The Complexity of Intersecting Finite Automata Having Few Final - PowerPoint PPT Presentation

Introduction Automata Intersection Problem Conclusion The Complexity of Intersecting Finite Automata Having Few Final States Michael Blondin 1 2 Andreas Krebs 3 Pierre McKenzie 1 1 DIRO, Université de Montréal 2 LSV, ENS Cachan 3 Universität Tübingen October 31, 2013 1 / 48

Definitions Introduction Motivation and Prior Work Automata Intersection Problem Complexity Classes Conclusion Our Results Definition An automaton is a 5-tuple: Ω (finite set of states ) Σ (finite alphabet ) δ : Ω × Σ → Ω ( transition function ) α ∈ Ω ( initial state ) F ⊆ Ω ( final states ) 2 / 48

Definitions Introduction Motivation and Prior Work Automata Intersection Problem Complexity Classes Conclusion Our Results Definition Transition monoid M ( A ) of A : �{ T σ : σ ∈ Σ }� where T σ ( γ ) = δ ( γ, σ ) . Example 0 0 1    α β γ ω γ α β T 011 = 0  β β γ ω 1 1 ω 0 , 1 3 / 48

Definitions Introduction Motivation and Prior Work Automata Intersection Problem Complexity Classes Conclusion Our Results Definition AutoInt b ( X ) (Automata nonemptiness intersection problem) Input : Automata A 1 , . . . , A k on alphabet Σ with M ( A i ) ∈ X and at most b final states. k � Question: Language ( A i ) � = ∅ ? i = 1 4 / 48

Definitions Introduction Motivation and Prior Work Automata Intersection Problem Complexity Classes Conclusion Our Results Definition AutoInt b ( ∪ m X ) (Generalized automata intersection problem) Input : Automata on alphabet Σ with A 1 , 1 , . . . , A k , m M ( A i , j ) ∈ X and at most b final states. k m � � Question: Language ( A i , j ) � = ∅ ? i = 1 j = 1 5 / 48

Definitions Introduction Motivation and Prior Work Automata Intersection Problem Complexity Classes Conclusion Our Results Kozen 77 AutoInt and AutoInt 1 are PSPACE − complete. Galil 76 AutoInt is NP − complete when Σ = { a } . 6 / 48

Definitions Introduction Motivation and Prior Work Automata Intersection Problem Complexity Classes Conclusion Our Results AutoInt interesting because generalizes: Definition Memb ( X ) (Membership problem) g , g 1 , . . . , g k : [ m ] → [ m ] such that � g 1 , . . . , g k � ∈ X . Input : g ∈ � g 1 , . . . , g k � ? Question: 7 / 48

Definitions Introduction Motivation and Prior Work Automata Intersection Problem Complexity Classes Conclusion Our Results AutoInt interesting because generalizes: Definition Memb ( X ) (Membership problem) g , g 1 , . . . , g k : [ m ] → [ m ] such that � g 1 , . . . , g k � ∈ X . Input : g ∈ � g 1 , . . . , g k � ? Question: Connections with graph isomorphism led to deep results on group problems. It is known that Memb ( Groups ) ∈ NC. 8 / 48

Definitions Introduction Motivation and Prior Work Automata Intersection Problem Complexity Classes Conclusion Our Results Definition AC k : languages accepted by Boolean circuits of poly size and depth O ( log k n ) . NC k : similar with gates of indegree 2. � NC k NC = AC = k ≥ 0 9 / 48

Definitions Introduction Motivation and Prior Work Automata Intersection Problem Complexity Classes Conclusion Our Results Definition L: languages accepted by log-space Turing machines. NL: languages accepted by log-space non deterministic Turing machines. Mod p L: languages S s.t. w ∈ S iff # accept paths ≡ 0 ( mod p ) for some NL machine. 10 / 48

Definitions Introduction Motivation and Prior Work Automata Intersection Problem Complexity Classes Conclusion Our Results Inclusion chain of complexity classes NL AC 0 NC 1 NC 2 L NC P NP PSPACE Mod p L 11 / 48

Definitions Introduction Motivation and Prior Work Automata Intersection Problem Complexity Classes Conclusion Our Results Inclusion chain of complexity classes NL AC 0 NC 1 NC 2 L NC P NP PSPACE Mod p L Contains: binary addition/substraction, star-free languages. Does not contain: parity/majority. Equals: FO(BIT), FO( + , × ) where variables = positions in words. 12 / 48

Definitions Introduction Motivation and Prior Work Automata Intersection Problem Complexity Classes Conclusion Our Results Inclusion chain of complexity classes NL AC 0 NC 1 NC 2 L NC P NP PSPACE Mod p L Contains: binary multiplication/division, regular languages, parity/majority. 13 / 48

Definitions Introduction Motivation and Prior Work Automata Intersection Problem Complexity Classes Conclusion Our Results Inclusion chain of complexity classes NL AC 0 NC 1 NC 2 L NC P NP PSPACE Mod p L Complete problems: undirected connectivity, 2 ⊕ SAT. Contains: problems defined in MSO on graphs of bounded tree-width. 14 / 48

Definitions Introduction Motivation and Prior Work Automata Intersection Problem Complexity Classes Conclusion Our Results Inclusion chain of complexity classes NL AC 0 NC 1 NC 2 L NC P NP PSPACE Mod p L Complete problems: directed connectivity, 2SAT, testing an automaton for emptiness. Equals: coNL. 15 / 48

