Lipschitz Robustness of Finite-state Transducers

Transcription

1 Lipschitz Robustness of Finite-state Transducers Roopsha Samanta IST Austria Joint work with Tom Henzinger and Jan Otop December 16, 2014 Roopsha Samanta Lipschitz Robustness of Finite-state Transducers 1 / 18

2 Problem Overview Computational systems in physical environments ) uncertain data Roopsha Samanta Lipschitz Robustness of Finite-state Transducers 2 / 18

3 Problem Overview Corrupt data from sensors in avionics software or medical devices Incomplete DNA strings in computational biology Wrongly spelled inputs to text processors Roopsha Samanta Lipschitz Robustness of Finite-state Transducers 2 / 18

4 Problem Overview Functional correctness not enough System behaviour must degrade smoothly given input perturbation We need continuity or robustness Roopsha Samanta Lipschitz Robustness of Finite-state Transducers 2 / 18

5 Informal definition Focus on finite-state transducers Robustness of transducers Lipschitz continuity Transducer T is K -Lipschitz robust if for all inputs s, t: d(s, t) < 1)d(T (s), T (t)) apple Kd(s, t) Roopsha Samanta Lipschitz Robustness of Finite-state Transducers 3 / 18

6 Example d(s, t) =Hamming distance(s, t) if s = t, else 1 T NR : a/a q 0 b/b T R : a/b q 0 b/a q 1 q 2 q 1 q 2 a/a a/b a/b a/b Inputs: d(a k+1, ba k )=1 Outputs: d(a k+1, b k+1 )=k + 1 Inputs: d(a k+1, ba k )=1 Outputs: d(b k+1, ab k )=1 Roopsha Samanta Lipschitz Robustness of Finite-state Transducers 4 / 18

7 Main contributions Undecidability of K -robustness of deterministic transducers Characterization of functional transducers with decidable K -robustness Consider distances computable by weighted automata Reduce K -robustness to emptiness problem for weighted automata Polynomial-time decision procedure Formalization/study of K -robustness of nondeterministic transducers All results hold for transducers over finite or infinite words Roopsha Samanta Lipschitz Robustness of Finite-state Transducers 5 / 18

8 Main contributions Undecidability of K -robustness of deterministic transducers Characterization of functional transducers with decidable K -robustness Consider distances computable by weighted automata Reduce K -robustness to emptiness problem for weighted automata Polynomial-time decision procedure Formalization/study of K -robustness of nondeterministic transducers All results hold for transducers over finite or infinite words Roopsha Samanta Lipschitz Robustness of Finite-state Transducers 5 / 18

9 Main contributions Undecidability of K -robustness of deterministic transducers Characterization of functional transducers with decidable K -robustness Consider distances computable by weighted automata Reduce K -robustness to emptiness problem for weighted automata Polynomial-time decision procedure Formalization/study of K -robustness of nondeterministic transducers All results hold for transducers over finite or infinite words Roopsha Samanta Lipschitz Robustness of Finite-state Transducers 5 / 18

10 Main contributions Undecidability of K -robustness of deterministic transducers Characterization of functional transducers with decidable K -robustness Consider distances computable by weighted automata Reduce K -robustness to emptiness problem for weighted automata Polynomial-time decision procedure Formalization/study of K -robustness of nondeterministic transducers All results hold for transducers over finite or infinite words Roopsha Samanta Lipschitz Robustness of Finite-state Transducers 5 / 18

11 Main contributions Undecidability of K -robustness of deterministic transducers Characterization of functional transducers with decidable K -robustness Consider distances computable by weighted automata Reduce K -robustness to emptiness problem for weighted automata Polynomial-time decision procedure Formalization/study of K -robustness of nondeterministic transducers All results hold for transducers over finite or infinite words Roopsha Samanta Lipschitz Robustness of Finite-state Transducers 5 / 18

12 Transducers (T ) Finite-state device with two tapes In each step, a transducer T reads an input letter (alphabet ) writes a finite word (alphabet ) nondeterministically changes state Output of T : defined only if run is accepting dom(t ) Transduction JT K dom(t ) Functional: at most one output word for every input word s 0 = JT K(s) Mealy machines: deterministic, letter-to-letter, all states accepting Roopsha Samanta Lipschitz Robustness of Finite-state Transducers 6 / 18

13 Transducers (T ) Finite-state device with two tapes In each step, a transducer T reads an input letter (alphabet ) writes a finite word (alphabet ) nondeterministically changes state Output of T : defined only if run is accepting dom(t ) Transduction JT K dom(t ) Functional: at most one output word for every input word s 0 = JT K(s) Mealy machines: deterministic, letter-to-letter, all states accepting Roopsha Samanta Lipschitz Robustness of Finite-state Transducers 6 / 18

