3b/+gfhjungqgdzzlglvcahgwsudtcco4lcgguc8h4dua3n1bphsthmnqj+hq8pazaqzes9frnivt5ycrxclkbhi1++as3ifkaotrmuk7npsy6kcczwwuqlghkxnsixusljvd7fzuktmzyocesbildzmrvycygmk9iqlbgvezsvetz66ymvyzlpuogslrweqiinj7g8y4aqzernlkfc3kryicrkjenzewll9eja1a1buo1u4vk/wbi4inmapnimhv1chohaexgm4rle4crzovz7nwswgtonmmf+b8/gdra4o</latexit> 1 a<latexit sha1_base64="gl1urywrwjdcnj6f9achk1cng4m=">aaab6nicbvbns8naej3ur1q/qh69lbbbumqomeif48v7qeowyk3bpzhnnij/qlecji1v/kzx/jts1bwx8mn6bywzekaiujet+o4w19y3nrejawd3b/+gfhjungqgdzzlglvcahgwsudtcco4lcgguc8h4dua3n1bphsthmnqj+hq8pazaqzqtev1xxq+4czjv4oalajka//nubxcynubomqnzdzmn1flobm4lfvsjqllyzrerqwsrqj9bh7qljxzzudcwnmshszv3xmzjbserihtjkgz6wvvjv7ndvmtxvszllqullfojavxmrk9jczcixmiikllclubyvsrbvlxqztsif4yy+vklat6l1ua/exlfpnhkcrtuauzsgdk6jdhtsgcqyg8ayv8oyi58v5dz4wrqunnzmg3a+fwdp5yn</latexit> 1 z<latexit sha1_base64="ao7yovg3txfgnebi4fv/eoii=">aaab6nicbvbns8naej34wetx1aoxxsj4kkkv9fj4rgi/yalm1i7dbmlurqih8glbw8+ou8+w/ctjlo64obx3szzmwlesg1cd1vzv1bx1js7bv3n7z3dsvhrwdzwqhgwi1i1a6prciknw43adqkqrohavjc6mfqtr1sax/lbjbizqqosmgivd/w8xqnsvtwzydlxclkghve6avbj1kaotrmuk7npsy6kcczwuuymghkrnsahusljvd7ezuctm1sp+esbildzmpvycygmk9jglbgvez1ivevz66qmvizlpuogtzrweqiinj9g/s5wqzewnlkfc3krykcrkjenaewfl9ejs1qxtuvvo8uyrxri4chmmjniehl1cdw6hdaxgm4ble4crzovz7nzmw1ecfoyi/sd5/aeqcom</latexit> 1 a<latexit sha1_base64="niruwwwhuyejnuz/tbo71yq8=">aaab6nicbvbns8naej3ur1q/qh69lbbbumqomeif48v7qeowyk3bpzhnnij/qlecji1v/kzx/jts1bwx8mn6bywzekaiujet+o4w19y3nrejawd3b/+gfhjungqgdzzlglvcahgwsudtcco4lcgguc8h4dua3n1bphsthmnqj+hq8pazaqzqu1frnivt5ycrxclkbhi1++as3ifkaotrmuk7npsy6kcczwwuqlghkxnsixusljvd7fzuktmzyocesbildzmrvycygmk9iqlbgvezsvetz66ymvyzlpuogslrweqiinj7g8y4aqzernlkfc3kryicrkjenzewll9eja1a1buo1u4vk/wbi4inmapnimhv1chohaexgm4rle4crzovz7nwswgtonmmf+b8/gdrayo</latexit> z<latexit sha1_base64="zzkjtcjfguoajipvjcjutanly=">aaab6nicbvbns8naej34wetx1aoxxsj4kkkv9fj4rgi/yalm1y7dbmlurkih8glbw8+ou8+w/ctjlo64obx3szzmwleikmuu63s7k6tr6xwdgqbu/s7uxdg6bjk414wwy1i3aq4fio3ukdk7urzggwst4lrzdrvxjtrkwecjxw6idjulbkfrp/qlx7zxkbswdgswtlydlyfhvlb66/zilevfijdwm47kj+hnvkjjkki3ntyhbeqhvgopohe3fjy7dujordinyaxtksqz9fderinjxlfgoyokq7otcx/ve6k4zwfczwkybwblwptstam79jxjoui4touwleythq6opq5toybglb68tjrvindeqd5dlgvxerwfoiytoamlqegt1chbjaywdo8wpsjnrfn3fmyt644+cwr/ihz+qmrjon</latexit> b<latexit sha1_base64="ambb7fjcw54wngon41/vohqvm=">aaab6nicbvbns8naej3ur1q/qh69lbbbumqomeif48v7qeowyk3bpzhnnij/qlecji1v/kzx/jts1bwx8mn6bywzekaiujet+o4w19y3nrejawd3b/+gfhjungqgdzzlglvcahgwsudtcco4lcgguc8h4dua3n1bphsthmnqj+hq8pazaqzerr/xlfrbpzkfxi5aqcorr98ldvelmqmmyofp3tcxfkav4uzgtnrlnsaujekqu5zkgqhs/mpu3jmlqejyvlgjjxf9knnj6egwm6jmpje9mfif11neo1nxcapqckwi8juebotd9kwbuyiyawuka4vzwwevwugztoyybglb+8slq1qndrrd1fvuo3erxfoiftoacrqaod9cajjaywjo8wpsjnbfn3flytbacfoyy/sd5/ahs74</latexit> b<latexit sha1_base64="zvwt5nu+mj/gcgd9djcmxqz+y=">aaab6nicbvbns8naej3ur1q/qh69lbbbulaqy9flx4rlpoq9lsj+3szsbsboqs+ho8efdeq7/im//gbzudtj4yelw3w8y8ibfcg9f9dgpr6xubw8xts7u3v5b+fcorenumwyxwmsqe1cngktsgw4edhkfnaoegbjm5n/+irk81g+memcfkshkoecuwol+6bf75crbtwdg6wslycvynhsl796g5ileurdbnw667mj8toqdgccp6veqjghbeyhlvugi1n81nzizqwxigctbpc5+nsio5hwkyiwnre1i73szct/vg5qwis/4zjjduqwbsmgpiyz4ma66qgtgxhdlf7aejaiiznhsjyeb/nlvdkuvb16txz3uwlc53eu4qro4rw8uiqg3eitwsbgcm/wcm+ocf6cd+djvpw8plj+an8wfuc4q</latexit> 3 a<latexit sha1_base64="kmvxvtnrvdvyhfr6amxaab5vho=">aaab6nicbvbns8naej3ur1q/qh69lbbbulaqy9flx4rlpoq9lsj+3szsbsboqs+ho8efdeq7/im//gbzudtj4yelw3w8y8ibfcg9f9dgpr6xubw8xts7u3v5b+fcorenumwyxwmsqe1cngktsgw4edhkfnaoegbjm5n/+irk81g+memcfkshkoecuwole9qv98svt+roqvajl5mk5gjy1+9qczsckvhgmrd9dze+blvhjob1iv1zhqnqzd7foqaytaz+antsmzvqykjjutachc/trujrsrtyzoiakv7zuj/xjc14zwfczmkbivblaptquxmzn+tavfijjhyqpni9lbcrlrrzmw6jruct/zykmnxql69wru7qdsu8zikcaknca4exeidbqejlwawhgd4htdhoc/ou/oxac4+cwx/ihz+qs7y</latexit> 3 z<latexit sha1_base64="xauhsnrws5733kvfckyzinjbile=">aaab6nicbvdltgjbeozff+il9ehlijhxrhbbri9elx4xyiobdzkdemhc7oxmztyecz/gxygewlvk3drahbsvppflvne6uibfcg9f9dnjr6xubw/ntws7u3v5b8fcoqenumwywwmsqhvcngktsgg4ethofnaoetolrzcxvalsjyzpygh9gb5cfn1fjp/qlx7rvlbtmdg6wslymlyfdvfb+6/zileurdbnw647mj8sdugc4etgvdvgncygosgoppbfqfzi/durorninyaxssum6u+jcyhkeb7yyogeplbyb+53vse175ey6t1kbkivhkoijyexvuckmrfjsyht3n5kjaqyoxnpbd8jzfxixnstmrlit3f6xadrzhhk7gfm7bguows3uoqembvamr/dmcoffexc+fq5j5s5hj9wn8aexknqa==</latexit> 3 = O(q,a,z) qq T (q, a, q )b t 1 (q) q Q O(q,a,z) qq,t(q, a, q )b t 1 (q) b<latexit sha1_base64="az+vsf6gsgih53caje9izzjol8=">aaab6nicbvbns8naej34wetx1aoxxsj4kkkt6lhoxwnf+wftkjvtpl6ytdivbcf4ixd4p49rd589+4bxq1gcdj/dmmjkxjfiydn1vz19y3nru7bt3n3bzgshr3tjxqxpsslrhubnrwkrrvokdjo4nmnaokbwfj5nffulaifg94ithfkshsoscubtsq9cv9utlt+loqvajl5my5gjs1+9qczsictkkhrt9dwe/yxqfezyabgxgp5qnqzd3rvuygb5ufoixnvhmqmnafjk5+nsio5exkyiwnrhfkvnzuj/xjff8nrheps5iotfowpjbitd9kidrnkcewukafvzwwedwuounaewll9eja1qxbusvo9r5fpnhkcbtuemlscdk6jdhtsgcqyg8ayv8ozi58v5dz4wrwtonmcf+b8/gdv94r</latexit> 4 States and actions in the (O)MD are discrete and finite but the belief evolves over a continuous space Results in a switched system with modes induced from the actions b t = f a (b t 1,z t ) <latexit sha1_base64="rzr/ttro+dsrj1h8vc+9tynx/i=">aaaccnicbvdlsgnbejynrxhfqx69jayhgobdkohfchrxgme8ibuwclsmmtwuyvejecvfgrxjwo4tuv8obfoenoikfduvvn91dxiy4asv6nnilivlk/nvwtr6xuawub3tufeikavtsesy5rhfba9zhtgi1ooli4enwnmbxi/95jtikfhhqxj1glil+q+pws5jr7ngv4evsucqtzoes5kzzyo+mhfxzje34csivbymwezkgrzai55pftjwgssbcoieq1bsugtkokccryqoakiswedkittunscbuj58mskhwuli5k6qsat9fdesgklhognowmcftxrjcx/vhyc/kun5wgcaavpdjgfcawrhuecu1wycmkocags61sx7rnjkojcjoee/bledkolo3tcuxrfi9yulioz1gerirueoim5qddurry/ogbin+jedhejy9pa87iznbrhxif7ammac=</latexit> Ufuk Topcu 4

15 rivacy in terms of the belief of the intruder A set of secret states: Q s Q <latexit sha1_base64="u/v3euvpb7boagdg/jw5jxcjfo=">aaab83icbvbns8naej3ur1q/qh69lbbbumqomeif48ta9oqtlsn+3szsbstors+je8efdeq3/gm//gbzudtj4yelw3w8y8mjxcoot+o4wnza3tnejuaw//4cofhzsnkmmgw+xrca6g1ldpvc8hqil76aaziuvboo7+d+54lrixl1ijoubzedkhejrtfkfrnvfjofhinp9ssvt+ouqnajl5mk5gjy1/+igfzzbuysy3pew6kwzrqfezywcnde8pg9mh71mqamxnmf3ccmxvhmqkngfjkf+ntismnjjnfoooki7qzcx/vf6gwwfsrnkcuxbrlkmbc5ggqgdccozxyqpkw9lbcrlrthjamkg3bw315nbrrve+qwmtev+p3erxfoinzuaqbqaod9cafjbi4rle4c3jnbfn3flythacfoyu/sd5/ag9xzf8</latexit> to represent the information we want to keep private, e.g., intent, target, goal, preference, etc. Ufuk Topcu 5

16 rivacy in terms of the belief of the intruder A set of secret states: Q s Q <latexit sha1_base64="u/v3euvpb7boagdg/jw5jxcjfo=">aaab83icbvbns8naej3ur1q/qh69lbbbumqomeif48ta9oqtlsn+3szsbstors+je8efdeq3/gm//gbzudtj4yelw3w8y8mjxcoot+o4wnza3tnejuaw//4cofhzsnkmmgw+xrca6g1ldpvc8hqil76aaziuvboo7+d+54lrixl1ijoubzedkhejrtfkfrnvfjofhinp9ssvt+ouqnajl5mk5gjy1/+igfzzbuysy3pew6kwzrqfezywcnde8pg9mh71mqamxnmf3ccmxvhmqkngfjkf+ntismnjjnfoooki7qzcx/vf6gwwfsrnkcuxbrlkmbc5ggqgdccozxyqpkw9lbcrlrthjamkg3bw315nbrrve+qwmtev+p3erxfoinzuaqbqaod9cafjbi4rle4c3jnbfn3flythacfoyu/sd5/ag9xzf8</latexit> to represent the information we want to keep private, e.g., intent, target, goal, preference, etc. rivacy is breached if the intruder becomes confident in that the system is in a secret state with a probability larger than a threshold at a time t: X qq s b t (q) > <latexit sha1_base64="bpv3rhz8td6+gzvi8bswp8stu=">aaacbxicbvc7sgnbfjnrxhfq5zadayhnme3cmojarvlbmwdsmgzncwmqzmnzozqljsgrnhakodv6nkqlttxw4xdovdx7txazqrtjffu5ldw19y38zmfred3z94/akookzgcmqiq6qiowktbum9kojue8ykqvdg+nfuubseujca/hmely1bcpbhpi/nsacs7qcjjwpy99uk8hvpdhbj9rhnylelttmzas4tnynfkkhm19el8ijjjjhptque6suymsmmjgjguvusrgeij6pgooqjyobjr7ygjjdkdysrncq1n6u+jfhglxjwwnrzpgvrpuj/xifr4vu3psjonbf4vihmgnqrneyce1qsrnnyeiqlnbdceasywck5gq3mwxlmzunby5x6rbf6ncwrbfgbjsacy5bfdybgmgadb7bm3gfb9at9wk9wx/z1pyvzryc7a+fwaq15hb</latexit> (sum over the secret states) Ufuk Topcu 5

17 rivacy verification b<latexit sha1_base64="xjd5v7unvnn+uddndprnub1vhrs=">aaab6nicbvbns8naej3ur1q/qh69lbbbumqomeif48v7qeowyk3bpzhnnij/qlecji1v/kzx/jts1bwx8mn6bywzekaiujet+o4w19y3nrejawd3b/+gfhjungqgdzzlglvcahgwsudtcco4lcgguc8h4dua3n1bphsthmnqj+hq8pazaqzetdfrnivt5ycrxclkbhi1++as3ifkaotrmuk7npsy6kcczwwuqlghkxnsixusljvd7fzuktmzyocesbildzmrvycygmk9iqlbgvezsvetz66ymvyzlpuogslrweqiinj7g8y4aqzernlkfc3kryicrkjenzewll9eja1a1buo1u4vk/wbi4inmapnimhv1chohaexgm4rle4crzovz7nwswgtonmmf+b8/gdp54n</latexit> b<latexit sha1_base64="ib8zac5wbd9/ovfzlt7xpersce=">aaab6nicbvbns8naej3ur1q/qh69lbbbumqomeif48v7qeowyk3bpzhnnij/qlecji1v/kzx/jts1bwx8mn6bywzekaiujet+o4w19y3nrejawd3b/+gfhjungqgdzzlglvcahgwsudtcco4lcgguc8h4dua3n1bphsthmnqj+hq8pazaqzes9frnivt5ycrxclkbhi1++as3ifkaotrmuk7npsy6kcczwwuqlghkxnsixusljvd7fzuktmzyocesbildzmrvycygmk9iqlbgvezsvetz66ymvyzlpuogslrweqiinj7g8y4aqzernlkfc3kryicrkjenzewll9eja1a1buo1u4vk/wbi4inmapnimhv1chohaexgm4rle4crzovz7nwswgtonmmf+b8/gdra4o</latexit> 1 b<latexit sha1_base64="ambb7fjcw54wngon41/vohqvm=">aaab6nicbvbns8naej3ur1q/qh69lbbbumqomeif48v7qeowyk3bpzhnnij/qlecji1v/kzx/jts1bwx8mn6bywzekaiujet+o4w19y3nrejawd3b/+gfhjungqgdzzlglvcahgwsudtcco4lcgguc8h4dua3n1bphsthmnqj+hq8pazaqzerr/xlfrbpzkfxi5aqcorr98ldvelmqmmyofp3tcxfkav4uzgtnrlnsaujekqu5zkgqhs/mpu3jmlqejyvlgjjxf9knnj6egwm6jmpje9mfif11neo1nxcapqckwi8juebotd9kwbuyiyawuka4vzwwevwugztoyybglb+8slq1qndrrd1fvuo3erxfoiftoacrqaod9cajjaywjo8wpsjnbfn3flytbacfoyy/sd5/ahs74</latexit> b<latexit sha1_base64="az+vsf6gsgih53caje9izzjol8=">aaab6nicbvbns8naej34wetx1aoxxsj4kkkt6lhoxwnf+wftkjvtpl6ytdivbcf4ixd4p49rd589+4bxq1gcdj/dmmjkxjfiydn1vz19y3nru7bt3n3bzgshr3tjxqxpsslrhubnrwkrrvokdjo4nmnaokbwfj5nffulaifg94ithfkshsoscubtsq9cv9utlt+loqvajl5my5gjs1+9qczsictkkhrt9dwe/yxqfezyabgxgp5qnqzd3rvuygb5ufoixnvhmqmnafjk5+nsio5exkyiwnrhfkvnzuj/xjff8nrheps5iotfowpjbitd9kidrnkcewukafvzwwedwuounaewll9eja1qxbusvo9r5fpnhkcbtuemlscdk6jdhtsgcqyg8ayv8ozi58v5dz4wrwtonmcf+b8/gdv94r</latexit> 4 b<latexit sha1_base64="zvwt5nu+mj/gcgd9djcmxqz+y=">aaab6nicbvbns8naej3ur1q/qh69lbbbulaqy9flx4rlpoq9lsj+3szsbsboqs+ho8efdeq7/im//gbzudtj4yelw3w8y8ibfcg9f9dgpr6xubw8xts7u3v5b+fcorenumwyxwmsqe1cngktsgw4edhkfnaoegbjm5n/+irk81g+memcfkshkoecuwol+6bf75crbtwdg6wslycvynhsl796g5ileurdbnw667mj8toqdgccp6veqjghbeyhlvugi1n81nzizqwxigctbpc5+nsio5hwkyiwnre1i73szct/vg5qwis/4zjjduqwbsmgpiyz4ma66qgtgxhdlf7aejaiiznhsjyeb/nlvdkuvb16txz3uwlc53eu4qro4rw8uiqg3eitwsbgcm/wcm+ocf6cd+djvpw8plj+an8wfuc4q</latexit> 3 Ufuk Topcu 6

