A form of feasible interpolation for constant depth Frege systems

Abstract

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 Chain_L,Φ,Ψ(n,m) holds: For any chain of finite L-structures C₁, …, C_m with the universe [n] one of the following conditions must fail:

• 1. C₁ ⊨ Φ,

• 2. C_i ≅ C_i+1, for i = 1, …, m-1,

• 3. C_m ⊨ Ψ.

For each fixed L and parameters n, m the principle Chain_L,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 c_L such that the following holds: If a constant depth Frege system in DeMorgan language proves Chain_L,Φ,Ψ(n, c_L· 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 2^log(s)O(1) and with bottom fan-in log(s)^O(1).