Approximate Satisfiability and Equivalence

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Approximate Satisfiability and Equivalence

• Michel de Rougemont
• University Paris II LRI
• Joint work with E. Fischer, Technion,
• F. Magniez, LRI ,
• LICS 2006

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Plan

• Tester on a class K approximation of decision
  problems
• Equality between two strings (trees) (e2, e)
  Tolerant tester , additive approximation of the
  Edit Distance with Moves.
• Membership is w in L ?
• Equivalence tester between two regular
  properties polynomial time algorithm (Exact
  Equivalence is PSPACE complete)
• Generalizations regular trees, context-free
  languages, infinite words,
• Current research probabilistic systems.

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Motivations

• 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 ?

L
B
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1. Testers on a class K

• Let F be a property on a class K of structures
  U
• An e -tester for F is a probabilistic algorithm
  A such that
• If U F, A accepts
• If U is e far from F, A rejects with high
  probability
• F is testable if there is a probabilistic
  algorithm A such that
• A is an e -tester for all e
• Time(A) is independent of nsize(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.
  (e1, e2)-Tolerant Tester

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Approximate Satisfiability and Equivalence

• Satisfiability T F
• Approximate Satisfiability T F
• Approximate Equivalence
• Image on a class K of trees

G
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Edit Distances with Moves

• Classical Edit DistanceInsertions, Deletions,
  Modifications
• Edit Distance with moves dist(w,w)
• 0111000011110011001
• 0111011110000011001
• 3. Edit Distance with Moves generalizes to
  Ordered Trees

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Tester for equality Block and Uniform statistics
W001010101110 length n, b.stat
consecutive subwords of length k, n/k
blocks u.stat any subwords of length k, n-k1
blocks, shingles

For k2, n/k6
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Tester for equality
Edit distance with moves. NP-complete problem,
but approximable in constant time with additive
error. Uniform statistics ( )
W001010101110 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). Corollar
y. Y(w)-Y(w) approximates dist(w,w)/n. Tester
1 If Y(w)-Y(w) lte. accept, else reject.
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Theorem 1 Soundness and Robustness

• Soundness e-close strings have close statistics
• Robustness e-far strings have far statistics
• We prove
• b.stat is robust
• u.stat is sound
• u.stat is robust (harder)

hard
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Robustness of b.stat
Robustness of b-stat
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Soundness of u.stat
Soundness of u-stat Simple edit Move
wA.B.C.D, wA.C.B.D Hence, for e2.n
operations, Remark b.stat is not
sound. Problem robustness of u.stat ? Harder!
We need an auxiliary distribution and two key
lemmas.
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Statistics on words
Block statistics b.stat

k

Uniform statistics u.stat
k

Block Uniform statistics bu.stat
t
k-t

K
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Uniform Statistics

A

B
Lemma 2
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Block Uniform Statistics

Lemma 1

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Robustness of the uniform Statistics
Lemma 2 Lemma 1
(Probabilistic method)

• Tolerant tester
• Theorem for two words w and w large enough, the
  tester
• Accepts if ww with probability 1
• Accepts if w,w are e2-close with probability 2/3
• Rejects if w,w are e-far with probability 2/3

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3. Testers for Membership and Equivalence

• Membership decide if
• Inclusion and Equivalence
• Equivalence tester

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Automata for Regular languages
A automaton with m states on S, Ak automaton
with m states on Sk. Basic property Proposit
ion Caratheodorys theorem in dimension d,
convex hull of N points can be decomposed into in
the union of convex hulls of d1 points. Large
loops can be decomposed. Small loops (less than
mA) suffice.

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Approximate Parikh mapping
Lemma For every X in H, w of size n s. t.
H is a fair representation of L
w
X .
b-stat(w)
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Example
Hstat(w) w in r is a union of
polytopes. 2 Polytopes for r.
Y(w)
Membership Tester
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Construction of H in polynomial time
Enumeration of all m loops Number of b-stat
of words of length m on Sk is less than
Some loops have same b-stat ABBA and
BBAA Construct H by matrix iteration

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Membership tester
Membership Tester for w in L (regular) 1.
Construction of the tester Precompute He 2.
Tester Compute Y(w) (approx. b.stat(w)).
Accept iff Y(w) is at distance less
than e to He 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.

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Equivalence Tester for regular properties
Time polynomial in mMax(A , B )
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3. Generalization Trees
(1,.)c

(1,(1,(1,.),1),.)c
T squeleton
T Ordered (extended) Tree of rank 2
W word with labels. Apply u.stat on W and
define u.stat(T).
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Infinite words

• Buchi Automata. Distance on infinite
  words
• Two words are e-close if
• A word is e-close to a language L if there exists
  w in L s. t. W and w are e-close.
• Statistics set of accumulation points of
• H compatible loops of connected components of
  accepting states
• Tester for Buchi Automata
• Compute HA and HB
• Reject if HA and HB are different.
• Approximate Model-Checking

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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?

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4. Probabilistic Systems

Probabilistic Automata Ma is a stochastic
matrix for letter a. If wa1 a1 . an then Mw
Ma1 . Man
PM Probabilistic Membership Is ut.Mw.vgt ? ?

• APM Approximate Probabilistic Membership Let P
  ut.Mw.vgt ?
• Decide if w satisfies P or if w is e -far
  from P.

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Approximations for Probabilistic Automata

• Approximate probabilities
• Introduce e around ?
• Approximate membership
• Approximate Equivalence (Tzeng 92) is harder than
  Equivalence.
• Approximate distances between states
• Generalization of bisimulation
• Desharnais et al., Van Breugel-Worell
• Our approach Approximation on the input
• http//www.lri.fr/mdr/verap

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Basic Decompositions in H
s3ccc
s4aa
s3bc
s1abc
s2ba
H1
H2 Waabaaaaababcabcabcabcabcabcbc close
to W(aa)3(ba)2(abc)6 N Samples approximate
ustat(W) close to ?1.ustat((aa))
?2.ustat((ba)) ?3.ustat((abc))
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Basic Decompositions in H1
s4aa
s3
s1abc
s2ab

• 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) rn for
  n large enough.
• Analyze all possible decompositions of ustat(w)
  in H

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Claim

• Hypothesis all simple loops are distinct.
• Input W of length n
• Claim Upper bound for ut.Mw.v for W close to
  W.
• ?1.ustat((aa)) ?2.ustat((ba))
  ?3.ustat((abc)) indicates densities ?1, ?2, ?3
  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.

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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 d1 summits, and obtain a bound Bi,
• Consider all possible links on Hi, let Ci the
  optimal bound,
• Let DMaxi Ci.Di

If ?lt D, Accept else Reject.
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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?

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Conclusion

• Tolerant Tester for Equality on strings under the
  Edit Distance with Moves
• Additive approximation in O(1) of the EDM
• 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)
• Generalization to trees, infinite words
• Probabilistic systems.