14 Transducers (T ) Finite-state device with two tapes In each step, a transducer T reads an input letter (alphabet ) writes a finite word (alphabet ) nondeterministically changes state Output of T : defined only if run is accepting dom(t ) Transduction JT K dom(t ) Functional: at most one output word for every input word s 0 = JT K(s) Mealy machines: deterministic, letter-to-letter, all states accepting Roopsha Samanta Lipschitz Robustness of Finite-state Transducers 6 / 18

15 Transducers (T ) Finite-state device with two tapes In each step, a transducer T reads an input letter (alphabet ) writes a finite word (alphabet ) nondeterministically changes state Output of T : defined only if run is accepting dom(t ) Transduction JT K dom(t ) Functional: at most one output word for every input word s 0 = JT K(s) Mealy machines: deterministic, letter-to-letter, all states accepting Roopsha Samanta Lipschitz Robustness of Finite-state Transducers 6 / 18

16 Transducers (T ) Finite-state device with two tapes In each step, a transducer T reads an input letter (alphabet ) writes a finite word (alphabet ) nondeterministically changes state Output of T : defined only if run is accepting dom(t ) Transduction JT K dom(t ) Functional: at most one output word for every input word s 0 = JT K(s) Mealy machines: deterministic, letter-to-letter, all states accepting Roopsha Samanta Lipschitz Robustness of Finite-state Transducers 6 / 18

17 Weighted automata Finite automaton A with weighted transitions (weights 2 Q) A (weighted) run is a sequence c( ) of weights Given value-function f, value of a run : f ( ) =f (c( )) Examples for value-functions: sum, discounted sum, limit-average Notation: SUM-WA, DISC -WA, LIMAVG-WA Value of a word s: L A(s) =inf 2Acc(s) f ( ) Roopsha Samanta Lipschitz Robustness of Finite-state Transducers 7 / 18

18 Weighted automata Finite automaton A with weighted transitions (weights 2 Q) A (weighted) run is a sequence c( ) of weights Given value-function f, value of a run : f ( ) =f (c( )) Examples for value-functions: sum, discounted sum, limit-average Notation: SUM-WA, DISC -WA, LIMAVG-WA Value of a word s: L A(s) =inf 2Acc(s) f ( ) Roopsha Samanta Lipschitz Robustness of Finite-state Transducers 7 / 18

19 Weighted automata Finite automaton A with weighted transitions (weights 2 Q) A (weighted) run is a sequence c( ) of weights Given value-function f, value of a run : f ( ) =f (c( )) Examples for value-functions: sum, discounted sum, limit-average Notation: SUM-WA, DISC -WA, LIMAVG-WA Value of a word s: L A(s) =inf 2Acc(s) f ( ) Roopsha Samanta Lipschitz Robustness of Finite-state Transducers 7 / 18

20 Weighted automata Finite automaton A with weighted transitions (weights 2 Q) A (weighted) run is a sequence c( ) of weights Given value-function f, value of a run : f ( ) =f (c( )) Examples for value-functions: sum, discounted sum, limit-average Notation: SUM-WA, DISC -WA, LIMAVG-WA Value of a word s: L A(s) =inf 2Acc(s) f ( ) Roopsha Samanta Lipschitz Robustness of Finite-state Transducers 7 / 18

21 Weighted automata Given an f -WA A and a threshold : Emptiness: 9 s : L A(s) <? Universality: 8 s : L A(s) <? Emptiness is decidable in polynomial time for SUM-, DISC -, LIMAVG-WA Universality is undecidable for SUM-WA. Roopsha Samanta Lipschitz Robustness of Finite-state Transducers 8 / 18

22 Similarity functions d : S S! Q [1such that 8x, y 2 S: d(x, y) 0 d(x, y) =d(y, x) Example: Manhattan distance d(s, t) = {s[i] 6= t[i]} Roopsha Samanta Lipschitz Robustness of Finite-state Transducers 9 / 18

23 Similarity functions Pairing: ab bcaa = a b b c # a # a Roopsha Samanta Lipschitz Robustness of Finite-state Transducers 9 / 18

24 Similarity functions Pairing: ab bcaa = a b b c # a # a d : S S! Q 1 is computed by a weighted automaton A if: 8s, t 2 S : d(s, t) =L A(s t) Roopsha Samanta Lipschitz Robustness of Finite-state Transducers 9 / 18

25 Similarity functions Pairing: ab bcaa = a b b c # a # a d : S S! Q 1 is computed by a weighted automaton A if: 8s, t 2 S : d(s, t) =L A(s t) Example: Manhattan distance SUM-WA A : a, b a b 0 q 0 a, b b a 1 L A( s[1] t[1] s[2] t[2]...) = {s[i] 6= t[i]} Roopsha Samanta Lipschitz Robustness of Finite-state Transducers 9 / 18

