Foundation of Diagnosis and Predictability in Probabilistic Systems
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1 Foundtion of Dignosis nd Preditility in Proilisti Systems Nthlie Bertrnd 1, Serge Hddd 2, Engel Lefuheux 1,2 1 Inri Rennes, Frne 2 LSV, ENS Chn & CNRS & Inri Sly, Frne De. 16th FSTTCS 14
2 Dignosis of disrete event systems Ojetive: tell whether fult f ourred, sed on oservtions. f f 1 f 2 f 3 q 0 u q 1 q 2 Foundtion of Dignosis nd Preditility in Proilisti Systems De. 16th FSTTCS 14-2/ 18
3 Dignosis of disrete event systems Ojetive: tell whether fult f ourred, sed on oservtions. q 0 f u f 1 f 2 f 3 q 1 q 2 + orret + fulty +? miguous Foundtion of Dignosis nd Preditility in Proilisti Systems De. 16th FSTTCS 14-2/ 18
4 Dignosis of disrete event systems Ojetive: tell whether fult f ourred, sed on oservtions. q 0 f u f 1 f 2 f 3 q 1 q 2 + orret + fulty +? miguous Dignosility: ll oserved sequenes re unmiguous. Foundtion of Dignosis nd Preditility in Proilisti Systems De. 16th FSTTCS 14-2/ 18
5 Dignosis of disrete event systems Ojetive: tell whether fult f ourred, sed on oservtions. q 0 f u f 1 f 2 f 3 q 1 q 2 + orret + fulty +? miguous Dignosility: ll oserved sequenes re unmiguous. Dignoser: ssigns verdits to oserved sequenes D : Σ o {,,?} Soundness: if fult is limed, fult ourred. Retivity: every fult will e deteted. Foundtion of Dignosis nd Preditility in Proilisti Systems De. 16th FSTTCS 14-2/ 18
6 Dignosis of disrete event systems Ojetive: tell whether fult f ourred, sed on oservtions. q 0 f u f 1 f 2 f 3 q 1 q 2 + orret + fulty +? miguous Dignosility: ll oserved sequenes re unmiguous. Dignoser: ssigns verdits to oserved sequenes D : Σ o {,,?} Soundness: if fult is limed, fult ourred. Retivity: every fult will e deteted. Dignosility nd dignoser synthesis in PTIME [Jing et l. TAC 2001] Foundtion of Dignosis nd Preditility in Proilisti Systems De. 16th FSTTCS 14-2/ 18
7 Dignosis of proilisti systems,1/2 f,1/2,1/2 f 1 f 2 f 3 q 0 u,1/2 q 1 q 2,1/2,1/2 Foundtion of Dignosis nd Preditility in Proilisti Systems De. 16th FSTTCS 14-3/ 18
8 Dignosis of proilisti systems,1/2 f,1/2,1/2 f 1 f 2 f 3 + miguous ut... q 0 u,1/2 q 1 q 2,1/2,1/2 Foundtion of Dignosis nd Preditility in Proilisti Systems De. 16th FSTTCS 14-3/ 18
9 Dignosis of proilisti systems,1/2 f,1/2,1/2 f 1 f 2 f 3 + miguous ut... q 0 u,1/2 q 1 q 2,1/2 lim n P(fn + u n ) = 0,1/2 Foundtion of Dignosis nd Preditility in Proilisti Systems De. 16th FSTTCS 14-3/ 18
10 Dignosis of proilisti systems,1/2 f,1/2,1/2 f 1 f 2 f 3 + miguous ut... q 0 u,1/2 q 1 q 2,1/2 lim n P(fn + u n ) = 0,1/2 Our ontriution Relevnt soundness nd retivity riteri in proilisti setting Deidility nd omplexity of dignosility Optiml dignoser onstrution Beyond dignosis: preditility nd predignosis Foundtion of Dignosis nd Preditility in Proilisti Systems De. 16th FSTTCS 14-3/ 18
11 Outline Dignosility Speifying dignosility Chrteristion Complexity Preditility nd predignosility Foundtion of Dignosis nd Preditility in Proilisti Systems De. 16th FSTTCS 14-4/ 18
12 Outline Dignosility Speifying dignosility Chrteristion Complexity Preditility nd predignosility Foundtion of Dignosis nd Preditility in Proilisti Systems De. 16th FSTTCS 14-5/ 18
13 All runs or fulty runs? u,1/2 f,1/2,1/2 q 1 q 0 f 1 f 2,1/2 Foundtion of Dignosis nd Preditility in Proilisti Systems De. 16th FSTTCS 14-6/ 18
