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Implicit proofs. (English) Zbl 1069.03053

Summary: 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 \(i\text{EF}\), implicit EF, corresponds to bounded arithmetic theory \(V^1_2\) and hence, in particular, polynomially simulates the quantified propositional calculus \(G\) and the \(\Pi^b_1\)-consequences of \(S^1_2\) proved with one use of exponentiation. Furthermore, the soundness of \(i\text{EF}\) is not provable in \(S^1_2\). An iteration of the construction yields a proof system corresponding to \(T_2 + \text{Exp}\) and, in principle, to much stronger theories.

MSC:

03F20 Complexity of proofs
03D15 Complexity of computation (including implicit computational complexity)
68Q15 Complexity classes (hierarchies, relations among complexity classes, etc.)
03F30 First-order arithmetic and fragments
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References:

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