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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 ariti ..."
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Cited by 807 (45 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.
An Illative Theory of Relations
, 1990
"... this paper we present a nonstandard logic for our structures. It is a typefree intensional logic, and is also in the tradition of Curry's illative logic [HS86]; see also [AczN, FM87, Smi84, MA88]. The logic has two judgments: that an object is a fact and that an object is a stateofa#airs (c ..."
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Cited by 15 (2 self)
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this paper we present a nonstandard logic for our structures. It is a typefree intensional logic, and is also in the tradition of Curry's illative logic [HS86]; see also [AczN, FM87, Smi84, MA88]. The logic has two judgments: that an object is a fact and that an object is a stateofa#airs (cf. truth and proposition). Objects are given using a variant of the traditional situation theory notation which is more standard, logically speaking, with explicit negation and quantification (see also [Bar87]). No metalinguistic apparatus is employed
Constructing type systems over an operational semantics
 Journal of Symbolic Computation
, 1992
"... Type theories in the sense of MartinLof and the NuPRL system are based on taking as primitive a typefree programming language given by an operational semantics, and defining types as partial equivalence relations on the set of closed terms. The construction of a type system is based on a general f ..."
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Cited by 13 (0 self)
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Type theories in the sense of MartinLof and the NuPRL system are based on taking as primitive a typefree programming language given by an operational semantics, and defining types as partial equivalence relations on the set of closed terms. The construction of a type system is based on a general form of inductive definition that may either be taken as acceptable in its own right, or further explicated in terms of other patterns of induction. One such account, based on a general theory of inductivelydefined relations, was given by Allen. An alternative account, based on an essentially settheoretic argument, is presented. 1
Categorical Properties of Logical Frameworks
, 1993
"... In this paper we give a new presentation of ELF which is wellsuited for semantic analysis. We introduce the notions of internal codability, internal definability, internal typed calculi and frame languages. These notions are central to our perspective of logical frameworks. We will argue that a ..."
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Cited by 1 (1 self)
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In this paper we give a new presentation of ELF which is wellsuited for semantic analysis. We introduce the notions of internal codability, internal definability, internal typed calculi and frame languages. These notions are central to our perspective of logical frameworks. We will argue that a logical framework is a typed calculus which formalizes the relationship between internal typed languages and frame languages. In the second half of the paper, we demonstrate the advantage of our logical framework by showing some categorical properties of it and of encodings in it. By doing so we hope to indicate a sensible model theory of encodings. Copyright c fl1993. All rights reserved. Reproduction of all or part of this work is permitted for educational or research purposes on condition that (1) this copyright notice is included, (2) proper attribution to the author or authors is made and (3) no commercial gain is involved. Technical Reports issued by the Department of Computer Sc...
OpenEndedness of Objects and Types in MartinL\"of’s Type Theory
"... This paper presents a comprehensive formulation of openendedness of types as well as objects in $Maltin L\ddot{o}f ’ s $ type theory. This formulation is a natural generalization of Allen’s nontypetheoretical Ieintelpretation of the theory, and demonstrates a structural extension of Howe’s formu ..."
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This paper presents a comprehensive formulation of openendedness of types as well as objects in $Maltin L\ddot{o}f ’ s $ type theory. This formulation is a natural generalization of Allen’s nontypetheoretical Ieintelpretation of the theory, and demonstrates a structural extension of Howe’s formulation of computational openendedness. Suppose that a language underlying the theory is specified as a method system, which consists of a preobject system as the computational part and a pretype system as the structural part. Then types and their objects are unifoImly and inductively constructed as a type system that is built from the method system and that can provide a semantics of the theory. The main theorem shows that the original inference rules concerning objects or types remain valid in any type system built from a deterministic and regular extension of the original method system. This result includes a prescription for the class of types that can be introduced into the theory, which prescription is useful for checking whether specific new types can be introduced. 1