20 <latexit sha1_base64="e8s1imn8+6ascd7bi8ngwqzryw=">aaacdxicbvdlssnafj34ruvdekmwiw6kukv7llgxmuv+4amhmnph6etgzeurid7jxv9y4umste3f+jzmilyeuha4517uvcelgrxsnl+pewv1bx1kz5ct7z1ff++ikoeeiriee95wacjibqllqx6mqcceay63vgy97v3wawnwls5icej8dckivyksnvj+ayxhbll3jqp5rntoi8qriusxopmrv8yaoywxskycvfcblqt/ooijagekjasrn8yy+mkmetkggrloxeqyzlgq+grguiahjnov8mme6umdd/iqkjptnxfeykohjgenurmtxtzxi7+5/ut6teclizxiieksv+wgwzgxkxobyijjnfmgeu3wrquayyyjvgguvgjx/8ilp1gvwwa1+fv5pnoo4sugqhaeqstafaqir1ejtrnadekiv6fv71j61n+191rqkftmh6a+j/4kzwb</latexit> rivacy verification b<latexit sha1_base64="xjd5v7unvnn+uddndprnub1vhrs=">aaab6nicbvbns8naej3ur1q/qh69lbbbumqomeif48v7qeowyk3bpzhnnij/qlecji1v/kzx/jts1bwx8mn6bywzekaiujet+o4w19y3nrejawd3b/+gfhjungqgdzzlglvcahgwsudtcco4lcgguc8h4dua3n1bphsthmnqj+hq8pazaqzetdfrnivt5ycrxclkbhi1++as3ifkaotrmuk7npsy6kcczwwuqlghkxnsixusljvd7fzuktmzyocesbildzmrvycygmk9iqlbgvezsvetz66ymvyzlpuogslrweqiinj7g8y4aqzernlkfc3kryicrkjenzewll9eja1a1buo1u4vk/wbi4inmapnimhv1chohaexgm4rle4crzovz7nwswgtonmmf+b8/gdp54n</latexit> b<latexit sha1_base64="ib8zac5wbd9/ovfzlt7xpersce=">aaab6nicbvbns8naej3ur1q/qh69lbbbumqomeif48v7qeowyk3bpzhnnij/qlecji1v/kzx/jts1bwx8mn6bywzekaiujet+o4w19y3nrejawd3b/+gfhjungqgdzzlglvcahgwsudtcco4lcgguc8h4dua3n1bphsthmnqj+hq8pazaqzes9frnivt5ycrxclkbhi1++as3ifkaotrmuk7npsy6kcczwwuqlghkxnsixusljvd7fzuktmzyocesbildzmrvycygmk9iqlbgvezsvetz66ymvyzlpuogslrweqiinj7g8y4aqzernlkfc3kryicrkjenzewll9eja1a1buo1u4vk/wbi4inmapnimhv1chohaexgm4rle4crzovz7nwswgtonmmf+b8/gdra4o</latexit> 1 b<latexit sha1_base64="ambb7fjcw54wngon41/vohqvm=">aaab6nicbvbns8naej3ur1q/qh69lbbbumqomeif48v7qeowyk3bpzhnnij/qlecji1v/kzx/jts1bwx8mn6bywzekaiujet+o4w19y3nrejawd3b/+gfhjungqgdzzlglvcahgwsudtcco4lcgguc8h4dua3n1bphsthmnqj+hq8pazaqzerr/xlfrbpzkfxi5aqcorr98ldvelmqmmyofp3tcxfkav4uzgtnrlnsaujekqu5zkgqhs/mpu3jmlqejyvlgjjxf9knnj6egwm6jmpje9mfif11neo1nxcapqckwi8juebotd9kwbuyiyawuka4vzwwevwugztoyybglb+8slq1qndrrd1fvuo3erxfoiftoacrqaod9cajjaywjo8wpsjnbfn3flytbacfoyy/sd5/ahs74</latexit> R(b ) Set of reachable belief states b<latexit sha1_base64="zvwt5nu+mj/gcgd9djcmxqz+y=">aaab6nicbvbns8naej3ur1q/qh69lbbbulaqy9flx4rlpoq9lsj+3szsbsboqs+ho8efdeq7/im//gbzudtj4yelw3w8y8ibfcg9f9dgpr6xubw8xts7u3v5b+fcorenumwyxwmsqe1cngktsgw4edhkfnaoegbjm5n/+irk81g+memcfkshkoecuwol+6bf75crbtwdg6wslycvynhsl796g5ileurdbnw667mj8toqdgccp6veqjghbeyhlvugi1n81nzizqwxigctbpc5+nsio5hwkyiwnre1i73szct/vg5qwis/4zjjduqwbsmgpiyz4ma66qgtgxhdlf7aejaiiznhsjyeb/nlvdkuvb16txz3uwlc53eu4qro4rw8uiqg3eitwsbgcm/wcm+ocf6cd+djvpw8plj+an8wfuc4q</latexit> 3 b<latexit sha1_base64="az+vsf6gsgih53caje9izzjol8=">aaab6nicbvbns8naej34wetx1aoxxsj4kkkt6lhoxwnf+wftkjvtpl6ytdivbcf4ixd4p49rd589+4bxq1gcdj/dmmjkxjfiydn1vz19y3nru7bt3n3bzgshr3tjxqxpsslrhubnrwkrrvokdjo4nmnaokbwfj5nffulaifg94ithfkshsoscubtsq9cv9utlt+loqvajl5my5gjs1+9qczsictkkhrt9dwe/yxqfezyabgxgp5qnqzd3rvuygb5ufoixnvhmqmnafjk5+nsio5exkyiwnrhfkvnzuj/xjff8nrheps5iotfowpjbitd9kidrnkcewukafvzwwedwuounaewll9eja1qxbusvo9r5fpnhkcbtuemlscdk6jdhtsgcqyg8ayv8ozi58v5dz4wrwtonmcf+b8/gdv94r</latexit> 4 Ufuk Topcu 6

21 <latexit sha1_base64="e8s1imn8+6ascd7bi8ngwqzryw=">aaacdxicbvdlssnafj34ruvdekmwiw6kukv7llgxmuv+4amhmnph6etgzeurid7jxv9y4umste3f+jzmilyeuha4517uvcelgrxsnl+pewv1bx1kz5ct7z1ff++ikoeeiriee95wacjibqllqx6mqcceay63vgy97v3wawnwls5icej8dckivyksnvj+ayxhbll3jqp5rntoi8qriusxopmrv8yaoywxskycvfcblqt/ooijagekjasrn8yy+mkmetkggrloxeqyzlgq+grguiahjnov8mme6umdd/iqkjptnxfeykohjgenurmtxtzxi7+5/ut6teclizxiieksv+wgwzgxkxobyijjnfmgeu3wrquayyyjvgguvgjx/8ilp1gvwwa1+fv5pnoo4sugqhaeqstafaqir1ejtrnadekiv6fv71j61n+191rqkftmh6a+j/4kzwb</latexit> <latexit sha1_base64="olsodby1fete/b56xfmtumvt7q=">aaacf3icbvdlsgnbejz1gemr6thlyhdijexgqs9cibeucwdsssyo+kkqyfzqkyclfefxvhhqxkve/btnkycawnbqvhxt3exhgiurc9jaxlldw9t5hf3nres3s7tdvleggdrajslz9qkdwebriuua7lkadxdlh15lfusepojreiejgnya9ke44yilrxcqkodhgv6e45hvwiezqxj1wbe8mlagmiwxofypwzraxctjbtjdhwv8oli5yeecitvkmobcxoplqizwlgesdrefmph3oabrsajsbtv4am8da6zq9sookzyovydsgig1cnzdmzr5r1m/m/rjni7cfmexglcykaleokwmtkzkmwul8bqjdshthj9q8kgvfkgosq8dsgef3mrnctl+7rcutkrvmuzohlkkbyrerhjoamsa1indclia3kil+tvedsejtfjfdq6zmxmdsgfgb/fzngvw==</latexit> rivacy verification b<latexit sha1_base64="xjd5v7unvnn+uddndprnub1vhrs=">aaab6nicbvbns8naej3ur1q/qh69lbbbumqomeif48v7qeowyk3bpzhnnij/qlecji1v/kzx/jts1bwx8mn6bywzekaiujet+o4w19y3nrejawd3b/+gfhjungqgdzzlglvcahgwsudtcco4lcgguc8h4dua3n1bphsthmnqj+hq8pazaqzetdfrnivt5ycrxclkbhi1++as3ifkaotrmuk7npsy6kcczwwuqlghkxnsixusljvd7fzuktmzyocesbildzmrvycygmk9iqlbgvezsvetz66ymvyzlpuogslrweqiinj7g8y4aqzernlkfc3kryicrkjenzewll9eja1a1buo1u4vk/wbi4inmapnimhv1chohaexgm4rle4crzovz7nwswgtonmmf+b8/gdp54n</latexit> b<latexit sha1_base64="ib8zac5wbd9/ovfzlt7xpersce=">aaab6nicbvbns8naej3ur1q/qh69lbbbumqomeif48v7qeowyk3bpzhnnij/qlecji1v/kzx/jts1bwx8mn6bywzekaiujet+o4w19y3nrejawd3b/+gfhjungqgdzzlglvcahgwsudtcco4lcgguc8h4dua3n1bphsthmnqj+hq8pazaqzes9frnivt5ycrxclkbhi1++as3ifkaotrmuk7npsy6kcczwwuqlghkxnsixusljvd7fzuktmzyocesbildzmrvycygmk9iqlbgvezsvetz66ymvyzlpuogslrweqiinj7g8y4aqzernlkfc3kryicrkjenzewll9eja1a1buo1u4vk/wbi4inmapnimhv1chohaexgm4rle4crzovz7nwswgtonmmf+b8/gdra4o</latexit> 1 b<latexit sha1_base64="ambb7fjcw54wngon41/vohqvm=">aaab6nicbvbns8naej3ur1q/qh69lbbbumqomeif48v7qeowyk3bpzhnnij/qlecji1v/kzx/jts1bwx8mn6bywzekaiujet+o4w19y3nrejawd3b/+gfhjungqgdzzlglvcahgwsudtcco4lcgguc8h4dua3n1bphsthmnqj+hq8pazaqzerr/xlfrbpzkfxi5aqcorr98ldvelmqmmyofp3tcxfkav4uzgtnrlnsaujekqu5zkgqhs/mpu3jmlqejyvlgjjxf9knnj6egwm6jmpje9mfif11neo1nxcapqckwi8juebotd9kwbuyiyawuka4vzwwevwugztoyybglb+8slq1qndrrd1fvuo3erxfoiftoacrqaod9cajjaywjo8wpsjnbfn3flytbacfoyy/sd5/ahs74</latexit> R(b ) Set of reachable belief states b<latexit sha1_base64="zvwt5nu+mj/gcgd9djcmxqz+y=">aaab6nicbvbns8naej3ur1q/qh69lbbbulaqy9flx4rlpoq9lsj+3szsbsboqs+ho8efdeq7/im//gbzudtj4yelw3w8y8ibfcg9f9dgpr6xubw8xts7u3v5b+fcorenumwyxwmsqe1cngktsgw4edhkfnaoegbjm5n/+irk81g+memcfkshkoecuwol+6bf75crbtwdg6wslycvynhsl796g5ileurdbnw667mj8toqdgccp6veqjghbeyhlvugi1n81nzizqwxigctbpc5+nsio5hwkyiwnre1i73szct/vg5qwis/4zjjduqwbsmgpiyz4ma66qgtgxhdlf7aejaiiznhsjyeb/nlvdkuvb16txz3uwlc53eu4qro4rw8uiqg3eitwsbgcm/wcm+ocf6cd+djvpw8plj+an8wfuc4q</latexit> 3 b<latexit sha1_base64="az+vsf6gsgih53caje9izzjol8=">aaab6nicbvbns8naej34wetx1aoxxsj4kkkt6lhoxwnf+wftkjvtpl6ytdivbcf4ixd4p49rd589+4bxq1gcdj/dmmjkxjfiydn1vz19y3nru7bt3n3bzgshr3tjxqxpsslrhubnrwkrrvokdjo4nmnaokbwfj5nffulaifg94ithfkshsoscubtsq9cv9utlt+loqvajl5my5gjs1+9qczsictkkhrt9dwe/yxqfezyabgxgp5qnqzd3rvuygb5ufoixnvhmqmnafjk5+nsio5exkyiwnrhfkvnzuj/xjff8nrheps5iotfowpjbitd9kidrnkcewukafvzwwedwuounaewll9eja1qxbusvo9r5fpnhkcbtuemlscdk6jdhtsgcqyg8ayv8ozi58v5dz4wrwtonmcf+b8/gdv94r</latexit> 4 Unsafe belief set B u = {b X qq s b(q) > } <latexit sha1_base64="kgrqoc+qrdi9qlkory3rwuwroiq=">aaacf3icbvbnswmxfmz6wetx1aoxybhqpeyqobdf9okxbvsl3bk8tdmnmluk6xq1vxxvwrxjwo4lvv/hvtoodgsgmfd4mqljzrrx3s9nbn5hcwk5t5jfxvvfcxsbdd1lchcaytikwqeoclnktymm5wykvbhjzehvrkx97r5vmkbwxw5ibhql6zacxkpboewlmdcl3mgutmt8n7xycisac+k7ga6cwsdq7o/s4iax4wfipurdzxjvqopogkpq+tbeukelyzwlrpubfppaami5xmet/rnabshy5twipbun1kx7kyvg+vnu5eyj5p8fj9vzgchooqjs5sqgnvzh4n9dmtoeltizj4zk8nook3bsijwqcbezostwosvafln/xaqhcoixveztcd55flsyx7r+xd6nhx4nrsrw7toj1uqh46qrfoglvqdrhgj7qc3p1hp1n5815/xmdcyy7o+gni9vrnigma==</latexit> Verify whether R(b ) \ B u = ; That is, privacy is not breached at any time t. Ufuk Topcu 6

22 How to attempt to verify the set emptiness? In general an undecidable problem. OMDs have been heavily studied in planning. Usually heuristics (or impractical exhaustive methods) Approximate results with no guarantee Hard to adapt to the problem in hand Abstraction-based techniques, i.e., discretization, have their intrinsic limitations The proposed approach: search for algebraic certificates that witness privacy Ufuk Topcu 7

23 <latexit sha1_base64="e8s1imn8+6ascd7bi8ngwqzryw=">aaacdxicbvdlssnafj34ruvdekmwiw6kukv7llgxmuv+4amhmnph6etgzeurid7jxv9y4umste3f+jzmilyeuha4517uvcelgrxsnl+pewv1bx1kz5ct7z1ff++ikoeeiriee95wacjibqllqx6mqcceay63vgy97v3wawnwls5icej8dckivyksnvj+ayxhbll3jqp5rntoi8qriusxopmrv8yaoywxskycvfcblqt/ooijagekjasrn8yy+mkmetkggrloxeqyzlgq+grguiahjnov8mme6umdd/iqkjptnxfeykohjgenurmtxtzxi7+5/ut6teclizxiieksv+wgwzgxkxobyijjnfmgeu3wrquayyyjvgguvgjx/8ilp1gvwwa1+fv5pnoo4sugqhaeqstafaqir1ejtrnadekiv6fv71j61n+191rqkftmh6a+j/4kzwb</latexit> <latexit sha1_base64="e8s1imn8+6ascd7bi8ngwqzryw=">aaacdxicbvdlssnafj34ruvdekmwiw6kukv7llgxmuv+4amhmnph6etgzeurid7jxv9y4umste3f+jzmilyeuha4517uvcelgrxsnl+pewv1bx1kz5ct7z1ff++ikoeeiriee95wacjibqllqx6mqcceay63vgy97v3wawnwls5icej8dckivyksnvj+ayxhbll3jqp5rntoi8qriusxopmrv8yaoywxskycvfcblqt/ooijagekjasrn8yy+mkmetkggrloxeqyzlgq+grguiahjnov8mme6umdd/iqkjptnxfeykohjgenurmtxtzxi7+5/ut6teclizxiieksv+wgwzgxkxobyijjnfmgeu3wrquayyyjvgguvgjx/8ilp1gvwwa1+fv5pnoo4sugqhaeqstafaqir1ejtrnadekiv6fv71j61n+191rqkftmh6a+j/4kzwb</latexit> Over-approximation of the reachable belief set If there exists a function V such that V (f a (b t 1,z)) V (b t 1 ) <, 8 V { B b V = {b B V (b) apple 1} (with additional standard technical assumptions), 8a A, z Z, b t 1 V, apple } Then, R(b ) V. 8 V = {b B V (b) apple 1} b<latexit sha1_base64="xjd5v7unvnn+uddndprnub1vhrs=">aaab6nicbvbns8naej3ur1q/qh69lbbbumqomeif48v7qeowyk3bpzhnnij/qlecji1v/kzx/jts1bwx8mn6bywzekaiujet+o4w19y3nrejawd3b/+gfhjungqgdzzlglvcahgwsudtcco4lcgguc8h4dua3n1bphsthmnqj+hq8pazaqzetdfrnivt5ycrxclkbhi1++as3ifkaotrmuk7npsy6kcczwwuqlghkxnsixusljvd7fzuktmzyocesbildzmrvycygmk9iqlbgvezsvetz66ymvyzlpuogslrweqiinj7g8y4aqzernlkfc3kryicrkjenzewll9eja1a1buo1u4vk/wbi4inmapnimhv1chohaexgm4rle4crzovz7nwswgtonmmf+b8/gdp54n</latexit> R(b ) Ufuk Topcu 8

