The Complexity of Transducer Synthesis from Multi-Sequential Specifications

Transcription

1 The Complexity of Transducer Synthesis from Multi-Sequential Specifications Léo Exibard 12 Emmanuel Filiot 13 Ismaël Jecker 1 Tuesday, August 28 th, Université libre de Bruxelles 2 LIS Aix-Marseille Université 3 FNRS

2 From verification to synthesis Reactive systems Environment Input Output System Interaction i 1 o 1 i 2 o 2 i 3 o 3 (I O) ω or (I O) Verification Check that a system satisfies a specification System Env = Specification Synthesis Generate a system from a specification? Env = Specification 1

3 From verification to synthesis Reactive systems Environment Input Output System Interaction i 1 o 1 i 2 o 2 i 3 o 3 (I O) ω or (I O) Verification Check that a system satisfies a specification System Env = Specification Synthesis Generate a system from a specification? Env = Specification 1

4 Synthesis? Environment = Specification Generate a system from a specification Implementing a specification Input words I Implementation M : I O Output words O Specification S I O M fulfils S, written M = S, if for all in I, (in, M(in)) S O S M I 2

5 The realisability and synthesis problem S = Class of specifications S I O M = Class of target implementations M : I O Synthesis problem from S to M Input: Output: Specification S S Implementation M M s.t. M = S if it exists No otherwise Realisability problem from S to M Corresponding decision problem 3

6 Finite transducers: automata with outputs b a b a a,b a Replace every letter with an a when there are at least two a s 4

7 Finite transducers: automata with outputs b a b a a,b a Replace every letter with an a when there are at least two a s a b a,b b b b 4 5 Replace every letter with a b when there is at least one b 4

8 Finite transducers: automata with outputs b a b a a,b a Replace every letter with an a when there are at least two a s a b a,b b b b 4 5 Replace every letter with a b when there is at least one b Sequential transducer The transition and output letter are determined by the input letter 4

9 Multi-sequential transducers b a b a a,b a b b 4 5 a b a,b b Running example D 1 D 2 Multi-sequential transducer Union of sequential transducers k T = i=1 D i Multi-sequentiality A relation is multi-sequential if it can be defined by a multi-sequential transducer Decidable for functions [?] Membership in PTime [?] 5

10 Transducer realisability problem Known results M = sequential transducers S Complexity MSO Nonelementary [?] LTL 2-ExpTime-c [?] Finite Transducers ExpTime-c 6

11 Transducer realisability problem Known results M = sequential transducers S Complexity MSO Nonelementary [?] LTL 2-ExpTime-c [?] Finite Transducers ExpTime-c Question: Class of transducers with better complexity? 6

12 Realisability of multi-sequential specifications S = Multi-seq. transducers Unions of sequential transducers T = k i=1 D i M = Seq. transducers Output letter and transition is determined by input letter b a b a a,b a b b 4 5 D 1 D 2 a b a,b b Theorem Sequential transducer synthesis from multi-sequential specifications is PSpace-complete. 7

13 Realisability of multi-sequential specifications Proof b a b a a,b a b b 4 5 D 1 D 2 a b a,b b 8

14 Realisability of multi-sequential specifications Proof b a b a a,b a b b 4 5 D 1 D 2 On input a, need to drop one transducer a b a,b b 8

15 Realisability of multi-sequential specifications Proof b a b a a,b a b b 4 5 a b a,b b D 1 D 2 On input a, need to drop one transducer Critical prefix u At least two runs on u disagree on their output 8

16 Realisability of multi-sequential specifications Proof b a b a a,b a b b 4 5 a b a,b b D 1 D 2 On input a, need to drop one transducer Critical prefix u At least two runs on u disagree on their output Residual property For all critical prefix u, there exists P {D 1,..., D k } s.t.: 1. All transducers in P produce the same output on u 2. The domain is still covered: u 1 dom(t ) = u 1 dom(d i ) i P 3. The residual specification u 1 D i is realisable i P 8

17 Realisability of multi-sequential specifications Proof Theorem Sequential transducer realisability from multi-sequential specifications is PSpace-complete. Easiness The residual property can be checked in PSpace. 9

18 Realisability of multi-sequential specifications Proof Theorem Sequential transducer realisability from multi-sequential specifications is PSpace-complete. Easiness The residual property can be checked in PSpace. Hardness Emptiness problem of the intersection of n DFAs S : w#σ wσ# if i, w L(A i ) (σ {a, b}) w#σ w#σ if i, w / L(A i ) / A i 2 b b 1 # # # σ σ # id Ai F 3 σ 4 σ A i 9

19 Asynchronicity b a b a a,b a b b 4 5 a b a,b b Our running example 10

20 Asynchronicity b a b a a,b a b b 4 5 Waiting two steps allows to determine whether: There is at least one b There are at least two a s a b a,b b Our running example 10

21 Asynchronicity b a b a a,b a b b 4 5 Waiting two steps allows to determine whether: There is at least one b There are at least two a s a b a,b b Our running example Asynchronous transducer At every transition, reads a letter, outputs a (possibly empty) word. 10