Definitions Introduction Motivation and Prior Work Automata Intersection Problem Complexity Classes Conclusion Our Results Inclusion chain of complexity classes NL AC 0 NC 1 NC 2 L NC P NP PSPACE Mod p L Complete problems: linear algebra mod p . Equals: coMod p L. 16 / 48

Definitions Introduction Motivation and Prior Work Automata Intersection Problem Complexity Classes Conclusion Our Results Inclusion chain of complexity classes NL AC 0 NC 1 NC 2 L NC P NP PSPACE Mod p L Contains: determinant, automata minimization. 17 / 48

Definitions Introduction Motivation and Prior Work Automata Intersection Problem Complexity Classes Conclusion Our Results Inclusion chain of complexity classes NL AC 0 NC 1 NC 2 L NC P NP PSPACE Mod p L Contains: membership in permutation groups. 18 / 48

Definitions Introduction Motivation and Prior Work Automata Intersection Problem Complexity Classes Conclusion Our Results Inclusion chain of complexity classes NL AC 0 NC 1 NC 2 L NC P NP PSPACE Mod p L Complete problems: circuit value problem, linear programming. 19 / 48

Definitions Introduction Motivation and Prior Work Automata Intersection Problem Complexity Classes Conclusion Our Results Main result: completeness results for AutoInt b ( X ) Maximum number of final states 1 with ∪ 2 1 2 3+ Σ = { a } L L NL NP Z 2 × · · · × Z 2 ⊕ L ⊕ L NP NP Z p × · · · × Z p Mod p L NP NP NP ∈ NC 3 Abelian groups NP NP NP Groups ∈ NC NP NP NP ∈ AC 0 J 1 NP NP NP *** Our classification. *** Beaudry 88. 20 / 48

AutoInt 2 ( X ) is NP − complete Introduction AutoInt 1 ( Abelian groups ) ∈ NC 3 Automata Intersection Problem AutoInt 2 ( Z 2 × · · · × Z 2 ) is ⊕ L − complete Conclusion Complexity of AutoInt 2 ( X ) Maximum number of final states 1 with ∪ 2 1 2 3+ Σ = { a } L L NL NP Z 2 × · · · × Z 2 ⊕ L ⊕ L NP NP Z p × · · · × Z p Mod p L NP NP NP ∈ NC 3 Abelian groups NP NP NP Groups ∈ NC NP NP NP ∈ AC 0 J 1 NP NP NP 21 / 48

AutoInt 2 ( X ) is NP − complete Introduction AutoInt 1 ( Abelian groups ) ∈ NC 3 Automata Intersection Problem AutoInt 2 ( Z 2 × · · · × Z 2 ) is ⊕ L − complete Conclusion Theorem AutoInt 2 ( X ) is hard for NP for any X beyond Z 2 × · · · × Z 2 . Proof sketch X �⊆ Z 2 × · · · × Z 2 implies aperiodic monoid or cyclic group Z q , q > 2, in X. Reduction from CIRCUIT–SAT to AutoInt 2 ( X ) in both cases. 22 / 48

AutoInt 2 ( X ) is NP − complete Introduction AutoInt 1 ( Abelian groups ) ∈ NC 3 Automata Intersection Problem AutoInt 2 ( Z 2 × · · · × Z 2 ) is ⊕ L − complete Conclusion Theorem AutoInt 2 ( X ) is hard for NP for any X beyond Z 2 × · · · × Z 2 . Proof sketch X �⊆ Z 2 × · · · × Z 2 implies aperiodic monoid or cyclic group Z q , q > 2, in X. Reduction from CIRCUIT–SAT to AutoInt 2 ( X ) in both cases. 23 / 48

AutoInt 2 ( X ) is NP − complete Introduction AutoInt 1 ( Abelian groups ) ∈ NC 3 Automata Intersection Problem AutoInt 2 ( Z 2 × · · · × Z 2 ) is ⊕ L − complete Conclusion Proof sketch: CIRCUIT–SAT reduces to AutoInt 2 ( Z q ) Given a circuit, we let Σ be the set of gates. ¬ ∧ ∨ Σ = {◦ 0 , ◦ 1 , ◦ 2 , ∧ 0 , ¬ 0 , ∨ 0 , ◦ 3 } 24 / 48

AutoInt 2 ( X ) is NP − complete Introduction AutoInt 1 ( Abelian groups ) ∈ NC 3 Automata Intersection Problem AutoInt 2 ( Z 2 × · · · × Z 2 ) is ⊕ L − complete Conclusion Proof sketch: CIRCUIT–SAT reduces to AutoInt 2 ( Z q ) Given a circuit, we let Σ be the set of gates. ¬ ∧ ∨ Σ = {◦ 0 , ◦ 1 , ◦ 2 , ∧ 0 , ¬ 0 , ∨ 0 , ◦ 3 } 25 / 48

AutoInt 2 ( X ) is NP − complete Introduction AutoInt 1 ( Abelian groups ) ∈ NC 3 Automata Intersection Problem AutoInt 2 ( Z 2 × · · · × Z 2 ) is ⊕ L − complete Conclusion Proof sketch: CIRCUIT–SAT reduces to AutoInt 2 ( Z q ) Given a circuit, we let Σ be the set of gates. ¬ ∧ ¬ ¬ ∧ ¬ Σ = {◦ 0 , ◦ 1 , ◦ 2 , ∧ 0 , ¬ 0 , ¬ 1 , ¬ 2 , ∧ 1 , ◦ 3 } 26 / 48