1. 1 Approximate Satisfiability and Equivalence Michel de Rougemont University Paris II & LRI Joint work with E. Fischer, Technion, F. Magniez, LRI, LICS 2006

2. 2 1.Tester on a class K: approximation of decision problems Equality between two strings (trees) : ( ε 2, ε) Tolerant tester, additive approximation of the Edit Distance with Moves. Membership: is w in L ? 3.Equivalence tester between two regular properties: polynomial time algorithm (Exact Equivalence is PSPACE complete) 4.Generalizations: regular trees, context-free languages, infinite words, 5.Current research: probabilistic systems. Plan

3. 3 Decisions on noisy inputs (distance to a language) Model Checking: can we approximate hard problems? Bounded MC, Abstraction, ….. Black-Box Checking: does B satisfies P ? Motivations B L

4. 4 Let F be a property on a class K of structures U: An ε -tester for F is a probabilistic algorithm A such that: If U |= F, A accepts If U is ε far from F, A rejects with high probability F is testable if there is a probabilistic algorithm A such that A is an ε -tester for all ε Time(A) is independent of n=size(U). Robust characterizations of polynomials, R. Rubinfeld, M. Sudan, 1994 Property Testing and its connection to Learning and Approximation. O. Goldreich, S. Goldwasser, D. Ron, 1996. Tester usually implies a linear time corrector. (ε 1, ε 2 )- Tolerant Tester 1. Testers on a class K

5. 5 1.Satisfiability : T |= F 2.Approximate Satisfiability T |= F 3.Approximate Equivalence Image on a class K of trees Approximate Satisfiability and Equivalence G

6. 6 1.Classical Edit Distance: Insertions, Deletions, Modifications 2.Edit Distance with moves : dist(w,w’) 0111000011110011001 0111011110000011001 3. Edit Distance with Moves generalizes to Ordered Trees Edit Distances with Moves

7. 7 Tester for equality: Block and Uniform statistics W=001010101110 length n, b.stat: consecutive subwords of length k, n/k blocks u.stat: any subwords of length k, n-k+1 blocks, shingles For k=2, n/k=6

8. 8 Tester for equality Edit distance with moves. NP-complete problem, but approximable in constant time with additive error. Uniform statistics ( ): W=001010101110 Theorem 1. |u.stat(w)-u.stat(w’)| approximates dist(w,w’)/n. Sample N subwords of length k, compute Y(w) and Y(w’): Lemma (Chernoff). Y(w) approximates u.stat(w). Corollary. |Y(w)-Y(w’)| approximates dist(w,w’)/n. Tester 1: If |Y(w)-Y(w’)| <ε. accept, else reject.

9. 9 Soundness: ε-close strings have close statistics Robustness: ε-far strings have far statistics We prove: 1.b.stat is robust 2.u.stat is sound 3.u.stat is robust (harder) Theorem 1 : Soundness and Robustness hard

10. 10 Robustness of b.stat Robustness of b-stat:

11. 11 Soundness of u.stat Soundness of u-stat: Simple edit: Move w=A.B.C.D, w’=A.C.B.D: Hence, for ε 2.n operations, Remark: b.stat is not sound. Problem: robustness of u.stat ? Harder! We need an auxiliary distribution and two key lemmas.

12. 12 Statistics on words k k K t k-t Block statistics: b.stat Uniform statistics: u.stat Block Uniform statistics: bu.stat

13. 13 Uniform Statistics A B Lemma 2:

14. 14 Block Uniform Statistics Lemma 1:

15. 15 Robustness of the uniform Statistics Lemma 2: Lemma 1: Tolerant tester: Theorem: for two words w and w’ large enough, the tester: 1.Accepts if w=w’ with probability 1 2.Accepts if w,w’ are ε 2 -close with probability 2/3 3.Rejects if w,w’ are ε-far with probability 2/3 (Probabilistic method)

16. 16 1.Membership: decide if 2.Inclusion and Equivalence Equivalence tester 3. Testers for Membership and Equivalence

17. 17 Automata for Regular languages A: automaton with m states on Σ, A k automaton with m states on Σ k. Basic property: Proposition: Caratheodory’s theorem: in dimension d, convex hull of N points can be decomposed into in the union of convex hulls of d+1 points. Large loops can be decomposed. Small loops (less than m=|A|) suffice.