24 <latexit sha1_base64="e8s1imn8+6ascd7bi8ngwqzryw=">aaacdxicbvdlssnafj34ruvdekmwiw6kukv7llgxmuv+4amhmnph6etgzeurid7jxv9y4umste3f+jzmilyeuha4517uvcelgrxsnl+pewv1bx1kz5ct7z1ff++ikoeeiriee95wacjibqllqx6mqcceay63vgy97v3wawnwls5icej8dckivyksnvj+ayxhbll3jqp5rntoi8qriusxopmrv8yaoywxskycvfcblqt/ooijagekjasrn8yy+mkmetkggrloxeqyzlgq+grguiahjnov8mme6umdd/iqkjptnxfeykohjgenurmtxtzxi7+5/ut6teclizxiieksv+wgwzgxkxobyijjnfmgeu3wrquayyyjvgguvgjx/8ilp1gvwwa1+fv5pnoo4sugqhaeqstafaqir1ejtrnadekiv6fv71j61n+191rqkftmh6a+j/4kzwb</latexit> <latexit sha1_base64="e8s1imn8+6ascd7bi8ngwqzryw=">aaacdxicbvdlssnafj34ruvdekmwiw6kukv7llgxmuv+4amhmnph6etgzeurid7jxv9y4umste3f+jzmilyeuha4517uvcelgrxsnl+pewv1bx1kz5ct7z1ff++ikoeeiriee95wacjibqllqx6mqcceay63vgy97v3wawnwls5icej8dckivyksnvj+ayxhbll3jqp5rntoi8qriusxopmrv8yaoywxskycvfcblqt/ooijagekjasrn8yy+mkmetkggrloxeqyzlgq+grguiahjnov8mme6umdd/iqkjptnxfeykohjgenurmtxtzxi7+5/ut6teclizxiieksv+wgwzgxkxobyijjnfmgeu3wrquayyyjvgguvgjx/8ilp1gvwwa1+fv5pnoo4sugqhaeqstafaqir1ejtrnadekiv6fv71j61n+191rqkftmh6a+j/4kzwb</latexit> Over-approximation of the reachable belief set If there exists a function V such that V (f a (b t 1,z)) V (b t 1 ) <, 8 V { B b V = {b B V (b) apple 1} (with additional standard technical assumptions), 8a A, z Z, b t 1 V, apple } Then, R(b ) V. 8 V = {b B V (b) apple 1} b<latexit sha1_base64="xjd5v7unvnn+uddndprnub1vhrs=">aaab6nicbvbns8naej3ur1q/qh69lbbbumqomeif48v7qeowyk3bpzhnnij/qlecji1v/kzx/jts1bwx8mn6bywzekaiujet+o4w19y3nrejawd3b/+gfhjungqgdzzlglvcahgwsudtcco4lcgguc8h4dua3n1bphsthmnqj+hq8pazaqzetdfrnivt5ycrxclkbhi1++as3ifkaotrmuk7npsy6kcczwwuqlghkxnsixusljvd7fzuktmzyocesbildzmrvycygmk9iqlbgvezsvetz66ymvyzlpuogslrweqiinj7g8y4aqzernlkfc3kryicrkjenzewll9eja1a1buo1u4vk/wbi4inmapnimhv1chohaexgm4rle4crzovz7nwswgtonmmf+b8/gdp54n</latexit> R(b ) B u Ufuk Topcu 8

25 <latexit sha1_base64="e8s1imn8+6ascd7bi8ngwqzryw=">aaacdxicbvdlssnafj34ruvdekmwiw6kukv7llgxmuv+4amhmnph6etgzeurid7jxv9y4umste3f+jzmilyeuha4517uvcelgrxsnl+pewv1bx1kz5ct7z1ff++ikoeeiriee95wacjibqllqx6mqcceay63vgy97v3wawnwls5icej8dckivyksnvj+ayxhbll3jqp5rntoi8qriusxopmrv8yaoywxskycvfcblqt/ooijagekjasrn8yy+mkmetkggrloxeqyzlgq+grguiahjnov8mme6umdd/iqkjptnxfeykohjgenurmtxtzxi7+5/ut6teclizxiieksv+wgwzgxkxobyijjnfmgeu3wrquayyyjvgguvgjx/8ilp1gvwwa1+fv5pnoo4sugqhaeqstafaqir1ejtrnadekiv6fv71j61n+191rqkftmh6a+j/4kzwb</latexit> <latexit sha1_base64="jzfr3hcfynghxvm1jph5quechoy=">aaab+xicbvdjsgnbekjw4zbqecvjugilzatbtgvhimybzihqgnu5m6vns7gmeix/ixymixvtb/6nnewgiq8khu9vuvuvsavxng+rclg5tbtng3tld/chhkh5+vjjjhkwieraqpq8bibmmubnvqijqkb7wbn/by5skj/gjnqtorxqq85azqo3k3blz6lfm1acy57aj+l6dtmponoqdeiusrmwajv6+fsczcwdnbleq6tqq9nernmcbpqzcptckbqfdy1phmrl55dyyvr+irmpklyk7n6eyknkvktkdcdedvdternx+8bqbdwy/nczppjnliuzgjohmyi4huusmxcqqyiq3txipjiybciqmrdc1zfxsatwda+qtyfrct1dxlgemzihcrhwa3w4hwygceynuev3qzcerhery9fa8fazpzch1if++skn8=</latexit> <latexit sha1_base64="nn4hgdpi3fazq3rqytlqgzt9oq=">aaab+xicbvbnswmxem3wr1q/vj16crahxsqmcnosefywx5auyzzdnqgzrnrkiupf/eiwdfvpvlvtns9aoudgcd7m8zmcxbtfg8b6ewsbm1vvcle3thxweuccnlrnikgtxsjwnzbqefxc3ajojmoofeoobo7+z+ewjk81g+mmkcfkshkg84o8zkgeugowgzfyjl3scnjaj3ljx9rba64tkpixynal3q9ewrqbnexqrbves4yfuwu4ezar9vincwvjoosupzjgo1scfkmx1iljwexsiunxqi/jziaatnqtszutsq95c/m/rpmzw6dcjqkbyzalbqnajsbzghcfkbgtchthf7kyjqigznqysdygsvrxowruquarwhq7ldzlhuurn6bxveee3qi7uuqm1eumt9ixezutos/ou/oxbc4+cwp+gn8wfuhpj+</latexit> <latexit sha1_base64="qhyyigigbss4htty9+oscofka=">aaab+xicbvdlsgnbeoynrxhfqx69dayhxsjuiugx4mvjbjmiyblmtnqtibmzydycmfeggiff/xjt/4+rxmschqkqm+6uibvcacf5tgobm1vbo8xdt7+wegrfxzsvkkmgbzyihl5gfcfgsfylwlfewligqalgtzo/mapebi/6emkxkqhmq85o9pivm3/zz69wkluowjfckub5edqjmhwsfukprhiazvf/x6ccsijdutvkmu66tay6nunamclnqzwpsyer1g19cyrqi8fh75lfwypu/crjqknzmrvydygik1iqltgve9vkvetz6y6vfyhqezxpgtfowzidohsxhintkwkwmouxycythqyopyaskgnbxx15nbrrvbderd1flrvumo4inme5vmcfajahtshbqzg8ayv8gbl1ov1bnswgvwcuyu/sd6/ahxopka</latexit> <latexit sha1_base64="e8s1imn8+6ascd7bi8ngwqzryw=">aaacdxicbvdlssnafj34ruvdekmwiw6kukv7llgxmuv+4amhmnph6etgzeurid7jxv9y4umste3f+jzmilyeuha4517uvcelgrxsnl+pewv1bx1kz5ct7z1ff++ikoeeiriee95wacjibqllqx6mqcceay63vgy97v3wawnwls5icej8dckivyksnvj+ayxhbll3jqp5rntoi8qriusxopmrv8yaoywxskycvfcblqt/ooijagekjasrn8yy+mkmetkggrloxeqyzlgq+grguiahjnov8mme6umdd/iqkjptnxfeykohjgenurmtxtzxi7+5/ut6teclizxiieksv+wgwzgxkxobyijjnfmgeu3wrquayyyjvgguvgjx/8ilp1gvwwa1+fv5pnoo4sugqhaeqstafaqir1ejtrnadekiv6fv71j61n+191rqkftmh6a+j/4kzwb</latexit> Over-approximation of the reachable belief set If there exists a function V such that V (f a (b t 1,z)) V (b t 1 ) <, 8 V { B b V = {b B V (b) apple 1} (with additional standard technical assumptions), 8a A, z Z, b t 1 V, apple } Then, R(b ) V. V a1 (b) apple 1 V a (b) apple 1 b<latexit sha1_base64="xjd5v7unvnn+uddndprnub1vhrs=">aaab6nicbvbns8naej3ur1q/qh69lbbbumqomeif48v7qeowyk3bpzhnnij/qlecji1v/kzx/jts1bwx8mn6bywzekaiujet+o4w19y3nrejawd3b/+gfhjungqgdzzlglvcahgwsudtcco4lcgguc8h4dua3n1bphsthmnqj+hq8pazaqzetdfrnivt5ycrxclkbhi1++as3ifkaotrmuk7npsy6kcczwwuqlghkxnsixusljvd7fzuktmzyocesbildzmrvycygmk9iqlbgvezsvetz66ymvyzlpuogslrweqiinj7g8y4aqzernlkfc3kryicrkjenzewll9eja1a1buo1u4vk/wbi4inmapnimhv1chohaexgm4rle4crzovz7nwswgtonmmf+b8/gdp54n</latexit> V a3 (b) apple 1 R(b ) B u Ufuk Topcu 8

26 <latexit sha1_base64="e8s1imn8+6ascd7bi8ngwqzryw=">aaacdxicbvdlssnafj34ruvdekmwiw6kukv7llgxmuv+4amhmnph6etgzeurid7jxv9y4umste3f+jzmilyeuha4517uvcelgrxsnl+pewv1bx1kz5ct7z1ff++ikoeeiriee95wacjibqllqx6mqcceay63vgy97v3wawnwls5icej8dckivyksnvj+ayxhbll3jqp5rntoi8qriusxopmrv8yaoywxskycvfcblqt/ooijagekjasrn8yy+mkmetkggrloxeqyzlgq+grguiahjnov8mme6umdd/iqkjptnxfeykohjgenurmtxtzxi7+5/ut6teclizxiieksv+wgwzgxkxobyijjnfmgeu3wrquayyyjvgguvgjx/8ilp1gvwwa1+fv5pnoo4sugqhaeqstafaqir1ejtrnadekiv6fv71j61n+191rqkftmh6a+j/4kzwb</latexit> <latexit sha1_base64="jzfr3hcfynghxvm1jph5quechoy=">aaab+xicbvdjsgnbekjw4zbqecvjugilzatbtgvhimybzihqgnu5m6vns7gmeix/ixymixvtb/6nnewgiq8khu9vuvuvsavxng+rclg5tbtng3tld/chhkh5+vjjjhkwieraqpq8bibmmubnvqijqkb7wbn/by5skj/gjnqtorxqq85azqo3k3blz6lfm1acy57aj+l6dtmponoqdeiusrmwajv6+fsczcwdnbleq6tqq9nernmcbpqzcptckbqfdy1phmrl55dyyvr+irmpklyk7n6eyknkvktkdcdedvdternx+8bqbdwy/nczppjnliuzgjohmyi4huusmxcqqyiq3txipjiybciqmrdc1zfxsatwda+qtyfrct1dxlgemzihcrhwa3w4hwygceynuev3qzcerhery9fa8fazpzch1if++skn8=</latexit> <latexit sha1_base64="nn4hgdpi3fazq3rqytlqgzt9oq=">aaab+xicbvbnswmxem3wr1q/vj16crahxsqmcnosefywx5auyzzdnqgzrnrkiupf/eiwdfvpvlvtns9aoudgcd7m8zmcxbtfg8b6ewsbm1vvcle3thxweuccnlrnikgtxsjwnzbqefxc3ajojmoofeoobo7+z+ewjk81g+mmkcfkshkg84o8zkgeugowgzfyjl3scnjaj3ljx9rba64tkpixynal3q9ewrqbnexqrbves4yfuwu4ezar9vincwvjoosupzjgo1scfkmx1iljwexsiunxqi/jziaatnqtszutsq95c/m/rpmzw6dcjqkbyzalbqnajsbzghcfkbgtchthf7kyjqigznqysdygsvrxowruquarwhq7ldzlhuurn6bxveee3qi7uuqm1eumt9ixezutos/ou/oxbc4+cwp+gn8wfuhpj+</latexit> <latexit sha1_base64="qhyyigigbss4htty9+oscofka=">aaab+xicbvdlsgnbeoynrxhfqx69dayhxsjuiugx4mvjbjmiyblmtnqtibmzydycmfeggiff/xjt/4+rxmschqkqm+6uibvcacf5tgobm1vbo8xdt7+wegrfxzsvkkmgbzyihl5gfcfgsfylwlfewligqalgtzo/mapebi/6emkxkqhmq85o9pivm3/zz69wkluowjfckub5edqjmhwsfukprhiazvf/x6ccsijdutvkmu66tay6nunamclnqzwpsyer1g19cyrqi8fh75lfwypu/crjqknzmrvydygik1iqltgve9vkvetz6y6vfyhqezxpgtfowzidohsxhintkwkwmouxycythqyopyaskgnbxx15nbrrvbderd1flrvumo4inme5vmcfajahtshbqzg8ayv8gbl1ov1bnswgvwcuyu/sd6/ahxopka</latexit> <latexit sha1_base64="e8s1imn8+6ascd7bi8ngwqzryw=">aaacdxicbvdlssnafj34ruvdekmwiw6kukv7llgxmuv+4amhmnph6etgzeurid7jxv9y4umste3f+jzmilyeuha4517uvcelgrxsnl+pewv1bx1kz5ct7z1ff++ikoeeiriee95wacjibqllqx6mqcceay63vgy97v3wawnwls5icej8dckivyksnvj+ayxhbll3jqp5rntoi8qriusxopmrv8yaoywxskycvfcblqt/ooijagekjasrn8yy+mkmetkggrloxeqyzlgq+grguiahjnov8mme6umdd/iqkjptnxfeykohjgenurmtxtzxi7+5/ut6teclizxiieksv+wgwzgxkxobyijjnfmgeu3wrquayyyjvgguvgjx/8ilp1gvwwa1+fv5pnoo4sugqhaeqstafaqir1ejtrnadekiv6fv71j61n+191rqkftmh6a+j/4kzwb</latexit> Over-approximation of the reachable belief set If there exists a function V such that V (f a (b t 1,z)) V (b t 1 ) <, 8 V { B b V = {b B V (b) apple 1} (with additional standard technical assumptions), 8a A, z Z, b t 1 V, apple } Then, R(b ) V. V a1 (b) apple 1 V max = {b B max V a(b) apple 1} aa <latexit sha1_base64="qgpbejb1kg4uaevfnvrz/5fayk=">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</latexit> V a (b) apple 1 b<latexit sha1_base64="xjd5v7unvnn+uddndprnub1vhrs=">aaab6nicbvbns8naej3ur1q/qh69lbbbumqomeif48v7qeowyk3bpzhnnij/qlecji1v/kzx/jts1bwx8mn6bywzekaiujet+o4w19y3nrejawd3b/+gfhjungqgdzzlglvcahgwsudtcco4lcgguc8h4dua3n1bphsthmnqj+hq8pazaqzetdfrnivt5ycrxclkbhi1++as3ifkaotrmuk7npsy6kcczwwuqlghkxnsixusljvd7fzuktmzyocesbildzmrvycygmk9iqlbgvezsvetz66ymvyzlpuogslrweqiinj7g8y4aqzernlkfc3kryicrkjenzewll9eja1a1buo1u4vk/wbi4inmapnimhv1chohaexgm4rle4crzovz7nwswgtonmmf+b8/gdp54n</latexit> V a3 (b) apple 1 R(b ) B u Ufuk Topcu 8

