Journal of Symbolic Logic

A form of feasible interpolation for constant depth Frege systems

Jan Krajíček

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Let L be a first-order language and Φ and Ψ two Σ11 L-sentences that cannot be satisfied simultaneously in any finite L-structure. Then obviously the following principle ChainL,Φ,Ψ(n,m) holds: For any chain of finite L-structures C1, …, Cm with the universe [n] one of the following conditions must fail:

  • 1. C1 ⊨ Φ,
  • 2. Ci ≅ Ci+1, for i = 1, …, m-1,
  • 3. Cm ⊨ Ψ.
For each fixed L and parameters n, m the principle ChainL,Φ,Ψ(n,m) can be encoded into a propositional DNF formula of size polynomial in n, m.

For any language L containing only constants and unary predicates we show that there is a constant cL such that the following holds: If a constant depth Frege system in DeMorgan language proves ChainL,Φ,Ψ(n, cL· n) by a size s proof then the class of finite L-structures with universe [n] satisfying Φ can be separated from the class of those L-structures on [n] satisfying Ψ by a depth 3 formula of size 2log(s)O(1) and with bottom fan-in log(s)O(1).

Article information

J. Symbolic Logic, Volume 75, Issue 2 (2010), 774-784.

First available in Project Euclid: 18 March 2010

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Krajíček, Jan. A form of feasible interpolation for constant depth Frege systems. J. Symbolic Logic 75 (2010), no. 2, 774--784. doi:10.2178/jsl/1268917504.

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