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DynamicallyTyped Computations for OrderSorted Equational Presentations (Extended Abstract)
 Proc. 21st International Colloquium on Automata, Languages, and Programming, volume 820 of Lecture Notes in Computer Science
, 1994
"... Equational presentations with ordered sorts encompass partially defined functions and subtyping information in an algebraic framework. In this work we address the problem of computing in ordersorted algebras, with very few restrictions on the allowed presentations. We adopt an algebraic framework w ..."
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Cited by 10 (8 self)
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Equational presentations with ordered sorts encompass partially defined functions and subtyping information in an algebraic framework. In this work we address the problem of computing in ordersorted algebras, with very few restrictions on the allowed presentations. We adopt an algebraic framework where equational, membership and existence formulas can be expressed. A complete deduction calculus is provided to incorporate the interaction between all these formulas. The notion of decorated terms is proposed to memorize local sort information, dynamically changed by a rewriting process. A completion procedure for equational presentations with ordered sorts computes a set of rewrite rules with which not only equational theorems of the form (t = t 0 ), but also typing theorems of the for...
Axiomatizing Permutation Equivalence
 Mathematical Structures in Computer Science
, 1994
"... We axiomatize permutation equivalence in term rewriting systems and Klop's orthogonal Combinatory Reduction Systems [Klop 1980]. The axioms for the former ones are provided by the general approach proposed by Meseguer [Meseguer 1992]. The latter ones need extra axioms modeling the interplay between ..."
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Cited by 8 (0 self)
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We axiomatize permutation equivalence in term rewriting systems and Klop's orthogonal Combinatory Reduction Systems [Klop 1980]. The axioms for the former ones are provided by the general approach proposed by Meseguer [Meseguer 1992]. The latter ones need extra axioms modeling the interplay between reductions and the operation of substitution. As a consequence of this work, the definition of permutation equivalence is rid of residual calculi, which are heavy in general. 1 Introduction 1. What does permutation equivalence mean? A wellknown syntactical property of the  calculus is the ChurchRosser theorem. It states that, if a term M reduces into N 1 and N 2 by firing two different redexes, then there exists a term P which is a reduct both of N 1 and N 2 . Graphically: P @ @ @R @ @ @R \Gamma \Gamma \Gamma\Psi \Gamma \Gamma \Gamma\Psi N 1 N 2 \Gamma \Gamma \Gamma\Psi @ @ @R M v oe u ae Actually the ChurchRosser property can be asserted in a stronger way. For this purpose, re...
Representations, Hierarchies, and Graphs of Institutions
, 1996
"... For the specification of abstract data types, quite a number of logical systems have been developed. In this work, we will try to give an overview over this variety. As a prerequisite, we first study notions of {\em representation} and embedding between logical systems, which are formalized as {\em ..."
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Cited by 5 (4 self)
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For the specification of abstract data types, quite a number of logical systems have been developed. In this work, we will try to give an overview over this variety. As a prerequisite, we first study notions of {\em representation} and embedding between logical systems, which are formalized as {\em institutions} here. Different kinds of representations will lead to a looser or tighter connection of the institutions, with more or less good possibilities of faithfully embedding the semantics and of reusing proof support. In the second part, we then perform a detailed ``empirical'' study of the relations among various wellknown institutions of total, ordersorted and partial algebras and firstorder structures (all with Horn style, i.e.\ universally quantified conditional, axioms). We thus obtain a {\em graph} of institutions, with different kinds of edges according to the different kinds of representations between institutions studied in the first part. We also prove some separation results, leading to a {\em hierarchy} of institutions, which in turn naturally leads to five subgraphs of the above graph of institutions. They correspond to five different levels of expressiveness in the hierarchy, which can be characterized by different kinds of conditional generation principles. We introduce a systematic notation for institutions of total, ordersorted and partial algebras and firstorder structures. The notation closely follows the combination of features that are present in the respective institution. This raises the question whether these combinations of features can be made mathematically precise in some way. In the third part, we therefore study the combination of institutions with the help of socalled parchments (which are certain algebraic presentations of institutions) and parchment morphisms. The present book is a revised version of the author's thesis, where a number of mathematical problems (pointed out by Andrzej Tarlecki) and a number of misuses of the English language (pointed out by Bernd KriegBr\"uckner) have been corrected. Also, the syntax of specifications has been adopted to that of the recently developed Common Algebraic Specification Language {\sc Casl} \cite{CASL/Summary,Mosses97TAPSOFT}.
