A framework for defining logics
Journal of the ACM (JACM)
Institutions: abstract model theory for specification and programming
Journal of the ACM (JACM)
Information and Computation
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WADT '99 Selected papers from the 14th International Workshop on Recent Trends in Algebraic Development Techniques
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Proceedings of the 1983 International FCT-Conference on Fundamentals of Computation Theory
Proof Systems for Institutional Logic
Journal of Logic and Computation
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IJCAI'85 Proceedings of the 9th international joint conference on Artificial intelligence - Volume 1
Combining type theory and untyped set theory
IJCAR'06 Proceedings of the Third international joint conference on Automated Reasoning
A practical module system for LF
Proceedings of the Fourth International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice
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AISC'10/MKM'10/Calculemus'10 Proceedings of the 10th ASIC and 9th MKM international conference, and 17th Calculemus conference on Intelligent computer mathematics
Representing model theory in a type-theoretical logical framework
Theoretical Computer Science
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We give a comprehensive formal representation of first-order logic using the recently developed module system for the Twelf implementation of the Edinburgh Logical Framework LF. The module system places strong emphasis on signature morphisms as the main primitive concept, which makes it particularly useful to reason about structural translations, which occur frequently in proof and model theory. Syntax and proof theory are encoded in the usual way using LF's higher order abstract syntax and judgments-as-types paradigm, but using the module system to treat all connectives and quantifiers independently. The difficulty is to reason about the model theory, for which the mathematical foundation in which the models are expressed must be encoded itself. We choose a variant of Martin-Lof's type theory as this foundation and use it to axiomatize first-order model theoretic semantics. Then we can encode the soundness proof as a signature morphism from the proof theory to the model theory. We extend our results to models given in terms of set theory using an encoding of Zermelo-Fraenkel set theory in LF and giving a signature morphism from Martin-Lof type theory into it. These encodings can be checked mechanically by Twelf. Our results demonstrate the feasibility of comprehensively formalizing large scale representation theorems and thus promise significant future applications.