14 All runs or fulty runs? u,1/2 f,1/2,1/2 q 1 q 0 f 1 f 2 + is miguous,1/2 Foundtion of Dignosis nd Preditility in Proilisti Systems De. 16th FSTTCS 14-6/ 18
15 All runs or fulty runs? u,1/2 f,1/2,1/2 q 1 q 0 f 1 f 2 + is miguous lim n P(fn ) = 0,1/2 Foundtion of Dignosis nd Preditility in Proilisti Systems De. 16th FSTTCS 14-6/ 18
16 All runs or fulty runs? u,1/2 f,1/2,1/2 q 1 q 0 f 1 f 2 + is miguous lim n P(fn ) = 0,1/2 lim n P(un ) = 1 2 Foundtion of Dignosis nd Preditility in Proilisti Systems De. 16th FSTTCS 14-6/ 18
17 All runs or fulty runs? u,1/2 f,1/2,1/2 q 1 q 0 f 1 f 2 + is miguous lim n P(fn ) = 0,1/2 lim n P(un ) = 1 2 Retivity speifitions: Detet fult, lmost surely. Foundtion of Dignosis nd Preditility in Proilisti Systems De. 16th FSTTCS 14-6/ 18
18 All runs or fulty runs? u,1/2 f,1/2,1/2 q 1 q 0 f 1 f 2 + is miguous lim n P(fn ) = 0,1/2 lim n P(un ) = 1 2 Retivity speifitions: Detet fult, lmost surely. Detet if run is fulty or orret, lmost surely. Foundtion of Dignosis nd Preditility in Proilisti Systems De. 16th FSTTCS 14-6/ 18
19 Infinite sequenes or their finite prefixes? u,1/2 u,1/2 f,1/2,1/2 q 1 q 0 q 2 f 1 f 2,1,1/2,1/2,1/2 Foundtion of Dignosis nd Preditility in Proilisti Systems De. 16th FSTTCS 14-7/ 18
20 Infinite sequenes or their finite prefixes? u,1/2 u,1/2 f,1/2,1/2 q 1 q 0 q 2 f 1 f 2 ω is orret.,1,1/2,1/2,1/2 Foundtion of Dignosis nd Preditility in Proilisti Systems De. 16th FSTTCS 14-7/ 18
21 Infinite sequenes or their finite prefixes? u,1/2 u,1/2 f,1/2,1/2 q 1 q 0 q 2 f 1 f 2,1,1/2,1/2,1/2 ω is orret. n is miguous nd P(q 0 u(q 1 ) n ) = 1 2. Foundtion of Dignosis nd Preditility in Proilisti Systems De. 16th FSTTCS 14-7/ 18
22 Infinite sequenes or their finite prefixes? u,1/2 u,1/2 f,1/2,1/2 q 1 q 0 q 2 f 1 f 2,1,1/2,1/2,1/2 ω is orret. n is miguous nd P(q 0 u(q 1 ) n ) = 1 2. Infinite sequenes re lmost surely non miguous. Foundtion of Dignosis nd Preditility in Proilisti Systems De. 16th FSTTCS 14-7/ 18
23 Infinite sequenes or their finite prefixes? u,1/2 u,1/2 f,1/2,1/2 q 1 q 0 q 2 f 1 f 2,1,1/2,1/2,1/2 ω is orret. n is miguous nd P(q 0 u(q 1 ) n ) = 1 2. Infinite sequenes re lmost surely non miguous. The proility of miguous prefixes tends to 0. Foundtion of Dignosis nd Preditility in Proilisti Systems De. 16th FSTTCS 14-7/ 18
24 Four dignosility notions Dignosility All runs Fulty runs Finite prefixes FA FF Infinite sequenes IA IF Foundtion of Dignosis nd Preditility in Proilisti Systems De. 16th FSTTCS 14-8/ 18
25 Four dignosility notions Dignosility All runs Fulty runs Finite prefixes FA FF Infinite sequenes IA IF Fous on IF in this tlk. Foundtion of Dignosis nd Preditility in Proilisti Systems De. 16th FSTTCS 14-8/ 18
26 Chrteristion of dignosility Speifition of IF-dignosility: Infinite sequenes, Fult dignosis u u f A q 1 q 0 q 2 f 1 f 2 Foundtion of Dignosis nd Preditility in Proilisti Systems De. 16th FSTTCS 14-9/ 18
27 Chrteristion of dignosility Speifition of IF-dignosility: Infinite sequenes, Fult dignosis u u f A q 1 q 0 q 2 f 1 f 2 Oserver: trks possile orret sttes fter given oserved sequene. O A {q 0} {q 1,q 2}, Foundtion of Dignosis nd Preditility in Proilisti Systems De. 16th FSTTCS 14-9/ 18
