Modal Logics with Presburger Constraints St´ ephane Demri LSV, ENS de Cachan, CNRS, INRIA Saclay LABRI – March 5th, 2009 Joint work with Denis Lugiez (LIF, Marseille)
Introduction Extended modal logic ( EML ) Space upper bounds EML vs other logics Conclusion Overview Introduction Space upper bounds Presburger constraints Consistent sets Regularity constraints Non-deterministic algorithm Motivations Results Extended modal logic ( EML ) Boundedness Lemma Definition EML vs other logics Simplifications Sheaves logic PDL over finite trees Conclusion St´ ephane Demri Modal Logics with Presburger Constraints
Introduction Extended modal logic ( EML ) Presburger constraints Space upper bounds Regularity constraints EML vs other logics Motivations Conclusion Presburger constraints ◮ Presburger arithmetic: ◮ First-order theory of � N , + , = � . ◮ Quantifier elimination. ◮ Satisfiability in 3-exptime . ◮ Presburger constraints on graphs/trees: ◮ Constraints in counter automata. ◮ Constraints on the number of event occurrences. [Bouajjani & Echahed & Habermehl, LICS 95] ◮ Constraints on XML documents. [Dal Zilio & Lugiez, RTA 03] [Seidl et al, ICALP 04] ◮ Graded modal logics ( ♦ � 3 p ). [Fine, NDJFL 72] ◮ Description logics (( � 3 R · C )). [Hollunder & Baader, KR 91] ◮ Hennessy-Milner Logic (HML). St´ ephane Demri Modal Logics with Presburger Constraints
Introduction Extended modal logic ( EML ) Presburger constraints Space upper bounds Regularity constraints EML vs other logics Motivations Conclusion Regularity constraints ◮ Regularity constraints in graphs: sequences of states/transitions belong to some regular language. ◮ Extended Temporal Logic (ETL). [Wolper, IC 83] ◮ Again, constraints for XML documents. [Dal Zilio & Lugiez, RTA 03, Seidl et al, ICALP 04] ◮ Propositional dynamic logic PDL. [Pratt 76] ◮ Description logics (ALCReg). [Baader, IJCAI 91] St´ ephane Demri Modal Logics with Presburger Constraints
Introduction Extended modal logic ( EML ) Presburger constraints Space upper bounds Regularity constraints EML vs other logics Motivations Conclusion Presburger and regularity constraints in graphs s s 1 | = φ 1 s 2 | = φ 2 s 3 | = φ 1 ∧ φ 2 s 4 | = φ 1 = ( ♯φ 1 = ♯φ 2 + 1) ∧ φ 1 φ ∗ 2 φ + s | 1 . St´ ephane Demri Modal Logics with Presburger Constraints
Introduction Extended modal logic ( EML ) Presburger constraints Space upper bounds Regularity constraints EML vs other logics Motivations Conclusion Logics that count in pspace ◮ Minimal graded modal logic. [Tobies, CADE 99] ◮ Majority logic. [Pacuit & Salame, KR 04] ◮ Rank-1 modal logics. [Schr¨ oder & Pattinson, LICS 06] ◮ Constraints on sets with cardinalities. [Kuncak & Manette & Rinard, Dagstuhl 05 ] St´ ephane Demri Modal Logics with Presburger Constraints
Introduction Extended modal logic ( EML ) Presburger constraints Space upper bounds Regularity constraints EML vs other logics Motivations Conclusion Other logics that count ◮ Graded µ -calculus. [Kupferman & Sattler & Vardi, CADE 02] ◮ Logic with fixpoint operators. [Seidl et al, ICALP 04] ◮ Sheaves logic is decidable. [Dal Zilio & Lugiez, RTA 03] ◮ CTL ⋆ with counting. [Moller & Rabinovich, IC 03] ◮ + jungle of logics: ◮ MSO logics. [Seidl & Schwentick & Muscholl, PODS 03] ◮ Spatial logics, description logics (with number restrictions), ◮ FO + counting quantifiers . . . St´ ephane Demri Modal Logics with Presburger Constraints
Introduction Extended modal logic ( EML ) Presburger constraints Space upper bounds Regularity constraints EML vs other logics Motivations Conclusion Goal ◮ To study languages with counting and regularity constraints. ◮ To provide conditions for satisfiability in pspace . ◮ pspace upper bound implies ◮ Presburger constraints in quantifier-free fragment, ◮ no general fixpoint operators. St´ ephane Demri Modal Logics with Presburger Constraints
Introduction Extended modal logic ( EML ) Definition Space upper bounds Simplifications EML vs other logics Conclusion Models Kripke-like structures M = � T , ( R r ) r ∈ Σ , ( < r s ) s ∈ T , l � ◮ T : set of nodes. ◮ l : labelling of nodes by propositional variables. ◮ R r : binary relation on T with finite-branching. ◮ < r s : total ordering on the set of R r successors of node s . St´ ephane Demri Modal Logics with Presburger Constraints
