On Positive TAGED with a Bounded Number of Constraints P.-C. Héam, Vincent Hugot, O. Kouchnarenko {pheam,vhugot,okouchna}@femto-st.fr University of Franche-Comté DGA & INRIA/CASSIS & FEMTO-ST (DISC) July 2, 2013
Introduction Expressivity Membership Rigidification Emptiness Finiteness Conclusions Appendix Introduction & Preliminaries 1 Effects on Expressive Power 2 The Membership Decision Problem 3 Rigidification of One Constraint 4 The Emptiness Decision Problem 5 The Finiteness Decision Problem 6 Conclusions & Future Works 7 Appendix: Taxonomy of Constraints 8 TAGE + with Bounded Constraints Vincent HUGOT FORWAL 2 / 27
Introduction Expressivity Membership Rigidification Emptiness Finiteness Conclusions Appendix Introduction & Preliminaries 1 Effects on Expressive Power 2 The Membership Decision Problem 3 Rigidification of One Constraint 4 The Emptiness Decision Problem 5 The Finiteness Decision Problem 6 Conclusions & Future Works 7 Appendix: Taxonomy of Constraints 8 TAGE + with Bounded Constraints Vincent HUGOT FORWAL 2 / 27
Introduction Expressivity Membership Rigidification Emptiness Finiteness Conclusions Appendix Tree Automata [Comon et al., 2008a] Introduced in the fifties; regular tree languages : model-checking: programs, protocols,. . . automated theorem-proving XML schema and (esp. variants) query languages . . . and so much more Doesn’t deal with comparisons and non-linearity : { f ( u, u ) | u ∈ T ( Σ ) } e.g. password verification { f ( u, v ) | u, v ∈ T ( Σ ) , u � = v } e.g. primary keys R ( ℓ ) , ℓ regular, R a TRS e.g. { g ( x ) → f ( x, x ) } ( T ( A )) TAGE + with Bounded Constraints Vincent HUGOT FORWAL 3 / 27
Introduction Expressivity Membership Rigidification Emptiness Finiteness Conclusions Appendix Tree Automata Bottom-Up, Non-Deterministic, Finite Tree Automaton A = � A , Q, F, ∆ � : finite ranked alphabet A Q finite set of states final states, F ⊆ Q F ∆ finite set of transitions Transition r ∈ ∆ : σ ( q 1 , . . . , q n ) → q σ ∈ A n q 1 , . . . , q n , q ∈ Q TAGE + with Bounded Constraints Vincent HUGOT FORWAL 4 / 27
Introduction Expressivity Membership Rigidification Emptiness Finiteness Conclusions Appendix Tree Automata Bottom-Up, Non-Deterministic, Finite A = { ∧ , ∨ / 2 , ¬ / 1 , ⊤ , ⊥ / 0 } , Q = { q 0 , q 1 } , F = { q 1 } , ∆ = � ⊤ → q 1 , � ⊥ → q 0 , ¬ ( q b ) → q ¬ b � � b, b ′ ∈ { 0, 1 } � � ∧ ( q b , q b ′ ) → q b ∧ b ′ , ∨ ( q b , q b ′ ) → q b ∨ b ′ t = ∧ ¬ ∨ ∧ ⊥ ¬ ⊥ ⊤ ⊥ TAGE + with Bounded Constraints Vincent HUGOT FORWAL 5 / 27
Introduction Expressivity Membership Rigidification Emptiness Finiteness Conclusions Appendix Tree Automata Bottom-Up, Non-Deterministic, Finite → ∗ → ∗ → ∗ ∧ ∧ ∧ ∆ ∆ ∆ ¬ ∨ ¬ ∨ ¬ ∨ ∧ ⊥ ¬ ∧ q 0 ¬ q 0 q 0 q 1 ⊥ ⊤ ⊥ q 0 q 1 q 0 → ∆ q 1 ∧ q 1 q 1 TAGE + with Bounded Constraints Vincent HUGOT FORWAL 5 / 27