28 Chrteristion of dignosility Speifition of IF-dignosility: Infinite sequenes, Fult dignosis u u f A q 1 q 0 q 2 f 1 f 2 Oserver: trks possile orret sttes fter given oserved sequene. O A {q 0} {q 1,q 2}, A is not IF-dignosle iff there exists stte (q, U) in BSCC of A O A with q fulty nd U. Foundtion of Dignosis nd Preditility in Proilisti Systems De. 16th FSTTCS 14-9/ 18
29 Dignoser synthesis For every IF-dignosle system with n orret sttes one n uild n IF-dignoser with t most 2 n sttes. Dignoser derived from oserver O A : emits in stte. Foundtion of Dignosis nd Preditility in Proilisti Systems De. 16th FSTTCS 14-10/ 18
30 Dignoser synthesis For every IF-dignosle system with n orret sttes one n uild n IF-dignoser with t most 2 n sttes. Dignoser derived from oserver O A : emits in stte. There is fmily (A n ) of IF-dignosle systems suh tht A n hs n + 1 orret sttes nd ny IF-dignoser needs 2 n sttes., q 0,,, q 1 q 2... q n A n f 0 f,,, f 1 f 2... f n Foundtion of Dignosis nd Preditility in Proilisti Systems De. 16th FSTTCS 14-10/ 18
31 Dignosility is in PSPACE Dignosility is deidle in PSPACE for proilisti systems. Foundtion of Dignosis nd Preditility in Proilisti Systems De. 16th FSTTCS 14-11/ 18
32 Dignosility is in PSPACE Dignosility is deidle in PSPACE for proilisti systems. Sketh of proof relies on the hrteristion on A O A voids uilding the produt uses Svith s theorem for pproprite guesses Foundtion of Dignosis nd Preditility in Proilisti Systems De. 16th FSTTCS 14-11/ 18
33 Dignosility is PSPACE-hrd L Σ is eventully universl if v Σ, v 1 L = Σ. The eventul universlity prolem for NFA is PSPACE-hrd. Foundtion of Dignosis nd Preditility in Proilisti Systems De. 16th FSTTCS 14-12/ 18
34 Dignosility is PSPACE-hrd L Σ is eventully universl if v Σ, v 1 L = Σ. The eventul universlity prolem for NFA is PSPACE-hrd. Dignosility is PSPACE-hrd. Redution from eventul universlity to dignosility. NFA q 0 u q f 0 f 0 Σ A not dignosle iff A O A ontins BSCC where eh stte hs the form (f 0, U) with U Foundtion of Dignosis nd Preditility in Proilisti Systems De. 16th FSTTCS 14-12/ 18
35 Comprison with non-proilisti disrete event systems Dignosility is PSPACE-omplete for proilisti systems. Foundtion of Dignosis nd Preditility in Proilisti Systems De. 16th FSTTCS 14-13/ 18
36 Comprison with non-proilisti disrete event systems Dignosility is PSPACE-omplete for proilisti systems. Dignosility is deidle in PTIME for non-proilisti systems. [Jing, Hung, Chndr, Kumr TAC 2001] Sketh of proof uild the twin-produt with opy restrited to orret sttes hek for SCC with fulty sttes in the first omponent Foundtion of Dignosis nd Preditility in Proilisti Systems De. 16th FSTTCS 14-13/ 18
37 Comprison with non-proilisti disrete event systems Dignosility is PSPACE-omplete for proilisti systems. Dignosility is deidle in PTIME for non-proilisti systems. [Jing, Hung, Chndr, Kumr TAC 2001] Sketh of proof uild the twin-produt with opy restrited to orret sttes hek for SCC with fulty sttes in the first omponent Erroneous dpttion to proilisti se in [Chen, Kumr TASE 2013]. Foundtion of Dignosis nd Preditility in Proilisti Systems De. 16th FSTTCS 14-13/ 18
38 Outline Dignosility Speifying dignosility Chrteristion Complexity Preditility nd predignosility Foundtion of Dignosis nd Preditility in Proilisti Systems De. 16th FSTTCS 14-14/ 18
39 Preditility Ojetive: tell whether fult will our, sed on oservtions. Foundtion of Dignosis nd Preditility in Proilisti Systems De. 16th FSTTCS 14-15/ 18
