The Effects of Bounding Syntactic Resources on Presburger LTL - PowerPoint PPT Presentation

The Effects of Bounding Syntactic Resources on Presburger LTL (extended abstract) S. Demri R. Gascon LSV, ENS Cachan, CNRS, INRIA TIME’07, June 28–30, 2007

Motivations Presburger LTL Contribution A pspace -complete problem An example of undecidable problem Conclusion Counter systems ◮ Verification of infinite-state systems by model-checking. ◮ Ubiquity of counter systems (CS) ◮ Embedded systems/protocols, Petri nets, . . . ◮ Programs with pointer variables. [Bardin et al, AVIS 06; Bouajjani et al, CAV 06] ◮ Broadcast protocols. [Leroux & Finkel, FSTTCS 02] ◮ Logics for data words. [Boja´ nczyk et al, LICS 06] ◮ (High) undecidability ◮ Checking safety properties for CS is undecidable. ◮ Checking liveness properties for CS is Σ 1 1 -hard. S. Demri, R. Gascon The Effects of Bounding Syntactic Resources on Presburger LTL

Motivations Presburger LTL Contribution A pspace -complete problem An example of undecidable problem Conclusion Taming counter systems ◮ Classes with decidable reachability problems ◮ Reversal-bounded CS. [Ibarra, JACM 78] ◮ Flat relational CS. [Comon & Jurski, CAV 98] ◮ Flat linear CS. [Boigelot, PhD 98; Finkel & Leroux, FSTTCS 02] ◮ Petri nets. [Kosaraju, STOC 82] ◮ Decision procedures ◮ Translation into Presburger arithmetic. [Ibarra, JACM 78, Comon & Jurski, CAV 98] ◮ Well-structured transition systems. [Finkel & Schnoebelen, TCS 01] ◮ Tools: Fast , Lash , TReX , . . . S. Demri, R. Gascon The Effects of Bounding Syntactic Resources on Presburger LTL

Motivations Presburger LTL Contribution Language A pspace -complete problem Problems An example of undecidable problem Conclusion Presburger arithmetic ◮ Decision ◮ First-order theory of � Z , 0 , + � . ◮ Decidability shown in [Presburger 29]. ◮ Quantifier elimination in presence of modulo constraints. ◮ Satisfiability in 3exptime . S. Demri, R. Gascon The Effects of Bounding Syntactic Resources on Presburger LTL

Motivations Presburger LTL Contribution Language A pspace -complete problem Problems An example of undecidable problem Conclusion Presburger arithmetic ◮ Decision ◮ First-order theory of � Z , 0 , + � . ◮ Decidability shown in [Presburger 29]. ◮ Quantifier elimination in presence of modulo constraints. ◮ Satisfiability in 3exptime . ◮ Fragments ◮ DL : E ::= x ∼ y + d | x ∼ d | E ∧ E | ¬ E . ( d ∈ Z , ∼∈ { <, >, = } ). ◮ DL + : DL + x ≡ k c , x ≡ k y + c ( c , k ∈ N ). ◮ QFP : E ::= � i ∈ I a i x i ∼ d | � i ∈ I a i x i ≡ k c | E ∧ E | ¬ E . ( a i ∈ Z ) S. Demri, R. Gascon The Effects of Bounding Syntactic Resources on Presburger LTL

Motivations Presburger LTL Contribution Language A pspace -complete problem Problems An example of undecidable problem Conclusion Syntax for CLTL ( L ) ◮ L is a fragment among DL , DL + , QFP . ◮ Formulae: φ ::= E [ x 1 ← X l 1 x j 1 , . . . , x n ← X l n x j n ] | φ ∧ φ | ¬ φ | X φ | φ U φ ( E ∈ L ) i times � �� � XX · · · X x interpreted as the value of x at the i th next ◮ position. ◮ Definitions ◮ One-step constraint: l 1 , . . . , l n ≤ 1. ◮ X-length of φ : maximal i such that X i x occurs in φ . S. Demri, R. Gascon The Effects of Bounding Syntactic Resources on Presburger LTL

Motivations Presburger LTL Contribution Language A pspace -complete problem Problems An example of undecidable problem Conclusion Semantics for Presburger LTL ◮ Models: ω -sequences of valuations of the form VAR → Z . S. Demri, R. Gascon The Effects of Bounding Syntactic Resources on Presburger LTL

Motivations Presburger LTL Contribution Language A pspace -complete problem Problems An example of undecidable problem Conclusion Semantics for Presburger LTL ◮ Models: ω -sequences of valuations of the form VAR → Z . ◮ Satisfaction relation: ◮ σ, i | = E [ x 1 ← X l 1 x j 1 ,..., x n ← X l n x j n ] iff ( σ ( i + l 1 )( x j 1 ) ,..., σ ( i + l n )( x j n )) | = E in PA, ◮ σ, i | = X φ iff σ, i + 1 | = φ , = φ U φ ′ iff there is j ≥ i such that σ, j | = φ ′ and for every ◮ σ, i | i ≤ k < j , we have σ, k | = φ . S. Demri, R. Gascon The Effects of Bounding Syntactic Resources on Presburger LTL