Combining Algebraic and SetTheoretic Specifications
 Recent Trends in Data Type Specification", Proc. 11th Workshop on Specification of Abstract Data Types joint with the 9th general COMPASS workshop
, 1996
"... Specification frameworks such as B and Z provide power sets and cartesian products as builtin type constructors, and employ a rich notation for defining (among other things) abstract data types using formulae of predicate logic and lambdanotation. In contrast, the socalled algebraic specificat ..."
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Cited by 3 (2 self)
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Specification frameworks such as B and Z provide power sets and cartesian products as builtin type constructors, and employ a rich notation for defining (among other things) abstract data types using formulae of predicate logic and lambdanotation. In contrast, the socalled algebraic specification frameworks often limit the type structure to sort constants and firstorder functionalities, and restrict formulae to (conditional) equations.
VALIDE  Formal Methods and Tools for Distributed System Design
, 1994
"... This document describes the research projects of the Formal Methods and Tools research groups of the department of TeleInformatics and Open Systems at the Faculty of Computers Science of the University of Twente. 1 Valide 1. Project name: Valide (version 1) 2. Project period: starting date: 1 Ja ..."
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This document describes the research projects of the Formal Methods and Tools research groups of the department of TeleInformatics and Open Systems at the Faculty of Computers Science of the University of Twente. 1 Valide 1. Project name: Valide (version 1) 2. Project period: starting date: 1 January 1994 duration: indefinite 3. Project leaders: prof. dr Ed Brinksma dr Henk Alblas 4. Coordinating department: INF/TIOS 5. Embedding: Valide is part of the VFresearch programme on TeleInformatics and Open Systems. As such it is part of the TIOS participation in the research programme of the Centre for TeleInformatics and Information Technology (CTIT). The subprojects participate in a number of international research programmes such as ESPRIT Basic Research Action and RACE, as well as national projects funded by NWO. The Valide subprojects are: ffl Leibniz (Transformational Design of Distributed Systems) ffl Popper (Validation of Distributed System Design) ffl Plato (Specificati...
R^n and G^nLogics
 HigherOrder Algebra, Logic, and Term Rewriting, volume 1074 of Lecture Notes in Computer Science
, 1996
"... This paper proposes a simple, settheoretic framework providing expressive typing, higherorder functions and initial models at the same time. Building upon Russell's ramified theory of types, we develop the theory of R logics, which are axiomatisable by an ordersorted equational Horn logic ..."
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This paper proposes a simple, settheoretic framework providing expressive typing, higherorder functions and initial models at the same time. Building upon Russell's ramified theory of types, we develop the theory of R logics, which are axiomatisable by an ordersorted equational Horn logic with a membership predicate, and of G logics, that provide in addition partial functions. The latter are therefore more adapted to the use in the program specification domain, while sharing interesting properties, like existence of an initial model, with R logics. Operational semantics of R logics presentations is obtained through ordersorted conditional rewriting.
R nandG nLogics
"... is permitted for educational or research use on condition that this copyright notice is included in any copy. See back inner page for a list of recent publications in the BRICS Report Series. Copies may be obtained by contacting: BRICS ..."
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is permitted for educational or research use on condition that this copyright notice is included in any copy. See back inner page for a list of recent publications in the BRICS Report Series. Copies may be obtained by contacting: BRICS
DOI: 10.1007/s1081700690657 Structures for abstract rewriting
, 2009
"... Abstract. When rewriting is used to generate convergent and complete rewrite systems in order to answer the validity problem for some theories, all the rewriting theories rely on a same set of notions, properties and methods. Rewriting techniques have mainly been used to answer the validity problem ..."
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Abstract. When rewriting is used to generate convergent and complete rewrite systems in order to answer the validity problem for some theories, all the rewriting theories rely on a same set of notions, properties and methods. Rewriting techniques have mainly been used to answer the validity problem of equational theories, that is to compute congruences. However, recently, they have been extended in order to be applied to other algebraic structures such as preorders and orders. In this paper, we investigate an abstract form of rewriting, by following the paradigm of “logicalsystem independency”. To achieve this purpose, we provide a few simple conditions (or axioms) under which rewriting (and then the set of classical properties and methods) can be modeled, understood, studied, proven and generalized. This enables us to extend rewriting techniques to other algebraic structures than congruences and preorders such as congruences closed under monotonicity and modusponens. Finally, we introduce convergent rewrite systems that enable one to describe deduction procedures for their corresponding theory, and propose a KnuthBendix style completion procedure in this abstract framework.