The Inclusion Problem for Weighted Automata on Infinite Trees - PDF document

See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/236590870 The Inclusion Problem for Weighted Automata on Infinite Trees (Slides) Data · August 2011 CITATIONS READS 0 30 2 authors: Stefan Borgwardt Rafael Peñaloza Technische Universität Dresden Università degli Studi di Milano-Bicocca 87 PUBLICATIONS 579 CITATIONS 161 PUBLICATIONS 1,232 CITATIONS SEE PROFILE SEE PROFILE Some of the authors of this publication are also working on these related projects: Repairing Description Logic Ontologies View project All content following this page was uploaded by Stefan Borgwardt on 22 May 2014. The user has requested enhancement of the downloaded file.

Institute of Theoretical Computer Science Chair of Automata Theory THE INCLUSION PROBLEM FOR WEIGHTED AUTOMATA ON INFINITE TREES Stefan Borgwardt Rafael Pe˜ naloza Debrecen, August 21, 2011

Introduction • Automata on infinite trees can recognize tree-shaped models • Emptiness test useful to decide satisfiability in logics • Inclusion test could be used to decide entailment • Here: generalization to lattice-weighted automata Debrecen, August 21, 2011 Inclusion for Weighted Automata 2

Lattices De Morgan lattice: • Bounded distributive lattice L = ( L, ⊕ , ⊗ , 0 , 1 ) • De Morgan negation − : L → L 1 1 a b c x = 1 − x a b c 0 0 Debrecen, August 21, 2011 Inclusion for Weighted Automata 3

Trees Infinite k -ary trees: • Nodes are identified by their positions in K ∗ , where K := { 1 , . . . , k } ε 1 2 11 12 21 22 . . . . . . . . . . . . . . . . . . . . . . . . Debrecen, August 21, 2011 Inclusion for Weighted Automata 4

Trees Infinite k -ary trees: • Nodes are identified by their positions in K ∗ , where K := { 1 , . . . , k } ε 1 2 11 12 21 22 . . . . . . . . . . . . . . . . . . . . . . . . • A labeled tree t ∈ Σ K ∗ is a function t : K ∗ → Σ Debrecen, August 21, 2011 Inclusion for Weighted Automata 4

Automata Tree automaton A = ( Q, Σ , I, ∆ , X ) : • states Q • input alphabet Σ • initial state set I ⊆ Q • transition relation ∆ ⊆ Q × Σ × Q k • acceptance condition X ⊆ Q ω Debrecen, August 21, 2011 Inclusion for Weighted Automata 5

Automata Tree automaton A = ( Q, Σ , I, ∆ , X ) : • states Q • input alphabet Σ • initial state set I ⊆ Q • transition relation ∆ ⊆ Q × Σ × Q k • acceptance condition X ⊆ Q ω Successful run r ∈ succ ( A ) ⊆ Q K ∗ : every path is in X Debrecen, August 21, 2011 Inclusion for Weighted Automata 5

Automata Tree automaton A = ( Q, Σ , I, ∆ , X ) : • states Q • input alphabet Σ • initial state set I ⊆ Q • transition relation ∆ ⊆ Q × Σ × Q k • acceptance condition X ⊆ Q ω Successful run r ∈ succ ( A ) ⊆ Q K ∗ : every path is in X Transition of r on t ∈ Σ K ∗ at u ∈ K ∗ : − − − → r ( t, u ) := ( r ( u ) , t ( u ) , r ( u 1 ) , . . . , r ( uk )) Debrecen, August 21, 2011 Inclusion for Weighted Automata 5

Automata Tree automaton A = ( Q, Σ , I, ∆ , X ) : • states Q • input alphabet Σ • initial state set I ⊆ Q • transition relation ∆ ⊆ Q × Σ × Q k • acceptance condition X ⊆ Q ω Successful run r ∈ succ ( A ) ⊆ Q K ∗ : every path is in X Transition of r on t ∈ Σ K ∗ at u ∈ K ∗ : − − − → r ( t, u ) := ( r ( u ) , t ( u ) , r ( u 1 ) , . . . , r ( uk )) − − − → r ( ε ) ∈ I ∧ ∀ ∃ t ∈ L ( A ) iff r ( t, u ) ∈ ∆ r ∈ succ ( A ) u ∈ K ∗ Debrecen, August 21, 2011 Inclusion for Weighted Automata 5

Automata Weighted tree automaton A = ( Q, Σ , S, in , wt , X ) : • states Q • input alphabet Σ • (finite) distributive lattice S • initial distribution in : Q → S • transition weight function wt : Q × Σ × Q k → S • acceptance condition X ⊆ Q ω Successful run r ∈ succ ( A ) ⊆ Q K ∗ : every path is in X Transition of r on t ∈ Σ K ∗ at u ∈ K ∗ : − − − → r ( t, u ) := ( r ( u ) , t ( u ) , r ( u 1 ) , . . . , r ( uk )) − − − → r ( ε ) ∈ I ∧ ∀ ∃ t ∈ L ( A ) iff r ( t, u ) ∈ ∆ r ∈ succ ( A ) u ∈ K ∗ Debrecen, August 21, 2011 Inclusion for Weighted Automata 5

Automata Weighted tree automaton A = ( Q, Σ , S, in , wt , X ) : • states Q • input alphabet Σ • (finite) distributive lattice S • initial distribution in : Q → S • transition weight function wt : Q × Σ × Q k → S • acceptance condition X ⊆ Q ω Successful run r ∈ succ ( A ) ⊆ Q K ∗ : every path is in X Transition of r on t ∈ Σ K ∗ at u ∈ K ∗ : − − − → r ( t, u ) := ( r ( u ) , t ( u ) , r ( u 1 ) , . . . , r ( uk )) wt ( − − − → � � ( �A� , t ) = in ( r ( ε )) ⊗ r ( t, u )) r ∈ succ ( A ) u ∈ K ∗ Debrecen, August 21, 2011 Inclusion for Weighted Automata 5

