Our main goal is automating termination proofs for programs in rewriting-based languages with features such as: (i) expressive type structures, (ii) conditional rules, (iii) matching modulo axioms, and (iv) contextsensitive rewriting. Specifically, we present a new operational termination method for membership equational programs with features (i)-(iv) that can be applied to programs in membership equational logic (MEL). The method first transforms a MEL program into a simpler, yet semantically equivalent, conditional order-sorted (OS) program. Subsequent trasformations make the OS-program unconditonal, and, finally, unsorted. In particular, we extend and generalize to this richer setting an order-sorted termination technique for unconditional OS programs proposed by Ölveczky and Lysne. An important advantage of our method is that it minimizes the use of conditional rules and produces simpler transformed programs whose termination is often easier to prove automatically.

The focus of this chapter is the incremental presentation of partial firstorder logic, seen as a powerful framework where the specification of most data types can be directly represented in the most natural way. Both model theory and logical deduction are described in full detail. Alternatives to partiality, like (variants of) error algebras and order-sortedness are also discussed, showing their uses and limitations. Moreover, both the total and the partial (positive) conditional fragment are investigated in detail, and in particular the existence of initial (free) models for such restricted logical paradigms is proved. Some more powerful algebraic frameworks are sketched at the end. Equational specifications introduced in last chapter, are a powerful tool to represent the most common data types used in programming languages and their semantics. Indeed, Bergstra and Tucker have shown in a series of papers (see [BT87] for a complete exposition of results) that a data type is semicompu...

A metalogical framework is a logic with an associated methodology that is used to represent other logics and to reason about their metalogical properties. We propose that logical frameworks can be good metalogical frameworks when their logics support reective reasoning and their theories always have initial models. We present a concrete realization of this idea in rewriting logic. Theories in rewriting logic always have initial models and this logic supports reective reasoning. This implies that inductive reasoning is valid when proving properties about the initial models of theories in rewriting logic, and that we can use reection to reason at the metalevel about these properties. In fact, we can uniformly reect induction principles for proving metatheorems about rewriting logic theories and their parameterized extensions. We show that this reective methodology provides an eective framework for dierent, non-trivial, kinds of formal metatheoretic reasoning; one can...

This paper studies the composition of modules that can hide information, over a very general class of logical systems called inclusive institutions. Two semantics are given for composition of such modules using five familiar operations, and a property called conservativity is shown necessary and sufficient for these semantics to agree. The first semantics extracts the visible properties of the result of composing the visible and hidden parts of modules, while the second uses only the visible properties of the components; the semantics agree when the visible consequences of hidden information are enough to determine the result of the composition. A number of "laws of software composition" are proved relating the composition operations. Inclusive institutions simplify many proofs.

: The benefits of the object, logic (or relational), functional, and constraint paradigms

. Rewriting is a general paradigm for expressing computations in various logics, and we focus here on rewriting techniques in equational logic. When used at the proof level, rewriting provides with a very powerful methodology for proving completeness results, a technique that is illustrated here. We also consider whether important properties of rewrite systems such as confluence and termination can be proved in a modular way. Finally, we stress the links between rewriting and tree automata. Previous surveys include [21; 18; 37; 12; 45; 46]. The present one owes much to [21]. Keywords. completion, confluence, critical pair, ground reducibility, inductive completion, local confluence, modularity, narrowing, order-sorted algebras, rewrite rule, rewriting, term algebra, termination, tree automata. 1 Introduction The use of equations is traditional in mathematics. Its use in computer science has culminated with the success of algebraic specifications, a method of specifying software by enc...

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This article presents an example of correct circuit design through interactive transformation. Interactive transformation differs from traditional hardware design transformation frameworks in that it focuses on the issue of finding suitable hardware architecture for the specified system and the issue of architecture correctness. The transformation framework divides every transformation in designs into two steps. The first step is to find a proper architecture implementation. Although the framework does not guarantee existence of such an implementation, nor its discovery, it does provide a characterization of architectural implementation so that the question "is this a correct implementation?" can be answered by equational rewriting. The framework allows a correct architecture implementation to be automatically incorporated with control descriptions to obtain a new system description. The significance of this transformation framework lies in the fact that it requires simpler mechanism o...

: This paper is an introduction to recent research on hidden algebra and its application to software engineering; it is intended to be informal and friendly, but still precise. We first review classical algebraic specification for traditional "Platonic" abstract data types like integers, vectors, matrices, and lists. Software engineering also needs changeable "abstract machines," recently called "objects," that can communicate concurrently with other objects through visible "attributes" and state-changing "methods." Hidden algebra is a new development in algebraic semantics designed to handle such systems. Equational theories are used in both cases, but the notion of satisfaction for hidden algebra is behavioral, in the sense that equations need only appear to be true under all possible experiments; this extra flexibility is needed to accommodate the clever implementations that software engineers often use to conserve space and/or time. The most important results in hidden algebra are ...

We study the complexity of reasoning in Kleene algebra and *-continuous Kleene algebra in the presence of extra equational assumptions E; that is, the complexity of deciding the validity of universal Horn formulas E ! s = t, where E is a finite set of equations. We obtain various levels of complexity based on the form of the assumptions E. Our main results are: for *- continuous Kleene algebra, ffl if E contains only commutativity assumptions pq = qp, the problem is \Pi 0 1 -complete; ffl if E contains only monoid equations, the problem is \Pi 0 2 -complete; ffl for arbitrary equations E, the problem is \Pi 1 1 - complete. The last problem is the universal Horn theory of the *-continuous Kleene algebras. This resolves an open question of Kozen (1994). 1 Introduction Kleene algebra (KA) is fundamental and ubiquitous in computer science. Since its invention by Kleene in 1956, it has arisen in various forms in program logic and semantics [17, 28], relational algebra [27, 32], aut...