Introduction Expressivity Membership Rigidification Emptiness Finiteness Conclusions Appendix Tree Automata Runs and Languages The reduction t → ∗ ∆ q 1 is captured by the run : q 1 decorated: ε ∧ q 1 q 1 q 1 1 ¬ q 1 2 ∨ q 1 q 0 q 0 q 1 11 ∧ q 0 21 ⊥ q 0 22 ¬ q 1 q 0 q 1 q 0 111 ⊥ q 0 112 ⊤ q 1 221 ⊥ q 0 TAGE + with Bounded Constraints Vincent HUGOT FORWAL 6 / 27
Introduction Expressivity Membership Rigidification Emptiness Finiteness Conclusions Appendix Tree Automata With Global Equality Constraints [Filiot et al., 2008] TAGE, TA = , Positive TAGED, A = � A , Q, F, ∆, ≅ � : � A , Q, F, ∆ � vanilla tree automaton ta ( A ) equality constraints , ≅ ⊆ Q 2 ≅ Constraint p ≅ q : run ρ of A on t : run of ta ( A ) on t satisfying ≅ : ∀ α, β ∈ P ( t ) ; ρ ( α ) ≅ ρ ( β ) ⇒ t | α = t | β accepting run : accepting for ta ( A ) TAGE + with Bounded Constraints Vincent HUGOT FORWAL 7 / 27
Introduction Expressivity Membership Rigidification Emptiness Finiteness Conclusions Appendix Tree Automata With Global Equality Constraints [Filiot et al., 2008] q ≅ ^ A = { a/ 0 , f/ 2 } , Q = { q, ^ q, q f } , F = { q f } , ^ q , and ∆ = { f (^ q, ^ q ) → q f , f ( q, q ) → q, f ( q, q ) → ^ q, a → q, a → ^ q } u = f and v = f f f f a a a a a a a TAGE + with Bounded Constraints Vincent HUGOT FORWAL 8 / 27
Introduction Expressivity Membership Rigidification Emptiness Finiteness Conclusions Appendix Tree Automata With Global Equality Constraints [Filiot et al., 2008] q ≅ ^ A = { a/ 0 , f/ 2 } , Q = { q, ^ q, q f } , F = { q f } , ^ q , and ∆ = { f (^ q, ^ q ) → q f , f ( q, q ) → q, f ( q, q ) → ^ q, a → q, a → ^ q } u, ρ u = f q f and v = f f ^ f ^ f a q q a q a q a q a q a a TAGE + with Bounded Constraints Vincent HUGOT FORWAL 8 / 27
Introduction Expressivity Membership Rigidification Emptiness Finiteness Conclusions Appendix Tree Automata With Global Equality Constraints [Filiot et al., 2008] q ≅ ^ A = { a/ 0 , f/ 2 } , Q = { q, ^ q, q f } , F = { q f } , ^ q , and ∆ = { f (^ q, ^ q ) → q f , f ( q, q ) → q, f ( q, q ) → ^ q, a → q, a → ^ q } u = f and v, ρ v = f q f f f f ^ a ^ q q a a a a a q a q TAGE + with Bounded Constraints Vincent HUGOT FORWAL 8 / 27
Introduction Expressivity Membership Rigidification Emptiness Finiteness Conclusions Appendix Tree Automata With Global Equality Constraints [Filiot et al., 2008] A = { ∧ , ∨ / 2 , ¬ / 1 , ⊤ , ⊥ / 0 } ⊎ X , Q = { q 0 , q 1 } ⊎ { v x | x ∈ X } and F = { q 1 } , new rules ⊤ → v x , ⊥ → v x , x ( q 0 , v x ) → q 1 , x ( v x , q 1 ) → q 0 for each x ∈ X , v x ≅ v x . ( x ∧ y ) ∨ ¬ x ≡ ∨ ∧ ¬ x y x ⊥ ⊤ ⊥ ⊤ ⊥ ⊤ TAGE + with Bounded Constraints Vincent HUGOT FORWAL 9 / 27
Introduction Expressivity Membership Rigidification Emptiness Finiteness Conclusions Appendix Tree Automata With Global Equality Constraints [Filiot et al., 2008] A = { ∧ , ∨ / 2 , ¬ / 1 , ⊤ , ⊥ / 0 } ⊎ X , Q = { q 0 , q 1 } ⊎ { v x | x ∈ X } and F = { q 1 } , new rules ⊤ → v x , ⊥ → v x , x ( q 0 , v x ) → q 1 , x ( v x , q 1 ) → q 0 for each x ∈ X , v x ≅ v x . ( x ∧ y ) ∨ ¬ x ≡ ∨ q 1 ∧ q 0 ¬ q 1 x q 0 y q 1 x q 0 ⊥ v x ⊤ q 1 ⊥ q 0 ⊤ v y ⊥ v x ⊤ q 1 TAGE + with Bounded Constraints Vincent HUGOT FORWAL 9 / 27
