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Logic Programming in a Fragment of Intuitionistic Linear Logic
"... When logic programming is based on the proof theory of intuitionistic logic, it is natural to allow implications in goals and in the bodies of clauses. Attempting to prove a goal of the form D ⊃ G from the context (set of formulas) Γ leads to an attempt to prove the goal G in the extended context Γ ..."
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Cited by 306 (40 self)
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When logic programming is based on the proof theory of intuitionistic logic, it is natural to allow implications in goals and in the bodies of clauses. Attempting to prove a goal of the form D ⊃ G from the context (set of formulas) Γ leads to an attempt to prove the goal G in the extended context Γ ∪ {D}. Thus during the bottomup search for a cutfree proof contexts, represented as the lefthand side of intuitionistic sequents, grow as stacks. While such an intuitionistic notion of context provides for elegant specifications of many computations, contexts can be made more expressive and flexible if they are based on linear logic. After presenting two equivalent formulations of a fragment of linear logic, we show that the fragment has a goaldirected interpretation, thereby partially justifying calling it a logic programming language. Logic programs based on the intuitionistic theory of hereditary Harrop formulas can be modularly embedded into this linear logic setting. Programming examples taken from theorem proving, natural language parsing, and data base programming are presented: each example requires a linear, rather than intuitionistic, notion of context to be modeled adequately. An interpreter for this logic programming language must address the problem of splitting contexts; that is, when attempting to prove a multiplicative conjunction (tensor), say G1 ⊗ G2, from the context ∆, the latter must be split into disjoint contexts ∆1 and ∆2 for which G1 follows from ∆1 and G2 follows from ∆2. Since there is an exponential number of such splits, it is important to delay the choice of a split as much as possible. A mechanism for the lazy splitting of contexts is presented based on viewing proof search as a process that takes a context, consumes part of it, and returns the rest (to be consumed elsewhere). In addition, we use collections of Kripke interpretations indexed by a commutative monoid to provide models for this logic programming language and show that logic programs admit a canonical model.
Principles and implementation of deductive parsing
 JOURNAL OF LOGIC PROGRAMMING
, 1995
"... We present a system for generating parsers based directly on the metaphor of parsing as deduction. Parsing algorithms can be represented directly as deduction systems, and a single deduction engine can interpret such deduction systems so as to implement the corresponding parser. The method generaliz ..."
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Cited by 165 (5 self)
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We present a system for generating parsers based directly on the metaphor of parsing as deduction. Parsing algorithms can be represented directly as deduction systems, and a single deduction engine can interpret such deduction systems so as to implement the corresponding parser. The method generalizes easily to parsers for augmented phrase structure formalisms, such as definiteclause grammars and other logic grammar formalisms, and has been used for rapid prototyping of parsing algorithms for a variety of formalisms including variants of treeadjoining grammars, categorial grammars, and lexicalized contextfree grammars.
Higherorder logic programming
 HANDBOOK OF LOGIC IN AI AND LOGIC PROGRAMMING, VOLUME 5: LOGIC PROGRAMMING. OXFORD (1998
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Specifying FillerGap Dependency Parsers in a LinearLogic Programming Language
 Proceedings of the Joint International Conference and Symposium on Logic Programming
, 1992
"... An aspect of the Generalized Phrase Structure Grammar formalism proposed by Gazdar, et al. is the introduction of the notion of "slashed categories " to handle the parsing of structures, such as relative clauses, which involve unbounded dependencies. This has been implemented in Definite Clause Gram ..."
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Cited by 27 (4 self)
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An aspect of the Generalized Phrase Structure Grammar formalism proposed by Gazdar, et al. is the introduction of the notion of "slashed categories " to handle the parsing of structures, such as relative clauses, which involve unbounded dependencies. This has been implemented in Definite Clause Grammars through the technique of gap threading, in which a difference list of extracted noun phrases (gaps) is maintained. However, this technique is cumbersome, and can result in subtle soundness problems in the implemented grammars. Miller and Pareschi have proposed a method of implementing gap threading at the logical level in intuitionistic logic. Unfortunately that implementation itself suffered from serious problems, which the authors recognized. This paper builds on work first presented with Miller in which we developed a fillergap dependency parser in Girard's linear logic. This implementation suffers from none of the pitfalls of either the traditional implementation, or the intuitioni...
