Rewriting Logic and Maude: Concepts and Applications

Rewriting logic expresses an essential equivalence between logic and computation. Systemstatesare in bijective correspondence withformulas, and concurrentcomputationsare in bijective correspondence withproofs.Given this equivalence between computation and logic, a rewriting logic axiom of the form t → t’ has two readings. Computationally, it means that a fragment of a system’s state that is an instance of the patterntcanchangeto the corresponding instance oft’concurrently with any other state changes; logically, it just means that we can derive the formulat’from the formulat.Rewriting logic is entirely neutral about the structure and properties of the formulas/statest.They are entirelyuser-definableas an algebraic data type satisfying certain equational axioms. Because of this ecumenical neutrality, rewriting logic has, from a logical viewpoint, good properties as alogical framework, in which many other logics can be naturally represented. And, computationally, it has also good properties as asemantic framework, in which many different system styles and models of concurrent computation and many different languages can be naturally expressed without any distorting encodings. The goal of this paper is to provide a relatively gentle introduction to rewriting logic, and to paint in broad strokes the main research directions that, since its introduction in 1990, have been pursued by a growing number of researchers in Europe, the US, and Japan. Key theoretical developments, as well as the main current applications of rewriting logic as a logical and semantic framework, and the work on formal reasoning to prove properties of specifications are surveyed.

Willard Van Orman Quine. On what there is. A reprint of XIX 134(1). From a logical point of view, by Willard Van Orman Quine, second, revised edition, Harvard University Press, Cambridge, Mass., 1961, and Harper Torchbooks, The Science Library, Harper & Row, New York and Evanston 1963, pp. 1–19. - Willard Van Orman Quine. Two dogmas of empiricism. A reprint of XIX 134(2). From a logical point of view, by Willard Van Orman Quine, second, revised edition, Harvard University Press, Cambridge, Mass., 1961, and Harper Torchbooks, The Science Library, Harper & Row, New York and Evanston 1963, pp. 20–46. - Willard Van Orman Quine. The problem of meaning in linguistics. A reprint of XIX 134(3). From a logical point of view, by Willard Van Orman Quine, second, revised edition, Harvard University Press, Cambridge, Mass., 1961, and Harper Torchbooks, The Science Library, Harper & Row, New York and Evanston 1963, pp. 47–64. - Willard Van Orman Quine. Identity, ostension, and hypostasis. A reprint of XIX 134(4). From a logical point of view, by Willard Van Orman Quine, second, revised edition, Harvard University Press, Cambridge, Mass., 1961, and Harper Torchbooks, The Science Library, Harper & Row, New York and Evanston 1963, pp. 65–79. - Willard Van Orman Quine. New foundations for mathematical logic. A reprint of XIX 134(5). From a logical point of view, by Willard Van Orman Quine, second, revised edition, Harvard University Press, Cambridge, Mass., 1961, and Harper Torchbooks, The Science Library, Harper & Row, New York and Evanston 1963, pp. 80–101. - Willard Van Orman Quine. Logic and the reification of universals. A revised reprint of XIX 135. From a logical point of view, by Willard Van Orman Quine, second, revised edition, Harvard University Press, Cambridge, Mass., 1961, and Harper Torchbooks, The Science Library, Harper & Row, New York and Evanston 1963, pp. 102–129. - Willard Van Orman Quine. Notes on the theory of reference. A reprint of XIX 136. From a logical point of view, by Willard Van Orman Quine, second, revised edition, Harvard University Press, Cambridge, Mass., 1961, and Harper Torchbooks, The Science Library, Harper & Row, New York and Evanston 1963, pp. 130–138. - Willard Van Orman Quine. Reference and modality. A revised reprint of XIX 137. From a logical point of view, by Willard Van Orman Quine, second, revised edition, Harvard University Press, Cambridge, Mass., 1961, and Harper Torchbooks, The Science Library, Harper & Row, New York and Evanston 1963, pp. 139–159. - Willard Van Orman Quine. Meaning and existential inference. A reprint of XIX 138. From a logical point of view, by Willard Van Orman Quine, second, revised edition, Harvard University Press, Cambridge, Mass., 1961, and Harper Torchbooks, The Science Library, Harper & Row, New York and Evanston 1963, pp. 160–167. - Volume 33 Issue 1

The task of designing and implementing a compiler can be a difficult and error-prone process. In this paper, we present a new approach based on the use of higher-order abstract syntax and term rewriting in a logical framework. All program transformations, from parsing to code generation, are cleanly isolated and specified as term rewrites. This has several advantages. The correctness of the compiler depends solely on a small set of rewrite rules that are written in the language of formal mathematics. In addition, the logical framework guarantees the preservation of scoping, and it automates many frequently-occurring tasks including substitution and rewriting strategies. As we show, compiler development in a logical framework can be easier than in a general-purpose language like ML, in part because of automation, and also because the framework provides extensive support for examination, validation, and debugging of the compiler transformations. The paper is organized around a case study, using the MetaPRL logical framework to compile an ML-like language to Intel x86 assembly. We also present a scoped formalization of x86 assembly in which all registers are immutable.

