On Proving Termination of Constrained Term Rewrite Systems by Eliminating Edges from Dependency Graphs

Tsubasa, Sakata; Nishida, Naoki; Sakabe, Toshiki

doi:10.1007/978-3-642-22531-4_9

Cited by 10 publications

(27 citation statements)

References 25 publications

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“…On the other hand, RI-based methods are procedures within RI frameworks to apply inference rules under specified strategies. In recent years, various RI-based methods for constrained rewriting (see, e.g., constrained TRSs [9,19], conditional and constrained TRSs [2], Z-TRSs [6], and logically constrained TRSs [11]) have been developed [2,20,6,12,8]. Constrained systems have built-in semantics for some function and predicate symbols and have been used as a computation model of not only functional but also imperative programs [4,7,9,5,21,12,8].…”

Section: Introductionmentioning

confidence: 99%

Transforming Proof Tableaux of Hoare Logic into Inference Sequences of Rewriting Induction

Mizutani

Nishida

2018

Electron. Proc. Theor. Comput. Sci.

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A proof tableau of Hoare logic is an annotated program with pre-and post-conditions, which corresponds to an inference tree of Hoare logic. In this paper, we show that a proof tableau for partial correctness can be transformed into an inference sequence of rewriting induction for constrained rewriting. We also show that the resulting sequence is a valid proof for an inductive theorem corresponding to the Hoare triple if the constrained rewriting system obtained from the program is terminating. Such a valid proof with termination of the constrained rewriting system implies total correctness of the program w.r.t. the Hoare triple. The transformation enables us to apply techniques for proving termination of constrained rewriting to proving total correctness of programs together with proof tableaux for partial correctness.Transforming Proof Tableaux of Hoare Logic into Inference Sequences of Rewriting Induction corresponding to the Hoare triple for the proof tableau if the LCTRS obtained from the program is terminating.Given a while program P and a proof tableau T P of a Hoare triple {ϕ P } P {ψ P } for partial correctness, we proceed as follows:1. We transform P into an equivalent LCTRS R P , and we prove termination of the LCTRS R P .2. We prepare rewrite rules R check to verify the post-condition ψ P in the proof tableau.3. We prepare a constrained equation e P corresponding to the Hoare triple {ϕ P } P {ψ P }.4. Starting with the equation e P , we transform the proof tableau into an inference sequence ({e P }, / 0) RI · · · RI ( / 0, H) of RI in a top-down fashion, where we do not prove termination in constructing the inference sequence of RI.

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Section: Introductionmentioning

confidence: 99%

Transforming Proof Tableaux of Hoare Logic into Inference Sequences of Rewriting Induction

Mizutani

Nishida

2018

Electron. Proc. Theor. Comput. Sci.

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“…[10]). In verifying programs with comparison operators over the integers via term rewriting, constrained rewriting is very useful to avoid very complicated rewrite rules for the comparison operators, and various formalizations of constrained rewriting have been proposed: constrained TRSs [11,4,24,23] (e.g., membership conditional TRSs [25]), constrained equational systems (CESs, for short) [5], integer TRSs (ITRSs, for short) [9], PA-based TRSs (Z-TRSs) [6] (simplified variants of CESs), and logically constrained TRSs (LCTRSs, for short) [17,18].…”

Section: Introductionmentioning

confidence: 99%

“…the survey of Zantema [26]). At present, the dependency pair (DP) method [2] and the DP framework [14] are key fundamentals for proving termination of TRSs, and they have been extended to several kinds of rewrite systems [5,1,6,9,23,20,10]. In the DP framework, termination problems are reduced to finiteness of DP problems which consist of sets of dependency pairs and rewrite rules.…”

Section: Introductionmentioning

confidence: 99%

“…We prove finiteness by applying sound DP processors to an input DP problem and then by decomposing the DP problem to smaller ones in the sense that all the DP sets of output DP problems are strict subsets of the DP set of the input problem. In the DP frameworks for constrained rewriting [5,23], the DP processors based on polynomial interpretations (the PI-based processors, for short) decompose a given DP problem by using a polynomial interpretation Pol that transforms dependency chains into bounded decreasing sequences of integers-roughly speaking, a dependency pair s → t [ [ ϕ ] ] is removed from the given problem if the integer arithmetic formula ϕ ⇒ Pol(s) > Pol(t) is valid. The processor in [23] can be considered a simplified version of that in [5] in the sense that for efficiency, Pol drops reducible positions-arguments of marked symbols, which may contain an uninterpreted function symbol when a dependency pair is instantiated-and then the rules in the given system can be ignored in applying the PI-based processor.…”

Section: Introductionmentioning

confidence: 99%

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Transforming Dependency Chains of Constrained TRSs into Bounded Monotone Sequences of Integers

Tomohiro

Nishida

Sakai

et al. 2018

Electron. Proc. Theor. Comput. Sci.

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In the dependency pair framework for proving termination of rewriting systems, polynomial interpretations are used to transform dependency chains into bounded decreasing sequences of integers, and they play an important role for the success of proving termination, especially for constrained rewriting systems. In this paper, we show sufficient conditions of linear polynomial interpretations for transforming dependency chains into bounded monotone (i.e., decreasing or increasing) sequences of integers. Such polynomial interpretations transform rewrite sequences of the original system into decreasing or increasing sequences independently of the transformation of dependency chains. When we transform rewrite sequences into increasing sequences, polynomial interpretations have non-positive coefficients for reducible positions of marked function symbols. We propose four DP processors parameterized by transforming dependency chains and rewrite sequences into either decreasing or increasing sequences of integers, respectively. We show that such polynomial interpretations make us succeed in proving termination of the McCarthy 91 function over the integers.

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“…Rewriting modulo SMT techniques can then be applied to increase the power of rewrite-based equational reasoning for (Σ, E) such as, for instance, inductive theorem proving [29,39,40], termination checking [28,61], and procedural verification [41]. However, the full power of rewriting modulo SMT, including its model checking capabilities, can be better exploited when applied to concurrent open systems.…”

Section: Introductionmentioning

confidence: 99%

Rewriting modulo SMT and open system analysis

Rocha

Meseguer

Muñoz

2017

Journal of Logical and Algebraic Methods in Programming

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Abstract. This paper proposes rewriting modulo SMT, a new technique that combines the power of SMT solving, rewriting modulo theories, and model checking. Rewriting modulo SMT is ideally suited to model and analyze reachability properties of infinite-state open systems, i.e., systems that interact with a nondeterministic environment. Such systems exhibit both internal nondeterminism, which is proper to the system, and external nondeterminism, which is due to the environment. In a reflective formalism, such as rewriting logic, rewriting modulo SMT can be reduced to standard rewriting. Hence, rewriting modulo SMT naturally extends rewriting-based reachability analysis techniques, which are available for closed systems, to open systems. The proposed technique is illustrated with the formal analysis of: (i) a real-time system that is beyond the scope of timed-automata methods and (ii) automatic detection of reachability violations in a synchronous language developed to support autonomous spacecraft operations.

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On Proving Termination of Constrained Term Rewrite Systems by Eliminating Edges from Dependency Graphs

Cited by 10 publications

References 25 publications

Transforming Proof Tableaux of Hoare Logic into Inference Sequences of Rewriting Induction

Transforming Proof Tableaux of Hoare Logic into Inference Sequences of Rewriting Induction

Transforming Dependency Chains of Constrained TRSs into Bounded Monotone Sequences of Integers

Rewriting modulo SMT and open system analysis

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