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New Generation Computing
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Information and Computation - Semantics of Data Types
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Programming in Martin-Lo¨f's type theory: an introduction
Programming in Martin-Lo¨f's type theory: an introduction
Logical frameworks
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Logical frameworks
Logical frameworks
Program synthesis using realizability
Theoretical Computer Science
An abstraction mechanism for symbolic expressions
Artificial intelligence and mathematical theory of computation
The influence of the designer on the design—J. McCarthy and LISP
Artificial intelligence and mathematical theory of computation
A framework for defining logics
Journal of the ACM (JACM)
A theory of binding structures and applications to rewriting
Theoretical Computer Science
Basic proof theory
A lambda-calculus for dynamic binding
Theoretical Computer Science - Special issue: theoretical aspects of coordination languages
Recursive functions of symbolic expressions and their computation by machine, Part I
Communications of the ACM
Theoretical Computer Science
A Calculus of Lambda Calculus Contexts
Journal of Automated Reasoning
Revised Report on the Algorithmic Language Scheme
Higher-Order and Symbolic Computation
Higher-Order and Symbolic Computation
TLCA '95 Proceedings of the Second International Conference on Typed Lambda Calculi and Applications
A Simply Typed Context Calculus with First-Class Environments
FLOPS '01 Proceedings of the 5th International Symposium on Functional and Logic Programming
CAL: A Computer Assisted Learning System for Computation and Logic
Computer Aided Systems Theory - EUROCAST 2001-Revised Papers
Classical Brouwer-Heyting-Kolmogorov Interpretation
ALT '97 Proceedings of the 8th International Conference on Algorithmic Learning Theory
Program synthesis through Gödel's interpretation
Proceedings of the International Conference on Mathematical Studies of Information Processing
Abstract Syntax and Variable Binding
LICS '99 Proceedings of the 14th Annual IEEE Symposium on Logic in Computer Science
Theory of Pairs, Part I: Provably Recursive Functions
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LISP 1.5 Programmer's Manual
Fundamenta Informaticae - Typed Lambda Calculi and Applications (TLCA'99)
Towards a mathematical theory of program synthesis
IJCAI'79 Proceedings of the 6th international joint conference on Artificial intelligence - Volume 2
Program synthesis from natural deduction proofs
IJCAI'79 Proceedings of the 6th international joint conference on Artificial intelligence - Volume 1
A framework for checking proofs naturally
Journal of Intelligent Information Systems
External and internal syntax of the λ-calculus
Journal of Symbolic Computation
A simple theory of expressions, judgments and derivations
ASIAN'04 Proceedings of the 9th Asian Computing Science conference on Advances in Computer Science: dedicated to Jean-Louis Lassez on the Occasion of His 5th Cycle Birthday
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We propose a computational and logical framework NF (Natural Framework) which is suitable for presenting mathematics formally. Our framework is an extendable framework since it is open-ended both computationally and logically in the sense of Martin-L枚f's theory or types. NF is developed in three steps. Firstly, we introduce a theory of expressions and schemata which is used to provide a universe for representing mathematical objects, in particular, judgments and derivations as well as other usual mathematical objects. Secondly, we develop a theory of judgments within the syntactic universe of expressions. Finally, we introduce the notions of derivation and derivation game and will show that we can develop mathematics as derivation games by regarding mathematics as an open-ended process of defining new concepts and deriving new judgments. Our theory is inspired by Martin-L枚f's theory of expressions and Edinburgh LF, but conceptually much simpler. Our theory is also influenced by Gentzen's natural deduction systems.