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115
The Foundation of a Generic Theorem Prover
 Journal of Automated Reasoning
, 1989
"... Isabelle [28, 30] is an interactive theorem prover that supports a variety of logics. It represents rules as propositions (not as functions) and builds proofs by combining rules. These operations constitute a metalogic (or `logical framework') in which the objectlogics are formalized. Isabell ..."
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Cited by 423 (47 self)
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Isabelle [28, 30] is an interactive theorem prover that supports a variety of logics. It represents rules as propositions (not as functions) and builds proofs by combining rules. These operations constitute a metalogic (or `logical framework') in which the objectlogics are formalized. Isabelle is now based on higherorder logic  a precise and wellunderstood foundation. Examples illustrate use of this metalogic to formalize logics and proofs. Axioms for firstorder logic are shown sound and complete. Backwards proof is formalized by metareasoning about objectlevel entailment. Higherorder logic has several practical advantages over other metalogics. Many proof techniques are known, such as Huet's higherorder unification procedure. Key words: higherorder logic, higherorder unification, Isabelle, LCF, logical frameworks, metareasoning, natural deduction Contents 1 History and overview 2 2 The metalogic M 4 2.1 Syntax of the metalogic ......................... 4 2.2 ...
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.
A Linear Logical Framework
, 1996
"... We present the linear type theory LLF as the forAppeared in the proceedings of the Eleventh Annual IEEE Symposium on Logic in Computer Science  LICS'96 (E. Clarke editor), pp. 264275, New Brunswick, NJ, July 2730 1996. mal basis for a conservative extension of the LF logical framework. ..."
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Cited by 215 (44 self)
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We present the linear type theory LLF as the forAppeared in the proceedings of the Eleventh Annual IEEE Symposium on Logic in Computer Science  LICS'96 (E. Clarke editor), pp. 264275, New Brunswick, NJ, July 2730 1996. mal basis for a conservative extension of the LF logical framework. LLF combines the expressive power of dependent types with linear logic to permit the natural and concise representation of a whole new class of deductive systems, namely those dealing with state. As an example we encode a version of MiniML with references including its type system, its operational semantics, and a proof of type preservation. Another example is the encoding of a sequent calculus for classical linear logic and its cut elimination theorem. LLF can also be given an operational interpretation as a logic programming language under which the representations above can be used for type inference, evaluation and cutelimination. 1 Introduction A logical framework is a formal system desig...
Logic Programming in the LF Logical Framework
, 1991
"... this paper we describe Elf, a metalanguage intended for environments dealing with deductive systems represented in LF. While this paper is intended to include a full description of the Elf core language, we only state, but do not prove here the most important theorems regarding the basic building b ..."
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Cited by 175 (50 self)
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this paper we describe Elf, a metalanguage intended for environments dealing with deductive systems represented in LF. While this paper is intended to include a full description of the Elf core language, we only state, but do not prove here the most important theorems regarding the basic building blocks of Elf. These proofs are left to a future paper. A preliminary account of Elf can be found in [26]. The range of applications of Elf includes theorem proving and proof transformation in various logics, definition and execution of structured operational and natural semantics for programming languages, type checking and type inference, etc. The basic idea behind Elf is to unify logic definition (in the style of LF) with logic programming (in the style of Prolog, see [22, 24]). It achieves this unification by giving types an operational interpretation, much the same way that Prolog gives certain formulas (Hornclauses) an operational interpretation. An alternative approach to logic programming in LF has been developed independently by Pym [28]. Here are some of the salient characteristics of our unified approach to logic definition and metaprogramming. First of all, the Elf search process automatically constructs terms that can represent objectlogic proofs, and thus a program need not construct them explicitly. This is in contrast to logic programming languages where executing a logic program corresponds to theorem proving in a metalogic, but a metaproof is never constructed or used and it is solely the programmer's responsibility to construct objectlogic proofs where they are needed. Secondly, the partial correctness of many metaprograms with respect to a given logic can be expressed and proved by Elf itself (see the example in Section 5). This creates the possibilit...
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.
Information Structure and the SyntaxPhonology Interface
, 1998
"... The paper proposes a theory relating syntax, semantics, and intonational prosody, and covering a wide range of English intonational tunes and their semantic interpretation in terms of focus and information structure. The theory is based on a version of combinatory categorial grammar which directly p ..."