18. 18 Approximate Parikh mapping Lemma: For every X in H, w of size n s. t. X. b-stat(w) w H is a fair representation of L

19. 19 Example Y(w) H={stat(w) : w in r } is a union of polytopes. 2 Polytopes for r. Membership Tester:

20. 20 Construction of H in polynomial time Enumeration of all m loops: Number of b-stat of words of length m on Σ k is less than : Some loops have same b-stat: ABBA and BBAA Construct H by matrix iteration:

21. 21 Membership tester Membership Tester for w in L (regular): 1. Construction of the tester: Precompute H ε 2. Tester: Compute Y(w) (approx. b.stat(w)). Accept iff Y(w) is at distance less than ε to H ε Construction: Time is Tester: query complexity in time complexity in Remark 1: Time complexity of previous testers was exponential in m. Remark 2: The same method works for L context-free.

22. 22 Equivalence Tester for regular properties Time polynomial in m=Max(|A |, |B |):

23. 23 3. Generalization: Trees (1,(1,(1,.),1),.)=c (1,.)=c T: Ordered (extended) Tree of rank 2 T’: squeleton W: word with labels. Apply u.stat on W and define u.stat(T).

24. 24 Infinite words Buchi Automata. Distance on infinite words: Two words are ε-close if A word is ε-close to a language L if there exists w’ in L s. t. W and w’ are ε-close. Statistics: set of accumulation points of H: compatible loops of connected components of accepting states Tester for Buchi Automata: Compute H A and H B Reject if H A and H B are different. Approximate Model-Checking

25. 25 Other Logics Equivalence of Context-free grammars is undecidable, Approximate Equivalence in exponential time. Consider formulas in different Logics (LFP, m- calculus,…..). Can Equivalence, Implication be approximated on a definable class K with a distance? Definability and approximation: can first-order definable classes of trees testable with the Edit distance?

26. 26 4. Probabilistic Systems Probabilistic Automata: M a is a stochastic matrix for letter a. If w=a 1 a 1 …. a n then M w =M a1 …. M an PM Probabilistic Membership: Is u t.M w.v> λ ? APM: Approximate Probabilistic Membership: Let P= u t.M w.v> λ Decide if w satisfies P or if w is ε -far from P.

27. 27 Approximations for Probabilistic Automata 1.Approximate probabilities Introduce ε around λ Approximate membership Approximate Equivalence (Tzeng 92) is harder than Equivalence. 2.Approximate distances between states Generalization of bisimulation Desharnais et al., Van Breugel-Worell 3.Our approach: Approximation on the input http://www.lri.fr/~mdr/verap

28. 28 Basic Decompositions in H s1=abc s2=ba s3=bc s4=aa s3=ccc H1 H2 W=aabaaaaababcabcabcabcabcabcbc close to W’=(aa) 3 (ba) 2 (abc) 6 N Samples approximate ustat(W) close to : λ 1.ustat((aa)*)+ λ 2.ustat((ba)*) + λ 3.ustat((abc)*)

29. 29 Basic Decompositions in H1 For each summit s, basic loop in A, let h(s,n)=Probability to follow s after n iterations of s Analyze all loops mutliple of s: h(s,n)= r n for n large enough. Analyze all possible decompositions of ustat(w) in H: s1=abc s2=ab s3 s4=aa

30. 30 Claim Hypothesis: all simple loops are distinct. Input W of length n Claim: Upper bound for u t.M w’.v for W’ close to W. λ1.ustat((aa)*)+ λ2.ustat((ba)*) + λ3.ustat((abc)*) indicates densities λ1, λ2, λ3 to follow aa, ba, abc on Hi. We need to connect loops aa, ba, abc by some inputs: there are finitely many possibilities. Let Ci the best probability.

31. 31 Tester for APM in O(1) Input W of length n Tester(W,k) Sample W with N(k) samples, Select H such that ustat is close, Decompose ustat on possible subpolytopes Hi with at most d+1 summits, and obtain a bound Bi, Consider all possible links on Hi, let Ci the optimal bound, Let D=Max i Ci.Di If λ< D, Accept else Reject.

32. 32 Non determinism and Probabilistic Can we combine both non determinism and probabilistic behaviors? Stationary distributions for a given scheduler are distributions on the states, for which there is also a polytope representation. Classical results exist about positional schedulers. Problem: does the projection of these distributions on ustat vectors keep the distances? Thesis of Mathieu Tracol. For large scale systems, evolutionary games also provide a statistical representation of the states. Can we predict approximate properties of the Equilibria?

33. 33 Conclusion 1.Tolerant Tester for Equality on strings under the Edit Distance with Moves Additive approximation in O(1) of the EDM 2.Equivalence tester for automata Polynomial time approximate algorithm (PSPACE-complete) Generalization to Buchi automata : approximate Model- Checking Context-Free Languages: exponential algorithm (exact problem is undecidable) 3.Generalization to trees, infinite words 4.Probabilistic systems.