27 <latexit sha1_base64="nw71nul4tvfztckauywxsyn+tm=">aaaccxicbvdlssnafj34ruvdelmsah1u5iqeh6mzlbfuajotjdnionunsmylqyrzu/bu3lhrx6x+48+ctllo64elh3u5d57/jhrqszr1hzxvvfcxsfbd3dvfzydtowsguklrywsxr9jwmhiwooqrrqxiij7jht8fxu79wtiwku3qljtfyobienkezks54jhy7uecowxmv1j/ufhjlwlx1nfnl8vredzdnlvswaas4toyclkkpmv9o8ijj6hcdenzs61yusksimjgsqktsbijeidtmrjxin519ksftrfrhealdoyiz9fdeiriue+7rzundctgbiv95vuqfntelyzwoeul5oibhuevwggvsugwyhnnebzu3wrxeamelq6vqeowf19eju1qxt6vvg8vso1ahkcbhimtuayuaqncaoaoauweat4bw8gu/gi/fufmxbv4x85gj8gfh5a4rcmis=</latexit> <latexit sha1_base64="nw71nul4tvfztckauywxsyn+tm=">aaaccxicbvdlssnafj34ruvdelmsah1u5iqeh6mzlbfuajotjdnionunsmylqyrzu/bu3lhrx6x+48+ctllo64elh3u5d57/jhrqszr1hzxvvfcxsfbd3dvfzydtowsguklrywsxr9jwmhiwooqrrqxiij7jht8fxu79wtiwku3qljtfyobienkezks54jhy7uecowxmv1j/ufhjlwlx1nfnl8vredzdnlvswaas4toyclkkpmv9o8ijj6hcdenzs61yusksimjgsqktsbijeidtmrjxin519ksftrfrhealdoyiz9fdeiriue+7rzundctgbiv95vuqfntelyzwoeul5oibhuevwggvsugwyhnnebzu3wrxeamelq6vqeowf19eju1qxt6vvg8vso1ahkcbhimtuayuaqncaoaoauweat4bw8gu/gi/fufmxbv4x85gj8gfh5a4rcmis=</latexit> Example: over-approximation of the reachable belief set If there exists a function V such that V (f a (b t 1,z)) V (b t 1 ) <, 8 V { B b V = {b B V (b) apple 1} (with additional standard technical assumptions), 8a A, z Z, b t 1 V, apple } Then, R(b ) V. <latexit sha1_base64="e8s1imn8+6ascd7bi8ngwqzryw=">aaacdxicbvdlssnafj34ruvdekmwiw6kukv7llgxmuv+4amhmnph6etgzeurid7jxv9y4umste3f+jzmilyeuha4517uvcelgrxsnl+pewv1bx1kz5ct7z1ff++ikoeeiriee95wacjibqllqx6mqcceay63vgy97v3wawnwls5icej8dckivyksnvj+ayxhbll3jqp5rntoi8qriusxopmrv8yaoywxskycvfcblqt/ooijagekjasrn8yy+mkmetkggrloxeqyzlgq+grguiahjnov8mme6umdd/iqkjptnxfeykohjgenurmtxtzxi7+5/ut6teclizxiieksv+wgwzgxkxobyijjnfmgeu3wrquayyyjvgguvgjx/8ilp1gvwwa1+fv5pnoo4sugqhaeqstafaqir1ejtrnadekiv6fv71j61n+191rqkftmh6a+j/4kzwb</latexit> B = {b X q b(q) =1} B = {b X q b(q) =1} Three-state OMD with two actions V a1 <latexit sha1_base64="whf4mqnmzgc+4oigkjdz/wyonjs=">aaab7nicbvbns8naej34wetx1aoxxsj4kkkv7lhgxwmf+wftcjvtpl6ytdivbcf4qxd4p49fd489+4bxq1gcdj/dmmjkxpliydn1vznza3tnt7rx3j84dqunjxtjjpxtsskynuhdrwkrrvodje6nmna4l74atu7nffelaieq94jtlfkxhskscubrstxknbmqaxq1twfydrxclkfaqg8juyjiyluuimqtf9z3rz6lgwssflqez4sllezrifusvjbnx88w5m3jplsgjem1livmovydyghszjubgvmcm1vvlv7n9tomgn4uvjohvy5kmokwytmfyddotldobwemi3srysnqaymbujlg4k3+vi66drr3nwt/nbtbtakoepwdhdwbr7cqhuoqvtydcbz3ifnyd1xpx35zuueumfwb87ndwu7j1g=</latexit> V a1 <latexit sha1_base64="whf4mqnmzgc+4oigkjdz/wyonjs=">aaab7nicbvbns8naej34wetx1aoxxsj4kkkv7lhgxwmf+wftcjvtpl6ytdivbcf4qxd4p49fd489+4bxq1gcdj/dmmjkxpliydn1vznza3tnt7rx3j84dqunjxtjjpxtsskynuhdrwkrrvodje6nmna4l74atu7nffelaieq94jtlfkxhskscubrstxknbmqaxq1twfydrxclkfaqg8juyjiyluuimqtf9z3rz6lgwssflqez4sllezrifusvjbnx88w5m3jplsgjem1livmovydyghszjubgvmcm1vvlv7n9tomgn4uvjohvy5kmokwytmfyddotldobwemi3srysnqaymbujlg4k3+vi66drr3nwt/nbtbtakoepwdhdwbr7cqhuoqvtydcbz3ifnyd1xpx35zuueumfwb87ndwu7j1g=</latexit> V<latexit sha1_base64="zsbnufrkwqw8otjm3m/iuuoliqc=">aaab+nicdvdlssnafj3uv6vvjdubovgkirpaouu4mzlbfuanotjdniontyymagl5lcufderv/izr9xlzqqmdh3u5z45fskokkb5ozxw1jct8rblz3dvfdvxrye3hkmenimmv84cnbgi1iv1ljycdhbiu+i31/dlh4/rvcbyjazlibuisuqdipfukqdxrygsu4xy1su9ler3uafxtoo81bcdbjqnxatluqu+nuhwgppuanrndx9fromzpsckjgrjiajmjddejcwm5jvrkkic8axnyfdrciveunkieg5ltkgqczviyrcqn83mhqkmq99nvkefb+9qvzlg6yyalkzjzjukggvdwupgzkgrq9wtdnbksvqzhtlrxikeiis9vwrzxw9v4+nzhlu37cun1m6t6iidy3aczoafmqanlkehdaegt+abifn7v571f61+vosvvthiefn4+avwhlli=</latexit> max V<latexit sha1_base64="zsbnufrkwqw8otjm3m/iuuoliqc=">aaab+nicdvdlssnafj3uv6vvjdubovgkirpaouu4mzlbfuanotjdniontyymagl5lcufderv/izr9xlzqqmdh3u5z45fskokkb5ozxw1jct8rblz3dvfdvxrye3hkmenimmv84cnbgi1iv1ljycdhbiu+i31/dlh4/rvcbyjazlibuisuqdipfukqdxrygsu4xy1su9ler3uafxtoo81bcdbjqnxatluqu+nuhwgppuanrndx9fromzpsckjgrjiajmjddejcwm5jvrkkic8axnyfdrciveunkieg5ltkgqczviyrcqn83mhqkmq99nvkefb+9qvzlg6yyalkzjzjukggvdwupgzkgrq9wtdnbksvqzhtlrxikeiis9vwrzxw9v4+nzhlu37cun1m6t6iidy3aczoafmqanlkehdaegt+abifn7v571f61+vosvvthiefn4+avwhlli=</latexit> max V a <latexit sha1_base64="hpufberltcfqbfon1oy5rbyfx+q=">aaab7nicbvbns8naej34wetx1aoxxsj4kkkv7lhgxwmf+wftcjvtpl6ytdivbcf4qxd4p49fd489+4bxq1gcdj/dmmjkxpliydn1vznza3tnt7rx3j84dqunjxtjjpxtsskynuhdrwkrrvodje6nmna4l74atu7nffelaieq94jtlfkxhskscubrstxknkjgkrvrbklkhxifaqkbvpb5wswtfgwc4vmum6npuin1ongkk+kw8yw1kjnte+5yqgnj54tzz+tskkmsjdqwqrjqf/kndzmgoem6y4nqvexz6cynfxcqdrdrthyuzrjggmz/6gqnogcmojzvrywwkbuzotkngrv9ev1qnxvota/egmmwucztghc7gcjy4hsbcqwvawgacz/akb7qvdjvzseydcmpzs7gd5zhw1aj1k=</latexit> V a <latexit sha1_base64="hpufberltcfqbfon1oy5rbyfx+q=">aaab7nicbvbns8naej34wetx1aoxxsj4kkkv7lhgxwmf+wftcjvtpl6ytdivbcf4qxd4p49fd489+4bxq1gcdj/dmmjkxpliydn1vznza3tnt7rx3j84dqunjxtjjpxtsskynuhdrwkrrvodje6nmna4l74atu7nffelaieq94jtlfkxhskscubrstxknkjgkrvrbklkhxifaqkbvpb5wswtfgwc4vmum6npuin1ongkk+kw8yw1kjnte+5yqgnj54tzz+tskkmsjdqwqrjqf/kndzmgoem6y4nqvexz6cynfxcqdrdrthyuzrjggmz/6gqnogcmojzvrywwkbuzotkngrv9ev1qnxvota/egmmwucztghc7gcjy4hsbcqwvawgacz/akb7qvdjvzseydcmpzs7gd5zhw1aj1k=</latexit> iece-wise affine V iece-wise cubic polynomial V Ufuk Topcu 9

28 <latexit sha1_base64="/gzmqsc7qujhs/hlc4wjslqx9by=">aaacc3icbzdnsgmxfixv1l9a/6ou3qsl4krmikc4krhxwaf/mbgtjppqzjkgsemvyrxlrvh3anbnin8gxmg1nobuhah/n3as3jo58z1v5zsvrg5lz5u7kzu7d/ud86mizkulbrhkpehhwldnb4yztnupojijoo1g49tz3ngsjmpwmasidbq8firrcxlh+flfqy5jjpmg15tbdudb/8aqoqafmw3udytjeiom4vhr33nte+ryguy4nvb6maypjmm8pl5fgroqg3x+8hsdwweayqnsewbn3d8bou6nisrnuywgenlbgauzfjrrqr8zmtxqc5emhkqyokcooisdqrbgyostwiqvmflofqgsefsbg1lex7xjlxfyhzkxdc+ve/wwtcvviytoivz8oakgnahtwgdaqn8akvzpz5rw7h4vrklshmmfoz8/kywa+g==</latexit> <latexit sha1_base64="/gzmqsc7qujhs/hlc4wjslqx9by=">aaacc3icbzdnsgmxfixv1l9a/6ou3qsl4krmikc4krhxwaf/mbgtjppqzjkgsemvyrxlrvh3anbnin8gxmg1nobuhah/n3as3jo58z1v5zsvrg5lz5u7kzu7d/ud86mizkulbrhkpehhwldnb4yztnupojijoo1g49tz3ngsjmpwmasidbq8firrcxlh+flfqy5jjpmg15tbdudb/8aqoqafmw3udytjeiom4vhr33nte+ryguy4nvb6maypjmm8pl5fgroqg3x+8hsdwweayqnsewbn3d8bou6nisrnuywgenlbgauzfjrrqr8zmtxqc5emhkqyokcooisdqrbgyostwiqvmflofqgsefsbg1lex7xjlxfyhzkxdc+ve/wwtcvviytoivz8oakgnahtwgdaqn8akvzpz5rw7h4vrklshmmfoz8/kywa+g==</latexit> <latexit sha1_base64="/gzmqsc7qujhs/hlc4wjslqx9by=">aaacc3icbzdnsgmxfixv1l9a/6ou3qsl4krmikc4krhxwaf/mbgtjppqzjkgsemvyrxlrvh3anbnin8gxmg1nobuhah/n3as3jo58z1v5zsvrg5lz5u7kzu7d/ud86mizkulbrhkpehhwldnb4yztnupojijoo1g49tz3ngsjmpwmasidbq8firrcxlh+flfqy5jjpmg15tbdudb/8aqoqafmw3udytjeiom4vhr33nte+ryguy4nvb6maypjmm8pl5fgroqg3x+8hsdwweayqnsewbn3d8bou6nisrnuywgenlbgauzfjrrqr8zmtxqc5emhkqyokcooisdqrbgyostwiqvmflofqgsefsbg1lex7xjlxfyhzkxdc+ve/wwtcvviytoivz8oakgnahtwgdaqn8akvzpz5rw7h4vrklshmmfoz8/kywa+g==</latexit> <latexit sha1_base64="/gzmqsc7qujhs/hlc4wjslqx9by=">aaacc3icbzdnsgmxfixv1l9a/6ou3qsl4krmikc4krhxwaf/mbgtjppqzjkgsemvyrxlrvh3anbnin8gxmg1nobuhah/n3as3jo58z1v5zsvrg5lz5u7kzu7d/ud86mizkulbrhkpehhwldnb4yztnupojijoo1g49tz3ngsjmpwmasidbq8firrcxlh+flfqy5jjpmg15tbdudb/8aqoqafmw3udytjeiom4vhr33nte+ryguy4nvb6maypjmm8pl5fgroqg3x+8hsdwweayqnsewbn3d8bou6nisrnuywgenlbgauzfjrrqr8zmtxqc5emhkqyokcooisdqrbgyostwiqvmflofqgsefsbg1lex7xjlxfyhzkxdc+ve/wwtcvviytoivz8oakgnahtwgdaqn8akvzpz5rw7h4vrklshmmfoz8/kywa+g==</latexit> <latexit sha1_base64="mr83cwkyqaiiyl1ktckic9g=">aaacc3icbzdlsgmxfib1futt6pln8eiucozrvbcfdy4rnabtichkbaewyjbmhjhel71idyjwx/cz/altc8lbftd4o/z4gto58z1v53cxubw9k5xt7s3f3b4vd4+awuzkujbrhkpuhhwldnbw4yztrupojijoo1eo7tp3nmksjmpmmacidba8firrcxlh+ftfqu5jistcjyxa6m6fv8bzqgyuayfmn15cks6gwhgotfc9ntzbjzrjhdflqzzqmmizwgowbu6odvlzyrnyz+iqwytxgc/9u5djrepxedjlbzqixs6m5nkutts7wmxfbdktawaoiml4owji9ggnrnihldxxywucx+apehvpgyw1/jtumtd7ek7vrvc6vew1wlfrvoqqhnca6x4me11oeegtacahje4bxengfn3flwuejbwexcwr/5hz9apmjmvs=</latexit> <latexit sha1_base64="mr83cwkyqaiiyl1ktckic9g=">aaacc3icbzdlsgmxfib1futt6pln8eiucozrvbcfdy4rnabtichkbaewyjbmhjhel71idyjwx/cz/altc8lbftd4o/z4gto58z1v53cxubw9k5xt7s3f3b4vd4+awuzkujbrhkpuhhwldnbw4yztrupojijoo1eo7tp3nmksjmpmmacidba8firrcxlh+ftfqu5jistcjyxa6m6fv8bzqgyuayfmn15cks6gwhgotfc9ntzbjzrjhdflqzzqmmizwgowbu6odvlzyrnyz+iqwytxgc/9u5djrepxedjlbzqixs6m5nkutts7wmxfbdktawaoiml4owji9ggnrnihldxxywucx+apehvpgyw1/jtumtd7ek7vrvc6vew1wlfrvoqqhnca6x4me11oeegtacahje4bxengfn3flwuejbwexcwr/5hz9apmjmvs=</latexit> <latexit sha1_base64="mr83cwkyqaiiyl1ktckic9g=">aaacc3icbzdlsgmxfib1futt6pln8eiucozrvbcfdy4rnabtichkbaewyjbmhjhel71idyjwx/cz/altc8lbftd4o/z4gto58z1v53cxubw9k5xt7s3f3b4vd4+awuzkujbrhkpuhhwldnbw4yztrupojijoo1eo7tp3nmksjmpmmacidba8firrcxlh+ftfqu5jistcjyxa6m6fv8bzqgyuayfmn15cks6gwhgotfc9ntzbjzrjhdflqzzqmmizwgowbu6odvlzyrnyz+iqwytxgc/9u5djrepxedjlbzqixs6m5nkutts7wmxfbdktawaoiml4owji9ggnrnihldxxywucx+apehvpgyw1/jtumtd7ek7vrvc6vew1wlfrvoqqhnca6x4me11oeegtacahje4bxengfn3flwuejbwexcwr/5hz9apmjmvs=</latexit> <latexit sha1_base64="mr83cwkyqaiiyl1ktckic9g=">aaacc3icbzdlsgmxfib1futt6pln8eiucozrvbcfdy4rnabtichkbaewyjbmhjhel71idyjwx/cz/altc8lbftd4o/z4gto58z1v53cxubw9k5xt7s3f3b4vd4+awuzkujbrhkpuhhwldnbw4yztrupojijoo1eo7tp3nmksjmpmmacidba8firrcxlh+ftfqu5jistcjyxa6m6fv8bzqgyuayfmn15cks6gwhgotfc9ntzbjzrjhdflqzzqmmizwgowbu6odvlzyrnyz+iqwytxgc/9u5djrepxedjlbzqixs6m5nkutts7wmxfbdktawaoiml4owji9ggnrnihldxxywucx+apehvpgyw1/jtumtd7ek7vrvc6vew1wlfrvoqqhnca6x4me11oeegtacahje4bxengfn3flwuejbwexcwr/5hz9apmjmvs=</latexit> <latexit sha1_base64="5yxtmtg++qwe9n+y5ycstza=">aaacenicbvdltsjafl3ff+kr6tlnrdbxrvogldeny4xetcbppkou5gwntyzuxls8bcu3ephudnu/qg/wz9wclocjkbnjxzb3lucrlolhacb6usbm1vverezthxweccnhrwnktaixksnwkskgectjxtnd4lkuio4lqbjo9yvzuhurfyoppqridwulgchasl5t1/or1iocexy785a8uqu3fmqkvelugvcrr8+6c/iekauaejxr1xcfrxoalzottwawfkppgmszdjnu4igql5snn6elowxqgeszqqo5+vciw5fsygwm3lmtezl4lovmdi6o5fq8nrlmehstqvzjahtjnsm8n7qgelknj8agolk5glerlhioklfdoou9zfkuk6q5tdx8a1eznvmzzuaclsgfkjcbsgdqqm8akv8gy9w+/wh/w5wc1zxcp/i19qsr7jzj</latexit> <latexit sha1_base64="5yxtmtg++qwe9n+y5ycstza=">aaacenicbvdltsjafl3ff+kr6tlnrdbxrvogldeny4xetcbppkou5gwntyzuxls8bcu3ephudnu/qg/wz9wclocjkbnjxzb3lucrlolhacb6usbm1vverezthxweccnhrwnktaixksnwkskgectjxtnd4lkuio4lqbjo9yvzuhurfyoppqridwulgchasl5t1/or1iocexy785a8uqu3fmqkvelugvcrr8+6c/iekauaejxr1xcfrxoalzottwawfkppgmszdjnu4igql5snn6elowxqgeszqqo5+vciw5fsygwm3lmtezl4lovmdi6o5fq8nrlmehstqvzjahtjnsm8n7qgelknj8agolk5glerlhioklfdoou9zfkuk6q5tdx8a1eznvmzzuaclsgfkjcbsgdqqm8akv8gy9w+/wh/w5wc1zxcp/i19qsr7jzj</latexit> <latexit sha1_base64="5yxtmtg++qwe9n+y5ycstza=">aaacenicbvdltsjafl3ff+kr6tlnrdbxrvogldeny4xetcbppkou5gwntyzuxls8bcu3ephudnu/qg/wz9wclocjkbnjxzb3lucrlolhacb6usbm1vverezthxweccnhrwnktaixksnwkskgectjxtnd4lkuio4lqbjo9yvzuhurfyoppqridwulgchasl5t1/or1iocexy785a8uqu3fmqkvelugvcrr8+6c/iekauaejxr1xcfrxoalzottwawfkppgmszdjnu4igql5snn6elowxqgeszqqo5+vciw5fsygwm3lmtezl4lovmdi6o5fq8nrlmehstqvzjahtjnsm8n7qgelknj8agolk5glerlhioklfdoou9zfkuk6q5tdx8a1eznvmzzuaclsgfkjcbsgdqqm8akv8gy9w+/wh/w5wc1zxcp/i19qsr7jzj</latexit> <latexit sha1_base64="5yxtmtg++qwe9n+y5ycstza=">aaacenicbvdltsjafl3ff+kr6tlnrdbxrvogldeny4xetcbppkou5gwntyzuxls8bcu3ephudnu/qg/wz9wclocjkbnjxzb3lucrlolhacb6usbm1vverezthxweccnhrwnktaixksnwkskgectjxtnd4lkuio4lqbjo9yvzuhurfyoppqridwulgchasl5t1/or1iocexy785a8uqu3fmqkvelugvcrr8+6c/iekauaejxr1xcfrxoalzottwawfkppgmszdjnu4igql5snn6elowxqgeszqqo5+vciw5fsygwm3lmtezl4lovmdi6o5fq8nrlmehstqvzjahtjnsm8n7qgelknj8agolk5glerlhioklfdoou9zfkuk6q5tdx8a1eznvmzzuaclsgfkjcbsgdqqm8akv8gy9w+/wh/w5wc1zxcp/i19qsr7jzj</latexit> rivacy verification using barrier certificates If there exists a function B such that (with additional standard technical assumptions) b T a1 B<latexit sha1_base64="vctrvbtrvpi6lblbhcvfihhl5o=">aaab9hicbvdlsgmxflpr1pfvzdugkvwvwaqomtsny4raeq8mkmtykxmttkem/q43lhrx68e48/mtlq1gobwzn3ck+ohwuujen8o8lg5tbtng3tld/chhuj5p6yhrllvojclv9ylmgkvwmtwi1ovi6evwmef3gv+z8qu5pf8nloyesezsr5wsoyvvh5izjgsktbmgrqrjhvzwg8ttycvcbhc1d+6g8jmormgiqi1j3xiyxemu4fwxe6ieaxyroyij1ljukznplf6hn+miqqxxeyj5p8el9vzgsuotz6nvjlkre9tlx6+xmodws7mme8mkxr4keofnhlmg8jarroywuko4jyrpmoicdwp5itwv398jpp16ruvbxcfpn/i6inag53ajltxahe6hcsg8at8apvaipedv6wi4wul5zcn+an8ab8csra==</latexit> u B b T a Ufuk Topcu 1

