Almost linear Büchi automata

Babiak, Tomáš; Řehák, Vojtěch; Strejček, Jan

doi:10.1017/s0960129511000399

Cited by 3 publications

(5 citation statements)

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“…Handling regular constraints: It is difficult to extend Theorem 5 to the case with regular constraints because they may introduce nestings of cycles (which breaks the flat control structure) even for regular-oriented word equation. However, we can show that restricting to regular constraints given by 1-weak NFA [2] (i.e. a dag of SCCs, each with at most one state) preserves the flat control structure.…”

Section: An Extension To Flat Control Structures and An Acceleration ...mentioning

confidence: 97%

“…which contradicts minimality of the cycle π . Therefore, we have n = m. Since f n = f n = f , by the equations in (1) and (2) we have that f i = f i for all i ∈ {0, . .…”

Section: 3mentioning

confidence: 99%

“…a dag of SCCs, each with at most one state) preserves the flat control structure. The class of 1-weak automata is in fact quite powerful, e.g., when considered as recognisers of languages of ω-words, they capture the subclass of LTL with operators F and G [2]. They have also been used to obtain a decidable extension of infinite-state concurrent systems in term rewriting systems, e.g., see [20,31].…”

Section: An Extension To Flat Control Structures and An Acceleration ...mentioning

confidence: 99%

“…Theorem 11 gives rise to a simple and sound (but not complete) technique for solving quadratic word equations with length constraints: given a quadratic word equation (E, S) with regular constraints, if the counter system C(E, S) is flat, each of whose simple cycle is 1-variable-reducing with unary Presburger guards, then apply the decision procedure from Theorem 11. In this section, we show completeness of this method for the class of regular-oriented word equations recently defined in [9], which can be extended with regular constraints given as 1-weak NFA [2]. A word equation is regular if each variable x ∈ V occurs at most once on each side of the equation.…”

mentioning

confidence: 99%

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Quadratic Word Equations with Length Constraints, Counter Systems, and Presburger Arithmetic with Divisibility

Lin

Majumdar

2018

Lecture Notes in Computer Science

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Word equations are a crucial element in the theoretical foundation of constraint solving over strings, which have received a lot of attention in recent years. A word equation relates two words over string variables and constants. Its solution amounts to a function mapping variables to constant strings that equate the left and right hand sides of the equation. While the problem of solving word equations is decidable, the decidability of the problem of solving a word equation with a length constraint (i.e., a constraint relating the lengths of words in the word equation) has remained a long-standing open problem. In this paper, we focus on the subclass of quadratic word equations, i.e., in which each variable occurs at most twice. We first show that the length abstractions of solutions to quadratic word equations are in general not Presburger-definable. We then describe a class of counter systems with Presburger transition relations which capture the length abstraction of a quadratic word equation with regular constraints. We provide an encoding of the effect of a simple loop of the counter systems in the theory of existential Presburger Arithmetic with divisibility (PAD). Since PAD is decidable, we get a decision procedure for quadratic words equations with length constraints for which the associated counter system is flat (i.e., all nodes belong to at most one cycle). We show a decidability result (in fact, also an NP algorithm with a PAD oracle) for a recently proposed NP-complete fragment of word equations called regular-oriented word equations, together with length constraints. Decidability holds when the constraints are additionally extended with regular constraints with a 1-weak control structure.

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Section: An Extension To Flat Control Structures and An Acceleration ...mentioning

confidence: 97%

“…which contradicts minimality of the cycle π . Therefore, we have n = m. Since f n = f n = f , by the equations in (1) and (2) we have that f i = f i for all i ∈ {0, . .…”

Section: 3mentioning

confidence: 99%

Section: An Extension To Flat Control Structures and An Acceleration ...mentioning

confidence: 99%

mentioning

confidence: 99%

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Quadratic Word Equations with Length Constraints, Counter Systems, and Presburger Arithmetic with Divisibility

Lin

Majumdar

2018

Lecture Notes in Computer Science

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“…Theorem 5.2 gives rise to a simple and sound (but not complete) technique for solving quadratic word equations with length constraints: given a quadratic word equation (E, S) with regular constraints, if the counter system C(E, S) is flat, each of whose simple cycle is 1-variable-reducing with unary Presburger guards, then apply the decision procedure from Theorem 5.2. In this section, we show completeness of this method for the class of regular-oriented word equations recently defined in [DMN17], which can be extended with regular constraints given as 1-weak NFA [BRS12]. A word equation is regular if each variable x ∈ V occurs at most once on each side of the equation.…”

Section: Solving Quadratic Word Equationsmentioning

confidence: 99%

Quadratic Word Equations with Length Constraints, Counter Systems, and Presburger Arithmetic with Divisibility

Lin¹,

Majumdar²

2021

Logical Methods in Computer Science

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Word equations are a crucial element in the theoretical foundation of constraint solving over strings. A word equation relates two words over string variables and constants. Its solution amounts to a function mapping variables to constant strings that equate the left and right hand sides of the equation. While the problem of solving word equations is decidable, the decidability of the problem of solving a word equation with a length constraint (i.e., a constraint relating the lengths of words in the word equation) has remained a long-standing open problem. We focus on the subclass of quadratic word equations, i.e., in which each variable occurs at most twice. We first show that the length abstractions of solutions to quadratic word equations are in general not Presburger-definable. We then describe a class of counter systems with Presburger transition relations which capture the length abstraction of a quadratic word equation with regular constraints. We provide an encoding of the effect of a simple loop of the counter systems in the existential theory of Presburger Arithmetic with divisibility (PAD). Since PAD is decidable (NP-hard and is in NEXP), we obtain a decision procedure for quadratic words equations with length constraints for which the associated counter system is flat (i.e., all nodes belong to at most one cycle). In particular, we show a decidability result (in fact, also an NP algorithm with a PAD oracle) for a recently proposed NP-complete fragment of word equations called regular-oriented word equations, when augmented with length constraints. We extend this decidability result (in fact, with a complexity upper bound of PSPACE with a PAD oracle) in the presence of regular constraints.

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Refining the Process Rewrite Systems Hierarchy via Ground Tree Rewrite Systems

Göller

Lin

2014

ACM Trans. Comput. Logic

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Abstract. In his seminal paper, R. Mayr introduced the well-known Process Rewrite Systems (PRS) hierarchy, which contains many well-studied classes of infinite systems including pushdown systems, Petri nets and PA-processes. A seperate development in the term rewriting community introduced the notion of Ground Tree Rewrite Systems (GTRS), which is a model that strictly extends pushdown systems while still enjoying desirable decidable properties. There have been striking similarities between the verification problems that have been shown decidable (and undecidable) over GTRS and over models in the PRS hierarchy such as PA and PAD processes. It is open to what extent PRS and GTRS are connected in terms of their expressive power. In this paper we pinpoint the exact connection between GTRS and models in the PRS hierarchy in terms of their expressive power with respect to strong, weak, and branching bisimulation. Among others, this connection allows us to give new insights into the decidability results for subclasses of PRS, e.g., simpler proofs of known decidability results of verifications problems on PAD.

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Almost linear Büchi automata

Cited by 3 publications

References 18 publications

Quadratic Word Equations with Length Constraints, Counter Systems, and Presburger Arithmetic with Divisibility

Quadratic Word Equations with Length Constraints, Counter Systems, and Presburger Arithmetic with Divisibility

Quadratic Word Equations with Length Constraints, Counter Systems, and Presburger Arithmetic with Divisibility

Refining the Process Rewrite Systems Hierarchy via Ground Tree Rewrite Systems

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