40 Preditility Ojetive: tell whether fult will our, sed on oservtions. q 3 q 0 q 1 q 2 f f 1 + orret + surely eventully fulty +.s. eventully fulty Foundtion of Dignosis nd Preditility in Proilisti Systems De. 16th FSTTCS 14-15/ 18
41 Preditility Ojetive: tell whether fult will our, sed on oservtions. q 3 q 0 q 1 q 2 f f 1 + orret + surely eventully fulty +.s. eventully fulty Two notions of soundness: sure: if fult is limed, fult will our lmost-sure: if fult is limed, fult will lmost-surely our Retivity: fult is deteted t lest k steps efore ourrene. Foundtion of Dignosis nd Preditility in Proilisti Systems De. 16th FSTTCS 14-15/ 18
42 Preditility Ojetive: tell whether fult will our, sed on oservtions. q 3 q 0 q 1 q 2 f f 1 + orret + surely eventully fulty +.s. eventully fulty surely 0-preditle lmost surely 1-preditle not 2-preditle Two notions of soundness: sure: if fult is limed, fult will our lmost-sure: if fult is limed, fult will lmost-surely our Retivity: fult is deteted t lest k steps efore ourrene. Foundtion of Dignosis nd Preditility in Proilisti Systems De. 16th FSTTCS 14-15/ 18
43 Preditility Ojetive: tell whether fult will our, sed on oservtions. q 3 q 0 q 1 q 2 f f 1 + orret + surely eventully fulty +.s. eventully fulty surely 0-preditle lmost surely 1-preditle not 2-preditle Two notions of soundness: sure: if fult is limed, fult will our lmost-sure: if fult is limed, fult will lmost-surely our Retivity: fult is deteted t lest k steps efore ourrene. Preditility is NLOGSPACE-omplete for proilisti systems. Foundtion of Dignosis nd Preditility in Proilisti Systems De. 16th FSTTCS 14-15/ 18
44 Predignosility Ojetive: detet nd foresee fults nlysing pst nd future f f 1 q 0 q 1 f q 2 + orret +.s. eventully fulty Foundtion of Dignosis nd Preditility in Proilisti Systems De. 16th FSTTCS 14-16/ 18
45 Predignosility Ojetive: detet nd foresee fults nlysing pst nd future f f 1 q 0 q 1 f q 2 + orret +.s. eventully fulty Soundness: If fult is limed, fult hppened or (lmost) surely will. Retivity: Fults re lmost surely limed. Foundtion of Dignosis nd Preditility in Proilisti Systems De. 16th FSTTCS 14-16/ 18
46 Predignosility Ojetive: detet nd foresee fults nlysing pst nd future f f 1 q 0 q 1 f q 2 + orret +.s. eventully fulty Soundness: If fult is limed, fult hppened or (lmost) surely will. Retivity: Fults re lmost surely limed. Predignosility is PSPACE-omplete. Foundtion of Dignosis nd Preditility in Proilisti Systems De. 16th FSTTCS 14-16/ 18
47 Conlusion: Foundtion of proilisti dignosis Summry of ontriutions Investigtion of semntil issues Tight omplexity ounds for dignosility nd dignoser synthesis prolems Introdution of predignosility Foundtion of Dignosis nd Preditility in Proilisti Systems De. 16th FSTTCS 14-17/ 18
48 Conlusion: Foundtion of proilisti dignosis Summry of ontriutions Investigtion of semntil issues Tight omplexity ounds for dignosility nd dignoser synthesis prolems Introdution of predignosility Future work Approximte dignosis (relxing soundness) Other prdigms relted to prtil oservtion (detetility, opity, et.) Spe nd time optimistion of oservtions Foundtion of Dignosis nd Preditility in Proilisti Systems De. 16th FSTTCS 14-17/ 18
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