## Category-based Semantics for Equational and Constraint Logic Programming (1994)

Citations: | 24 - 10 self |

### BibTeX

@TECHREPORT{Diaconescu94category-basedsemantics,

author = {Razvan Diaconescu},

title = {Category-based Semantics for Equational and Constraint Logic Programming},

institution = {},

year = {1994}

}

### Years of Citing Articles

### OpenURL

### Abstract

This thesis proposes a general framework for equational logic programming, called categorybased equational logic by placing the general principles underlying the design of the programming language Eqlog and formulated by Goguen and Meseguer into an abstract form. This framework generalises equational deduction to an arbitrary category satisfying certain natural conditions; completeness is proved under a hypothesis of quantifier projectivity, using a semantic treatment that regards quantifiers as models rather than variables, and regards valuations as model morphisms rather than functions. This is used as a basis for a model theoretic category-based approach to a paramodulation-based operational semantics for equational logic programming languages. Category-based equational logic in conjunction with the theory of institutions is used to give mathematical foundations for modularisation in equational logic programming. We study the soundness and completeness problem for module imports i...

### Citations

921 |
Categories for the working mathematician
- Lane
- 1998
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785 | Constraint Logic Programming
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- 1987
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Citation Context ... Unfortunately, the users of Prolog-like systems still need to supply a lot of control information. During the 1980's, the constraint programming paradigm gradually grew out of logic programming (see =-=[58]-=-). This brought a new perspective on logic programming, in which the concept of unification is generalised to the concept of constraint solving [16, 15]. However, Lassez showed that constraint logic p... |

476 |
Institutions: Abstract model theory for specication and programming
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- 1992
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Citation Context ...theory of algebraic specifications and abstract data types. The theory of algebraic specifications entered a completely new era with the discovery of the theory of institutions by Goguen and Burstall =-=[33]-=-, transcending its origins in equational logic to encompass a wide variety of logic systems, including first order logic, Horn clause logic, higher order logic, infinitary logic, dynamic logic, intuit... |

440 |
Can programming be liberated from the von neumann style? A functional style and its algebra of programs
- Backus
- 1978
(Show Context)
Citation Context ...gic) and most of the actual implementations which are far from having a logic-based semantics. In general, they tend in the direction of imperative programming, which can be confusing and inefficient =-=[2]-=-. (The argument is that the denotational semantics of imperative programs is complex and complicated, with the ultimate practical consequence being that the debugging is very hard.) As Goguen and Mese... |

364 |
Confluent reductions: Abstract properties and applications to term rewriting systems: Abstract properties and applications to term rewriting systems
- Huet
- 1980
(Show Context)
Citation Context ...ial] algebra of ground terms T \Sigma for an MSA signature \Sigma. In Section 4.5.3 we explain the relationship between A-confluence and the notion of confluence modulo an equivalence as presented in =-=[55, 61, 20, 30]-=-. The following establishes the crucial link between the confluence of \Gamma and the transitivity of the paramodulation relation induced by \Gamma: Proposition 4.44 Consider a model A and \Gamma a co... |

332 | An efficient unification algorithm
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- 1982
(Show Context)
Citation Context ...g as an inference rule without side conditions, [54] introduces a new representation for goals consisting of a skeleton part 39 Originating from Martelli and Montanari's work on syntactic unification =-=[71]-=-. 40 Obtained by renaming all variables in the rule with new names. 48 (just goals in the ordinary sense, i.e., pairs of terms) and an environment part (the accumulation of the computed substitutions)... |

273 |
An Initial Algebra Approach to the Specification, Correctness and Implementation of Abstract Data Types
- Goguen, Thatcher, et al.
- 1978
(Show Context)
Citation Context ...ature \Sigma in the sense of Definition 2.41, the interpretations of an overloaded operation symbol oe 2 \Sigma w 1 ;s 1 "\Sigma w 2 ;s 2 in an algebra A need not necessarily agree on elements 28=-= See [43]-=- for a discussion of the difficulties with handling errors in MSA. 29 Given a poset (S ; ), let j denote the transitive and symmetric closure of . Then j is an equivalence relation whose equivalence c... |

