Completeness of Context-Sensitive Rewriting

Restrictions of rewriting may turn normal forms of some terms unreachable, leading to incomplete computations. Context-sensitive rewriting (csr) is the restriction of rewriting that only permits reductions on arguments selected by a replacement map μ, which associates a subset μ(f ) of argument indices with each function symbol f . Hendrix and Meseguer defined an algebraic semantics for Term Rewriting Systems (TRSs) executing csr that can be used to reason about programs written in programming languages like CafeOBJ and Maude, where such replacement restrictions can be specified in programs. Semantic completeness of csr was also defined. In this paper we show that canonical replacement maps, which play a prominent role in simulating rewriting computations with csr, are necessary for completeness in important classes of TRSs. © 2014 Elsevier B.V. All rights reserved.Supported by NSF CNS 13-19109, MINECO project TIN2010-21062-C02-02, GV (Generalitat Valenciana) Grants BEST/2014/026 and PROMETEO/2011/052.Lucas Alba, S. (2015). Completeness of Context-Sensitive Rewriting. Information Processing Letters. 115(2):87-92. https://doi.org/10.1016/j.ipl.2014.07.004S8792115

Termination of Rewriting with and Automated Synthesis of Forbidden Patterns

We introduce a modified version of the well-known dependency pair framework that is suitable for the termination analysis of rewriting under forbidden pattern restrictions. By attaching contexts to dependency pairs that represent the calling contexts of the corresponding recursive function calls, it is possible to incorporate the forbidden pattern restrictions in the (adapted) notion of dependency pair chains, thus yielding a sound and complete approach to termination analysis. Building upon this contextual dependency pair framework we introduce a dependency pair processor that simplifies problems by analyzing the contextual information of the dependency pairs. Moreover, we show how this processor can be used to synthesize forbidden patterns suitable for a given term rewriting system on-the-fly during the termination analysis.Comment: In Proceedings IWS 2010, arXiv:1012.533

Expression-based aliasing for OO-languages

Alias analysis has been an interesting research topic in verification and optimization of programs. The undecidability of determining whether two expressions in a program may reference to the same object is the main source of the challenges raised in alias analysis. In this paper we propose an extension of a previously introduced alias calculus based on program expressions, to the setting of unbounded program executions s.a. infinite loops and recursive calls. Moreover, we devise a corresponding executable specification in the K-framework. An important property of our extension is that, in a non-concurrent setting, the corresponding alias expressions can be over-approximated in terms of a notion of regular expressions. This further enables us to show that the associated K-machinery implements an algorithm that always stops and provides a sound over-approximation of the "may aliasing" information, where soundness stands for the lack of false negatives. As a case study, we analyze the integration and further applications of the alias calculus in SCOOP. The latter is an object-oriented programming model for concurrency, recently formalized in Maude; K-definitions can be compiled into Maude for execution

Finite Model Finding for Parameterized Verification

In this paper we investigate to which extent a very simple and natural "reachability as deducibility" approach, originated in the research in formal methods in security, is applicable to the automated verification of large classes of infinite state and parameterized systems. The approach is based on modeling the reachability between (parameterized) states as deducibility between suitable encodings of states by formulas of first-order predicate logic. The verification of a safety property is reduced to a pure logical problem of finding a countermodel for a first-order formula. The later task is delegated then to the generic automated finite model building procedures. In this paper we first establish the relative completeness of the finite countermodel finding method (FCM) for a class of parameterized linear arrays of finite automata. The method is shown to be at least as powerful as known methods based on monotonic abstraction and symbolic backward reachability. Further, we extend the relative completeness of the approach and show that it can solve all safety verification problems which can be solved by the traditional regular model checking.Comment: 17 pages, slightly different version of the paper is submitted to TACAS 201

A Combination Framework for Complexity

In this paper we present a combination framework for polynomial complexity analysis of term rewrite systems. The framework covers both derivational and runtime complexity analysis. We present generalisations of powerful complexity techniques, notably a generalisation of complexity pairs and (weak) dependency pairs. Finally, we also present a novel technique, called dependency graph decomposition, that in the dependency pair setting greatly increases modularity. We employ the framework in the automated complexity tool TCT. TCT implements a majority of the techniques found in the literature, witnessing that our framework is general enough to capture a very brought setting

Soundness of Unravelings for Conditional Term Rewriting Systems via Ultra-Properties Related to Linearity

Unravelings are transformations from a conditional term rewriting system (CTRS, for short) over an original signature into an unconditional term rewriting systems (TRS, for short) over an extended signature. They are not sound w.r.t. reduction for every CTRS, while they are complete w.r.t. reduction. Here, soundness w.r.t. reduction means that every reduction sequence of the corresponding unraveled TRS, of which the initial and end terms are over the original signature, can be simulated by the reduction of the original CTRS. In this paper, we show that an optimized variant of Ohlebusch's unraveling for a deterministic CTRS is sound w.r.t. reduction if the corresponding unraveled TRS is left-linear or both right-linear and non-erasing. We also show that soundness of the variant implies that of Ohlebusch's unraveling. Finally, we show that soundness of Ohlebusch's unraveling is the weakest in soundness of the other unravelings and a transformation, proposed by Serbanuta and Rosu, for (normal) deterministic CTRSs, i.e., soundness of them respectively implies that of Ohlebusch's unraveling.Comment: 49 pages, 1 table, publication in Special Issue: Selected papers of the "22nd International Conference on Rewriting Techniques and Applications (RTA'11)

Canonized Rewriting and Ground AC Completion Modulo Shostak Theories : Design and Implementation

AC-completion efficiently handles equality modulo associative and commutative function symbols. When the input is ground, the procedure terminates and provides a decision algorithm for the word problem. In this paper, we present a modular extension of ground AC-completion for deciding formulas in the combination of the theory of equality with user-defined AC symbols, uninterpreted symbols and an arbitrary signature disjoint Shostak theory X. Our algorithm, called AC(X), is obtained by augmenting in a modular way ground AC-completion with the canonizer and solver present for the theory X. This integration rests on canonized rewriting, a new relation reminiscent to normalized rewriting, which integrates canonizers in rewriting steps. AC(X) is proved sound, complete and terminating, and is implemented to extend the core of the Alt-Ergo theorem prover.Comment: 30 pages, full version of the paper TACAS'11 paper "Canonized Rewriting and Ground AC-Completion Modulo Shostak Theories" accepted for publication by LMCS (Logical Methods in Computer Science