Motivations Presburger LTL Contribution Language A pspace -complete problem Problems An example of undecidable problem Conclusion Semantics for Presburger LTL ◮ Models: ω -sequences of valuations of the form VAR → Z . ◮ Satisfaction relation: ◮ σ, i | = E [ x 1 ← X l 1 x j 1 ,..., x n ← X l n x j n ] iff ( σ ( i + l 1 )( x j 1 ) ,..., σ ( i + l n )( x j n )) | = E in PA, ◮ σ, i | = X φ iff σ, i + 1 | = φ , = φ U φ ′ iff there is j ≥ i such that σ, j | = φ ′ and for every ◮ σ, i | i ≤ k < j , we have σ, k | = φ . x = X 2 x x = X 3 x S. Demri, R. Gascon The Effects of Bounding Syntactic Resources on Presburger LTL

Motivations Presburger LTL Contribution Language A pspace -complete problem Problems An example of undecidable problem Conclusion Fragments CLTL l k ( L ) ◮ CLTL l k ( L ) is the fragment of CLTL ( L ) with ◮ atomic formulae built from constraints in L , ◮ formulae use variables from { x 1 , . . . , x k } , ◮ the term X i x can occur only if i ≤ l . ◮ Examples ◮ x 1 = X 8 x 2 + 1 belongs to CLTL 8 2 ( DL ), ◮ X 2 x 1 ≡ 4 2 belongs to CLTL 2 1 ( DL + ) ∩ CLTL 2 1 ( QFP ), ◮ XXX(5X x 1 + 2 x 2 ≥ 27) belongs to CLTL 1 2 ( QFP ). S. Demri, R. Gascon The Effects of Bounding Syntactic Resources on Presburger LTL

Motivations Presburger LTL Contribution Language A pspace -complete problem Problems An example of undecidable problem Conclusion k -variable L -automata ◮ Definition: → q ′ for one-step constraint E in L . E ◮ Transitions of the form q − X x > y +1 x =0 ∧ y =0 ⊤ → q ′ , q 0 Examples: q − − − − − − − − − → q , q − → q . ◮ Standard B¨ uchi acceptance condition. ◮ Accepting runs of the form N → Q × Z k . ◮ σ realizes E 0 · E 1 · · · iff for every i , we have σ, i | = E i . S. Demri, R. Gascon The Effects of Bounding Syntactic Resources on Presburger LTL

Motivations Presburger LTL Contribution Language A pspace -complete problem Problems An example of undecidable problem Conclusion k - Z -counter automata ◮ Restriction of k -variable DL -automaton with constraints � � E test i ∧ E update i i ∈{ 1 ... k } i ∈{ 1 ... k } with ◮ E test i ∈ {⊤} ∪ { x i ∼ 0 | ∼∈ { <, >, = , � = }} , ◮ E update i ∈ { X x i = x i + u | u ∈ Z } ◮ Initial values of the counters are zero. ◮ Simple Z -counter automata: updates in { 0 , − 1 , 1 } . S. Demri, R. Gascon The Effects of Bounding Syntactic Resources on Presburger LTL

Motivations Presburger LTL Contribution Language A pspace -complete problem Problems An example of undecidable problem Conclusion Model checking problems ◮ Model-checking CLTL l k ( L ) formulae over a class C of automata: ◮ Input: a k -variable automaton A in C and a formula in CLTL l k ( L ). ◮ Question: Is there a model σ that realizes a word accepted by A and such that σ, 0 | = φ ? ◮ Model-checking CLTL 1 3 ( DL ) over the class of 3- N -automata is Σ 1 1 -complete. [Alur & Henzinger, JACM 94] S. Demri, R. Gascon The Effects of Bounding Syntactic Resources on Presburger LTL

Motivations Presburger LTL Contribution A standard undecidability result A pspace -complete problem Summary An example of undecidable problem Conclusion CLTL 1 3 ( DL ) satisfiability is Σ 1 1 -complete ◮ Reduction from the recurring problem for nondeterministic Minsky machines. ◮ Σ 1 1 -hardness from [Alur & Henzinger, JACM 94]. ◮ The instruction “ n : C 1 := C 1 + 1; goto either n ′ or n ′′ ” is encoded by G( x inst = n ⇒ (X x 1 = x 1 +1 ∧ X x 2 = x 2 ∧ (X x inst = n ′ ∨ X x inst = n ′′ ))) ◮ Recurring condition: GF( x inst = 1). S. Demri, R. Gascon The Effects of Bounding Syntactic Resources on Presburger LTL

Motivations Presburger LTL Contribution A standard undecidability result A pspace -complete problem Summary An example of undecidable problem Conclusion Taxonomy of subproblems ◮ Problems: ◮ satisfiability, ◮ model-checking L -automata, ◮ model-checking Z -counter automata. S. Demri, R. Gascon The Effects of Bounding Syntactic Resources on Presburger LTL

Motivations Presburger LTL Contribution A standard undecidability result A pspace -complete problem Summary An example of undecidable problem Conclusion Taxonomy of subproblems ◮ Problems: ◮ satisfiability, ◮ model-checking L -automata, ◮ model-checking Z -counter automata. ◮ Fragments: DL , DL + , QFP . S. Demri, R. Gascon The Effects of Bounding Syntactic Resources on Presburger LTL

Motivations Presburger LTL Contribution A standard undecidability result A pspace -complete problem Summary An example of undecidable problem Conclusion Taxonomy of subproblems ◮ Problems: ◮ satisfiability, ◮ model-checking L -automata, ◮ model-checking Z -counter automata. ◮ Fragments: DL , DL + , QFP . ◮ Bounding syntactic resources: X-length, number of variables. S. Demri, R. Gascon The Effects of Bounding Syntactic Resources on Presburger LTL