229 |
Abstract and Concrete Categories
- Adámek, Herrlich, et al.
- 1990
(Show Context)
Citation Context ..., in the sense that xj; f ] U = f . x xFU xF AU A xj // f B B B B B B !! f ] U ----w w w w w w w f ] --- --- --- --- --- --- Notice that (A; U) can be regarded as a concrete category (in the sense of =-=[60]-=-) over the category of domains. The condition that U preserves pullbacks relates to the fact that congruences are equivalences; this will become more transparent later. Notice that U automatically pre... |

218 |
Unification revisited
- Lassez, Maher, et al.
- 1986
(Show Context)
Citation Context ... 33 whereas normal equational logic does not. Our terminology is influenced by Lassez who replaced the traditional logic programming terminology of computed answer substitution by that of solved form =-=[67]-=-. The modern terminology has the advantage to allow more flexibility for the representations of solutions (i.e., staying away of from the traditional representations of solution forms as substitutions... |

208 | Order-sorted algebra I: Equational deduction for multiple inheritance, overloading, exceptions and partial operations
- Goguen
- 1992
(Show Context)
Citation Context ...te by Goguen and Meseguer before mid 1980's [37], but because of its inability to handle erorrs, it was replaced by order sorted equational logic (which is many sorted equational logic with subtyping =-=[41]-=-) as the modern logical system underlying the theory of algebraic specifications and abstract data types. The theory of algebraic specifications entered a completely new era with the discovery of the ... |

195 |
The Situation in Logic
- Barwise
- 1989
(Show Context)
Citation Context ...bstract model theory. By abstract model theory (abbreviated AMT) we mean far more than the respective tradition in logic which abstracts the Tarskian approach to cover other logical systems 11 (e.g., =-=[6, 5]-=-). Our category-based framework is very close in spirit to the theory of institutions [33] in the sense that ffl it abstracts Tarski's classic semantic definition of truth [93], based on a relation of... |

189 | Toposes, Triples, and Theories
- Barr, Wells
- 1985
(Show Context)
Citation Context ...tbooks. 15 As opposed to to the syntactic perspective that regards terms as tree-like syntactic constructs. 16 Not to be confused to the Computing concept of module! 6 nicely), the theory of sketches =-=[4], and the -=-recently developed theory of "abstract algebraic institutions" [89, 91]. However, no uniform proof theory has previously been developed for all these equational logics. It could be argued th... |

185 |
Initial Algebra Semantics and Continuous Algebras
- Goguen, Thatcher, et al.
- 1977
(Show Context)
Citation Context ...ming. For example, the construction of initial models, and more generally of free models of logic programs [39], follows immediately from the well known construction of initial and free algebras (see =-=[41, 45]-=-, etc). The same principle applies to free extensions along theory morphisms, which were suggested in [39] as a semantic basis for constraint logic programming. Recall (e.g., from [33]) that a (many s... |

175 |
On the structure of abstract algebras
- Birkhoff
- 1935
(Show Context)
Citation Context ...tems [21]. 32 3.3 Completeness Our approach to the completeness of category-based equational deduction follows the traditional approach (probably originating with Birkhoff's work on universal algebra =-=[10]-=-), in that the central concept is the congruence determined by an arbitrary collection \Gamma of [conditional] equations on an arbitrary model A. The key to the completeness result is to regard this c... |

168 |
General logics
- Meseguer
- 1989
(Show Context)
Citation Context ...) ! Mod(T 0 ). This property, which we call exactness, seems to have first arisen in [85], and is also used in the pioneering work of Tarlecki [91] on abstract algebraic institutions, and of Meseguer =-=[72]-=- on categorical logics 63 . 61 Meaning that M j= \Sigma e for any \Sigma-model M that satisfies all sentences in E . 62 In the case of the usual logical systems, this corresponds exactly to the power ... |

