Abstract. This paper gives an overviewof the Maude 2.0 system. We emphasize the full generality with which rewriting logic and membership equational logic are supported, operational semantics issues, the new built-in modules, the more general Full Maude module algebra, the new META-LEVEL module, the LTL model checker, and newimplementation techniques yielding substantial performance improvements in rewriting modulo. We also comment on Maude’s formal tool environment and on applications. 1

Abstract. Formal semantic definitions of concurrent languages, when specified in a well-suited semantic framework and supported by generic and efficient formal tools, can be the basis of powerful software analysis tools. Such tools can be obtained for free from the semantic definitions; in our experience in just the few weeks required to define a language’s semantics even for large languages like Java. By combining, yet distinguishing, both equations and rules, rewriting logic semantic definitions unify both the semantic equations of equational semantics (in their higher-order denotational version or their first-order algebraic counterpart) and the semantic rules of SOS. Several limitations of both SOS and equational semantics are thus overcome within this unified framework. By using a high-performance implementation of rewriting logic such as Maude, a language’s formal specification can be automatically transformed into an efficient interpreter. Furthermore, by using Maude’s breadth first search command, we also obtain for free a semi-decision procedure for finding failures of safety properties; and by using Maude’s LTL model checker, we obtain, also for free, a decision procedure for LTL properties of finite-state programs. These possibilities, and the competitive performance of the analysis tools thus obtained, are illustrated by means of a concurrent Caml-like language; similar experience with Java (source and JVM) programs is also summarized. 1

Rewriting logic is a flexible and expressive logical framework that unifies denotational semantics and SOS in a novel way, avoiding their respective limitations and allowing very succinct semantic definitions. The fact that a rewrite theory’s axioms include both equations and rewrite rules provides a very useful “abstraction knob” to find the right balance between abstraction and observability in semantic definitions. Such semantic definitions are directly executable as interpreters in a rewriting logic language such as Maude, whose generic formal tools can be used to endow those interpreters with powerful program analysis capabilities.

Abstract. Abstraction reduces the problem of whether an infinite state system satisfies version. The most common abstractions are quotients of the original system. We present a simple method of defining quotient abstractions by means of equations collapsing the set of states. Our method yields the minimal quotient system together with a set of proof obligations that guarantee its executability and can be discharged with tools such as those in the Maude formal environment.

Abstract. Computational systems are often represented by means of Kripke structures, and related using simulations. We propose rewriting logic as a flexible and executable framework in which to formally specify these mathematical models, and introduce a particular and elegant way of representing simulations in it: theoroidal maps. A categorical viewpoint is very natural in the study of these structures and we show how to organize Kripke structures in categories that afterwards are lifted to the rewriting logic’s level. We illustrate the use of theoroidal maps with two applications: predicate abstraction and the study of fairness constraints. 1

One can distinguish two specification levels: a system specification level, in which the computational system of interest is specified; and a property specification level, in which the relevant properties are specified. These lectures present an approach to executable system specification based on equational logic for deterministic systems and on rewriting logic for concurrent systems that is seamlessly integrated with a property specification level using first-order, inductive, and temporal logics. This integration is directly supported by formal verification tools in the formal environment of the Maude rewriting logic language. We show how this approach and the supporting tools can be applied to the specification and verification of a wide variety of programs, that can be either declarative or imperative, and either deterministic or concurrent.

We show that the generalized variant of rewriting logic where the underlying equational specifications are membership equational theories, and where the rules are conditional and can have equations, memberships and rewrites in the conditions is reflective. We also show that membership equational logic, many-sorted equational logic, and Horn logic with equality are likewise reflective. These results provide logical foundations for reflective languages and tools based on these logics, and in particular for the Maude language itself. 1

verification and validation.

Abstract. Rewriting logic is a flexible and general logic to specify concurrent systems. To prove properties about concurrent systems in temporal logic, it is very useful to use simulations that relate the transitions and atomic predicates of a system to those of a potentially much simpler one; then, if the simpler system satisfies a property ϕ in a suitable temporal logic we are guaranteed that the more complex system does too. In this paper, the suitability of rewriting logic as a formal framework not only to specify concurrent systems but also to specify simulations is explored in depth. For this, increasingly more general notions of simulation (allowing stuttering) are first defined for Kripke structures, and suitable temporal logics allowing properties to be reflected back by such simulations are characterized. The paper then proves various representability results à la Bergstra and Tucker, showing that recursive Kripke structures and recursive simulation maps (resp. r.e. simulation relations) can always be specified in a finitary way in rewriting logic. Using simulations typically requires both model checking and theorem proving, since their correctness requires discharging proof obligations. In this regard, rewriting logic, by containing equational logic as a sublogic and having equationally-based inductive theorem proving at its disposal, is shown to be particularly well-suited for verifying the correctness of simulations.

Abstract: We present a novel approach to the verification of functional-logic programs. For our verification purposes, equational reasoning is not valid due to the presence of non-deterministic and partial functions. Our approach transforms functionallogic programs into Maude theories and then uses the Rewriting Logic logical framework to verify properties of the transformed programs. We propose an inductive proving method based on the length of the computation on the Rewriting Logic framework to cope with the non-deterministic and non-terminating aspects of the programs. We illustrate the application of the method on various examples, where we analyze the sequence of steps to be performed by the proof in order to get expertise for the automatization of the process. Then, since the proposed transformation process is also amenable of automatization, we will obtain a tool for proving properties of CRWL programs. Another advantage of our methodology, that distinguish it from other approaches, is that it does not confuse the original functional-logic program with the subjects we want to talk about in the properties, but it allows the equational definition of observations on top of the transformed programs which simplifies the obtained proofs.