Abstract
The paper considers a commonly used axiomatization of the classical propositional logic and studies how different axiom schemata in this system contribute to proof complexity of the logic. The existence of a polynomial bound on proof complexity of every statement provable in this logic is a well-known open question.
The axiomatization consists of three schemata. We show that any statement provable using unrestricted number of axioms from the first of the three schemata and polynomially-bounded in size set of axioms from the other schemata, has a polynomially-bounded proof complexity. In addition, it is also established, that any statement, provable using unrestricted number of axioms from the remaining two schemata and polynomially-bounded in size set of axioms from the first scheme, also has a polynomially-bounded proof complexity.
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Naumov, P. Upper bounds on complexity of Frege proofs with limited use of certain schemata. Arch. Math. Logic 45, 431–446 (2006). https://doi.org/10.1007/s00153-005-0325-8
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DOI: https://doi.org/10.1007/s00153-005-0325-8