166 |
Canonical forms and unification
- Hullot
- 1980
(Show Context)
Citation Context ...oduced by Slagle [86]. Later, narrowing was used as a basis for semantic unification (i.e., unification modulo a set of rules) algorithms. Basic narrowing appeared for the first time in Hullot's work =-=[57]-=-. The completeness result for innermost narrowing in the context of canonical term rewriting systems is originally due to Fribourg [22]. Holldobler's thesis [54] gives a systematic presentation of the... |

161 |
The semantics of Clear, a specification language
- Burstall, Goguen
- 1980
(Show Context)
Citation Context ..., infinitary logic, dynamic logic, intuitionistic logic, order sorted logic, temporal logic, etc. Today, nearly 15 years after the first insights given by the work on the specification language Clear =-=[13]-=-, the spirit of abstract model theory (in its institutional form) is a significant part of the culture of algebraic specification. The language OBJ [46] played a major role in the development of algeb... |

161 |
Solving equations in abstract algebras: A rulebased survey of unification
- Jouannaud, Kirchner
- 1991
(Show Context)
Citation Context ... of a signature plus some variables. In the context of many sorted equational logic the importance of explicit quantification was emphasized for the first time by Goguen and Meseguer [37]. The survey =-=[62]-=- shows that explicit qunatification adds a key syntactic information in the case of constraints and unification. In this way, the quantifier becomes part of the concept of equation. Although terms are... |

145 |
Completion of a set of rules modulo a set of equations
- Jouannaud, Kirchner
- 1986
(Show Context)
Citation Context ...ial] algebra of ground terms T \Sigma for an MSA signature \Sigma. In Section 4.5.3 we explain the relationship between A-confluence and the notion of confluence modulo an equivalence as presented in =-=[55, 61, 20, 30]-=-. The following establishes the crucial link between the confluence of \Gamma and the transitivity of the paramodulation relation induced by \Gamma: Proposition 4.44 Consider a model A and \Gamma a co... |

144 |
Putting Theories Together to Make Specifications
- Burstall, Goguen
- 1977
(Show Context)
Citation Context ... of module is essentially syntactic: it concerns texts that describe systems. The earliest work that we know on specification modules is by Goguen and Burstall, for their specification language Clear =-=[12, 13]-=-, the semantics of which is based on institutions. 59 This approach to modules has been applied to various logic-based languages, particularly OBJ [46], Eqlog [38], foops [40, 47] (which combines the ... |

115 |
Module algebra
- Bergstra, Heering, et al.
- 1990
(Show Context)
Citation Context ...in [23], and others occur in the proofs in [13]. Some recent results on the formal properties of module composition over institutions appear in [29]. The module algebra of Bergstra, Heering and Klint =-=[9]-=- attempts to capture the horizontal structure of modules with equations among certain basic operations on modules, including sum, renaming, and information hiding. These equations, together with const... |

100 |
EQLOG: equality, types and generic modules for logic programming
- Goguen, Meseguer
(Show Context)
Citation Context ...ogramming. It belongs to the tradition of equational and constraint logic programming started by Goguen and Meseguer in their pioneering work on the programming language Eqlog during the mid-eighties =-=[38, 39]-=-. Eqlog has been implemented in Oxford by the author of this thesis as an extension of the SRI implementation of OBJ3. 3 1.1 The Equational Logic Programming Paradigm 1.1.1 A historical perspective Eq... |

99 |
Unifying functional, object-oriented and relational programming with logical semantics
- Goguen, Meseguer
- 1987
(Show Context)
Citation Context ...ation language Clear [12, 13], the semantics of which is based on institutions. 59 This approach to modules has been applied to various logic-based languages, particularly OBJ [46], Eqlog [38], foops =-=[40, 47]-=- (which combines the functional and object paradigms), and fooplog [40] (which combines functional, logic and object paradigms); it could also be applied to any pure logic-based programming language, ... |