29 <latexit sha1_base64="kcsgcjmfcldg1ukqobq7hrcfew=">aaach3icbvdlsgmxfm3uv6vqks3wsk4kjnv1gwpg5cv7am7zcikaruayyzjhaem45e48vfcufbe3vvtntzaoubwogce8k9x48e1dbeyu3srqvphflgxt7+zuffcmjqmfwungopqtximecsnycdyo1imrl4grx8fxubzypxko7acsw5abpl3osvgjk944xvgyhc4damcqpeuku9+mnth4oigchnhn97lmvcqfsatupvyzzzxsgveycjjrqhrpx/hz7iydjoekonxhssojkqbp4klbtfwlcjraasy6gkaddzjyvxsdg6wfzkxks8ez9vzgqqotx4jvj6al6zuk/3mdgpx3ytlkaym6fyjfmxsh3hafu5xxsiissgekm5uxxrifkfgkiyepzfymukwsk7z+xk7xmpwsvqykmjdixokymuurxdodpqiiqest6rx/wi/vmfvpf89gclecoj+wjj8wrkrg</latexit> <latexit sha1_base64="w1zcdyndogfcgilsitkkn9xejoe=">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</latexit> <latexit sha1_base64="/gzmqsc7qujhs/hlc4wjslqx9by=">aaacc3icbzdnsgmxfixv1l9a/6ou3qsl4krmikc4krhxwaf/mbgtjppqzjkgsemvyrxlrvh3anbnin8gxmg1nobuhah/n3as3jo58z1v5zsvrg5lz5u7kzu7d/ud86mizkulbrhkpehhwldnb4yztnupojijoo1g49tz3ngsjmpwmasidbq8firrcxlh+flfqy5jjpmg15tbdudb/8aqoqafmw3udytjeiom4vhr33nte+ryguy4nvb6maypjmm8pl5fgroqg3x+8hsdwweayqnsewbn3d8bou6nisrnuywgenlbgauzfjrrqr8zmtxqc5emhkqyokcooisdqrbgyostwiqvmflofqgsefsbg1lex7xjlxfyhzkxdc+ve/wwtcvviytoivz8oakgnahtwgdaqn8akvzpz5rw7h4vrklshmmfoz8/kywa+g==</latexit> <latexit sha1_base64="/gzmqsc7qujhs/hlc4wjslqx9by=">aaacc3icbzdnsgmxfixv1l9a/6ou3qsl4krmikc4krhxwaf/mbgtjppqzjkgsemvyrxlrvh3anbnin8gxmg1nobuhah/n3as3jo58z1v5zsvrg5lz5u7kzu7d/ud86mizkulbrhkpehhwldnb4yztnupojijoo1g49tz3ngsjmpwmasidbq8firrcxlh+flfqy5jjpmg15tbdudb/8aqoqafmw3udytjeiom4vhr33nte+ryguy4nvb6maypjmm8pl5fgroqg3x+8hsdwweayqnsewbn3d8bou6nisrnuywgenlbgauzfjrrqr8zmtxqc5emhkqyokcooisdqrbgyostwiqvmflofqgsefsbg1lex7xjlxfyhzkxdc+ve/wwtcvviytoivz8oakgnahtwgdaqn8akvzpz5rw7h4vrklshmmfoz8/kywa+g==</latexit> <latexit sha1_base64="/gzmqsc7qujhs/hlc4wjslqx9by=">aaacc3icbzdnsgmxfixv1l9a/6ou3qsl4krmikc4krhxwaf/mbgtjppqzjkgsemvyrxlrvh3anbnin8gxmg1nobuhah/n3as3jo58z1v5zsvrg5lz5u7kzu7d/ud86mizkulbrhkpehhwldnb4yztnupojijoo1g49tz3ngsjmpwmasidbq8firrcxlh+flfqy5jjpmg15tbdudb/8aqoqafmw3udytjeiom4vhr33nte+ryguy4nvb6maypjmm8pl5fgroqg3x+8hsdwweayqnsewbn3d8bou6nisrnuywgenlbgauzfjrrqr8zmtxqc5emhkqyokcooisdqrbgyostwiqvmflofqgsefsbg1lex7xjlxfyhzkxdc+ve/wwtcvviytoivz8oakgnahtwgdaqn8akvzpz5rw7h4vrklshmmfoz8/kywa+g==</latexit> <latexit sha1_base64="/gzmqsc7qujhs/hlc4wjslqx9by=">aaacc3icbzdnsgmxfixv1l9a/6ou3qsl4krmikc4krhxwaf/mbgtjppqzjkgsemvyrxlrvh3anbnin8gxmg1nobuhah/n3as3jo58z1v5zsvrg5lz5u7kzu7d/ud86mizkulbrhkpehhwldnb4yztnupojijoo1g49tz3ngsjmpwmasidbq8firrcxlh+flfqy5jjpmg15tbdudb/8aqoqafmw3udytjeiom4vhr33nte+ryguy4nvb6maypjmm8pl5fgroqg3x+8hsdwweayqnsewbn3d8bou6nisrnuywgenlbgauzfjrrqr8zmtxqc5emhkqyokcooisdqrbgyostwiqvmflofqgsefsbg1lex7xjlxfyhzkxdc+ve/wwtcvviytoivz8oakgnahtwgdaqn8akvzpz5rw7h4vrklshmmfoz8/kywa+g==</latexit> <latexit sha1_base64="mr83cwkyqaiiyl1ktckic9g=">aaacc3icbzdlsgmxfib1futt6pln8eiucozrvbcfdy4rnabtichkbaewyjbmhjhel71idyjwx/cz/altc8lbftd4o/z4gto58z1v53cxubw9k5xt7s3f3b4vd4+awuzkujbrhkpuhhwldnbw4yztrupojijoo1eo7tp3nmksjmpmmacidba8firrcxlh+ftfqu5jistcjyxa6m6fv8bzqgyuayfmn15cks6gwhgotfc9ntzbjzrjhdflqzzqmmizwgowbu6odvlzyrnyz+iqwytxgc/9u5djrepxedjlbzqixs6m5nkutts7wmxfbdktawaoiml4owji9ggnrnihldxxywucx+apehvpgyw1/jtumtd7ek7vrvc6vew1wlfrvoqqhnca6x4me11oeegtacahje4bxengfn3flwuejbwexcwr/5hz9apmjmvs=</latexit> <latexit sha1_base64="mr83cwkyqaiiyl1ktckic9g=">aaacc3icbzdlsgmxfib1futt6pln8eiucozrvbcfdy4rnabtichkbaewyjbmhjhel71idyjwx/cz/altc8lbftd4o/z4gto58z1v53cxubw9k5xt7s3f3b4vd4+awuzkujbrhkpuhhwldnbw4yztrupojijoo1eo7tp3nmksjmpmmacidba8firrcxlh+ftfqu5jistcjyxa6m6fv8bzqgyuayfmn15cks6gwhgotfc9ntzbjzrjhdflqzzqmmizwgowbu6odvlzyrnyz+iqwytxgc/9u5djrepxedjlbzqixs6m5nkutts7wmxfbdktawaoiml4owji9ggnrnihldxxywucx+apehvpgyw1/jtumtd7ek7vrvc6vew1wlfrvoqqhnca6x4me11oeegtacahje4bxengfn3flwuejbwexcwr/5hz9apmjmvs=</latexit> <latexit sha1_base64="mr83cwkyqaiiyl1ktckic9g=">aaacc3icbzdlsgmxfib1futt6pln8eiucozrvbcfdy4rnabtichkbaewyjbmhjhel71idyjwx/cz/altc8lbftd4o/z4gto58z1v53cxubw9k5xt7s3f3b4vd4+awuzkujbrhkpuhhwldnbw4yztrupojijoo1eo7tp3nmksjmpmmacidba8firrcxlh+ftfqu5jistcjyxa6m6fv8bzqgyuayfmn15cks6gwhgotfc9ntzbjzrjhdflqzzqmmizwgowbu6odvlzyrnyz+iqwytxgc/9u5djrepxedjlbzqixs6m5nkutts7wmxfbdktawaoiml4owji9ggnrnihldxxywucx+apehvpgyw1/jtumtd7ek7vrvc6vew1wlfrvoqqhnca6x4me11oeegtacahje4bxengfn3flwuejbwexcwr/5hz9apmjmvs=</latexit> <latexit sha1_base64="mr83cwkyqaiiyl1ktckic9g=">aaacc3icbzdlsgmxfib1futt6pln8eiucozrvbcfdy4rnabtichkbaewyjbmhjhel71idyjwx/cz/altc8lbftd4o/z4gto58z1v53cxubw9k5xt7s3f3b4vd4+awuzkujbrhkpuhhwldnbw4yztrupojijoo1eo7tp3nmksjmpmmacidba8firrcxlh+ftfqu5jistcjyxa6m6fv8bzqgyuayfmn15cks6gwhgotfc9ntzbjzrjhdflqzzqmmizwgowbu6odvlzyrnyz+iqwytxgc/9u5djrepxedjlbzqixs6m5nkutts7wmxfbdktawaoiml4owji9ggnrnihldxxywucx+apehvpgyw1/jtumtd7ek7vrvc6vew1wlfrvoqqhnca6x4me11oeegtacahje4bxengfn3flwuejbwexcwr/5hz9apmjmvs=</latexit> <latexit sha1_base64="5yxtmtg++qwe9n+y5ycstza=">aaacenicbvdltsjafl3ff+kr6tlnrdbxrvogldeny4xetcbppkou5gwntyzuxls8bcu3ephudnu/qg/wz9wclocjkbnjxzb3lucrlolhacb6usbm1vverezthxweccnhrwnktaixksnwkskgectjxtnd4lkuio4lqbjo9yvzuhurfyoppqridwulgchasl5t1/or1iocexy785a8uqu3fmqkvelugvcrr8+6c/iekauaejxr1xcfrxoalzottwawfkppgmszdjnu4igql5snn6elowxqgeszqqo5+vciw5fsygwm3lmtezl4lovmdi6o5fq8nrlmehstqvzjahtjnsm8n7qgelknj8agolk5glerlhioklfdoou9zfkuk6q5tdx8a1eznvmzzuaclsgfkjcbsgdqqm8akv8gy9w+/wh/w5wc1zxcp/i19qsr7jzj</latexit> <latexit sha1_base64="5yxtmtg++qwe9n+y5ycstza=">aaacenicbvdltsjafl3ff+kr6tlnrdbxrvogldeny4xetcbppkou5gwntyzuxls8bcu3ephudnu/qg/wz9wclocjkbnjxzb3lucrlolhacb6usbm1vverezthxweccnhrwnktaixksnwkskgectjxtnd4lkuio4lqbjo9yvzuhurfyoppqridwulgchasl5t1/or1iocexy785a8uqu3fmqkvelugvcrr8+6c/iekauaejxr1xcfrxoalzottwawfkppgmszdjnu4igql5snn6elowxqgeszqqo5+vciw5fsygwm3lmtezl4lovmdi6o5fq8nrlmehstqvzjahtjnsm8n7qgelknj8agolk5glerlhioklfdoou9zfkuk6q5tdx8a1eznvmzzuaclsgfkjcbsgdqqm8akv8gy9w+/wh/w5wc1zxcp/i19qsr7jzj</latexit> <latexit sha1_base64="5yxtmtg++qwe9n+y5ycstza=">aaacenicbvdltsjafl3ff+kr6tlnrdbxrvogldeny4xetcbppkou5gwntyzuxls8bcu3ephudnu/qg/wz9wclocjkbnjxzb3lucrlolhacb6usbm1vverezthxweccnhrwnktaixksnwkskgectjxtnd4lkuio4lqbjo9yvzuhurfyoppqridwulgchasl5t1/or1iocexy785a8uqu3fmqkvelugvcrr8+6c/iekauaejxr1xcfrxoalzottwawfkppgmszdjnu4igql5snn6elowxqgeszqqo5+vciw5fsygwm3lmtezl4lovmdi6o5fq8nrlmehstqvzjahtjnsm8n7qgelknj8agolk5glerlhioklfdoou9zfkuk6q5tdx8a1eznvmzzuaclsgfkjcbsgdqqm8akv8gy9w+/wh/w5wc1zxcp/i19qsr7jzj</latexit> <latexit sha1_base64="5yxtmtg++qwe9n+y5ycstza=">aaacenicbvdltsjafl3ff+kr6tlnrdbxrvogldeny4xetcbppkou5gwntyzuxls8bcu3ephudnu/qg/wz9wclocjkbnjxzb3lucrlolhacb6usbm1vverezthxweccnhrwnktaixksnwkskgectjxtnd4lkuio4lqbjo9yvzuhurfyoppqridwulgchasl5t1/or1iocexy785a8uqu3fmqkvelugvcrr8+6c/iekauaejxr1xcfrxoalzottwawfkppgmszdjnu4igql5snn6elowxqgeszqqo5+vciw5fsygwm3lmtezl4lovmdi6o5fq8nrlmehstqvzjahtjnsm8n7qgelknj8agolk5glerlhioklfdoou9zfkuk6q5tdx8a1eznvmzzuaclsgfkjcbsgdqqm8akv8gy9w+/wh/w5wc1zxcp/i19qsr7jzj</latexit> rivacy verification using barrier certificates If there exists a function B such that (with additional standard technical assumptions) b T a1 B<latexit sha1_base64="vctrvbtrvpi6lblbhcvfihhl5o=">aaab9hicbvdlsgmxflpr1pfvzdugkvwvwaqomtsny4raeq8mkmtykxmttkem/q43lhrx68e48/mtlq1gobwzn3ck+ohwuujen8o8lg5tbtng3tld/chhuj5p6yhrllvojclv9ylmgkvwmtwi1ovi6evwmef3gv+z8qu5pf8nloyesezsr5wsoyvvh5izjgsktbmgrqrjhvzwg8ttycvcbhc1d+6g8jmormgiqi1j3xiyxemu4fwxe6ieaxyroyij1ljukznplf6hn+miqqxxeyj5p8el9vzgsuotz6nvjlkre9tlx6+xmodws7mme8mkxr4keofnhlmg8jarroywuko4jyrpmoicdwp5itwv398jpp16ruvbxcfpn/i6inag53ajltxahe6hcsg8at8apvaipedv6wi4wul5zcn+an8ab8csra==</latexit> u B 8 B B(b) >, 8b B u b T a 8 B 8 B B(b) <. 8b B apple 8 B(f a (b t 1,z)) B(b t 1,z) apple 8a A, z Z, t Z 1 Then b t / B u 8t Z 1. Ufuk Topcu 1