Scoping Constructs In Logic Programming: Implementation Problems And Their Solution
, 1995
"... Machine (WAM). The provision of implications in goals results in the possibility of program clauses being added to the program for the purpose of solving specific subgoals. A naive scheme based on asserting and retracting program clauses does not suffice for implementing such additions for two reaso ..."
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Cited by 21 (9 self)
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Machine (WAM). The provision of implications in goals results in the possibility of program clauses being added to the program for the purpose of solving specific subgoals. A naive scheme based on asserting and retracting program clauses does not suffice for implementing such additions for two reasons. First, it is necessary to also support the resurrection of an earlier existing program in the face of backtracking. Second, the possibility for implication goals to be surrounded by quantifiers requires a consideration of the parameterization of program clauses by bindings for their free variables. Devices for supporting these additional requirements are described as also is the integration of these devices into the WAM. Further extensions to the machine are outlined for handling higherorder additions to the language. The ideas Work on this paper has been partially supported by NSF Grants CCR8905825 and CCR 9208465. Address correspondence to Gopalan Nadathur, Department of Compute...
HigherOrder Linear Logic Programming of Categorial Deduction’, Report de Recerca LSI–94–42–R, Departament de Llenguatges i
 Sistemes Informàtics, Universitat Politècnica de Catalunya Morrill, Glyn: 1994b, Type Logical Grammar: Categorial Logic of Signs
"... We show how categorial deduction can be implemented in higherorder (linear) logic programming, thereby realising parsing as deduction for the associative and nonassociative Lambek calculi. This provides a method of solution to the parsing problem of Lambek categorial grammar applicable to a variet ..."
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Cited by 13 (4 self)
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We show how categorial deduction can be implemented in higherorder (linear) logic programming, thereby realising parsing as deduction for the associative and nonassociative Lambek calculi. This provides a method of solution to the parsing problem of Lambek categorial grammar applicable to a variety of its extensions. The present work deals with the parsing problem for Lambek calculus and its extensions as developed
Higher Order Babel: Language and Implementation
 In editors Proc. 5th International Workshop on Extensions of Logic Programming ELP'96
, 1996
"... . We present the functional logic language Higher Order Babel which provides higher order unification for parameter passing and solving equations. When searching for a function which solves an equation, not only "polynomial functions" but also defined functions are taken into account. In contrast to ..."
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Cited by 11 (2 self)
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. We present the functional logic language Higher Order Babel which provides higher order unification for parameter passing and solving equations. When searching for a function which solves an equation, not only "polynomial functions" but also defined functions are taken into account. In contrast to all other programming languages which support higher order unification HOBabel replaces the expensive fireduction by the much more efficient combinator reduction. Moreover, HOBabel is more homogeneous since it does not distinguish functions which only represent data structures and defined functions which are equipped with the full execution mechanism of the language. 1 Introduction In comparison to purely logic programming languages, integrated functional logic programming languages allow a more efficient implementation since functions are deterministic and this determinism can be exploited to reduce the search space. On the other hand, functional logic languages have more expressive po...
Elimination of Negation in a Logical Framework
, 2000
"... Logical frameworks with a logic programming interpretation such as hereditary Harrop formulae (HHF) [15] cannot express directly negative information, although negation is a useful specification tool. Since negationasfailure does not fit well in a logical framework, especially one endowed with ..."
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Cited by 10 (3 self)
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Logical frameworks with a logic programming interpretation such as hereditary Harrop formulae (HHF) [15] cannot express directly negative information, although negation is a useful specification tool. Since negationasfailure does not fit well in a logical framework, especially one endowed with hypothetical and parametric judgements, we adapt the idea of elimination of negation introduced in [21] for Horn logic to a fragment of higherorder HHF. This entails finding a middle ground between the Closed World Assumption usually associated with negation and the Open World Assumption typical of logical frameworks; the main technical idea is to isolate a set of programs where static and dynamic clauses do not overlap.
Difference Lists and Difference Bags for Logic Programming of Categorial Deduction
"... We show how difference lists can be used for systematically compiled linear clauses for Lambek categorial grammar and its generalisations, in analogy with standard Horn clauses for CF grammar. We also consider use of difference bags for partitioning of linear sequents, and methods for ambiguity and ..."
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Cited by 4 (2 self)
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We show how difference lists can be used for systematically compiled linear clauses for Lambek categorial grammar and its generalisations, in analogy with standard Horn clauses for CF grammar. We also consider use of difference bags for partitioning of linear sequents, and methods for ambiguity and polymorphism.