CafeOBJ: Logical Foundations and Methodologies

CafeOBJ is an executable industrial strength multi-logic algebraic specification language which is a modern successor of OBJ and incorporates several new algebraic specification paradigms. In this paper we survey its logical foundations and present some of its methodologies

Equational logic and rewriting

International audienceIn this survey, we do not address higher order logics nor type theory, but rather restrict to first-order concepts. We focus on equational logic and its relation to rewriting logic and we consider their impact on automated deduction and programming languages

Order-Sorted Equality Enrichments Modulo Axioms

Built-in equality and inequality predicates based on comparison of canonical forms in algebraic specifications are frequently used because they are handy and efficient. However, their use places algebraic specifications with initial algebra semantics beyond the pale of theorem proving tools based, for example, on explicit or inductionless induction techniques, and of other formal tools for checking key properties such as confluence, termination, and sufficient completeness. Such specifications would instead be amenable to formal analysis if an equationally-defined equality predicate enriching the algebraic data types were to be added to them. Furthermore, having an equationally-defined equality predicate is very useful in its own right, particularly in inductive theorem proving. Is it possible to effectively define a theory transformation epsilon bar right arrow epsilon(similar to) that extends an algebraic specification epsilon to a specification epsilon(similar to) having an equationally-defined equality predicate? This paper answers this question in the affirmative for a broad class of order-sorted conditional specifications epsilon that are sort-decreasing, ground confluent, and operationally terminating modulo axioms B and have a subsignature of constructors. The axioms B can consist of associativity, or commutativity, or associativity-commutativity axioms, so that the constructors are free modulo B. We prove that the transformation epsilon bar right arrow epsilon(similar to) preserves all the just-mentioned properties of epsilon. The transformation has been automated in Maude using reflection and is used as a component in many Maude formal tools. (C) 2014 Elsevier B.V. All rights reserved.This work has been supported in part by NSF Grants CCF 09-05584 and CNS 13-19109, the EU (FEDER) and the Spanish MINECO under Grants TIN 2010-21062-C02 and TIN 2013-45732-C4-1-P, and by the Generalitat Valenciana, ref. PROMETEO/2011/052. Raul Gutierrez is also partially supported by a Juan de la Cierva Fellowship from the Spanish MINECO, ref. JCI-2012-13528.Gutiérrez Gil, R.; Meseguer, J.; Rocha, C. (2015). Order-Sorted Equality Enrichments Modulo Axioms. Science of Computer Programming. 99:235-261. https://doi.org/10.1016/j.scico.2014.07.003S2352619

Rn and Gn Logics

This paper proposes a simple, set-theoretic framework providingexpressive typing, higher-order functions and initial models atthe same time. Building upon Russell's ramified theory of types, we developthe theory of Rn-logics, which are axiomatisable by an order-sortedequational Horn logic with a membership predicate, and of Gn-logics,that provide in addition partial functions. The latter are therefore moreadapted to the use in the program specification domain, while sharing interesting properties, like existence of an initial model, with Rn-logics. Operational semantics of Rn-/Gn-logics presentations is obtained throughorder-sorted conditional rewriting