Analysis of Rewriting-Based Systems as First-Order Theories

Lucas, Salvador

doi:10.1007/978-3-319-94460-9_11

Cited by 4 publications

(21 citation statements)

References 25 publications

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“…In [14], some answers to these questions are given: a sentence ϕ which is an Existentially Closed Boolean Combination of Atoms (ECBCA for short) does not hold in the initial model I S of a theory S consisting of a set of ground atoms if we find a model A of S ∪ {¬ϕ} [14,Corollary 2]. This is useful in program analysis when considering programs P that are given a theory P representing its operational semantics so that the execution of P is described as a set I P of (ground) atoms A which can be proved from P (i.e., I P is the initial model of P in the usual first-order sense; in the following, we often refer to it as its canonical model [11,Section 1.5]).…”

Section: Introductionmentioning

confidence: 99%

“…The next sentences describe the propagation of rewriting steps to (arguments of) symbols s, add and mul. The second column describes the rules of R. More details can be found in [14,Section 4]. In the initial or least Herbrand model I R of R, → and → * are interpreted as the sets (→) IR and (→ * ) IR of all pairs (s, t) of ground terms s and t such that s → R t and s → * R t, respectively.…”

Section: Introductionmentioning

confidence: 99%

“…Our approach in [14] relies on the notion of preservation of a formula under homomorphisms h between interpretations. Roughly speaking, a homomorphism [14] rely on this fact. In this paper we extend [14] to deal with more general program properties.…”

Section: Introductionmentioning

confidence: 99%

“…Roughly speaking, a homomorphism [14] rely on this fact. In this paper we extend [14] to deal with more general program properties. Homomorphisms preserve other first-order sentences if further requirements are imposed: (i) positive sentences (where connective '¬' is absent) are preserved under surjective homomorphisms and (ii) arbitrary sentences are preserved under embeddings [11,Theorem 2.4.3].…”

Section: Introductionmentioning

confidence: 99%

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Proving Program Properties as First-Order Satisfiability

Lucas

2019

Logic-Based Program Synthesis and Transformation

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Program semantics can often be expressed as a (many-sorted) first-order theory S, and program properties as sentences ϕ which are intended to hold in the canonical model of such a theory, which is often incomputable. Recently, we have shown that properties ϕ expressed as the existential closure of a boolean combination of atoms can be disproved by just finding a model of S and the negation ¬ϕ of ϕ. Furthermore, this idea works quite well in practice due to the existence of powerful tools for the automatic generation of models for (many-sorted) first-order theories. In this paper we extend our previous results to arbitrary properties, expressed as sentences without any special restriction. Consequently, one can prove a program property ϕ by just finding a model of an appropriate theory (including S and possibly something else) and an appropriate first-order formula related to ϕ. Beyond its possible theoretical interest, we show that our results can also be of practical use in several respects.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Proving Program Properties as First-Order Satisfiability

Lucas

2019

Logic-Based Program Synthesis and Transformation

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“…In fact, confluence provers for CTRSs, ConCon [28], CO3 [20], and CoScart [9], based on e.g., transformations of CTRSs into TRSs or reachability analysis for infeasibility of conditional critical pairs, failed to prove confluence of R 1 (see Confluence Competition 2016, 2017, and 2018, 1 327.trs). In addition, a semantic approach in [16,15] cannot prove confluence of R 1 using AGES [12], a tool for generating logical models of order-sorted first-order theories-non-existence of t 1 ,t 2 above cannot be proved via its web interface with default parameters. Timbuk 3.2 [7], which is based on tree automata techniques [6], cannot prove infeasibility of x < y true, y < x true w.r.t.…”

Section: Introductionmentioning

confidence: 99%

On Transforming Narrowing Trees into Regular Tree Grammars Generating Ranges of Substitutions

Nishida

Maeda

2019

Electron. Proc. Theor. Comput. Sci.

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The grammar representation of a narrowing tree for a syntactically deterministic conditional term rewriting system and a pair of terms is a regular tree grammar that generates expressions for substitutions obtained by all possible innermost-narrowing derivations that start with the pair and end with particular non-narrowable terms. In this paper, under a certain syntactic condition, we show a transformation of the grammar representation of a narrowing tree into another regular tree grammar that overapproximately generates the ranges of ground substitutions generated by the grammar representation. In our previous work, such a transformation is restricted to the ranges w.r.t. a given single variable, and thus, the usefulness is limited. We extend the previous transformation by representing the range of a ground substitution as a tuple of terms, which is obtained by the coding for finite trees.We show a precise definition of the transformation and prove that the language of the transformed regular tree grammar is an overapproximation of the ranges of ground substitutions generated by the grammar representation. We leave an experiment to evaluate the usefulness of the transformation as future work.

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Automatically Proving and Disproving Feasibility Conditions

Gutiérrez

Lucas

2020

Automated Reasoning

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In the realm of term rewriting, given terms s and t, a reachability condition s → * t is called feasible if there is a substitution σ such that σ(s) rewrites into σ(t) in zero or more steps; otherwise, it is called infeasible. Checking infeasibility of (sequences of) reachability conditions is important in the analysis of computational properties of rewrite systems like confluence or (operational) termination. In this paper, we generalize this notion of feasibility to arbitrary n-ary relations on terms defined by first-order theories. In this way, properties of computational systems whose operational semantics can be given as a first-order theory can be investigated. We introduce a framework for proving feasibility/infeasibility, and a new tool, infChecker, which implements it.

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Analysis of Rewriting-Based Systems as First-Order Theories

Cited by 4 publications

References 25 publications

Proving Program Properties as First-Order Satisfiability

Proving Program Properties as First-Order Satisfiability

On Transforming Narrowing Trees into Regular Tree Grammars Generating Ranges of Substitutions

Automatically Proving and Disproving Feasibility Conditions

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