98 |
Universal Algebra
- Grätzer
- 1979
(Show Context)
Citation Context ... is in fact the coproduct A $$ ` xF 0 between A $$ and the free \Sigma 0 -model xF 0 . When \Sigma = \Sigma 0 are unsorted algebraic signatures, this is a well known construction in universal algebra =-=[48]-=-. However, the best known example still comes from linear algebra: Example 6.2 Let X be a set of variables. R[X ] is the ring of polynomials over X and with real numbers as coefficients. In this examp... |

95 | Equations and rewrite rules - a survey
- Huet, Oppen
- 1980
(Show Context)
Citation Context ...s Equational deduction modulo a set of axioms (abbreviated ELM ) becomes vital when dealing with non-orientable equations in the context of rewriting. A detailed exposition of the subject is given in =-=[30, 56, 20, 63]-=-. Although in practice non-orientable axioms are mostly unconditional 32 , there is no theoretical reason to exclude the case of equational deduction modulo a set of conditional equations. Definition ... |

92 | SLOG: A logic programming language interpreter based on clausal superposition and rewriting - Fribourg - 1985 |

86 |
Computing in Systems Described by Equations
- O’DONNELL
- 1977
(Show Context)
Citation Context ...ed on Horn clause logic and equational (i.e., functional) programming based on equational logic, i.e., the logic of substituting equals for equals. One of the earliest contributions to this field was =-=[76]-=-. As Goguen and Meseguer repeatedly pointed out [38, 39], the best way to achieve this goal should be to unify the two logics involved. However, because equational logic is more fundamental than Horn ... |

85 | Logical support for modularisation
- Diaconescu, Goguen, et al.
- 1991
(Show Context)
Citation Context ..., this work has first order logic built into its choice of the constructors for signatures and sentences. However, Bergstra et al. abstract some interesting general principles from this special case. =-=[21]-=- develops a module algebra in the context of the theory of institutions. In [21] it is shown that all reasonable institutions support certain simple operations on theories; what properties ensure that... |

85 | Towards an algebraic semantics for the object paradigm
- GOGUEN, DIACONESCU
- 1994
(Show Context)
Citation Context ...is only indirectly related to linear algebra. In the same way, institutions may be relevant to an area only remotely related to equational logic programming, such as semantics for the object paradigm =-=[29, 11, 35]-=-. A uniform treatment of the model theory of classical equational logic is now possible due to the comprehensive development of categorical universal algebra; without any claim of completeness, I ment... |

79 |
Reusing and Interconnecting Software Components
- Goguen
- 1986
(Show Context)
Citation Context ... paradigms), and fooplog [40] (which combines functional, logic and object paradigms); it could also be applied to any pure logic-based programming language, such as (pure) Lisp and (pure) Prolog. In =-=[26]-=-, this is even extended to imperative programming. The module system of (Standard) ML [50] has also been strongly influenced by this work on Clear. Clear introduced the ideas that a specification modu... |

73 |
Graph expressions and graph rewritings
- Bauderon, B
(Show Context)
Citation Context ....e., for rewriting or narrowing). The most notorious cases are associativity (A), commutativity (C) and their combination (AC). Also, graph rewriting is a particular case of rewriting modulo a theory =-=[7]-=-. By taking a semantic perspective on computations modulo a theory we introduce the more abstract concept of paramodulation modulo a model morphism and use it for showing that computing in the quotien... |

72 |
Types as theories
- Goguen
- 1991
(Show Context)
Citation Context ...is only indirectly related to linear algebra. In the same way, institutions may be relevant to an area only remotely related to equational logic programming, such as semantics for the object paradigm =-=[29, 11, 35]-=-. A uniform treatment of the model theory of classical equational logic is now possible due to the comprehensive development of categorical universal algebra; without any claim of completeness, I ment... |

69 |
Algebraic Theories
- Manes
- 1974
(Show Context)
Citation Context ...the comprehensive development of categorical universal algebra; without any claim of completeness, I mention the so-called Lawvere algebraic theories, either in classical form [69] or in monadic form =-=[70]-=- (although neither of these fits order sorted algebra 12 Devoted to modularisation issues. 13 The precise definition is given in Chapter 5. 14 Sometimes called language or vocabulary in classical logi... |

