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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 377 (110 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.
A logic programming language with lambdaabstraction, function variables, and simple unification
 Extensions of Logic Programming. Springer Lecture Notes in Artificial Intelligence
, 1990
"... A meta programming language must be able to represent and manipulate such syntactic structures as programs, formulas, types, and proofs. A common characteristic of all these structures is that they involve notions of abstractions, scope, bound and free variables, substitution instances, and equality ..."
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Cited by 288 (25 self)
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A meta programming language must be able to represent and manipulate such syntactic structures as programs, formulas, types, and proofs. A common characteristic of all these structures is that they involve notions of abstractions, scope, bound and free variables, substitution instances, and equality up to alphabetic changes of bound variables.
Unification under a mixed prefix
 Journal of Symbolic Computation
, 1992
"... Unification problems are identified with conjunctions of equations between simply typed λterms where free variables in the equations can be universally or existentially quantified. Two schemes for simplifying quantifier alternation, called Skolemization and raising (a dual of Skolemization), are pr ..."
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Cited by 123 (13 self)
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Unification problems are identified with conjunctions of equations between simply typed λterms where free variables in the equations can be universally or existentially quantified. Two schemes for simplifying quantifier alternation, called Skolemization and raising (a dual of Skolemization), are presented. In this setting where variables of functional type can be quantified and not all types contain closed terms, the naive generalization of firstorder Skolemization has several technical problems that are addressed. The method of searching for preunifiers described by Huet is easily extended to the mixed prefix setting, although solving flexibleflexible unification problems is undecidable since types may be empty. Unification problems may have numerous incomparable unifiers. Occasionally, unifiers share common factors and several of these are presented. Various optimizations on the general unification search problem are as discussed. 1.
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.
An Extension to ML to Handle Bound Variables in Data Structures
, 1990
"... Most conventional programming languages have direct methods for representing firstorder terms (say, via concrete datatypes in ML). If it is necessary to represent structures containing bound variables, such as λterms, formulas, types, or proofs, these must first be mapped into firstorder terms, a ..."
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Cited by 36 (1 self)
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Most conventional programming languages have direct methods for representing firstorder terms (say, via concrete datatypes in ML). If it is necessary to represent structures containing bound variables, such as λterms, formulas, types, or proofs, these must first be mapped into firstorder terms, and then a significant number of auxiliary procedures must be implemented to manage bound variable names, check for free occurrences, do substitution, test for equality modulo alphaconversion, etc. We shall show how the applicative core of the ML programming language can be enhanced so that λterms can be represented more directly and so that the enhanced language, called MLλ, provides a more elegant method of manipulating bound variables within data structures. In fact, the names of bound variables will not be accessible to the MLλ programmer. This extension to ML involves the following: introduction of the new type constructor ’a => ’b for the type of λterms formed by abstracting a parameter of type ’a out of a term of type ’b; a very restricted and simple form of higherorder pattern matching; a method for extending a given data structure with a new constructor; and, a method for extending function definitions to handle such new constructors. We present several examples of MLλ programs.
Nominal logic programming
, 2006
"... Nominal logic is an extension of firstorder logic which provides a simple foundation for formalizing and reasoning about abstract syntax modulo consistent renaming of bound names (that is, αequivalence). This article investigates logic programming based on nominal logic. This technique is especial ..."
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Cited by 23 (8 self)
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Nominal logic is an extension of firstorder logic which provides a simple foundation for formalizing and reasoning about abstract syntax modulo consistent renaming of bound names (that is, αequivalence). This article investigates logic programming based on nominal logic. This technique is especially wellsuited for prototyping type systems, proof theories, operational semantics rules, and other formal systems in which bound names are present. In many cases, nominal logic programs are essentially literal translations of “paper” specifications. As such, nominal logic programming provides an executable specification language for prototyping, communicating, and experimenting with formal systems. We describe some typical nominal logic programs, and develop the modeltheoretic, prooftheoretic, and operational semantics of such programs. Besides being of interest for ensuring the correct behavior of implementations, these results provide a rigorous foundation for techniques for analysis and reasoning about nominal logic programs, as we illustrate via two examples.