Automata Weighted tree automaton A = ( Q, Σ , S, in , wt , X ) : • states Q • input alphabet Σ • (finite) distributive lattice S • initial distribution in : Q → S • transition weight function wt : Q × Σ × Q k → S • acceptance condition X ⊆ Q ω Successful run r ∈ succ ( A ) ⊆ Q K ∗ : every path is in X Transition of r on t ∈ Σ K ∗ at u ∈ K ∗ : − − − → r ( t, u ) := ( r ( u ) , t ( u ) , r ( u 1 ) , . . . , r ( uk )) wt ( − − − → � � ( �A� , t ) = in ( r ( ε )) ⊗ r ( t, u )) r ∈ succ ( A ) u ∈ K ∗ Acceptance conditions: looping, B¨ uchi, co-B¨ uchi, parity Debrecen, August 21, 2011 Inclusion for Weighted Automata 5

Automata (2) PT IME -problems for B¨ uchi automata: • Infimum of two automata: ( �C� , t ) = ( �A� , t ) ⊗ ( �B� , t ) • Supremum of two automata: ( �C� , t ) = ( �A� , t ) ⊕ ( �B� , t ) • Computing the behavior � t ∈ Σ K ∗ ( �A� , t ) [Baader, Pe˜ naloza 2010] Debrecen, August 21, 2011 Inclusion for Weighted Automata 6

Description Logics The description logic ALC Syntax concept name A ∈ N C r ∈ N R role name Debrecen, August 21, 2011 Inclusion for Weighted Automata 7

Description Logics interpretation I = ( · I , ∆ I ) The description logic ALC Syntax A I ⊆ ∆ I concept name A ∈ N C r I ⊆ ∆ I × ∆ I r ∈ N R role name Debrecen, August 21, 2011 Inclusion for Weighted Automata 7

Description Logics interpretation I = ( · I , ∆ I ) The description logic ALC Syntax A I ⊆ ∆ I concept name A ∈ N C r I ⊆ ∆ I × ∆ I r ∈ N R role name ∆ I ⊤ top concept bottom concept ⊥ ∅ C I ∩ D I conjunction C ⊓ D C I ∪ D I disjunction C ⊔ D negation ¬ C C I Debrecen, August 21, 2011 Inclusion for Weighted Automata 7

Description Logics interpretation I = ( · I , ∆ I ) The description logic ALC Syntax A I ⊆ ∆ I concept name A ∈ N C r I ⊆ ∆ I × ∆ I r ∈ N R role name ∆ I ⊤ top concept bottom concept ⊥ ∅ C I ∩ D I conjunction C ⊓ D C I ∪ D I disjunction C ⊔ D negation ¬ C C I { x | ∃ y : ( x, y ) ∈ r I ∧ y ∈ C I } existential restriction ∃ r.C { x | ∀ y : ( x, y ) ∈ r I → y ∈ C I } universal restriction ∀ r.C Debrecen, August 21, 2011 Inclusion for Weighted Automata 7

Description Logics interpretation I = ( · I , ∆ I ) The description logic ALC Syntax A I ⊆ ∆ I concept name A ∈ N C r I ⊆ ∆ I × ∆ I r ∈ N R role name ∆ I ⊤ top concept bottom concept ⊥ ∅ C I ∩ D I conjunction C ⊓ D C I ∪ D I disjunction C ⊔ D negation ¬ C C I { x | ∃ y : ( x, y ) ∈ r I ∧ y ∈ C I } existential restriction ∃ r.C { x | ∀ y : ( x, y ) ∈ r I → y ∈ C I } universal restriction ∀ r.C C I ⊆ D I terminological axiom C ⊑ D Debrecen, August 21, 2011 Inclusion for Weighted Automata 7

Description Logics interpretation I = ( · I , ∆ I ) The description logic ALC Syntax A I ⊆ ∆ I concept name A ∈ N C r I ⊆ ∆ I × ∆ I r ∈ N R role name ∆ I ⊤ top concept bottom concept ⊥ ∅ C I ∩ D I conjunction C ⊓ D C I ∪ D I disjunction C ⊔ D negation ¬ C C I { x | ∃ y : ( x, y ) ∈ r I ∧ y ∈ C I } existential restriction ∃ r.C { x | ∀ y : ( x, y ) ∈ r I → y ∈ C I } universal restriction ∀ r.C C I ⊆ D I terminological axiom C ⊑ D • Consistency of a TBox T (set of axioms): Is there a model of T ? • Satisfiability of C w.r.t. T : Is there a model I of T with C I � = ∅ ? • Subsumption C ⊑ T D : Does C I ⊆ D I hold in all models I of T ? Debrecen, August 21, 2011 Inclusion for Weighted Automata 7

ë Tree Models • ALC has the tree model property • Satisfiability of C w.r.t. T can be reduced to emptiness of A C, T Debrecen, August 21, 2011 Inclusion for Weighted Automata 8

Tree Models • ALC has the tree model property • Satisfiability of C w.r.t. T can be reduced to emptiness of A C, T • Behavior computation can be used for axiom pinpointing ë identifying the axioms of T that are responsible for a contradiction Debrecen, August 21, 2011 Inclusion for Weighted Automata 8

Tree Models • ALC has the tree model property • Satisfiability of C w.r.t. T can be reduced to emptiness of A C, T • Behavior computation can be used for axiom pinpointing ë identifying the axioms of T that are responsible for a contradiction • C ⊑ T D iff C ⊓ ¬ D is unsatisfiable Debrecen, August 21, 2011 Inclusion for Weighted Automata 8