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OrderSorted Dependency Pairs
, 2008
"... Types (or sorts) are pervasive in computer science and in rewritingbased programming languages, which often support subtypes (subsorts) and subtype polymorphism. Programs in these languages can be modeled as ordersorted term rewriting systems (OSTRSs). Often, termination of such programs heavily d ..."
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Cited by 4 (2 self)
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Types (or sorts) are pervasive in computer science and in rewritingbased programming languages, which often support subtypes (subsorts) and subtype polymorphism. Programs in these languages can be modeled as ordersorted term rewriting systems (OSTRSs). Often, termination of such programs heavily depends on sort information. But few techniques are currently available for proving termination of OSTRSs; and they often fail for interesting OSTRSs. In this paper we generalize the dependency pairs approach to prove termination of OSTRSs. Preliminary experiments suggest that this technique can succeed where existing ones fail, yielding easier and simpler termination proofs.
A Model for I/O in Equational Languages with Don't Care NonDeterminism
 In [DHO95
, 1995
"... Existing models for I/O in sideeffect free languages focus on functional languages, which are usually based on a largely deterministic reduction strategy, allowing for a strict sequentialization of I/O operations. In concurrent logic programming languages a model is used which allows for don't ca ..."
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Cited by 2 (0 self)
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Existing models for I/O in sideeffect free languages focus on functional languages, which are usually based on a largely deterministic reduction strategy, allowing for a strict sequentialization of I/O operations. In concurrent logic programming languages a model is used which allows for don't care nondeterminism; the sequentialization of I/O is extensional rather than intensional. We apply this model to equational languages, which are closely related to functional languages, but exhibit don't care nondeterminism. The semantics are formulated as constrained narrowing, a relation that contains the rewrite relation, and is contained in the narrowing relation. We present constrained narrowing and some of its properties; a constructive method to transform conventional term rewriting systems to constrained narrowing systems; and a discussion on requirements for an implementation. CR Subject Classification (1991): D.1.1 [Programming Techniques]: Applicative (Functional) Programm...
Polymorphically ordersorted types in OBJ3
, 1997
"... . OBJ3 [GWM + 93] is a functional programming language with firstorder function types. OBJ3 has two special features: overloading of function symbols and the possibility to order the sorts. This ordering is induced by set inclusion on the carrier sets. We call the feature to be able to order ..."
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Cited by 2 (1 self)
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. OBJ3 [GWM + 93] is a functional programming language with firstorder function types. OBJ3 has two special features: overloading of function symbols and the possibility to order the sorts. This ordering is induced by set inclusion on the carrier sets. We call the feature to be able to order the sorts inclusion set subtyping. The algebraic semantics of OBJ3 is based on the theory of ordersorted algebras [GM89]. Furthermore, OBJ3 allows parameterized programming [Gog90]. However, the concepts of higherorder functions and parametric polymorphism are only emulated by parameters of OBJ3 modules. In this paper we show how to extend OBJ3 by parametric polymorphism in an elegant way. We call this extended language OBJP. In the second part of the paper we describe the operational semantics of OBJP. The operational semantics is a translation of OBJP programs into programs without overloading and subtypes. Here, we improve the approaches of Goguen, Jouannaud, and Mesegu...
Algebraic System Specification and Development: Survey and Annotated Bibliography  Second Edition 
, 1997
"... Data Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4.5.4 Special Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4.6 Semantics of Programming Languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.6.1 Semantics of Ada . . . ..."
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Data Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4.5.4 Special Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4.6 Semantics of Programming Languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.6.1 Semantics of Ada . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.6.2 Action Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.7 Specification Languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.7.1 Early Algebraic Specification Languages . . . . . . . . . . . . . . . . . . . . . . . . 53 4.7.2 Recent Algebraic Specification Languages . . . . . . . . . . . . . . . . . . . . . . . 55 4.7.3 The Common Framework Initiative. . . . . . . . . . . . . . . . . . . . . . . . . . . 56 5 Methodology 57 5.1 Development Phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 5.1.1 Applica...