30 <latexit sha1_base64="kcsgcjmfcldg1ukqobq7hrcfew=">aaach3icbvdlsgmxfm3uv6vqks3wsk4kjnv1gwpg5cv7am7zcikaruayyzjhaem45e48vfcufbe3vvtntzaoubwogce8k9x48e1dbeyu3srqvphflgxt7+zuffcmjqmfwungopqtximecsnycdyo1imrl4grx8fxubzypxko7acsw5abpl3osvgjk944xvgyhc4damcqpeuku9+mnth4oigchnhn97lmvcqfsatupvyzzzxsgveycjjrqhrpx/hz7iydjoekonxhssojkqbp4klbtfwlcjraasy6gkaddzjyvxsdg6wfzkxks8ez9vzgqqotx4jvj6al6zuk/3mdgpx3ytlkaym6fyjfmxsh3hafu5xxsiissgekm5uxxrifkfgkiyepzfymukwsk7z+xk7xmpwsvqykmjdixokymuurxdodpqiiqest6rx/wi/vmfvpf89gclecoj+wjj8wrkrg</latexit> <latexit sha1_base64="w1zcdyndogfcgilsitkkn9xejoe=">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</latexit> <latexit sha1_base64="hzubufrnucjlnhz+1iagibnm3a=">aaab73icdvdlsgmxfm34rvvdekmwis6gtj1aotckhxjsoj9qduutjppqzozmckizehuhghift/x51/y6atokihlhzouzd77/fjzprg6mnawv1b39jmbew3d3b39gshhvjzlqfol4jls+vpqzqvuaau67saq49dnt+jorzo/cu6lyjg71nkzeieecbyxgbaruo+sfwuuibouisi9qlbjbgchgqoqunyyuq+65cxjzcicjzqdwnt/gjekpeitjpxqosjwxoqlzottwb6fkbpjmsejjnu4jaql53fo4onrhncijkmhizz9fteikolpqfvokosx+q3l4l/eb1ebzuvzsjonbvksshionqrzj6hqyypxxqccasmvshgwojityr5uix5/c/m7bdvidm7cyrxjcmhjsejkaehveedximmaaecohgat+dzurmerrfrddg6yi1njsawg+fetiorg==</latexit> <latexit sha1_base64="hzubufrnucjlnhz+1iagibnm3a=">aaab73icdvdlsgmxfm34rvvdekmwis6gtj1aotckhxjsoj9qduutjppqzozmckizehuhghift/x51/y6atokihlhzouzd77/fjzprg6mnawv1b39jmbew3d3b39gshhvjzlqfol4jls+vpqzqvuaau67saq49dnt+jorzo/cu6lyjg71nkzeieecbyxgbaruo+sfwuuibouisi9qlbjbgchgqoqunyyuq+65cxjzcicjzqdwnt/gjekpeitjpxqosjwxoqlzottwb6fkbpjmsejjnu4jaql53fo4onrhncijkmhizz9fteikolpqfvokosx+q3l4l/eb1ebzuvzsjonbvksshionqrzj6hqyypxxqccasmvshgwojityr5uix5/c/m7bdvidm7cyrxjcmhjsejkaehveedximmaaecohgat+dzurmerrfrddg6yi1njsawg+fetiorg==</latexit> <latexit sha1_base64="hzubufrnucjlnhz+1iagibnm3a=">aaab73icdvdlsgmxfm34rvvdekmwis6gtj1aotckhxjsoj9qduutjppqzozmckizehuhghift/x51/y6atokihlhzouzd77/fjzprg6mnawv1b39jmbew3d3b39gshhvjzlqfol4jls+vpqzqvuaau67saq49dnt+jorzo/cu6lyjg71nkzeieecbyxgbaruo+sfwuuibouisi9qlbjbgchgqoqunyyuq+65cxjzcicjzqdwnt/gjekpeitjpxqosjwxoqlzottwb6fkbpjmsejjnu4jaql53fo4onrhncijkmhizz9fteikolpqfvokosx+q3l4l/eb1ebzuvzsjonbvksshionqrzj6hqyypxxqccasmvshgwojityr5uix5/c/m7bdvidm7cyrxjcmhjsejkaehveedximmaaecohgat+dzurmerrfrddg6yi1njsawg+fetiorg==</latexit> <latexit sha1_base64="hzubufrnucjlnhz+1iagibnm3a=">aaab73icdvdlsgmxfm34rvvdekmwis6gtj1aotckhxjsoj9qduutjppqzozmckizehuhghift/x51/y6atokihlhzouzd77/fjzprg6mnawv1b39jmbew3d3b39gshhvjzlqfol4jls+vpqzqvuaau67saq49dnt+jorzo/cu6lyjg71nkzeieecbyxgbaruo+sfwuuibouisi9qlbjbgchgqoqunyyuq+65cxjzcicjzqdwnt/gjekpeitjpxqosjwxoqlzottwb6fkbpjmsejjnu4jaql53fo4onrhncijkmhizz9fteikolpqfvokosx+q3l4l/eb1ebzuvzsjonbvksshionqrzj6hqyypxxqccasmvshgwojityr5uix5/c/m7bdvidm7cyrxjcmhjsejkaehveedximmaaecohgat+dzurmerrfrddg6yi1njsawg+fetiorg==</latexit> <latexit sha1_base64="/gzmqsc7qujhs/hlc4wjslqx9by=">aaacc3icbzdnsgmxfixv1l9a/6ou3qsl4krmikc4krhxwaf/mbgtjppqzjkgsemvyrxlrvh3anbnin8gxmg1nobuhah/n3as3jo58z1v5zsvrg5lz5u7kzu7d/ud86mizkulbrhkpehhwldnb4yztnupojijoo1g49tz3ngsjmpwmasidbq8firrcxlh+flfqy5jjpmg15tbdudb/8aqoqafmw3udytjeiom4vhr33nte+ryguy4nvb6maypjmm8pl5fgroqg3x+8hsdwweayqnsewbn3d8bou6nisrnuywgenlbgauzfjrrqr8zmtxqc5emhkqyokcooisdqrbgyostwiqvmflofqgsefsbg1lex7xjlxfyhzkxdc+ve/wwtcvviytoivz8oakgnahtwgdaqn8akvzpz5rw7h4vrklshmmfoz8/kywa+g==</latexit> <latexit sha1_base64="/gzmqsc7qujhs/hlc4wjslqx9by=">aaacc3icbzdnsgmxfixv1l9a/6ou3qsl4krmikc4krhxwaf/mbgtjppqzjkgsemvyrxlrvh3anbnin8gxmg1nobuhah/n3as3jo58z1v5zsvrg5lz5u7kzu7d/ud86mizkulbrhkpehhwldnb4yztnupojijoo1g49tz3ngsjmpwmasidbq8firrcxlh+flfqy5jjpmg15tbdudb/8aqoqafmw3udytjeiom4vhr33nte+ryguy4nvb6maypjmm8pl5fgroqg3x+8hsdwweayqnsewbn3d8bou6nisrnuywgenlbgauzfjrrqr8zmtxqc5emhkqyokcooisdqrbgyostwiqvmflofqgsefsbg1lex7xjlxfyhzkxdc+ve/wwtcvviytoivz8oakgnahtwgdaqn8akvzpz5rw7h4vrklshmmfoz8/kywa+g==</latexit> <latexit sha1_base64="/gzmqsc7qujhs/hlc4wjslqx9by=">aaacc3icbzdnsgmxfixv1l9a/6ou3qsl4krmikc4krhxwaf/mbgtjppqzjkgsemvyrxlrvh3anbnin8gxmg1nobuhah/n3as3jo58z1v5zsvrg5lz5u7kzu7d/ud86mizkulbrhkpehhwldnb4yztnupojijoo1g49tz3ngsjmpwmasidbq8firrcxlh+flfqy5jjpmg15tbdudb/8aqoqafmw3udytjeiom4vhr33nte+ryguy4nvb6maypjmm8pl5fgroqg3x+8hsdwweayqnsewbn3d8bou6nisrnuywgenlbgauzfjrrqr8zmtxqc5emhkqyokcooisdqrbgyostwiqvmflofqgsefsbg1lex7xjlxfyhzkxdc+ve/wwtcvviytoivz8oakgnahtwgdaqn8akvzpz5rw7h4vrklshmmfoz8/kywa+g==</latexit> <latexit sha1_base64="/gzmqsc7qujhs/hlc4wjslqx9by=">aaacc3icbzdnsgmxfixv1l9a/6ou3qsl4krmikc4krhxwaf/mbgtjppqzjkgsemvyrxlrvh3anbnin8gxmg1nobuhah/n3as3jo58z1v5zsvrg5lz5u7kzu7d/ud86mizkulbrhkpehhwldnb4yztnupojijoo1g49tz3ngsjmpwmasidbq8firrcxlh+flfqy5jjpmg15tbdudb/8aqoqafmw3udytjeiom4vhr33nte+ryguy4nvb6maypjmm8pl5fgroqg3x+8hsdwweayqnsewbn3d8bou6nisrnuywgenlbgauzfjrrqr8zmtxqc5emhkqyokcooisdqrbgyostwiqvmflofqgsefsbg1lex7xjlxfyhzkxdc+ve/wwtcvviytoivz8oakgnahtwgdaqn8akvzpz5rw7h4vrklshmmfoz8/kywa+g==</latexit> <latexit sha1_base64="mr83cwkyqaiiyl1ktckic9g=">aaacc3icbzdlsgmxfib1futt6pln8eiucozrvbcfdy4rnabtichkbaewyjbmhjhel71idyjwx/cz/altc8lbftd4o/z4gto58z1v53cxubw9k5xt7s3f3b4vd4+awuzkujbrhkpuhhwldnbw4yztrupojijoo1eo7tp3nmksjmpmmacidba8firrcxlh+ftfqu5jistcjyxa6m6fv8bzqgyuayfmn15cks6gwhgotfc9ntzbjzrjhdflqzzqmmizwgowbu6odvlzyrnyz+iqwytxgc/9u5djrepxedjlbzqixs6m5nkutts7wmxfbdktawaoiml4owji9ggnrnihldxxywucx+apehvpgyw1/jtumtd7ek7vrvc6vew1wlfrvoqqhnca6x4me11oeegtacahje4bxengfn3flwuejbwexcwr/5hz9apmjmvs=</latexit> <latexit sha1_base64="mr83cwkyqaiiyl1ktckic9g=">aaacc3icbzdlsgmxfib1futt6pln8eiucozrvbcfdy4rnabtichkbaewyjbmhjhel71idyjwx/cz/altc8lbftd4o/z4gto58z1v53cxubw9k5xt7s3f3b4vd4+awuzkujbrhkpuhhwldnbw4yztrupojijoo1eo7tp3nmksjmpmmacidba8firrcxlh+ftfqu5jistcjyxa6m6fv8bzqgyuayfmn15cks6gwhgotfc9ntzbjzrjhdflqzzqmmizwgowbu6odvlzyrnyz+iqwytxgc/9u5djrepxedjlbzqixs6m5nkutts7wmxfbdktawaoiml4owji9ggnrnihldxxywucx+apehvpgyw1/jtumtd7ek7vrvc6vew1wlfrvoqqhnca6x4me11oeegtacahje4bxengfn3flwuejbwexcwr/5hz9apmjmvs=</latexit> <latexit sha1_base64="mr83cwkyqaiiyl1ktckic9g=">aaacc3icbzdlsgmxfib1futt6pln8eiucozrvbcfdy4rnabtichkbaewyjbmhjhel71idyjwx/cz/altc8lbftd4o/z4gto58z1v53cxubw9k5xt7s3f3b4vd4+awuzkujbrhkpuhhwldnbw4yztrupojijoo1eo7tp3nmksjmpmmacidba8firrcxlh+ftfqu5jistcjyxa6m6fv8bzqgyuayfmn15cks6gwhgotfc9ntzbjzrjhdflqzzqmmizwgowbu6odvlzyrnyz+iqwytxgc/9u5djrepxedjlbzqixs6m5nkutts7wmxfbdktawaoiml4owji9ggnrnihldxxywucx+apehvpgyw1/jtumtd7ek7vrvc6vew1wlfrvoqqhnca6x4me11oeegtacahje4bxengfn3flwuejbwexcwr/5hz9apmjmvs=</latexit> <latexit sha1_base64="mr83cwkyqaiiyl1ktckic9g=">aaacc3icbzdlsgmxfib1futt6pln8eiucozrvbcfdy4rnabtichkbaewyjbmhjhel71idyjwx/cz/altc8lbftd4o/z4gto58z1v53cxubw9k5xt7s3f3b4vd4+awuzkujbrhkpuhhwldnbw4yztrupojijoo1eo7tp3nmksjmpmmacidba8firrcxlh+ftfqu5jistcjyxa6m6fv8bzqgyuayfmn15cks6gwhgotfc9ntzbjzrjhdflqzzqmmizwgowbu6odvlzyrnyz+iqwytxgc/9u5djrepxedjlbzqixs6m5nkutts7wmxfbdktawaoiml4owji9ggnrnihldxxywucx+apehvpgyw1/jtumtd7ek7vrvc6vew1wlfrvoqqhnca6x4me11oeegtacahje4bxengfn3flwuejbwexcwr/5hz9apmjmvs=</latexit> <latexit sha1_base64="5yxtmtg++qwe9n+y5ycstza=">aaacenicbvdltsjafl3ff+kr6tlnrdbxrvogldeny4xetcbppkou5gwntyzuxls8bcu3ephudnu/qg/wz9wclocjkbnjxzb3lucrlolhacb6usbm1vverezthxweccnhrwnktaixksnwkskgectjxtnd4lkuio4lqbjo9yvzuhurfyoppqridwulgchasl5t1/or1iocexy785a8uqu3fmqkvelugvcrr8+6c/iekauaejxr1xcfrxoalzottwawfkppgmszdjnu4igql5snn6elowxqgeszqqo5+vciw5fsygwm3lmtezl4lovmdi6o5fq8nrlmehstqvzjahtjnsm8n7qgelknj8agolk5glerlhioklfdoou9zfkuk6q5tdx8a1eznvmzzuaclsgfkjcbsgdqqm8akv8gy9w+/wh/w5wc1zxcp/i19qsr7jzj</latexit> <latexit sha1_base64="5yxtmtg++qwe9n+y5ycstza=">aaacenicbvdltsjafl3ff+kr6tlnrdbxrvogldeny4xetcbppkou5gwntyzuxls8bcu3ephudnu/qg/wz9wclocjkbnjxzb3lucrlolhacb6usbm1vverezthxweccnhrwnktaixksnwkskgectjxtnd4lkuio4lqbjo9yvzuhurfyoppqridwulgchasl5t1/or1iocexy785a8uqu3fmqkvelugvcrr8+6c/iekauaejxr1xcfrxoalzottwawfkppgmszdjnu4igql5snn6elowxqgeszqqo5+vciw5fsygwm3lmtezl4lovmdi6o5fq8nrlmehstqvzjahtjnsm8n7qgelknj8agolk5glerlhioklfdoou9zfkuk6q5tdx8a1eznvmzzuaclsgfkjcbsgdqqm8akv8gy9w+/wh/w5wc1zxcp/i19qsr7jzj</latexit> <latexit sha1_base64="5yxtmtg++qwe9n+y5ycstza=">aaacenicbvdltsjafl3ff+kr6tlnrdbxrvogldeny4xetcbppkou5gwntyzuxls8bcu3ephudnu/qg/wz9wclocjkbnjxzb3lucrlolhacb6usbm1vverezthxweccnhrwnktaixksnwkskgectjxtnd4lkuio4lqbjo9yvzuhurfyoppqridwulgchasl5t1/or1iocexy785a8uqu3fmqkvelugvcrr8+6c/iekauaejxr1xcfrxoalzottwawfkppgmszdjnu4igql5snn6elowxqgeszqqo5+vciw5fsygwm3lmtezl4lovmdi6o5fq8nrlmehstqvzjahtjnsm8n7qgelknj8agolk5glerlhioklfdoou9zfkuk6q5tdx8a1eznvmzzuaclsgfkjcbsgdqqm8akv8gy9w+/wh/w5wc1zxcp/i19qsr7jzj</latexit> <latexit sha1_base64="5yxtmtg++qwe9n+y5ycstza=">aaacenicbvdltsjafl3ff+kr6tlnrdbxrvogldeny4xetcbppkou5gwntyzuxls8bcu3ephudnu/qg/wz9wclocjkbnjxzb3lucrlolhacb6usbm1vverezthxweccnhrwnktaixksnwkskgectjxtnd4lkuio4lqbjo9yvzuhurfyoppqridwulgchasl5t1/or1iocexy785a8uqu3fmqkvelugvcrr8+6c/iekauaejxr1xcfrxoalzottwawfkppgmszdjnu4igql5snn6elowxqgeszqqo5+vciw5fsygwm3lmtezl4lovmdi6o5fq8nrlmehstqvzjahtjnsm8n7qgelknj8agolk5glerlhioklfdoou9zfkuk6q5tdx8a1eznvmzzuaclsgfkjcbsgdqqm8akv8gy9w+/wh/w5wc1zxcp/i19qsr7jzj</latexit> rivacy verification using barrier certificates If there exists a function B such that (with additional standard technical assumptions) B(b) = b T a1 B<latexit sha1_base64="vctrvbtrvpi6lblbhcvfihhl5o=">aaab9hicbvdlsgmxflpr1pfvzdugkvwvwaqomtsny4raeq8mkmtykxmttkem/q43lhrx68e48/mtlq1gobwzn3ck+ohwuujen8o8lg5tbtng3tld/chhuj5p6yhrllvojclv9ylmgkvwmtwi1ovi6evwmef3gv+z8qu5pf8nloyesezsr5wsoyvvh5izjgsktbmgrqrjhvzwg8ttycvcbhc1d+6g8jmormgiqi1j3xiyxemu4fwxe6ieaxyroyij1ljukznplf6hn+miqqxxeyj5p8el9vzgsuotz6nvjlkre9tlx6+xmodws7mme8mkxr4keofnhlmg8jarroywuko4jyrpmoicdwp5itwv398jpp16ruvbxcfpn/i6inag53ajltxahe6hcsg8at8apvaipedv6wi4wul5zcn+an8ab8csra==</latexit> u B 8 B B(b) >, 8b B u b T a 8 B 8 B B(b) <. 8b B apple 8 B(f a (b t 1,z)) B(b t 1,z) apple 8a A, z Z, t Z 1 Then b t / B u 8t Z 1. Ufuk Topcu 1