Introduction Extended modal logic ( EML ) Definition Space upper bounds Simplifications EML vs other logics Conclusion Syntax ::= p | ¬ φ | φ ∧ φ | t ∼ b | t ≡ k c | A ( r , φ 1 , . . . , φ n ) φ t ::= a × ♯ r φ | t + a × ♯ r φ ( a ∈ Z , b ∈ N , k , c ∈ N ) ◮ p : propositional variable, r : relation symbol. ◮ A : finite-state automaton. ◮ ∼∈ { <, >, = } . ◮ ♯ r φ : number of nodes accessible by r satisfying φ . BINARY ENCODING OF INTEGERS St´ ephane Demri Modal Logics with Presburger Constraints
Introduction Extended modal logic ( EML ) Definition Space upper bounds Simplifications EML vs other logics Conclusion The satisfaction relation def i a i R ♯ ◮ M , s | = � ⇔ � i a i ♯ r i φ i ∼ b r i ,φ i ( s ) ∼ b with r i ,φ i ( s ) = card ( { s ′ ∈ T : � s , s ′ � ∈ R r i , M , s ′ | R ♯ = φ i } ) . def ◮ M , s | = A ( r , φ 1 , . . . , φ n ) ⇔ there is a i 1 · · · a i α ∈ L ( A ) s.t. ◮ R r ( s ) = s 1 < . . . < s α , ◮ for every j ∈ { 1 , . . . , α } , M , s j | = φ i j . St´ ephane Demri Modal Logics with Presburger Constraints
Introduction Extended modal logic ( EML ) Definition Space upper bounds Simplifications EML vs other logics Conclusion Examples ◮ ♦ φ ≈ ♯ r φ � 1 � φ ≈ ♯ r ¬ φ = 0 ♦ � n φ ≈ ♯ r φ � n . ◮ There are as many words accessible by r 1 satisfying φ 1 as worlds accessible by r 2 satisfying φ 2 : ♯ r 1 φ 1 = ♯ r 2 φ 2 . = A ( φ 1 , φ 2 ) with L ( A ) = a b ∗ a | s s 1 | = φ 1 s 2 | = φ 2 s 3 | = φ 2 s 4 | = φ 1 St´ ephane Demri Modal Logics with Presburger Constraints
Introduction Extended modal logic ( EML ) Definition Space upper bounds Simplifications EML vs other logics Conclusion Tree model property Lemma: φ has a model iff φ has a (unranked, ordered) tree model. ′ r s ) s ∈ T , l � → � T ′ , ( S r ) r ∈ Σ , ( < s ) s ∈ T ′ , l ′ � Unfold � T , ( R r ) r ∈ Σ , ( < r ◮ T ′ : set of finite sequences s r 1 s 1 . . . r k s k . ◮ ( s r 1 s 1 . . . r n s n ) S r ( s r 1 s 1 . . . r n s n r n +1 s n +1 ) iff � s n , s n +1 � ∈ R r and r = r n +1 . ◮ l ′ ( s r 1 s 1 . . . r n s n ) = l ( s n ). ′ r ◮ ordering < s ′ on sequences induced by orderings on last elements. St´ ephane Demri Modal Logics with Presburger Constraints
Introduction Extended modal logic ( EML ) Definition Space upper bounds Simplifications EML vs other logics Conclusion Reduction to one relation Lemme There is logspace reduction from EML satisfiability into EML satisfiability restricted to a single relation symbol. r 1 r 2 ⇐ p 1 p 2 ⇒ r 2 r 3 r 1 r 2 p 2 p 3 p 1 p 2 φ is transformed into ψ uni ∧ ψ subst : ◮ ψ uni states that a unique p i is true at each (non root) node, ◮ ψ subst is obtained from φ by replacing ◮ # r i ϕ by #( ϕ ∧ p i ) ◮ A ( r i , ϕ 1 , . . . ) by A ′ ( r , ¬ p i , ϕ 1 ∧ p i , . . . ) ( A ′ variant of A ). St´ ephane Demri Modal Logics with Presburger Constraints
Introduction Consistent sets Extended modal logic ( EML ) Non-deterministic algorithm Space upper bounds Results EML vs other logics Boundedness Lemma Conclusion n -maximally consistent sets ◮ Fischer/Ladner closure cl ( φ ): ◮ closure under subformulae and negation, ◮ t ∼ b ∈ cl ( φ ) implies t ∼ ′ b ∈ cl ( φ ) for ∼ ′ ∈ { <, >, = } , ◮ t ≡ k c ∈ cl ( φ ) implies t ≡ K c ′ ∈ cl ( φ ) for c ′ ∈ { 0 , . . . , K − 1 } ( K lcm). ◮ Relative closure of level n : cl ( n , φ ) ◮ cl (0 , φ ) = cl ( φ ), ◮ if # ψ occurs in cl ( n , φ ) then ψ ∈ cl ( n + 1 , φ ), ◮ similar condition for automata-based subformulae. ◮ Introduction of n -maximal consistency (extension from Hintikka sets). St´ ephane Demri Modal Logics with Presburger Constraints
Introduction Consistent sets Extended modal logic ( EML ) Non-deterministic algorithm Space upper bounds Results EML vs other logics Boundedness Lemma Conclusion M -bounded models ◮ nb ( d ): number of d -maximal consistent sets (wrt φ ). ◮ nb ( d ) is exponential in | φ | . ◮ �M , s � is M -bounded for φ : root depth d · · · · · · < M × nb ( d + 1) 1 2 St´ ephane Demri Modal Logics with Presburger Constraints
Introduction Consistent sets Extended modal logic ( EML ) Non-deterministic algorithm Space upper bounds Results EML vs other logics Boundedness Lemma Conclusion Ladner-like algorithms ◮ Modal logic K in pspace . [Ladner, SIAM 77] ◮ Nondeterministic algorithm that do not rely on automata, tableaux, sequents etc. See also [Spaan, 93] ◮ Correctness is partly based on the tree model property. St´ ephane Demri Modal Logics with Presburger Constraints
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