Introduction Expressivity Membership Rigidification Emptiness Finiteness Conclusions Appendix TA versus RTA versus TA = Closure, Complexity and Decidability RTA ( p ≅ p ) TA = TA ∪ PTime PTime PTime ∩ PTime ExpTime ExpTime ∅ ∅ ¬ ExpTime NP -c (a) t ∈ L ( A ) ? NP -c PTime L ( A ) = ∅ ? linear-time linear-time ExpTime -c |L ( A ) | ∈ N ? ExpTime -c PTime PTime L ( A ) = T ( Σ ) ? ExpTime -c undecidable undecidable L ( A ) ⊆ L ( B ) ? ExpTime -c undecidable undecidable L ( � i A i ) = ∅ ? ExpTime -c ExpTime -c ExpTime -c (a) SAT solver approach: [Héam et al., 2010]. TAGE + with Bounded Constraints Vincent HUGOT FORWAL 10 / 27
Introduction Expressivity Membership Rigidification Emptiness Finiteness Conclusions Appendix TA versus RTA versus TA = Closure, Complexity and Decidability RTA ( p ≅ p ) TA = TA ∪ PTime PTime PTime ∩ PTime ExpTime ExpTime ∅ ∅ ¬ ExpTime NP -c (a) t ∈ L ( A ) ? NP -c PTime L ( A ) = ∅ ? linear-time linear-time ExpTime-c |L ( A ) | ∈ N ? PTime ExpTime-c PTime L ( A ) = T ( Σ ) ? ExpTime -c undecidable undecidable L ( A ) ⊆ L ( B ) ? ExpTime -c undecidable undecidable L ( � i A i ) = ∅ ? ExpTime -c ExpTime -c ExpTime -c (a) SAT solver approach: [Héam et al., 2010]. TAGE + with Bounded Constraints Vincent HUGOT FORWAL 10 / 27
Introduction Expressivity Membership Rigidification Emptiness Finiteness Conclusions Appendix TA = versus TA = k Restriction on the kind of constraints ⇒ lower complexity (RTA) Restriction on the number of constraints ⇒ ? TA = k A = � Σ, Q, F, ∆, ≅ � : TA = A � Σ, Q, F, ∆, ≅ � ≅ such that Card ( ≅ ) � k TAGE + with Bounded Constraints Vincent HUGOT FORWAL 11 / 27
Introduction Expressivity Membership Rigidification Emptiness Finiteness Conclusions Appendix TA = versus TA = k Restriction on the kind of constraints ⇒ lower complexity (RTA) Restriction on the number of constraints ⇒ ? TA = k A = � Σ, Q, F, ∆, ≅ � : TA = A � Σ, Q, F, ∆, ≅ � ≅ such that Card ( ≅ ) � k Expressive power? k + 1 ⊂ · · · ⊂ TA = = TA = TA = 0 ⊂ TA = 1 ⊂ · · · ⊂ TA = k ⊂ TA = � TA = k k ∈ N so ∀ k � 0, L ( TA = TA = � � k ) ⊆ L . Are the inclusions strict ? Up k + 1 to some rank? Is there a k ∈ N such that L ( TA = k ) = L ( TA = ) ? TAGE + with Bounded Constraints Vincent HUGOT FORWAL 11 / 27
Introduction Expressivity Membership Rigidification Emptiness Finiteness Conclusions Appendix Introduction & Preliminaries 1 Effects on Expressive Power 2 The Membership Decision Problem 3 Rigidification of One Constraint 4 The Emptiness Decision Problem 5 The Finiteness Decision Problem 6 Conclusions & Future Works 7 Appendix: Taxonomy of Constraints 8 TAGE + with Bounded Constraints Vincent HUGOT FORWAL 11 / 27
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