Twenty years of rewriting logic

In this paper we present PLATO, a tool designed for theory presentation construction, theorem proving and formula derivation. It was motivated by equational algebraic theory manipulation needs, nevertheless its aim is software development by program calculation. PLATO has applications in algebra boolean algebra, relation algebra, etc. and mathematical logic as a general framework for theorem proving under different logics . In the field of software development, we show its application in program calculi construction and in program development under these calculi. We will consider a programming calculus as being a theory presentation within a given logic. Thus, the task of constructing new calculi can be thought of as one of building new theory presentations. This is true for, in particular, programming calculi based on Relation Algebras, Functional Programming, Fork Algebras, Type Theory and the like. We will consider a theory presentation as a pair , where T is a similarity type, i.e., declaration of operation symbols with their arity, and 2 is a finite set of formulas constituting an axiomatization for that theory.

In this article, an “algebra” (sometimes also called a “universal algebra”) is a nonempty set X together with a set of “basic operations”; each operation is a finitary total function on X , i.e., a function ƒ : X n → X for some n ∈ {1, 2, …}. While our operations always have finite arity, we allow infinitely many basic operations in our algebras. Clone theory investigates the structure of all algebras on a given set. It turns out that several questions originating in algebra can be solved (or at least attacked) with set-theoretic methods. This article collects a few examples of this approach, based on recent results of Pinsker, Shelah and the author. Background . A large part of universal algebra investigates the relations between the basic operations of an algebra; here we take a different point of view—all term operations are considered to live an an equal level; thus we will, for example, not distinguish between a Boolean algebra and the corresponding Boolean ring, because each ring operation can be written as composition of operations of the Boolean algebra, and conversely. This point of view is motivated by the observation that many properties of algebras depend only on the clone of its term functions or polynomials, without any need to distinguish basic operations; for example, an equivalence relation is a congruence relation iff it is closed under all unary polynomials.

Preface: Volume 58, Issue 2

In this paper we propose a certification technique for non-interference of Java programs based on rewriting logic, a very general logical and semantic framework efficiently implemented in the high-level programming language Maude. Non---interference is a semantic program property that prevents illicit information flow to happen. Starting from a basic specification of the semantics of Java written in Maude, we develop an information---flow extension of this operational Java semantics which allows us to observe non-interference of Java programs. Then we develop in Maude an abstract, finite-state version of the information-flow operational semantics which supports finite program verification. As a by---product of the verification, a certificate of non-interference is delivered which consists of a set of (abstract) rewriting proofs that can be easily checked by the code consumer using a standard rewriting logic engine.

The open calculus of constructions integrates key features of Martin-Löf's type theory, the calculus of constructions, membership equational logic, and rewriting logic into a single uniform language. The two key ingredients are dependent function types and conditional rewriting modulo equational theories. We explore the open calculus of constructions as a uniform framework for programming, specification and interactive verification in an equational higher-order style. By having equational logic and rewriting logic as executable sublogics we preserve the advantages of a first-order semantic and logical framework and we provide a foundation for a broad spectrum of applications ranging from what could be called executable mathematics, involving symbolic computations and logical proofs, to software and system engineering applications, involving symbolic execution and analysis of nondeterministic and concurrent systems.

This paper discusses some intuitions about models of concurrency. Several examples of concurrent systems are presented to illustrate different kinds of concurrent computational structures. I argue that supporting large-scale concurrent systems requires building on some surprisingly simple programming primitives. However, some models of concurrency cannot support these primitives. I describe the actor model which incorporates these primitives and illustrate its potential use in supporting different programming paradigms. Developments in concurrent computers are briefly discussed; specifically, I argue that because of their scalability, multicomputers represent the most promising development in computer architecture. Multicomputers provide a powerful realization of the actor model. Finally, I sketch the relation between different programming paradigms and actors and argue for the advantages of multiparadigm programming systems obtained by embedding different programming languages in an actor base.

Preface: Volume 36

Action refinement provides a mechanism to design a complex reactive system hierarchically. This paper is devoted to action refinement from a logical point of view, and to combining the hierarchical implementation of a complex system with the hierarchical specification of the system in order to verify it in an easy way. To this end, we use a TCSP-like language with an action refinement operator as a modeling language, and an extension of the modal μ-calculus, called FLC (Fixpoint Logic with Chop) [18], as a specification language. Specifications in FLC can be refined via a mapping that takes as arguments an abstract specification φ for the process P, an action a of P and a specification ψ for the process Q that may refine a and produces a refined specification. We prove under some syntactical conditions: if Q ⊨ ψ then P ⊨ φ iff. P[a⇝Q] satisfies the refined specification. Therefore our approach supports ‘a priori’ verification in system design and can be used to decrease substantially the complexity of verification.Keywordsaction refinementmodal logicsspecificationverificationreactive systems

