Proofs and types
Hilbert's Twenty-Fourth Problem
Journal of Automated Reasoning
Models for the Logic of Proofs
LFCS '97 Proceedings of the 4th International Symposium on Logical Foundations of Computer Science
On the Complexity of Explicit Modal Logics
Proceedings of the 14th Annual Conference of the EACSL on Computer Science Logic
On the complexity of the reflected logic of proofs
Theoretical Computer Science - Clifford lectures and the mathematical foundations of programming semantics
IEEE Transactions on Computers
Naming proofs in classical propositional logic
TLCA'05 Proceedings of the 7th international conference on Typed Lambda Calculi and Applications
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The problem of the identity criteria for proofs can be traced to Hilbert and Prawitz. One of the approaches, which uses the concept of generality of proofs, was suggested in 1968 by Lambek. Following his ideas, we propose a language and a logic to represent Hilbert-style proofs for classical propositional logic by adapting the Logic of Proofs (LP) introduced by Artemov in 1994. We prove that proof polynomials, the objects representing Hilbert derivations in LP, are sufficient to realize all propositional derivations, with or without hypotheses. We also show that proof polynomials respect the ideas of generality and provide an algorithm for determining whether two given proof polynomials represent the same proof. These results naturally extend similar properties of combinatory logic demonstrated by Hindley. The language of LPallows us to formally capture more structure of Hilbert-style proofs. In particular, we show how the well-known phenomenon of proof composition in classical logic manifests itself in the case of Hilbert proofs.