31 <latexit sha1_base64="kcsgcjmfcldg1ukqobq7hrcfew=">aaach3icbvdlsgmxfm3uv6vqks3wsk4kjnv1gwpg5cv7am7zcikaruayyzjhaem45e48vfcufbe3vvtntzaoubwogce8k9x48e1dbeyu3srqvphflgxt7+zuffcmjqmfwungopqtximecsnycdyo1imrl4grx8fxubzypxko7acsw5abpl3osvgjk944xvgyhc4damcqpeuku9+mnth4oigchnhn97lmvcqfsatupvyzzzxsgveycjjrqhrpx/hz7iydjoekonxhssojkqbp4klbtfwlcjraasy6gkaddzjyvxsdg6wfzkxks8ez9vzgqqotx4jvj6al6zuk/3mdgpx3ytlkaym6fyjfmxsh3hafu5xxsiissgekm5uxxrifkfgkiyepzfymukwsk7z+xk7xmpwsvqykmjdixokymuurxdodpqiiqest6rx/wi/vmfvpf89gclecoj+wjj8wrkrg</latexit> <latexit sha1_base64="c4zqyp5gqrlwe5guk9ddvsjv6ny=">aaacfnicbzdlssnafiz6q3ww7zs3ayqgupstekibtdukzqg7qhtkatdujkwsxeqkhv4dktoq7cevwz/alnlzzaoax//fw6c+bym6ks68virayurw/knwtbzu7e+b+qutgiscssieiy6hjeuspe3ffkedwfacejyvdhtng8/ucfzfdbuokzogach8xnbsluuevs6cvsviflxrnnns6ukvvcrlmktzm6d/ygrqhu91v3v9icqbdrxhwmqubcxksbfqjha6kfqsswnmrnhauxpdhfdpplrj+huo33krk/ukgz+3sjxygu48dtkwfwq7mytclwaytzue3uf6lk7iwthqnyfwc+firwjaeeozqyniywycky/gcgqcyubrkg7exu/grwrftirfbvyu8p6ysmxneazbliagtxbhzpa4bge4qvejsfjzxg3uajosboyq/mj5/amuna=</latexit> <latexit sha1_base64="c4zqyp5gqrlwe5guk9ddvsjv6ny=">aaacfnicbzdlssnafiz6q3ww7zs3ayqgupstekibtdukzqg7qhtkatdujkwsxeqkhv4dktoq7cevwz/alnlzzaoax//fw6c+bym6ks68virayurw/knwtbzu7e+b+qutgiscssieiy6hjeuspe3ffkedwfacejyvdhtng8/ucfzfdbuokzogach8xnbsluuevs6cvsviflxrnnns6ukvvcrlmktzm6d/ygrqhu91v3v9icqbdrxhwmqubcxksbfqjha6kfqsswnmrnhauxpdhfdpplrj+huo33krk/ukgz+3sjxygu48dtkwfwq7mytclwaytzue3uf6lk7iwthqnyfwc+firwjaeeozqyniywycky/gcgqcyubrkg7exu/grwrftirfbvyu8p6ysmxneazbliagtxbhzpa4bge4qvejsfjzxg3uajosboyq/mj5/amuna=</latexit> <latexit sha1_base64="c4zqyp5gqrlwe5guk9ddvsjv6ny=">aaacfnicbzdlssnafiz6q3ww7zs3ayqgupstekibtdukzqg7qhtkatdujkwsxeqkhv4dktoq7cevwz/alnlzzaoax//fw6c+bym6ks68virayurw/knwtbzu7e+b+qutgiscssieiy6hjeuspe3ffkedwfacejyvdhtng8/ucfzfdbuokzogach8xnbsluuevs6cvsviflxrnnns6ukvvcrlmktzm6d/ygrqhu91v3v9icqbdrxhwmqubcxksbfqjha6kfqsswnmrnhauxpdhfdpplrj+huo33krk/ukgz+3sjxygu48dtkwfwq7mytclwaytzue3uf6lk7iwthqnyfwc+firwjaeeozqyniywycky/gcgqcyubrkg7exu/grwrftirfbvyu8p6ysmxneazbliagtxbhzpa4bge4qvejsfjzxg3uajosboyq/mj5/amuna=</latexit> <latexit sha1_base64="c4zqyp5gqrlwe5guk9ddvsjv6ny=">aaacfnicbzdlssnafiz6q3ww7zs3ayqgupstekibtdukzqg7qhtkatdujkwsxeqkhv4dktoq7cevwz/alnlzzaoax//fw6c+bym6ks68virayurw/knwtbzu7e+b+qutgiscssieiy6hjeuspe3ffkedwfacejyvdhtng8/ucfzfdbuokzogach8xnbsluuevs6cvsviflxrnnns6ukvvcrlmktzm6d/ygrqhu91v3v9icqbdrxhwmqubcxksbfqjha6kfqsswnmrnhauxpdhfdpplrj+huo33krk/ukgz+3sjxygu48dtkwfwq7mytclwaytzue3uf6lk7iwthqnyfwc+firwjaeeozqyniywycky/gcgqcyubrkg7exu/grwrftirfbvyu8p6ysmxneazbliagtxbhzpa4bge4qvejsfjzxg3uajosboyq/mj5/amuna=</latexit> <latexit sha1_base64="uqheb9m6gtkxyb1syjgzus5m4=">aaacfnicbzdlssnafiz6q3ww7zs3ayqgupstekibtdukzqg7qhtkatdujkwsxeqkhv4dktoq7cevwz/alnlzzaoax//fw6c+bym6ks68virayurw/knwtbzu7e+b+qutgiscssieiy6hjeuspe3ffkedwfacejyvdhtng8/ucfzfdbuokzogach8xnbsluuevs6cvspny49xzgfxycoh1yxafwsm9b/sdiqqqe6a371+rjkahopwlgxxtmllpfgorjidfhqjpdemizygxyhdqhtn1e3sqnt7yi6ffqndm/br4kdkcedpyqcrovzmpubslnbwsqcbk/ssvkyj4qgzh6bn3ckijttcwzoetxsqzmbnofqgsibszkn1nq7dilxfyhvrviwxx7vlqsxwu95eeytqamnlxade6gdkg8aj8akvxpxzrwbh/rnjhthmifgz8/y4sc7w==</latexit> <latexit sha1_base64="uqheb9m6gtkxyb1syjgzus5m4=">aaacfnicbzdlssnafiz6q3ww7zs3ayqgupstekibtdukzqg7qhtkatdujkwsxeqkhv4dktoq7cevwz/alnlzzaoax//fw6c+bym6ks68virayurw/knwtbzu7e+b+qutgiscssieiy6hjeuspe3ffkedwfacejyvdhtng8/ucfzfdbuokzogach8xnbsluuevs6cvspny49xzgfxycoh1yxafwsm9b/sdiqqqe6a371+rjkahopwlgxxtmllpfgorjidfhqjpdemizygxyhdqhtn1e3sqnt7yi6ffqndm/br4kdkcedpyqcrovzmpubslnbwsqcbk/ssvkyj4qgzh6bn3ckijttcwzoetxsqzmbnofqgsibszkn1nq7dilxfyhvrviwxx7vlqsxwu95eeytqamnlxade6gdkg8aj8akvxpxzrwbh/rnjhthmifgz8/y4sc7w==</latexit> <latexit sha1_base64="uqheb9m6gtkxyb1syjgzus5m4=">aaacfnicbzdlssnafiz6q3ww7zs3ayqgupstekibtdukzqg7qhtkatdujkwsxeqkhv4dktoq7cevwz/alnlzzaoax//fw6c+bym6ks68virayurw/knwtbzu7e+b+qutgiscssieiy6hjeuspe3ffkedwfacejyvdhtng8/ucfzfdbuokzogach8xnbsluuevs6cvspny49xzgfxycoh1yxafwsm9b/sdiqqqe6a371+rjkahopwlgxxtmllpfgorjidfhqjpdemizygxyhdqhtn1e3sqnt7yi6ffqndm/br4kdkcedpyqcrovzmpubslnbwsqcbk/ssvkyj4qgzh6bn3ckijttcwzoetxsqzmbnofqgsibszkn1nq7dilxfyhvrviwxx7vlqsxwu95eeytqamnlxade6gdkg8aj8akvxpxzrwbh/rnjhthmifgz8/y4sc7w==</latexit> <latexit sha1_base64="uqheb9m6gtkxyb1syjgzus5m4=">aaacfnicbzdlssnafiz6q3ww7zs3ayqgupstekibtdukzqg7qhtkatdujkwsxeqkhv4dktoq7cevwz/alnlzzaoax//fw6c+bym6ks68virayurw/knwtbzu7e+b+qutgiscssieiy6hjeuspe3ffkedwfacejyvdhtng8/ucfzfdbuokzogach8xnbsluuevs6cvspny49xzgfxycoh1yxafwsm9b/sdiqqqe6a371+rjkahopwlgxxtmllpfgorjidfhqjpdemizygxyhdqhtn1e3sqnt7yi6ffqndm/br4kdkcedpyqcrovzmpubslnbwsqcbk/ssvkyj4qgzh6bn3ckijttcwzoetxsqzmbnofqgsibszkn1nq7dilxfyhvrviwxx7vlqsxwu95eeytqamnlxade6gdkg8aj8akvxpxzrwbh/rnjhthmifgz8/y4sc7w==</latexit> <latexit sha1_base64="/gzmqsc7qujhs/hlc4wjslqx9by=">aaacc3icbzdnsgmxfixv1l9a/6ou3qsl4krmikc4krhxwaf/mbgtjppqzjkgsemvyrxlrvh3anbnin8gxmg1nobuhah/n3as3jo58z1v5zsvrg5lz5u7kzu7d/ud86mizkulbrhkpehhwldnb4yztnupojijoo1g49tz3ngsjmpwmasidbq8firrcxlh+flfqy5jjpmg15tbdudb/8aqoqafmw3udytjeiom4vhr33nte+ryguy4nvb6maypjmm8pl5fgroqg3x+8hsdwweayqnsewbn3d8bou6nisrnuywgenlbgauzfjrrqr8zmtxqc5emhkqyokcooisdqrbgyostwiqvmflofqgsefsbg1lex7xjlxfyhzkxdc+ve/wwtcvviytoivz8oakgnahtwgdaqn8akvzpz5rw7h4vrklshmmfoz8/kywa+g==</latexit> <latexit sha1_base64="/gzmqsc7qujhs/hlc4wjslqx9by=">aaacc3icbzdnsgmxfixv1l9a/6ou3qsl4krmikc4krhxwaf/mbgtjppqzjkgsemvyrxlrvh3anbnin8gxmg1nobuhah/n3as3jo58z1v5zsvrg5lz5u7kzu7d/ud86mizkulbrhkpehhwldnb4yztnupojijoo1g49tz3ngsjmpwmasidbq8firrcxlh+flfqy5jjpmg15tbdudb/8aqoqafmw3udytjeiom4vhr33nte+ryguy4nvb6maypjmm8pl5fgroqg3x+8hsdwweayqnsewbn3d8bou6nisrnuywgenlbgauzfjrrqr8zmtxqc5emhkqyokcooisdqrbgyostwiqvmflofqgsefsbg1lex7xjlxfyhzkxdc+ve/wwtcvviytoivz8oakgnahtwgdaqn8akvzpz5rw7h4vrklshmmfoz8/kywa+g==</latexit> <latexit sha1_base64="/gzmqsc7qujhs/hlc4wjslqx9by=">aaacc3icbzdnsgmxfixv1l9a/6ou3qsl4krmikc4krhxwaf/mbgtjppqzjkgsemvyrxlrvh3anbnin8gxmg1nobuhah/n3as3jo58z1v5zsvrg5lz5u7kzu7d/ud86mizkulbrhkpehhwldnb4yztnupojijoo1g49tz3ngsjmpwmasidbq8firrcxlh+flfqy5jjpmg15tbdudb/8aqoqafmw3udytjeiom4vhr33nte+ryguy4nvb6maypjmm8pl5fgroqg3x+8hsdwweayqnsewbn3d8bou6nisrnuywgenlbgauzfjrrqr8zmtxqc5emhkqyokcooisdqrbgyostwiqvmflofqgsefsbg1lex7xjlxfyhzkxdc+ve/wwtcvviytoivz8oakgnahtwgdaqn8akvzpz5rw7h4vrklshmmfoz8/kywa+g==</latexit> <latexit sha1_base64="/gzmqsc7qujhs/hlc4wjslqx9by=">aaacc3icbzdnsgmxfixv1l9a/6ou3qsl4krmikc4krhxwaf/mbgtjppqzjkgsemvyrxlrvh3anbnin8gxmg1nobuhah/n3as3jo58z1v5zsvrg5lz5u7kzu7d/ud86mizkulbrhkpehhwldnb4yztnupojijoo1g49tz3ngsjmpwmasidbq8firrcxlh+flfqy5jjpmg15tbdudb/8aqoqafmw3udytjeiom4vhr33nte+ryguy4nvb6maypjmm8pl5fgroqg3x+8hsdwweayqnsewbn3d8bou6nisrnuywgenlbgauzfjrrqr8zmtxqc5emhkqyokcooisdqrbgyostwiqvmflofqgsefsbg1lex7xjlxfyhzkxdc+ve/wwtcvviytoivz8oakgnahtwgdaqn8akvzpz5rw7h4vrklshmmfoz8/kywa+g==</latexit> <latexit sha1_base64="mr83cwkyqaiiyl1ktckic9g=">aaacc3icbzdlsgmxfib1futt6pln8eiucozrvbcfdy4rnabtichkbaewyjbmhjhel71idyjwx/cz/altc8lbftd4o/z4gto58z1v53cxubw9k5xt7s3f3b4vd4+awuzkujbrhkpuhhwldnbw4yztrupojijoo1eo7tp3nmksjmpmmacidba8firrcxlh+ftfqu5jistcjyxa6m6fv8bzqgyuayfmn15cks6gwhgotfc9ntzbjzrjhdflqzzqmmizwgowbu6odvlzyrnyz+iqwytxgc/9u5djrepxedjlbzqixs6m5nkutts7wmxfbdktawaoiml4owji9ggnrnihldxxywucx+apehvpgyw1/jtumtd7ek7vrvc6vew1wlfrvoqqhnca6x4me11oeegtacahje4bxengfn3flwuejbwexcwr/5hz9apmjmvs=</latexit> <latexit sha1_base64="mr83cwkyqaiiyl1ktckic9g=">aaacc3icbzdlsgmxfib1futt6pln8eiucozrvbcfdy4rnabtichkbaewyjbmhjhel71idyjwx/cz/altc8lbftd4o/z4gto58z1v53cxubw9k5xt7s3f3b4vd4+awuzkujbrhkpuhhwldnbw4yztrupojijoo1eo7tp3nmksjmpmmacidba8firrcxlh+ftfqu5jistcjyxa6m6fv8bzqgyuayfmn15cks6gwhgotfc9ntzbjzrjhdflqzzqmmizwgowbu6odvlzyrnyz+iqwytxgc/9u5djrepxedjlbzqixs6m5nkutts7wmxfbdktawaoiml4owji9ggnrnihldxxywucx+apehvpgyw1/jtumtd7ek7vrvc6vew1wlfrvoqqhnca6x4me11oeegtacahje4bxengfn3flwuejbwexcwr/5hz9apmjmvs=</latexit> <latexit sha1_base64="mr83cwkyqaiiyl1ktckic9g=">aaacc3icbzdlsgmxfib1futt6pln8eiucozrvbcfdy4rnabtichkbaewyjbmhjhel71idyjwx/cz/altc8lbftd4o/z4gto58z1v53cxubw9k5xt7s3f3b4vd4+awuzkujbrhkpuhhwldnbw4yztrupojijoo1eo7tp3nmksjmpmmacidba8firrcxlh+ftfqu5jistcjyxa6m6fv8bzqgyuayfmn15cks6gwhgotfc9ntzbjzrjhdflqzzqmmizwgowbu6odvlzyrnyz+iqwytxgc/9u5djrepxedjlbzqixs6m5nkutts7wmxfbdktawaoiml4owji9ggnrnihldxxywucx+apehvpgyw1/jtumtd7ek7vrvc6vew1wlfrvoqqhnca6x4me11oeegtacahje4bxengfn3flwuejbwexcwr/5hz9apmjmvs=</latexit> <latexit sha1_base64="mr83cwkyqaiiyl1ktckic9g=">aaacc3icbzdlsgmxfib1futt6pln8eiucozrvbcfdy4rnabtichkbaewyjbmhjhel71idyjwx/cz/altc8lbftd4o/z4gto58z1v53cxubw9k5xt7s3f3b4vd4+awuzkujbrhkpuhhwldnbw4yztrupojijoo1eo7tp3nmksjmpmmacidba8firrcxlh+ftfqu5jistcjyxa6m6fv8bzqgyuayfmn15cks6gwhgotfc9ntzbjzrjhdflqzzqmmizwgowbu6odvlzyrnyz+iqwytxgc/9u5djrepxedjlbzqixs6m5nkutts7wmxfbdktawaoiml4owji9ggnrnihldxxywucx+apehvpgyw1/jtumtd7ek7vrvc6vew1wlfrvoqqhnca6x4me11oeegtacahje4bxengfn3flwuejbwexcwr/5hz9apmjmvs=</latexit> <latexit sha1_base64="5yxtmtg++qwe9n+y5ycstza=">aaacenicbvdltsjafl3ff+kr6tlnrdbxrvogldeny4xetcbppkou5gwntyzuxls8bcu3ephudnu/qg/wz9wclocjkbnjxzb3lucrlolhacb6usbm1vverezthxweccnhrwnktaixksnwkskgectjxtnd4lkuio4lqbjo9yvzuhurfyoppqridwulgchasl5t1/or1iocexy785a8uqu3fmqkvelugvcrr8+6c/iekauaejxr1xcfrxoalzottwawfkppgmszdjnu4igql5snn6elowxqgeszqqo5+vciw5fsygwm3lmtezl4lovmdi6o5fq8nrlmehstqvzjahtjnsm8n7qgelknj8agolk5glerlhioklfdoou9zfkuk6q5tdx8a1eznvmzzuaclsgfkjcbsgdqqm8akv8gy9w+/wh/w5wc1zxcp/i19qsr7jzj</latexit> <latexit sha1_base64="5yxtmtg++qwe9n+y5ycstza=">aaacenicbvdltsjafl3ff+kr6tlnrdbxrvogldeny4xetcbppkou5gwntyzuxls8bcu3ephudnu/qg/wz9wclocjkbnjxzb3lucrlolhacb6usbm1vverezthxweccnhrwnktaixksnwkskgectjxtnd4lkuio4lqbjo9yvzuhurfyoppqridwulgchasl5t1/or1iocexy785a8uqu3fmqkvelugvcrr8+6c/iekauaejxr1xcfrxoalzottwawfkppgmszdjnu4igql5snn6elowxqgeszqqo5+vciw5fsygwm3lmtezl4lovmdi6o5fq8nrlmehstqvzjahtjnsm8n7qgelknj8agolk5glerlhioklfdoou9zfkuk6q5tdx8a1eznvmzzuaclsgfkjcbsgdqqm8akv8gy9w+/wh/w5wc1zxcp/i19qsr7jzj</latexit> <latexit sha1_base64="5yxtmtg++qwe9n+y5ycstza=">aaacenicbvdltsjafl3ff+kr6tlnrdbxrvogldeny4xetcbppkou5gwntyzuxls8bcu3ephudnu/qg/wz9wclocjkbnjxzb3lucrlolhacb6usbm1vverezthxweccnhrwnktaixksnwkskgectjxtnd4lkuio4lqbjo9yvzuhurfyoppqridwulgchasl5t1/or1iocexy785a8uqu3fmqkvelugvcrr8+6c/iekauaejxr1xcfrxoalzottwawfkppgmszdjnu4igql5snn6elowxqgeszqqo5+vciw5fsygwm3lmtezl4lovmdi6o5fq8nrlmehstqvzjahtjnsm8n7qgelknj8agolk5glerlhioklfdoou9zfkuk6q5tdx8a1eznvmzzuaclsgfkjcbsgdqqm8akv8gy9w+/wh/w5wc1zxcp/i19qsr7jzj</latexit> <latexit sha1_base64="5yxtmtg++qwe9n+y5ycstza=">aaacenicbvdltsjafl3ff+kr6tlnrdbxrvogldeny4xetcbppkou5gwntyzuxls8bcu3ephudnu/qg/wz9wclocjkbnjxzb3lucrlolhacb6usbm1vverezthxweccnhrwnktaixksnwkskgectjxtnd4lkuio4lqbjo9yvzuhurfyoppqridwulgchasl5t1/or1iocexy785a8uqu3fmqkvelugvcrr8+6c/iekauaejxr1xcfrxoalzottwawfkppgmszdjnu4igql5snn6elowxqgeszqqo5+vciw5fsygwm3lmtezl4lovmdi6o5fq8nrlmehstqvzjahtjnsm8n7qgelknj8agolk5glerlhioklfdoou9zfkuk6q5tdx8a1eznvmzzuaclsgfkjcbsgdqqm8akv8gy9w+/wh/w5wc1zxcp/i19qsr7jzj</latexit> rivacy verification using barrier certificates If there exists a function B such that (with additional standard technical assumptions) B a1 (T,b T )= co{b a1,b a }(T,b T )= <latexit sha1_base64="4idnrl7xjcromsmsucgiv7atcs=">aaaccxicdvdlsgmxfm3uv6vqks3wsjukgvmwtq6eerdukzqf3sgizombwjmqziryjbbn/6kgxekuu3k3ztokknrgcg/n3etyjxsykqsufiztfwnza3sdm5nd//ih941bnbxdhp4oaffoaiqrj1svdsycgg5ar5lin9d3av+v1bwgun/i6ch8tmsny4qrvjkthziw4pyti8disotujlzs7jrcj+4k55e6ky/o5ytgzazwof7w9bphgikx69vkfrpksveak7sd/ls1cndkev9ihoqygnoo7rhxstejsc6kbakrnqejgsrqi48io15cksazpyzgoocqfakx6venghlczd1xtxpitsvvlxx/8oarhdfsmphjimlw+niwzlanny4ihygiwbk4iwp+qvee8rr1iq8hiqhk9l4f+kz5anstm8qraarvucwxactkergkaomuaateexyhahhsatenbutuftrxtdjma1c4x+aht7r87zk/</latexit> b T a1 B<latexit sha1_base64="vctrvbtrvpi6lblbhcvfihhl5o=">aaab9hicbvdlsgmxflpr1pfvzdugkvwvwaqomtsny4raeq8mkmtykxmttkem/q43lhrx68e48/mtlq1gobwzn3ck+ohwuujen8o8lg5tbtng3tld/chhuj5p6yhrllvojclv9ylmgkvwmtwi1ovi6evwmef3gv+z8qu5pf8nloyesezsr5wsoyvvh5izjgsktbmgrqrjhvzwg8ttycvcbhc1d+6g8jmormgiqi1j3xiyxemu4fwxe6ieaxyroyij1ljukznplf6hn+miqqxxeyj5p8el9vzgsuotz6nvjlkre9tlx6+xmodws7mme8mkxr4keofnhlmg8jarroywuko4jyrpmoicdwp5itwv398jpp16ruvbxcfpn/i6inag53ajltxahe6hcsg8at8apvaipedv6wi4wul5zcn+an8ab8csra==</latexit> u B B a (T,b T )= b T a Then b t / B u 8t Z 1. Ufuk Topcu 1

