Visibly Pushdown Languages and Term Rewriting

Chabin, Jacques; Réty, Pierre

doi:10.1007/978-3-540-74621-8_17

Cited by 5 publications

(10 citation statements)

References 9 publications

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“…Future Work. Regular relations can be further extended to other tree language classes (see, e.g., [9,10,20,25,33,39]). Another intriguing prospect is to automate the inference of inductive invariants represented in first-order languages with regular language membership predicates [15].…”

Section: Comparison Of the Expressive Powermentioning

confidence: 99%

Beyond the elementary representations of program invariants over algebraic data types

Kostyukov¹,

Mordvinov²,

Fedyukovich

2021

Proceedings of the 42nd ACM SIGPLAN International Conference on Programming Language Design and Implementation

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First-order logic is a natural way of expressing properties of computation. It is traditionally used in various program logics for expressing the correctness properties and certificates. Although such representations are expressive for some theories, they fail to express many interesting properties of algebraic data types (ADTs). In this paper, we explore three different approaches to represent program invariants of ADTmanipulating programs: tree automata, and first-order formulas with or without size constraints. We compare the expressive power of these representations and prove the negative definability of both first-order representations using the pumping lemmas. We present an approach to automatically infer program invariants of ADT-manipulating programs by a reduction to a finite model finder. The implementation called RInGen has been evaluated against state-of-the-art invariant synthesizers and has been experimentally shown to be competitive. In particular, program invariants represented by automata are capable of expressing more complex properties of computation and their automatic construction is often less expensive.

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Section: Comparison Of the Expressive Powermentioning

confidence: 99%

Beyond the elementary representations of program invariants over algebraic data types

Kostyukov¹,

Mordvinov²,

Fedyukovich

2021

Proceedings of the 42nd ACM SIGPLAN International Conference on Programming Language Design and Implementation

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“…The problem of Theorem 10, membership modulo, becomes decidable with some further syntactical restrictions on R based on the theory of visibly pushdown automata (VPA) [15]. VPA define a subset of context-free languages closed under intersection, and were generalized to tree recognizers in [16,17]. The idea in these works is that the signature Σ is partitioned into Σ = Σ c ⊎ Σ r ⊎ Σ ℓ and the operation performed by the VPA on the stack depends on the current symbol in input: if it is a call symbol of Σ c , the VPA can only do a push, for a return symbol of Σ r it can do a pop and it must leave the stack untouched for a local symbol of Σ ℓ .…”

Section: Rewrite Closurementioning

confidence: 99%

“…In [16], Chabin and Rety show that the class of visibly pushdown tree automata (VPTA) languages is closed under rewriting with so called linear contextfree visibly TRS. We use a similar definition in order to characterize a class of TRS for which membership modulo is decidable.…”

Section: Rewrite Closurementioning

confidence: 99%

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Rigid Tree Automata

Jacquemard

Klay²,

Vacher

2009

Language and Automata Theory and Applications

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International audienceWe introduce the class of Rigid Tree Automata (RTA), an extension of standard bottom-up automata on ranked trees with distinguished states called rigid. Rigid states define a restriction on the computation of RTA on trees: RTA can test for equality in subtrees reaching the same rigid state. RTA are able to perform local and global tests of equality between subtrees, non-linear tree pattern matching, and restricted disequality tests as well. Properties like determinism, boolean closure, and several decision problems are studied in detail. In particular, the emptiness problem is shown decidable in linear time for RTA whereas membership of a given tree to the language of a given RTA is NP-complete. Our main result is the decidability of whether a given tree belongs to the rewrite closure of a RTA language under a restricted family of term rewriting systems, whereas this closure is not a RTA language. This result, one of the first on rewrite closure of languages of tree automata with constraints, is enabling the extension of model checking procedures based on finite tree automata techniques. Finally, a comparison of RTA with several classes of tree automata with local and global equality tests, and with dag automata is also provided

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“…For the RMC, we use the fact that the intersection of the languages of a CF tree grammar G and a TA A is the language of a CF tree grammar whose size is the product of the respective sizes of G and A. ⊓ ⊔ Note however that joinability, the problem to decide, given two ground terms s, t ∈ T (Σ) and a TRS R, whether there exists u ∈ T (Σ) such that s − − → * R u ← − − * R t, is undecidable for CF TRS [5], because the emptiness of intersection is undecidable for CF tree languages.…”

Section: Cf and Cs Tree Languages And Trsmentioning

confidence: 99%

Controlled Term Rewriting

Jacquemard

Kojima

Sakai

2011

Frontiers of Combining Systems

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Abstract. Motivated by the problem of verification of imperative tree transformation programs, we study the combination, called controlled term rewriting systems (CntTRS), of term rewriting rules with constraints selecting the possible rewrite positions. These constraints are specified, for each rewrite rule, by a selection automaton which defines a set of positions in a term based on tree automata computations. We show that reachability is PSPACE-complete for so-called monotonic CntTRS, such that the size of every left-hand-side of every rewrite rule is larger or equal to the size of the corresponding right-hand-side, and also for the class of context-free non-collapsing CntTRS, which transform Context-Free (CF) tree language into CF tree languages. When allowing size-reducing rules, reachability becomes undecidable, even for flat CntTRS (both sides of rewrite rules are of depth at most one) when restricting to words (i.e. function symbols have arity at most one), and for ground CntTRS (rewrite rules have no variables). We also consider a restricted version of the control such that a position is selected if the sequence of symbols on the path from that position to the root of the tree belongs to a given regular language. This restriction enables decision results in the above cases.

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Visibly Pushdown Languages and Term Rewriting

Cited by 5 publications

References 9 publications

Beyond the elementary representations of program invariants over algebraic data types

Beyond the elementary representations of program invariants over algebraic data types

Rigid Tree Automata

Controlled Term Rewriting

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