26 Robust transducers Given: a functional transducer T, with JT K similarity function d :! Q 1 similarity function d :! Q 1 constant K 2 Q with K > 0 T is defined to be K -Lipschitz robust w.r.t d, d if: 8s, t 2 dom(t ): d (s, t) < 1)d (JT K(s), JT K(t)) apple Kd (s, t). Roopsha Samanta Lipschitz Robustness of Finite-state Transducers 10 / 18

27 Robust transducers Given: a functional transducer T, with JT K similarity function d :! Q 1 similarity function d :! Q 1 constant K 2 Q with K > 0 T is defined to be K -Lipschitz robust w.r.t d, d if: 8s, t 2 dom(t ): d (s, t) < 1)d (JT K(s), JT K(t)) apple Kd (s, t). Roopsha Samanta Lipschitz Robustness of Finite-state Transducers 10 / 18

28 Problem definition Given: a functional transducer T, with JT K similarity function d :! Q 1 similarity function d :! Q 1 constant K 2 Q with K > 0 Check if T is K -Lipschitz robust w.r.t d, d. Roopsha Samanta Lipschitz Robustness of Finite-state Transducers 11 / 18

29 Undecidability K -robustness of functional transducers is undecidable. 1-robustness of deterministic transducers w.r.t. generalized Manhattan distances is undecidable. Proof hint: Reduction from PCP. Roopsha Samanta Lipschitz Robustness of Finite-state Transducers 12 / 18

30 From K -robustness to robustness T is robust w.r.t. d, d if 9K : T is K -robust w.r.t. d, d. Let d, d be generalized Manhattan distances. 1 Robustness of T is decidable in CO-NP. 2 One can compute K T such that T is robust iff T is K T -robust. Roopsha Samanta Lipschitz Robustness of Finite-state Transducers 13 / 18

31 Synchronized transducers T with JT K!! is synchronized iff: 9 automaton A T over recognizing {s JT K(s) :s 2 dom(t )}. Synchronicity of a functional transducer can be decided in polynomial time Example: Mealy machines are synchronized transducers. q a a0! q 0 2A T iff (q, a, a 0, q 0 ) 2 T Roopsha Samanta Lipschitz Robustness of Finite-state Transducers 14 / 18

32 Robustness of synchronized transducers If d, d are similarity functions computed by functional f -WA A d, A d, K -robustness of synchronized T w.r.t. d, d is decidable in PTIME. Proof hint: Construct A over s t s 0 t 0 : Ā K d Ā L T Ā R T Ā 1 d A T accepts s s 0 A T accepts t t 0 L Ad (s t) L Ad (s 0 t 0 ) 9w : L A(w) < 0 iff T is not K -robust w.r.t. d, d Roopsha Samanta Lipschitz Robustness of Finite-state Transducers 15 / 18

33 Example Recall: Mealy machines are synchronized transducers Manhattan distances are computable by functional SUM-WA K -robustness of Mealy machines w.r.t. Manhattan distances is decidable. Roopsha Samanta Lipschitz Robustness of Finite-state Transducers 16 / 18

34 Nondeterministic transducers For a nondeterministic transducer T, JT K(s) 1 Roopsha Samanta Lipschitz Robustness of Finite-state Transducers 17 / 18

35 Nondeterministic transducers Given: a transducer T, with JT K, similarity function d :! Q 1 set-similarity function D : 2! Q 1 constant K 2 Q with K > 0 T is defined to be K -robust w.r.t d, D if: 8s, t 2 dom(t ): d (s, t) < 1)D (JT K(s), JT K(t)) apple Kd (s, t). Roopsha Samanta Lipschitz Robustness of Finite-state Transducers 17 / 18

36 Nondeterministic transducers Set-similarity functions D(A, B) induced by d Hausdorff: max{ sup s2a Inf-inf: inf inf d(s, t) s2a t2b Sup-sup: sup s2a sup d(s, t) t2b inf d(s, t), sup t2b s2b inf d(s, t) } t2a Roopsha Samanta Lipschitz Robustness of Finite-state Transducers 17 / 18

37 Nondeterministic transducers Set-similarity functions D(A, B) induced by d Hausdorff: max{ sup s2a Inf-inf: inf inf d(s, t) Undecidable s2a t2b Sup-sup: sup s2a sup d(s, t) t2b inf d(s, t), sup t2b s2b inf d(s, t) } Undecidable t2a K -robustness Decidable Roopsha Samanta Lipschitz Robustness of Finite-state Transducers 17 / 18

38 Thank you. Roopsha Samanta Lipschitz Robustness of Finite-state Transducers 18 / 18