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Cited by 114 (5 self)
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The paper proposes a theory relating syntax, semantics, and intonational prosody, and covering a wide range of English intonational tunes and their semantic interpretation in terms of focus and information structure. The theory is based on a version of combinatory categorial grammar which directly pairs phonological and logical forms without intermediary representational levels.
Ellipsis and higherorder unification
 Linguistics and Philosophy
, 1991
"... We present a new method for characterizing the interpretive possibilities generated by elliptical constructions in natural language. Unlike previous analyses, which postulate ambiguity of interpretation or derivation in the full clause source of the ellipsis, our analysis requires no such hidden amb ..."
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Cited by 110 (1 self)
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We present a new method for characterizing the interpretive possibilities generated by elliptical constructions in natural language. Unlike previous analyses, which postulate ambiguity of interpretation or derivation in the full clause source of the ellipsis, our analysis requires no such hidden ambiguity. Further, the analysis follows relatively directly from an abstract statement of the ellipsis interpretation problem. It predicts correctly a wide range of interactions between ellipsis and other semantic phenomena such as quantifier scope and bound anaphora. Finally, although the analysis itself is stated nonprocedurally, it admits of a direct computational method for generating interpretations. This article is available through the Computation and Language EPrint Archive as cmplg/9503008, and also appears in Linguistics and Philosophy 14(4):399–452. cmplg/9503008 Ellipsis and HigherOrder Unification 1
Reasoning with higherorder abstract syntax in a logical framework
, 2008
"... Logical frameworks based on intuitionistic or linear logics with highertype quantification have been successfully used to give highlevel, modular, and formal specifications of many important judgments in the area of programming languages and inference systems. Given such specifications, it is natu ..."
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Cited by 91 (24 self)
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Logical frameworks based on intuitionistic or linear logics with highertype quantification have been successfully used to give highlevel, modular, and formal specifications of many important judgments in the area of programming languages and inference systems. Given such specifications, it is natural to consider proving properties about the specified systems in the framework: for example, given the specification of evaluation for a functional programming language, prove that the language is deterministic or that evaluation preserves types. One challenge in developing a framework for such reasoning is that higherorder abstract syntax (HOAS), an elegant and declarative treatment of objectlevel abstraction and substitution, is difficult to treat in proofs involving induction. In this paper, we present a metalogic that can be used to reason about judgments coded using HOAS; this metalogic is an extension of a simple intuitionistic logic that admits higherorder quantification over simply typed λterms (key ingredients for HOAS) as well as induction and a notion of definition. The latter concept of definition is a prooftheoretic device that allows certain theories to be treated as “closed ” or as defining fixed points. We explore the difficulties of formal metatheoretic analysis of HOAS encodings by considering encodings of intuitionistic and linear logics, and formally derive the admissibility of cut for important subsets of these logics. We then propose an approach to avoid the apparent tradeoff between the benefits of higherorder abstract syntax and the ability to analyze the resulting encodings. We illustrate this approach through examples involving the simple functional and imperative programming languages PCF and PCF:=. We formally derive such properties as unicity of typing, subject reduction, determinacy of evaluation, and the equivalence of transition semantics and natural semantics presentations of evaluation.
PROOFS IN HIGHERORDER LOGIC
, 1983
"... Expansion trees are defined as generalizations of Herbrand instances for formulas in a nonextensional form of higherorder logic based on Church’s simple theory of types. Such expansion trees can be defined with or without the use of skolem functions. These trees store substitution terms and either ..."
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Cited by 72 (14 self)
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Expansion trees are defined as generalizations of Herbrand instances for formulas in a nonextensional form of higherorder logic based on Church’s simple theory of types. Such expansion trees can be defined with or without the use of skolem functions. These trees store substitution terms and either critical variables or skolem terms used to instantiate quantifiers in the original formula and those resulting from instantiations. An expansion tree is called an expansion tree proof (ETproof) if it encodes a tautology, and, in the form not using skolem functions, an “imbedding ” relation among the critical variables be acyclic. The relative completeness result for expansion tree proofs not using skolem functions, i.e. if A is provable in higherorder logic then A has such an expansion tree proof, is based on Andrews ’ formulation of Takahashi’s proof of the cutelimination theorem for higherorder logic. If the occurrences of skolem functions in instantiation terms are restricted appropriately, the use of skolem functions in place of critical variables is equivalent to the requirement that the imbedding relation is acyclic. This fact not only resolves the open question of what