61 |
Completeness of many-sorted equational logic
- Goguen, Meseguer
- 1985
(Show Context)
Citation Context ... Many sorted equational logic became the main logical system underlying the theory of algebraic specifications and abstract data types. It was proved complete by Goguen and Meseguer before mid 1980's =-=[37]-=-, but because of its inability to handle erorrs, it was replaced by order sorted equational logic (which is many sorted equational logic with subtyping [41]) as the modern logical system underlying th... |

57 |
Compiling Logic Programs with Equality
- Hanus
- 1990
(Show Context)
Citation Context ... The main consequence in this case is to speed up of the computation of innermost redeces. 53 Many other implementations of narrowing embed some sort of constructor discipline, notably the ALF system =-=[49]-=-. 54 Actually, the Eqlog system implements a stronger version of this: before a narrowing step is performed, the system tries to find the outermost occurrence at which the corresponding operators are ... |

54 |
Foundations of equational logic programming
- Hölldobler
- 1989
(Show Context)
Citation Context ...ed for the first time in Hullot's work [57]. The completeness result for innermost narrowing in the context of canonical term rewriting systems is originally due to Fribourg [22]. Holldobler's thesis =-=[54]-=- gives a systematic presentation of the current state of art of this field including also interesting historical references. Our presentation of the completeness of different refinements of paramodula... |

49 |
Unified algebras and institutions
- Mosses
- 1989
(Show Context)
Citation Context ...sorted sets and functions. The forgetful functor U : Alg \Sigma;E ! Set S forgets both the axioms and the algebraic structure of the algebras. Example 2.47 The logic of Mosses's unified algebras from =-=[75]-=- can be regarded as equational logic modulo a conditional theory. All unified specifications of a given unified signature contain a core essentially consisting of Horn clauses. Unified algebras appear... |

46 |
José Meseguer, Kokichi Futatsugi and JeanPierre Jouannaud. Introducing OBJ
- Goguen, Winkler
- 2000
(Show Context)
Citation Context ...by the work on the specification language Clear [13], the spirit of abstract model theory (in its institutional form) is a significant part of the culture of algebraic specification. The language OBJ =-=[46]-=- played a major role in the development of algebraic specification and, more generally, of declarative programming. It began as an algebraic specification language at UCLA about 1976, and has been fur... |

45 |
Axioms for Abstract Model Theory
- Barwise
- 1974
(Show Context)
Citation Context ...bstract model theory. By abstract model theory (abbreviated AMT) we mean far more than the respective tradition in logic which abstracts the Tarskian approach to cover other logical systems 11 (e.g., =-=[6, 5]-=-). Our category-based framework is very close in spirit to the theory of institutions [33] in the sense that ffl it abstracts Tarski's classic semantic definition of truth [93], based on a relation of... |

43 |
R.: A study in the foundations of programming methodology: Specifications, institutions, charters and parchments
- Goguen
- 1986
(Show Context)
Citation Context ...nclude that unsorted logics are unnatural for many applications in Computing Science. It is not hard to see that any chartered institution is exact. 64 Charters were introduced by Goguen and Burstall =-=[32]-=- as a general way to produce institutions. The basic intuition is that the syntax of a logical system is an initial algebra. Because it appears that most institutions of interest in Computing Science ... |

43 |
Order-Sorted Algebra Solves the Constructor-Selector, Multiple Representation and
- Goguen, Meseguer
- 1987
(Show Context)
Citation Context ...ady programmed constraint solvers, because they will represent data in various ways. Order sorted algebra provides a precise and systematic equational theory for this, based on initial semantics (see =-=[73]-=- for a detailed discussion, [34] and [73] for some further examples). Eqlog also supports loose specifications through its so-called theories, and provides views for asserting the satisfaction of theo... |

42 |
Models and equality for logical programming
- Goguen, Meseguer
- 1987
(Show Context)
Citation Context ...sed semantics for equational and constraint logic programming in the style of the language Eqlog, by placing the general principles underlying 7 A major advantage of Prolog is its good compiler. 8 In =-=[39] Goguen and Meseguer-=- use the terminology "generalised" instead of "extensible". 4 the design of the programming language Eqlog and formulated by Goguen and Meseguer in [38, 39] into an abstract form. ... |