In this paper, we have analyzed a number of solutions to the antinomy between divine foreknowledge and human freedom. If we assume that God is temporal, then a sort of backwards causation of past divine beliefs by future human acts must be acknowledged. Since this solution runs into difficulties, we consider the prospects of the view according to which God is outside time. A timeless and omniscient God seems to imply a B-theory of time and, at least at first glance, seems to jeopardize human freedom. Therefore, we have examined what happens when a non-standard A-theory of time like Fragmentalism is assumed. We demonstrate that in this case the prospects of a timeless view of God are much better: both human freedom and divine knowledge of the results of human choices are preserved if this metaphysics of time is adopted. The costs of this solution are, however, very high. From the logical point of view, it rejects bivalence; from the metaphysical point of view, the world is regarded as fragmentary and incoherent. However, if one is ready to accept these costs, this solution is one of the most successful in the search for a solution to the centuries-old problem of the conciliation between divine foreknowledge and human freedom.

Metaphor Revisited Dennis Sobolev (bio) Already for several generations, the problem of Metaphor has been the constant focus of different disciplines: from literary studies and analytical philosophy to cognitive psychology and linguistics. For literary criticism, however, it has been especially significant. In order to evaluate this significance, one has to bear in mind that since the Prague group and French structuralism, and until the rise of different types of poststructuralist research, literary scholars had been working within the “literature as a language” paradigm.1 And since the same type of structuralist research considered metaphor as one of the two pivotal operations on which the whole system of language is based,2 it was used as a model for an extremely wide range of structures and functions empirically found in literature. Having this in mind, one may surmise that the significance of the problem of metaphor was doomed to become downplayed when the linguistic paradigm was replaced with “postmodern” ones. Indeed, the poststructural paradigms of literary analysis tend to view “the space of literature” as a product of the superposition of different cultural forces, structures, and partial determinations and thus tend to deny the linguistic model the role of a single, privileged hermeneutic strategy. However, from the theoretical point of view, this tendency is not an inevitable result of the replacement of analytical paradigms. On the contrary, from the logical point of view, within a paradigm that tends to view culture as a field of partial and heterogeneous determinations, the significance of metaphor as a model must only grow. One of the operations central to the very being of culture, when viewed as such a field of heterogeneous determinations, is that of the creation of synthetic relations. Such operations of synthesis range from the narrative synthesis of heterogeneous historical and personal materials to the basic operations of the cultural construction of existential space and time. Therefore, being one of the simplest and most exhaustively studied operations of synthesis, metaphor may serve as a good case study and thus as a model of the analysis of the operations of synthesis in general. However, as will be shown, this exemplum demonstrates the high level of complexity of problems of cultural analysis. The studies of metaphor [End Page 903] carried out over the last decades have shown the extreme complexity of this seemingly simple operation of synthesis; metaphor has revealed itself to be an embodiment of analytical difficulties, as well as a source of irresolvable contradictions between different approaches. Although most scholars who have written about metaphor agree that it is central to the understanding of poetic language, this seems to be the only subject upon which a relative consensus has been achieved. None of the theories propounded in the field has been able to convince the majority of the academic community; and for every theory that has been proposed, numerous counterexamples have been found. In light of this situation, two mutually exclusive hypotheses become possible. One can either state that the failure to find a sufficiently cogent solution to this problem is just a temporary difficulty, and thus the philosophers of language and literary critics should continue looking for it. Or, conversely, one can suggest that the whole discussion has been misguided and that there is no such thing as the structure of metaphor, or, similarly, that there is no such thing as metaphor in the traditional sense of the term. I believe, however, that both radical positions are misguided. This essay aims to demonstrate that metaphor, even though it is formally identifiable, is not a single unified structure, but rather a field of heterogeneous possibilities, which is organized along several independent axes and is limited by border parameters. Correspondingly, it is the culturally constructed possibility of “identifying” metaphor that is responsible for the illusion of the existence of its singular essence, while it is the heterogeneity and multiplicity of metaphorical functions that is responsible for both its actual applications and the proliferation of the theories of metaphor. The first—“preliminary”—problem of metaphorical discourse is terminological. In order to account for the structure of metaphor, different sets of concepts have been proposed; the most known of them are I. A. Richards’s...

The paper addresses the problem of test suite derivation from a formal specification of a distributed concurrent software system given as a collection of labeled transition systems. It presents a new concurrency model, called behavior machine, and its construction algorithm. Further, the paper outlines how test derivation can be based on the new concurrency model in order to derive test suites that still exhibit true concurrency between test events. A toolset is presented to support the generation of concurrent test suites from specifications given in the formal description technique LOTOS. Finally, some comments on requirements for the design of a distributed test architecture are given.KeywordsDistributed concurrent software systemsconformance testingtest derivationlabeled transition systemsLOTOSPetri nets.