Reducing Total Correctness to Partial Correctness by a Transformation of the Language Semantics

Buruiana, Andrei-Sebastian; Ciobâcă, Ștefan

doi:10.4204/eptcs.289.1

Cited by 4 publications

(4 citation statements)

References 25 publications

(53 reference statements)

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“…The APR framework in this paper originates from Refs. [4], [7], [20], [21], where neither decidability nor disproofs has been discussed. A leading verification example in Ref.…”

Section: On Decidability Of Apr Problemsmentioning

confidence: 99%

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A Sufficient Condition of Logically Constrained Term Rewrite Systems for Decidability of All-path Reachability Problems with Constant Destinations

Kojima,

Nishida

2024

Journal of Information Processing

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An all-path reachability (APR, for short) problem of a logically constrained term rewrite system (LCTRS, for short) is a pair of constrained terms representing state sets. An APR problem is demonically valid if every finite execution path from any state in the first set to an irreducible state includes a state in the second set. We have proposed a framework to reduce the non-occurrence of specified error states in a transition system represented by an LCTRS to an APR problem with irreducible constant destinations. In this paper, by focusing on quasi-termination of constrained narrowing, we characterize a class of LCTRSs of which APR problems with constant destinations are decidable. Quasi-termination of a (constrained) term w.r.t. narrowing is the finiteness of the set of reachable narrowed (constrained) terms from the initial (constrained) term up to variable renaming (and equivalence of constraints). To this end, we first introduce an inference rule for disproofs to a proof system for APR problems with constant destinations and formulate (dis)proof trees of APR problems in the style of cyclic proofs. Secondly, to prove correctness of disproof trees, we adapt constrained narrowing to LCTRSs and show soundness of constrained narrowing of LCTRSs w.r.t. constrained rewriting. Thirdly, we show a sufficient condition of LCTRSs for quasi-termination of constrained narrowing starting from linear constrained terms that have no nesting of defined symbols: Rewrite rules are right-linear and nesting-free w.r.t. defined symbols, and the application of rewrite rules for (mutually) recursive defined symbols does not increase the height of resulting constrained terms of narrowing. Finally, we show that if a constrained term is quasi-terminating w.r.t. narrowing of an LCTRS over a sort-wise finite signature with a decidable theory, then the APR problem consisting of the constrained term and an irreducible constant is decidable.

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“…The APR framework in this paper originates from Refs. [4], [7], [20], [21], where neither decidability nor disproofs has been discussed. A leading verification example in Ref.…”

Section: On Decidability Of Apr Problemsmentioning

confidence: 99%

“…The goal of Ref. [4] is to re-* 13 The detail of our implementation of c-DCC can be seen in Ref. [21].…”

Section: On Decidability Of Apr Problemsmentioning

confidence: 99%

A Sufficient Condition of Logically Constrained Term Rewrite Systems for Decidability of All-path Reachability Problems with Constant Destinations

Kojima,

Nishida

2024

Journal of Information Processing

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“…As explained in the introduction, our proof system cannot establish the other direction, G −1 , B −1 ⊢ 0 G −1 , intuitively because it cannot prove that the termination of F (one phase) implies the termination of f (two phases). The full simulation relation would require an operationally-based termination argument [9] for the second phase of f , which we leave for future work.…”

Section: Finally Rulementioning

confidence: 99%

“…establishes partial equivalence of f and F and half of what is necessary for full equivalence. As future work, in order to enable the complete proof of full equivalence, we will add termination measures to the proof system as in [9] -the termination measure for the second phase of f will enable F ≺ f to be proven. The names of the four files corresponding to this example are of the form example1*.rmt.…”

Section: B Proofsmentioning

confidence: 99%

Operationally-based Program Equivalence Proofs using LCTRSs

Ciobâcă,

Lucanu,

Buruiană

2020

Preprint

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We propose an operationally-based deductive proof method for program equivalence. It is based on encoding the language semantics as logically constrained term rewriting systems (LCTRSs) and the two programs as terms. The main feature of our method is its flexibility. We illustrate this flexibility in two applications, which are novel. For the first application, we show how to encode low-level details such as stack size in the language semantics and how to prove equivalence between two programs operating at different levels of abstraction. For our running example, we show how our method can prove equivalence between a recursive function operating with an unbounded stack and its tail-recursive optimized version operating with a bounded stack. This type of equivalence checking can be used to ensure that new, undesirable behavior is not introduced by a more concrete level of abstraction. For the second application, we show how to formalize read-sets and writesets of symbolic expressions and statements by extending the operational semantics in a conservative way. This enables the relational verification of program schemas, which we exploit to prove correctness of compiler optimizations, some of which cannot be proven by existing tools. Our method requires an extension of standard LCTRSs with axiomatized symbols. We also present a prototype implementation that proves the feasibility of both applications that we propose.

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From Starvation Freedom to All-Path Reachability Problems in Constrained Rewriting

Kojima

Nishida

2023

Practical Aspects of Declarative Languages

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Reducing Total Correctness to Partial Correctness by a Transformation of the Language Semantics

Cited by 4 publications

References 25 publications

A Sufficient Condition of Logically Constrained Term Rewrite Systems for Decidability of All-path Reachability Problems with Constant Destinations

A Sufficient Condition of Logically Constrained Term Rewrite Systems for Decidability of All-path Reachability Problems with Constant Destinations

Operationally-based Program Equivalence Proofs using LCTRSs

From Starvation Freedom to All-Path Reachability Problems in Constrained Rewriting

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