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 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).
"A form of feasible interpolation for constant depth Frege systems." J. Symbolic Logic 75 (2) 774 - 784, June 2010. https://doi.org/10.2178/jsl/1268917504