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Uniform proofs as a foundation for logic programming
 ANNALS OF PURE AND APPLIED LOGIC
, 1991
"... A prooftheoretic characterization of logical languages that form suitable bases for Prologlike programming languages is provided. This characterization is based on the principle that the declarative meaning of a logic program, provided by provability in a logical system, should coincide with its ..."
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Cited by 385 (111 self)
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A prooftheoretic characterization of logical languages that form suitable bases for Prologlike programming languages is provided. This characterization is based on the principle that the declarative meaning of a logic program, provided by provability in a logical system, should coincide with its operational meaning, provided by interpreting logical connectives as simple and fixed search instructions. The operational semantics is formalized by the identification of a class of cutfree sequent proofs called uniform proofs. A uniform proof is one that can be found by a goaldirected search that respects the interpretation of the logical connectives as search instructions. The concept of a uniform proof is used to define the notion of an abstract logic programming language, and it is shown that firstorder and higherorder Horn clauses with classical provability are examples of such a language. Horn clauses are then generalized to hereditary Harrop formulas and it is shown that firstorder and higherorder versions of this new class of formulas are also abstract logic programming languages if the inference rules are those of either intuitionistic or minimal logic. The programming language significance of the various generalizations to firstorder Horn clauses is briefly discussed.
HigherOrder Horn Clauses
 JOURNAL OF THE ACM
, 1990
"... A generalization of Horn clauses to a higherorder logic is described and examined as a basis for logic programming. In qualitative terms, these higherorder Horn clauses are obtained from the firstorder ones by replacing firstorder terms with simply typed #terms and by permitting quantification ..."
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Cited by 62 (21 self)
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A generalization of Horn clauses to a higherorder logic is described and examined as a basis for logic programming. In qualitative terms, these higherorder Horn clauses are obtained from the firstorder ones by replacing firstorder terms with simply typed #terms and by permitting quantification over all occurrences of function symbols and some occurrences of predicate symbols. Several prooftheoretic results concerning these extended clauses are presented. One result shows that although the substitutions for predicate variables can be quite complex in general, the substitutions necessary in the context of higherorder Horn clauses are tightly constrained. This observation is used to show that these higherorder formulas can specify computations in a fashion similar to firstorder Horn clauses. A complete theorem proving procedure is also described for the extension. This procedure is obtained by interweaving higherorder unification with backchaining and goal reductions, and constitutes a higherorder generalization of SLDresolution. These results have a practical realization in the higherorder logic programming language called λProlog.
A Direct Semantic Characterization of RELFUN
 Proceedings of the 3rd International Workshop on Extensions of Logic Programming, ELP '92, Bologna 1992, volume 660 of LNAI
, 1998
"... Introduction RELFUN is a logic language primarily extended by callbyvalue (eager) functions that may be nonground, nondeterministic, varyingarity, and higherorder. These functions are defined by extended Horn clauses having a `foot' premise for value returning. This extension can also be ..."
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Cited by 3 (3 self)
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Introduction RELFUN is a logic language primarily extended by callbyvalue (eager) functions that may be nonground, nondeterministic, varyingarity, and higherorder. These functions are defined by extended Horn clauses having a `foot' premise for value returning. This extension can also be viewed as (directed) conditional equations permitting `extra' variables in conditions, which may accumulate partial results. It entails the following syntactic changes of PROLOG: Footed clauses: Starting with DATALOG, ":"rules may be augmented by an ampersand infix, "&", between the normal body premises and the foot premise; facts (empty bodies), by a joined infix, ":&". Active expressions: Proceeding to PROLOG, passive structures are rewritten using square brackets, "[...]", reserving round parentheses, "(...)", for RELFUN's active callbyvalue expressions (per
HigherOrder Logic as the Basis for Logic Programming
, 1989
"... : A generalization of Horn clauses to a higherorder logic is described and examined as a basis for logic programming. In qualitative terms, these higherorder Horn clauses are obtained from the firstorder ones by replacing firstorder terms with simply typed terms and by permitting quantification ..."
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: A generalization of Horn clauses to a higherorder logic is described and examined as a basis for logic programming. In qualitative terms, these higherorder Horn clauses are obtained from the firstorder ones by replacing firstorder terms with simply typed terms and by permitting quantification over all occurrences of function symbols and some occurrences of predicate symbols. Several prooftheoretic results concerning these extended clauses are presented. One result shows that although the substitutions for predicate variables can be quite complex in general, the substitutions necessary in the context of higherorder Horn clauses are tightly constrained. This observation is used to show that these higherorder formulas can specify computations in a fashion similar to firstorder Horn clauses. A complete theorem proving procedure is also described for the extension. This procedure is obtained by interweaving higherorder unification with backchaining and goal reductions, and const...
A Extensional HigherOrder Logic Programming
"... We propose a purely extensional semantics for higherorder logic programming. In this semantics program predicates denote sets of ordered tuples, and two predicates are equal iff they are equal as sets. Moreover, every program has a unique minimum Herbrand model which is the greatest lower bound of ..."
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We propose a purely extensional semantics for higherorder logic programming. In this semantics program predicates denote sets of ordered tuples, and two predicates are equal iff they are equal as sets. Moreover, every program has a unique minimum Herbrand model which is the greatest lower bound of all Herbrand models of the program and the least fixedpoint of an immediate consequence operator. We also propose an SLDresolution proof system which is proven sound and complete with respect to the minimum Herbrand model semantics. In other words, we provide a purely extensional theoretical framework for higherorder logic programming which generalizes the familiar theory of classical (firstorder) logic programming.