39 |
Order sorted algebra
- Goguen
- 1978
(Show Context)
Citation Context ... ! A (t) in A. Call A reachable iff ! A is surjective, i.e., iff each element of A is "named" by some ground term. 2.3.2 Order sorted algebra The first paper on order sorted algebra (abbrevi=-=ated OSA) [25]-=- says that its main motivation is to provide a better way of treating errors in abstract data types; 28 another motivation is that the use of subsorts can greatly speed up certain theorem proving prob... |

37 |
Categorical foundations for general systems theory
- Goguen
- 1973
(Show Context)
Citation Context ...deas that a specification module determines a theory, and that such theories can be put together using colimits; these ideas have their origin in some earlier work by Goguen on General Systems Theory =-=[23, 36]-=-. Clear provided operations for summing, renaming, extending, hiding, importing and (in the case of generics) applying theories. Theories in turn denote classes of models. The earliest work that we kn... |

36 |
CAT: a system for the structured elaboration of correct programs from structured specifications
- Goguen, Burstall
(Show Context)
Citation Context ...g, importing and (in the case of generics) applying theories. Theories in turn denote classes of models. The earliest work that we know giving a calculus of modules is also due to Goguen and Burstall =-=[31]. Building-=- on Clear, they studied laws for horizontal structuring relationships, and vertical implementing (also called "refinement") relationships, concluding that the axioms of a 2-category should b... |

36 |
Decision procedures for simple equational theories with permutative axioms: Complete sets of permutative reductions
- Lankford, Ballantyne
- 1977
(Show Context)
Citation Context ...sses in the case of a quotienting morphism defined by a theory. The notion of rewriting on the congruence classes (called class-rewriting in the survey [20]) was introduced by Lankford and Ballantyne =-=[65]-=- for permutative congruences, that is congruences for which each congruence class is finite. For example, associativity and/or commutativity gives rise to permutative congruences. Let \Sigma be an alg... |

33 |
Theorem Proving and Algebra
- Goguen
- 1994
(Show Context)
Citation Context ...s Equational deduction modulo a set of axioms (abbreviated ELM ) becomes vital when dealing with non-orientable equations in the context of rewriting. A detailed exposition of the subject is given in =-=[30, 56, 20, 63]-=-. Although in practice non-orientable axioms are mostly unconditional 32 , there is no theoretical reason to exclude the case of equational deduction modulo a set of conditional equations. Definition ... |

32 |
and Răzvan Diaconescu. An Oxford survey of order sorted algebra
- Goguen
- 1994
(Show Context)
Citation Context ...e type of the logical variable Lst, the term car(Lst) is not well typed because car is defined only on 52 Inherited from the OBJ3 system; for a detailed discussion on retracts and their semantics see =-=[34]-=-. 66 the subsort NList of the non-empty lists. However, during the computation process the order sorted unification function changes the sort of Lst to NList. This could be easily noticed in the first... |

29 |
Higher-order functions considered unnecessary for higher-order programming
- Goguen
- 1990
(Show Context)
Citation Context ... module expression actually builds a software system from the given components. 22 Thus parameterised programming in Eqlog gives significant support for large programs through module composition, and =-=[28]-=- shows that it also provides the power of higher order functions. The semantics of module importation is given by conservative extensions of theories in Horn clause logic with equality [39]. The stron... |

29 |
Lawvere, Functorial Semantics of Algebraic Theories
- William
- 1963
(Show Context)
Citation Context ... is now possible due to the comprehensive development of categorical universal algebra; without any claim of completeness, I mention the so-called Lawvere algebraic theories, either in classical form =-=[69]-=- or in monadic form [70] (although neither of these fits order sorted algebra 12 Devoted to modularisation issues. 13 The precise definition is given in Chapter 5. 14 Sometimes called language or voca... |