32 <latexit sha1_base64="7cttgnqck8auojhvyqvduxko4f8=">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</latexit> <latexit sha1_base64="vqoleg+axbwbfseewac8xere=">aaab7nicbvdlsgnbeoz1gemr6thlybdijexgwvyegbeecwdkixmtibjknnzdazxces+woshrbz6d78gyfjhjsxokgo6qa7k4ilmoi6387a+sbm1nzuj7+7t39wwdg6bpooyw3wcqj3q6o4vio3kcbkrdjzwkysn4kxrczv/xetrgresbjzqdpuycebrsqgh6xhi5teoeiw3tnikveyuoqm9v7hq9uwbjyhuxsyzqeg6ofuocst7ndxdy8rgdmg7lioacuon83on5nwqftkitcfzk7+nkhpamwkdgxnshfklrz+j/xsxbq9voh4gs5yotfgqsjmjsd9ixmjoueso8lestiiasrqjps3ixjll6+szqxsxzyr91ffwjwliwencayl8oaaanahdwgagze8wyu8obhz4rw7h4vwnseboye/cd5/agqhju4=</latexit> How to search for V or B? Useful features of verification problems in the belief space Belief dynamics are rational. Belief set is a unit simplex. b t (q) = O(q,a,z) qq T (q, a, q )b t 1 (q) q Q O(q,a,z) qq,t(q, a, q )b t 1 (q) Ufuk Topcu 11

33 <latexit sha1_base64="7cttgnqck8auojhvyqvduxko4f8=">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</latexit> <latexit sha1_base64="vqoleg+axbwbfseewac8xere=">aaab7nicbvdlsgnbeoz1gemr6thlybdijexgwvyegbeecwdkixmtibjknnzdazxces+woshrbz6d78gyfjhjsxokgo6qa7k4ilmoi6387a+sbm1nzuj7+7t39wwdg6bpooyw3wcqj3q6o4vio3kcbkrdjzwkysn4kxrczv/xetrgresbjzqdpuycebrsqgh6xhi5teoeiw3tnikveyuoqm9v7hq9uwbjyhuxsyzqeg6ofuocst7ndxdy8rgdmg7lioacuon83on5nwqftkitcfzk7+nkhpamwkdgxnshfklrz+j/xsxbq9voh4gs5yotfgqsjmjsd9ixmjoueso8lestiiasrqjps3ixjll6+szqxsxzyr91ffwjwliwencayl8oaaanahdwgagze8wyu8obhz4rw7h4vwnseboye/cd5/agqhju4=</latexit> How to search for V or B? Useful features of verification problems in the belief space Belief dynamics are rational. Belief set is a unit simplex. b t (q) = O(q,a,z) qq T (q, a, q )b t 1 (q) q Q O(q,a,z) qq,t(q, a, q )b t 1 (q) Optimization-based search for V or B: Restrict V or B to be polynomials of fixed, finite degree Formulate the search as polynomial optimization Relax to sum-ofsquares optimization problems Solve as semidefinite programming problems Ufuk Topcu 11

34 <latexit sha1_base64="u3pqlk5cy6ezgldzskciulfs=">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</latexit> <latexit sha1_base64="hq7sakh938qotn4wh/f4hpruo=">aaaclxicbvdlsgmxfm3uv6vqks3wskvmpmfrrxbtcuk9ghdmpwj3bgrmmmseuvorfollt/orlgrx68kfmnnowba7n3mvnox7emdk/wgl1ty3nrfs5md3b39g+zhuvfsssrkieyqyiniwjpmmtnmjckin9og7hn/myjlyqfwymerbqtobewliogjerlixmmse/p/narf4ozclfwor1itwdcwlnbdsvwjnxgy+bskjfxibonotqhfuv++nqhilgmjcqamwye6qapgefknfjrsmga+jrlqebckra4moct4zsgeb4+yfgk/vvxtjeeqnhg8mbei+wvqscauxgwmvyp197o9zkeuaxqqq+6suke4qq63ggses1hhgcrzitapa8sidyfj+4i1sk3q55ngl5/4yv7mz95rgj+gu5zgdrlaf3aeqqigcntalekvv1r1bn1ax7rldxfoub/yh3/auijpds=</latexit> <latexit sha1_base64="hq7sakh938qotn4wh/f4hpruo=">aaaclxicbvdlsgmxfm3uv6vqks3wskvmpmfrrxbtcuk9ghdmpwj3bgrmmmseuvorfollt/orlgrx68kfmnnowba7n3mvnox7emdk/wgl1ty3nrfs5md3b39g+zhuvfsssrkieyqyiniwjpmmtnmjckin9og7hn/myjlyqfwymerbqtobewliogjerlixmmse/p/narf4ozclfwor1itwdcwlnbdsvwjnxgy+bskjfxibonotqhfuv++nqhilgmjcqamwye6qapgefknfjrsmga+jrlqebckra4moct4zsgeb4+yfgk/vvxtjeeqnhg8mbei+wvqscauxgwmvyp197o9zkeuaxqqq+6suke4qq63ggses1hhgcrzitapa8sidyfj+4i1sk3q55ngl5/4yv7mz95rgj+gu5zgdrlaf3aeqqigcntalekvv1r1bn1ax7rldxfoub/yh3/auijpds=</latexit> <latexit sha1_base64="hq7sakh938qotn4wh/f4hpruo=">aaaclxicbvdlsgmxfm3uv6vqks3wskvmpmfrrxbtcuk9ghdmpwj3bgrmmmseuvorfollt/orlgrx68kfmnnowba7n3mvnox7emdk/wgl1ty3nrfs5md3b39g+zhuvfsssrkieyqyiniwjpmmtnmjckin9og7hn/myjlyqfwymerbqtobewliogjerlixmmse/p/narf4ozclfwor1itwdcwlnbdsvwjnxgy+bskjfxibonotqhfuv++nqhilgmjcqamwye6qapgefknfjrsmga+jrlqebckra4moct4zsgeb4+yfgk/vvxtjeeqnhg8mbei+wvqscauxgwmvyp197o9zkeuaxqqq+6suke4qq63ggses1hhgcrzitapa8sidyfj+4i1sk3q55ngl5/4yv7mz95rgj+gu5zgdrlaf3aeqqigcntalekvv1r1bn1ax7rldxfoub/yh3/auijpds=</latexit> <latexit sha1_base64="hq7sakh938qotn4wh/f4hpruo=">aaaclxicbvdlsgmxfm3uv6vqks3wskvmpmfrrxbtcuk9ghdmpwj3bgrmmmseuvorfollt/orlgrx68kfmnnowba7n3mvnox7emdk/wgl1ty3nrfs5md3b39g+zhuvfsssrkieyqyiniwjpmmtnmjckin9og7hn/myjlyqfwymerbqtobewliogjerlixmmse/p/narf4ozclfwor1itwdcwlnbdsvwjnxgy+bskjfxibonotqhfuv++nqhilgmjcqamwye6qapgefknfjrsmga+jrlqebckra4moct4zsgeb4+yfgk/vvxtjeeqnhg8mbei+wvqscauxgwmvyp197o9zkeuaxqqq+6suke4qq63ggses1hhgcrzitapa8sidyfj+4i1sk3q55ngl5/4yv7mz95rgj+gu5zgdrlaf3aeqqigcntalekvv1r1bn1ax7rldxfoub/yh3/auijpds=</latexit> <latexit sha1_base64="qqi8qc/6wyumyhy3w1klmz3us8k=">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</latexit> <latexit sha1_base64="qqi8qc/6wyumyhy3w1klmz3us8k=">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</latexit> <latexit sha1_base64="qqi8qc/6wyumyhy3w1klmz3us8k=">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</latexit> <latexit sha1_base64="qqi8qc/6wyumyhy3w1klmz3us8k=">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</latexit> Example Compute the minimum value γ such that b t (q )+b t (q 3 ) apple, 8t O a1 = O a (i, j) =O(q i,a,z j ) 3 3.7,.3.8,. 4.5,.55,O a = 4.6,.45..8,..,.8 (degree of B) B(b) =.169b(q 1 ) 3.938b(q ) + 98b(q 3 ).97b(q 1 )b(q ) b(q )b(q 3 ).7b(q 1 ).45b(q ) Ufuk Topcu 1

35 Wrap-up Summary: Verification in belief space as search for algebraic certificates for hybrid system dynamics Next: Verification Synthesis (O)MDs artial-information, two-player games rotecting the integrity of mission-critical information Ufuk Topcu erformance prediction in machine teaching 13

Pokazać jeszcze