Higher Order Logic
 In Handbook of Logic in Artificial Intelligence and Logic Programming
, 1994
"... Contents 1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 2 The expressive power of second order Logic : : : : : : : : : : : 3 2.1 The language of second order logic : : : : : : : : : : : : : 3 2.2 Expressing size : : : : : : : : : : : : : : : : : : : : : : : : 4 2.3 Definin ..."
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Cited by 18 (0 self)
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Contents 1 Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2 2 The expressive power of second order Logic : : : : : : : : : : : 3 2.1 The language of second order logic : : : : : : : : : : : : : 3 2.2 Expressing size : : : : : : : : : : : : : : : : : : : : : : : : 4 2.3 Defining data types : : : : : : : : : : : : : : : : : : : : : 6 2.4 Describing processes : : : : : : : : : : : : : : : : : : : : : 8 2.5 Expressing convergence using second order validity : : : : : : : : : : : : : : : : : : : : : : : : : 9 2.6 Truth definitions: the analytical hierarchy : : : : : : : : 10 2.7 Inductive definitions : : : : : : : : : : : : : : : : : : : : : 13 3 Canonical semantics of higher order logic : : : : : : : : : : : : 15 3.1 Tarskian semantics of second order logic : : : : : : : : : 15 3.2 Function and re
Higherorder Unification with Dependent Function Types
 3rd Int. Conf. Rewriting Techniques and Applications, LNCS 355
, 1989
"... Roughly fifteen years ago, Huet developed a complete semidecision algorithm for unification in the simply typed calculus ( ! ). In spite of the undecidability of this problem, his algorithm is quite usable in practice. Since then, many important applications have come about in such areas as theorem ..."
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Cited by 17 (0 self)
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Roughly fifteen years ago, Huet developed a complete semidecision algorithm for unification in the simply typed calculus ( ! ). In spite of the undecidability of this problem, his algorithm is quite usable in practice. Since then, many important applications have come about in such areas as theorem proving, type inference, program transformation, and machine learning. Another development is the discovery that by enriching ! to include dependent function types, the resulting calculus ( \Pi ) forms the basis of a very elegant and expressive Logical Framework, encompassing the syntax, rules, and proofs for a wide class of logics. This paper presents an algorithm in the spirit of Huet's, for unification in \Pi . This algorithm gives us the best of both worlds: the automation previously possible in ! , and the greatly enriched expressive power of \Pi . It can be used to considerable advantage in many of the current applications of Huet's algorithm, and has important new applications as w...
The Ergo Support System: An integrated set of tools for prototyping integrated environments
 SCHOOL OF COMPUTER SCIENCE, CARNEGIE MELLON UNIVERSITY, PITTSBURGH
, 1988
"... The Ergo Support System (ESS) is an engineering framework for experimentation and prototyping to support the application of formal methods to program development, ranging from program analysis and derivation to prooftheoretic approaches. The ESS is a growing suite of tools that are linked together ..."
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Cited by 14 (3 self)
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The Ergo Support System (ESS) is an engineering framework for experimentation and prototyping to support the application of formal methods to program development, ranging from program analysis and derivation to prooftheoretic approaches. The ESS is a growing suite of tools that are linked together by means of a set of abstract interfaces. The principal engineering challenge is the design of abstract interfaces that are semantically rich and yet flexible enough to permit experimentation with a wide variety of formallybased program and proof development paradigms and associated languages. As part of the design of ESS, several abstract interface designs have been developed that provide for more effective component integration while preserving flexibility and the potential for scaling. A benefit of the open architecture approach of ESS is the ability to mix formal and informal approaches in the same environment architecture. The ESS has already been applied in a number of formal methods experiments.