Proof Analysis by Resolution

Abstract

Proof analysis of existing proofs is one of the main sources of scientific progress in mathematics: new concepts can be obtained e.g. by denoting explicit definitions in proof parts and axiomatizing them as new mathematical objects in their own right (The development of the concept of integral is a well known example.) All forms of proof analysis are intended to make informations implicit in a proof explicit i.e.visible. Logical proof analysis is mainly concerned with the implicit constructive content of more or less formalized proofs. The following are major examples for logical proof analysis: p] Formal proofs of (∀x)(∃y)P(x, y) in computational contexts can be unwinded to proofs of (∀x)P(x, π(x)) for suitable programs π (see [5]) p] Herbrand disjunctions can be extracted from proofs of prenex formulas. Such disjunctions always exist in the case of first-order logic by Herbrand’s famous theorem, but can be extracted from many proofs in other systems either (c.f.Luckhardt’s analysis of the proof of Roth’s theorem [7]). Suitable Herbrand disjunctions can be used to improve bounds or to reduce parametrical dependencies.

Interpolants can be constructed from proofs of A → B ¹. Interpolation is the main tool to make implicit de?nitions explicit by Beth’s theorem².

In this paper, we concentrate on automatizable logical proof analysis in first-order logic by means of incooperating resolution.

An interpolant for A → B is a formula I such that (a) A → I and I → B are provable and (b) I contains only predicate and function symbols common to A and B.

P is defined implicitely by Σ(P) iff Σ(P)∪Σ (P′)⊢(∀x)(P(x)↔P′(x)), P is defined explicitely by Σ(P) iff Σ(P)⊢(∀x)(P(x)↔L(x)) for some L not containing P).