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Mechanizing Proof Theory: Resource-Aware Logics and Proof-Transformations to extract implicit information

Abstract: Few systems for mechanical proof-checking have been used so far to transform formal proofs rather than to formalize informal arguments and to verify correctness. The unwinding of proofs, namely, the process of applying lemmata and extracting explicit values for the parameters within a proof, is an obvious candidate for mechanization. It corresponds to the procedures of Cut-elimination and functional interpretation in proof-theory and allows the extraction of the constructive content of a proof, sometimes yielding information useful in mathematics and in computing.

Resource-aware logics restrict the number of times an assumption may be used in a proof and are of interest for proof-checking not inly in relation to their decidability or computational complexity, but also because they efficiently solve the practical problem of representing the structure of relevance in a derivation. In particular, in Direct Logic only one subformula-occurrence of the input is allowed, and the connections established during a successful proof-verification can be represented on the input without altering it. In addition, the values for the parameters obtained from unwinding are read off directly.

In Linear Logic, where classical logic is regarded as the limit of a resource-aware logic, long-standing issues in proof-theory have been successfully attacked. We are particularly interested in the system of proof-nets as a multiple-conclusion Natural Deduction system for Linear Logic.

In Part I of this thesis we present a new set of tools that provide a systematic and uniform approach to different resource-aware logics. In particular, we obtain uniqueness of the normal form for Multiplicative and Additive Linear Logic (sections 3 and 4) and an extension of Direct Logic of interest for nonmonotonic reasoning (section 8). In Part II we study Herbrand's Theorem in Linear Logic and the No Counterexample Interpretation in a fragment of Peano Arithmetic (section 10). As an application to Ramsey Theory we give a parametric form of the Ramsey Theorem, that generalizes the Infinite, the Finite and the Ramsey-Paris-Harrington Theorems for a fixed exponent (sections 10-13).

PhD Thesis - Price £8.50

LFCS report ECS-LFCS-91-165 (also published as CST-80-91)