Implicit proofs

Abstract

We describe a general method how to construct from a propositional proof system P a possibly much stronger proof system iP. The system iP operates with exponentially long P-proofs described “implicitly” by polynomial size circuits.

As an example we prove that proof system iEF, implicit EF, corresponds to bounded arithmetic theory V¹₂ and hence, in particular, polynomially simulates the quantified propositional calculus G and the Π^b₁-consequences of S¹₂ proved with one use of exponentiation. Furthermore, the soundness of iEF is not provable in S¹₂. An iteration of the construction yields a proof system corresponding to T₂ + Exp and, in principle, to much stronger theories.