Lazy Type Checking in Functional Logic Programming
"... In this paper we propose a lazy functional and logic language with a type system combining parametric and inclusion polymorphism. Programs in this language consist of an specication of parametric and ordered types together with a set of type declarations for data constructors and functions and a ..."
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In this paper we propose a lazy functional and logic language with a type system combining parametric and inclusion polymorphism. Programs in this language consist of an specication of parametric and ordered types together with a set of type declarations for data constructors and functions and a set of conditional constructor based rewriting rules describing the behaviour of functions, where the conditional part includes data and type conditions. In order to dene the semantics of the language we present a typed rewriting logic by means of two inference calculi, which allow to prove the validity of welltyped formulas w.r.t. a program. The operational semantics of the language is presented by means of a typed lazy narrowing calculus for goal solving which combines equational solving, lazy unication, sharing and type checking. In order to keep up the laziness of the language a lazy type checking is achieved during the goal solving process. We state soundness and complete...
On Modularity of Completeness in OrderSorted Term Rewriting Systems
"... . In this paper, we extend the results on the modularity of confluence and termination of singlesorted TRSs[3][6][7] to ordersorted ones. Ordersorted TRSs build a good framework for handling overloaded functions and subtypes. For proving modularity of completeness of ordersorted TRSs, we first tr ..."
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. In this paper, we extend the results on the modularity of confluence and termination of singlesorted TRSs[3][6][7] to ordersorted ones. Ordersorted TRSs build a good framework for handling overloaded functions and subtypes. For proving modularity of completeness of ordersorted TRSs, we first transform a TRS with overloaded functions to a nonoverloaded one, and then we demonstrate that our transformation preserves completeness. 1 Introduction Term rewriting system (TRS) can be viewed as a framework for implementing functional programming languages. Ordersorted TRS is a sorted model in which programs manipulate several kinds of objects, thus it is more suitable than the homogeneous one to treat computable objects in software systems. Moreover, ordersorted TRS has a good framework to handle partially defined functions and inclusion of sorts. Hence, its importance is constantly increasing. Indeed, many executable specification languages like OBJ3[4] have an operational semantics b...
Termination of OrderSorted Rewriting with NonMinimal Signatures Yoshinobu KAWABE
"... this paper, we extend the Gnaedig's results [2][3] on termination of ordersorted rewriting. Gnaedig required a condition for ordersorted signatures, called minimality, for the termination proof. We get rid of this restriction by introducing a transformation from a TRS with an arbitrary ordersorted ..."
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this paper, we extend the Gnaedig's results [2][3] on termination of ordersorted rewriting. Gnaedig required a condition for ordersorted signatures, called minimality, for the termination proof. We get rid of this restriction by introducing a transformation from a TRS with an arbitrary ordersorted signature to another TRS with a minimal signature, and proving that this transformation preserves termination.
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
Incremental Checking of WellFounded Recursive Speci cations Modulo Axioms ⋆
"... Abstract. We introduce the notion of wellfounded recursive ordersorted equational logic (OS) theories modulo axioms. Such theories de ne functions by wellfounded recursion and are inherently terminating. Moreover, for wellfounded recursive theories important properties such as con uence and su c ..."
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Abstract. We introduce the notion of wellfounded recursive ordersorted equational logic (OS) theories modulo axioms. Such theories de ne functions by wellfounded recursion and are inherently terminating. Moreover, for wellfounded recursive theories important properties such as con uence and su cient completeness are modular for socalled fair extensions. This enables us to incrementally check these properties for hierarchies of such theories that occur naturally in modular rulebased functional programs. Wellfounded recursive OS theories modulo axioms contain only commutativity and associativitycommutativity axioms. In order to support arbitrary combinations of associativity, commutativity and identity axioms, we show how to eliminate identity and (under certain conditions) associativity (without commutativity) axioms by theory transformations in the last part of the paper. 1