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A Framework for Defining Logics
 JOURNAL OF THE ASSOCIATION FOR COMPUTING MACHINERY
, 1993
"... The Edinburgh Logical Framework (LF) provides a means to define (or present) logics. It is based on a general treatment of syntax, rules, and proofs by means of a typed calculus with dependent types. Syntax is treated in a style similar to, but more general than, MartinLof's system of arities. T ..."
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Cited by 696 (39 self)
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The Edinburgh Logical Framework (LF) provides a means to define (or present) logics. It is based on a general treatment of syntax, rules, and proofs by means of a typed calculus with dependent types. Syntax is treated in a style similar to, but more general than, MartinLof's system of arities. The treatment of rules and proofs focuses on his notion of a judgement. Logics are represented in LF via a new principle, the judgements as types principle, whereby each judgement is identified with the type of its proofs. This allows for a smooth treatment of discharge and variable occurrence conditions and leads to a uniform treatment of rules and proofs whereby rules are viewed as proofs of higherorder judgements and proof checking is reduced to type checking. The practical benefit of our treatment of formal systems is that logicindependent tools such as proof editors and proof checkers can be constructed.
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 291 (24 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. LLF c ..."
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Cited by 217 (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...
Accomplishments and Research Challenges in MetaProgramming
 In 2nd Int. Workshop on Semantics, Applications, and Implementation of Program Generation, LNCS 2196
, 2000
"... this paper into several sections. As an overview, in Section 2, I try and classify metaprograms into groups. The purpose of this is to provide a common vocabulary which we can use to describe metaprogramming systems in the rest of the paper ..."
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Cited by 72 (7 self)
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this paper into several sections. As an overview, in Section 2, I try and classify metaprograms into groups. The purpose of this is to provide a common vocabulary which we can use to describe metaprogramming systems in the rest of the paper
Unification and AntiUnification in the Calculus of Constructions
 In Sixth Annual IEEE Symposium on Logic in Computer Science
, 1991
"... We present algorithms for unification and antiunification in the Calculus of Constructions, where occurrences of free variables (the variables subject to instantiation) are restricted to higherorder patterns, a notion investigated for the simplytyped calculus by Miller. Most general unifiers and ..."
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Cited by 61 (15 self)
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We present algorithms for unification and antiunification in the Calculus of Constructions, where occurrences of free variables (the variables subject to instantiation) are restricted to higherorder patterns, a notion investigated for the simplytyped calculus by Miller. Most general unifiers and least common antiinstances are shown to exist and are unique up to a simple equivalence. The unification algorithm is used for logic program execution and type and term reconstruction in the current implementation of Elf and has shown itself to be practical. The main application of the antiunification algorithm we have in mind is that of proof generalization. 1 Introduction Higherorder logic with an embedded simplytyped  calculus has been used as the basis for a number of theorem provers (for example [1, 19]) and the programming language Prolog [16]. Central to these systems is an implementation of Huet's preunification algorithm for the simplytyped calculus [12] which has shown it...
A SemiFunctional Implementation of a HigherOrder Logic Programming Language
 Topics in Advanced Language Implementation
, 1991
"... ions *) and varbind = Varbind of string * term (* Variable binders , Type *) In the implementation of the term language and the type checker, we have two constants type and pi. And, yes, type is a type, though this could be avoided by introducing universes (see [16]) without any changes to the code ..."
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Cited by 35 (0 self)
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ions *) and varbind = Varbind of string * term (* Variable binders , Type *) In the implementation of the term language and the type checker, we have two constants type and pi. And, yes, type is a type, though this could be avoided by introducing universes (see [16]) without any changes to the code of the unifier. As is customary, we use A ! B as an abbreviation for \Pix : A: B if x does not occur free in B. Also, however, \Pix : A: B is an abbreviation for the application pi A (x : A: B). In our formulation, then, the constant pi has type \PiA : type: ((A ! type) ! type). As an example consider a predicate constant eq of type \PiA : type: A ! A ! o (where o is the type of formulas as indicated in Section 9). The single clause eqAM M: correctly models equality, that is, a goal of the form eq AM N will succeed if M and N are unifiable. The fact that unification now has to branch can be seen by considering the goal eq int (F 1 1) 1 which has three solutions for the functional logic var...
Implementing the MetaTheory of Deductive Systems
 Proceedings of the 11th International Conference on Automated Deduction
, 1992
"... . We exhibit a methodology for formulating and verifying metatheorems about deductive systems in the Elf language, an implementation of the LF Logical Framework with an operational semantics in the spirit of logic programming. It is based on the mechanical verification of properties of transformatio ..."
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Cited by 32 (9 self)
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. We exhibit a methodology for formulating and verifying metatheorems about deductive systems in the Elf language, an implementation of the LF Logical Framework with an operational semantics in the spirit of logic programming. It is based on the mechanical verification of properties of transformations between deductions, which relies on type reconstruction and schemachecking. The latter is justified by induction principles for closed LF objects, which can be constructed over a given signature. We illustrate our technique through several examples, the most extensive of which is an interpretation of classical logic in minimal logic through a continuationpassingstyle transformation on proofs. 1 Introduction Formal deductive systems have become an important tool in computer science. They are used to specify logics, type systems, operational semantics and other aspects of languages. The role of such specifications is threefold. Firstly, inference rules serve as a highlevel notation w...
A Combinatory Logic Approach to Higherorder Eunification
 in Proceedings of the Eleventh International Conference on Automated Deduction, SpringerVerlag LNAI 607
, 1992
"... Let E be a firstorder equational theory. A translation of typed higherorder Eunification problems into a typed combinatory logic framework is presented and justified. The case in which E admits presentation as a convergent term rewriting system is treated in detail: in this situation, a modifi ..."
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Cited by 9 (3 self)
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Let E be a firstorder equational theory. A translation of typed higherorder Eunification problems into a typed combinatory logic framework is presented and justified. The case in which E admits presentation as a convergent term rewriting system is treated in detail: in this situation, a modification of ordinary narrowing is shown to be a complete method for enumerating higherorder Eunifiers. In fact, we treat a more general problem, in which the types of terms contain type variables. 1 Introduction Investigation of the interaction between firstorder and higherorder equational reasoning has emerged as an active line of research. The collective import of a recent series of papers, originating with [Bre88] and including (among others) [Bar90], [BG91a], [BG91b], [Dou92], [JO91] and [Oka89], is that when various typed calculi are enriched by firstorder equational theories, the validity problem is wellbehaved, and furthermore that the respective computational approaches to ...
HigherOrder Dynamic Pattern Unification for Dependent Types and Records
"... Abstract. While higherorder pattern unification for λ Πcalculus is decidable and unique unifiers exists, we face several challenges in practice: 1) the pattern fragment itself is too restrictive for many applications; this is typically addressed by solving subproblems which satisfy the pattern re ..."
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Cited by 3 (2 self)
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Abstract. While higherorder pattern unification for λ Πcalculus is decidable and unique unifiers exists, we face several challenges in practice: 1) the pattern fragment itself is too restrictive for many applications; this is typically addressed by solving subproblems which satisfy the pattern restriction eagerly but delay solving subproblems which are nonpatterns until we have accumulated more information. This leads to a dynamic pattern unification. 2) Many systems implement λ ΠΣ calculus and hence the known pattern unification algorithms for λ Π are too restrictive. In this paper, we present a constraintbased unification algorithm for λ ΠΣcalculus which solves a richer class of patterns than currently possible; in particular it takes into account type isomorphisms to translate unification problems containing Σtypes into problems only involving Πtypes. We prove correctness of our algorithm and discuss its application. 1