Topic
Natural deduction
About: Natural deduction is a research topic. Over the lifetime, 2536 publications have been published within this topic receiving 53482 citations.
Papers published on a yearly basis
Papers
More filters
Book•
[...]
TL;DR: In this paper, the Curry-Howard isomorphism and the normalisation theorem of a natural deduction system T coherence spaces have been studied in the context of linear logic and linear logic semantics.
Abstract: Sense, denotation and semantics natural deduction the Curry-Howard isomorphism the normalisation theorem Godel's system T coherence spaces denotational semantics of T sums in natural deduction system F coherence semantics of the sum cut elimination (Hauptsatz) strong normalisation for F representation theorem semantics of System F what is linear logic?
1,771 citations
TL;DR: All standard Hilbert type systems and natural deduction systems are equivalent, up to application of a polynomial, as far as minimum proof length goes, and extended Frege systems are introduced, which allow introduction of abbreviations for formulas.
Abstract: We are interested in studying the length of the shortest proof of a propositional tautology in various proof systems as a function of the length of the tautology. The smallest upper bound known for this function is exponential, no matter what the proof system. A question we would like to answer (but have not been able to) is whether this function has a polynomial bound for some proof system. (This question is motivated below.) Our results here are relative results.In §§2 and 3 we indicate that all standard Hilbert type systems (or Frege systems, as we call them) and natural deduction systems are equivalent, up to application of a polynomial, as far as minimum proof length goes. In §4 we introduce extended Frege systems, which allow introduction of abbreviations for formulas. Since these abbreviations can be iterated, they eliminate the need for a possible exponential growth in formula length in a proof, as is illustrated by an example (the pigeonhole principle). In fact, Theorem 4.6 (which is a variation of a theorem of Statman) states that with a penalty of at most a linear increase in the number of lines of a proof in an extended Frege system, no line in the proof need be more than a constant times the length of the formula proved.
941 citations
01 Jan 1972
TL;DR: In this paper, a notational system for lambda calculus is developed, where occurrences of variables are indicated by integers giving the "distance" to the binding λ instead of a name attached to that λ. This convention is known to cause considerable trouble in cases of substitution.
Abstract: In ordinary lambda calculus the occurrences of a bound variable are made recognizable by the use of one and the same (otherwise irrelevant) name at all occurrences. This convention is known to cause considerable trouble in cases of substitution. In the present paper a different notational system is developed, where occurrences of variables are indicated by integers giving the “distance” to the binding λ instead of a name attached to that λ. The system is claimed to be efficient for automatic formula manipulation as well as for metalingual discussion. As an example the most essential part of a proof of the Church-Rosser theorem is presented in this namefree calculus.
919 citations
01 Jan 1973
TL;DR: In this paper, Kripke models are used to define inductive definitions, trees and ordinals for intuitionistic formal systems, and normalization theorems for systems of natural deduction.
Abstract: Intuitionistic formal systems.- Models and computability.- Realizability and functional interpretations.- Normalization theorems for systems of natural deduction.- Applications of Kripke models.- Iterated inductive definitions, trees and ordinals.- Erratum.
809 citations