22 Asynchronicity b a b a a,b a b b 4 5 Waiting two steps allows to determine whether: There is at least one b There are at least two a s a b a,b b a,b b Our running example Asynchronous transducer At every transition, reads a letter, outputs a (possibly empty) word. 5 b b a ε 8 a,b a b bb a 6 7 An asynchronous implementation 10

23 Asynchronous transducer realisability problem Known results M = Unambiguous functional transducers Feasible for any asynchronous specification [?] M = Sequential transducers S (async. transducers) Complexity Nondeterministic Undecidable [?] Finite-valued 3-ExpTime [?] Multi-sequential PSpace-c 11

24 Asynchronous transducer realisability problem Known results M = Unambiguous functional transducers Feasible for any asynchronous specification [?] M = Sequential transducers S (async. transducers) Complexity Nondeterministic Undecidable [?] Finite-valued 3-ExpTime [?] Multi-sequential PSpace-c 11

25 Asynchronous transducer realisability problem Proof b ε a ε b a b b Delay del(u 1, u 2 ) = (l 1 u 1, l 1 u 2 ) l = u 1 u 2 12

26 Asynchronous transducer realisability problem Proof b ε a ε b a b b Delay del(u 1, u 2 ) = (l 1 u 1, l 1 u 2 ) del(a, ε) = (a, ε) 12

27 Asynchronous transducer realisability problem Proof b ε a ε b a b b Delay del(u 1, u 2 ) = (l 1 u 1, l 1 u 2 ) del(a, ε) = (a, ε) = del(aa, a) = (a, ε) 12

28 Asynchronous transducer realisability problem Proof b ε a b b a b b Delay del(u 1, u 2 ) = (l 1 u 1, l 1 u 2 ) del(a, b) = (a, b) del(aa, ba) = (aa, ba) 12

29 Asynchronous transducer realisability problem Proof b ε a b b a b b Critical loop Triple (u, v, X ) s.t.: Delay del(u 1, u 2 ) = (l 1 u 1, l 1 u 2 ) del(a, b) = (a, b) v β i del(aa, ba) = (aa, ba) u α 1. For all D i X, i p i q i 2. For all D i / X, no run on u 3. For two transducers D i, D j X, delays accumulate: del(α i, α j ) del(α i β i, α j β j ) 12

30 Asynchronous transducer realisability problem Proof Recursive characterisation T = k i=1 D i is realisable iff for all critical loops (u, v, X ), there exists Y X s.t.: 1. Delays do not accumulate: D i, D j Y, del(α i, α j ) = del(α i β i, α j β j ) 2. The domain is still covered: u 1 dom(t ) = i P u 1 dom(d i ) 3. The residual specification (u, l) 1 D i is realisable i Y l longest common prefix of the α i s Can be easily checked in ExpTime 13

31 Asynchronous transducer realisability problem Proof Theorem Asynchronous sequential transducer synthesis from multi-sequential specifications is PSpace-complete. PSpace-easiness: a non-recursive characterisation Witness of non-satisfaction Unfolding of the recursive characterisation Reformulation of delay difference Can be found in PSpace PSpace-hardness Similar to the synchronous case 14

32 Conclusion Multi-sequential specifications Membership decidable in PTime Sequential realisability is PSpace-c both in synchronous and asynchronous cases Improvement of the general case: synchronous = ExpTime-c asynchronous = undecidable Synthesis game Practical synthesis algorithm Suitable for any type of specification defined by transducers (might not terminate) 15

33 The synthesis game 1 : ɛ 5 : 1 : ɛ 4 : ɛ b 1 : a 5 : b b b 1 : 5 : ɛ a a a a 2 : ɛ 4 : b 2 : ɛ 5 : 2 : a 4 : b b b a 2 : 4 : ɛ 2 : 5 : ɛ b a a a 3 : ɛ 4 : 3 : ɛ 5 : b b b a b a a 3 : 4 : ɛ b 3 : 5 : ɛ b a a 16

34 Bibliography i Büchi, J. R. and Landweber, L. H. (1969). Solving Sequential Conditions by Finite-State Strategies. Transactions of the American Mathematical Society, 138: Carayol, A. and Löding, C. (2014). Uniformization in Automata Theory. In Proceedings of the 14th Congress of Logic, Methodology and Philosophy of Science Nancy, July 19-26, 2011, pages , London. College Publications. 17

35 Bibliography ii Choffrut, C. and Schützenberger, M. P. (1986). Décomposition de fonctions rationnelles. In 2nd Annual Symposium on Theoretical Aspects of Computer Science, STACS, pages Filiot, E., Jecker, I., Löding, C., and Winter, S. (2016). On equivalence and uniformisation problems for finite transducers. In 43rd International Colloquium on Automata, Languages, and Programming, ICALP 2016, July 11-15, Rome, Italy, pages 125:1 125:14. 18

36 Bibliography iii Jecker, I. and Filiot, E. (2015). Multi-sequential word relations. In Proceedings of the 19th International Conference on Developments in Language Theory, DLT 2015, Liverpool, UK, July 27-30, pages Kobayashi, K. (1969). Classification of formal languages by functional binary transductions. Information and Control, 15(1):

37 Bibliography iv Pnueli, A. and Rosner, R. (1989). On the synthesis of a reactive module. In Proceedings of the 16th ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, POPL 89, pages , New York, NY, USA. ACM. 20