A category-based equational logic semantics to constraint programming

Diaconescu, Răzvan

doi:10.1007/3-540-61629-2_44

Cited by 17 publications

(52 citation statements)

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“…The results in this paper show that there is also complete deduction within this framework. We are not aware of other similar categorical completeness results in the literature, except previous work by the author [35] where only unconditional axioms where supported and some interesting results by Diaconescu [14] within his category-based equational logic, where equations were regarded as parallel pairs of arrows and his five inference rules were the typical ones for equational deduction.…”

Section: Discussionmentioning

confidence: 65%

“…To show it complete, C also needs to have directed colimits and to be E-co-well-powered, and some appropriate notions of finiteness for arrows in E need to be introduced. A related variant by Diaconescu [14], called categorybased equational logic, considers equations as pairs of arrows, one for each term, and then gives a set of deduction rules that resembles that of equational logics.…”

Section: Introductionmentioning

confidence: 99%

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Complete Categorical Deduction for Satisfaction as Injectivity

Roşu

2006

Algebra, Meaning, and Computation

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Abstract. Birkhoff (quasi-)variety categorical axiomatizability results have fascinated many scientists by their elegance, simplicity and generality. The key factor leading to their generality is that equations, conditional or not, can be regarded as special morphisms or arrows in a special category, where their satisfaction becomes injectivity, a simple and abstract categorical concept. A natural and challenging next step is to investigate complete deduction within the same general and elegant framework. We present a categorical deduction system for equations as arrows and show that, under appropriate finiteness requirements, it is complete for satisfaction as injectivity. A straightforward instantiation of our results yields complete deduction for several equational logics, in which conditional equations can be derived as well at no additional cost, as opposed to the typical method using the theorems of constants and of deduction. At our knowledge, this is a new result in equational logics.

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Section: Discussionmentioning

confidence: 65%

Section: Introductionmentioning

confidence: 99%

Complete Categorical Deduction for Satisfaction as Injectivity

Roşu

2006

Algebra, Meaning, and Computation

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“…OBJ3 does not have so-called "logical variables," the values for which are supplied by the system through "solving" systems of constraints, although its extension to Eqlog does [71,72,25].…”

Section: Discussionmentioning

confidence: 99%

“…OBJ has been extended in many directions, including logic (or relational) programming (the Eqlog system [71,72,25]), object oriented programming (the FOOPS system [74,87]), object oriented specification (OOZE [3]), requirements tracing (TOOR [130]), higher-order functional programming [97,110], and LOTOS-style specification for communication protocols [127,128].…”

Section: A Brief History Of Objmentioning

confidence: 99%

“…With its module database and its ability to incorporate Lisp code, this provides a very flexible and extensible environment that is convenient for specification and rapid prototyping, as well as for building new systems, such as experimental languages and theorem proving environments. For example, OBJ3 has been used for building FOOPS, an object oriented specification and programming system [74,87], the Eqlog system [71,72,25] for equational logic (or relational) programming, OOZE [3], an object oriented specification language influenced by Z [142], the 20B] metalogical framework theorem prover [81], and TOOR [130], a system for tracing requirements.OB] has been used for many applications, including debugging algebraic specifications [77], rapid prototyping [69], defining programming languages in a way that directly yields an interpreter (see Appendix Section C.2, as well Introducing OBl 5 as [79] and some elegant work of Peter Mosses [120,121]), specifying software systems (e.g., the GKS graphics kernel system [28], an Ada configuration manager [40], the MacIntosh QuickDraw program [126], and OBJ itself [20]), hardware specification, simulation, and verification (see [144] and Section 4.8), speci fication and verification of imperative programs [66], specification of user interface designs [60,58], and theorem proving [53,66,59]; several of these were done under a UK government grant. OBJ has also been combined with Petri nets, thus allowing structured data in tokens [5], and was used to verify compilers for parallel programming languages in the ESPRIT sponsored PROcos project [137,138].…”

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confidence: 99%

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Introducing OBJ

Goguen

Winkler

Meseguer

et al. 2000

Advances in Formal Methods

150

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This is an introduction to the philosophy and use of OBJ, emphasizing its operational semantics, with aspects of its history and its logical semantics. Release 2 of OB13 is described in detail, with many examples. OBJ is a wide spectrum first-order functional language that is rigorously based on (order sorted) equationallogic and parameterized programming, supporting a declarative style that facilitates verification and allows OBJ to be used as a theorem prover.Order sorted algebra provides a notion of subsort that rigorously supports mUltiple inheritance, exception handling and overloading. Parameterized programming gives powerful support for design, verification, reuse, and maintenance, using two kinds of module: objects to encapsulate executable code, and in particular to define abstract data types by initial algebra semantics; and (loose) theories to specify both syntactic and semantic properties of modules. SOFTWARE ENGINEERING WITH OBlof module can be parameterized, where actual parameters may be modules. For parameter instantiation, a view binds the formal entities in an interface theory to actual entities in a module, and also asserts that the target module satisfies the semantic conditions of the interface theory. Module expressions allow complex combinations of already defined modules, including sums, instantiations, and transformations; moreover, evaluating a module expression actually constructs the described software (sub)system from the given components. Default views can greatly reduce the effort of instantiating modules, by allowing obvious correspondences to be left out. We argue that first-order parameterized programming includes much of the power of higher-order programming, in a form that is often more convenient.Although OBI executable code normally consists of equations that are interpreted as rewrite rules, OBJ3 objects can also encapsulate Lisp code, e.g., to provide efficient built-in data types, or to augment the system with new capabilities; we describe the syntax of this facility, and provide some examples. In addition, OBI provides rewriting modulo associative, commutative and/or identity equations, as well as user-definable evaluation strategies that allow lazy, eager, and mixed evaluation strategies on an operator-by-operator basis; memoization is also available on an operator-by-operator basis. In addition, OBJ3 supports the application of equations one at a time, either forwards or backwards; this is needed for equational theorem proving. Finally, OBI provides user-definable mixfix syntax, which supports the notational conventions of particular application domains. INTRODUCTIONThis paper motivates and describes the use of OB], based on Release 2.04 of the OBJ3 system. OBJ3 is the latest in a series of OB] systems, each of which has been rigorously based upon equational logic; however, the semantic basis of OB] is not developed here in full detail. The OBJ3 system is implemented in Common Lisp, and is based on ideas from order sorted algebra and parameterized programming. ...

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A Framework for Operational Equational Specifications with Pre-defined Structures

Avenhaus

Becker

1999

Journal of Symbolic Computation

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We present a general framework for studying equational specifications with pre-defined structures. The axioms of the specifications are to define new structures in addition to the given ones. In particular, they may define a new operator only partially over some given domain. Our approach allows one to assign easily semantics to such specifications in a denotational and operational fashion. In order to enable functional-style computations, we introduce a semantically enriched notion of term rewriting. This rewrite relation also allows us to infer the consistency of the specification. For the latter purpose one has to show confluence modulo the given structures. We outline how to obtain criteria easily for confluence and termination of the rewrite relation of discourse by generalizing results of the classical syntactic rewrite theory.

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A category-based equational logic semantics to constraint programming

Cited by 17 publications

References 16 publications

Complete Categorical Deduction for Satisfaction as Injectivity

Complete Categorical Deduction for Satisfaction as Injectivity

Introducing OBJ

A Framework for Operational Equational Specifications with Pre-defined Structures

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