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
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