Proofs and Computations

Abstract

Driven by the question, 'What is the computational content of a (formal) proof?', this book studies fundamental interactions between proof theory and computability. It provides a unique self-contained text for advanced students and researchers in mathematical logic and computer science. Part I covers basic proof theory, computability and Gödel's theorems. Part II studies and classifies provable recursion in classical systems, from fragments of Peano arithmetic up to Π11–CA0. Ordinal analysis and the (Schwichtenberg–Wainer) subrecursive hierarchies play a central role and are used in proving the 'modified finite Ramsey' and 'extended Kruskal' independence results for PA and Π11–CA0. Part III develops the theoretical underpinnings of the first author's proof assistant MINLOG. Three chapters cover higher-type computability via information systems, a constructive theory TCF of computable functionals, realizability, Dialectica interpretation, computationally significant quantifiers and connectives and polytime complexity in a two-sorted, higher-type arithmetic with linear logic.

Some Remarks on Proof-Theoretic Semantics

“…SN, for the full language: a straightforward extension of the proof of [37] for implication 4 ; also, the proofs for implication "directly carry over" [12] to a system with conjunctions and disjunctions. An argument (using the ordinary elimination rule for implication) is given in [35] for the rules for implication and existential quantification, with the virtue of illustrating in detail how to handle GE rules where the Tait-Martin-Löf method of induction on types familiar from [11] is not available. See also [13].…”

Extracting a DPLL Algorithm

“…We highlight a few aspects that are important to understand the optimizations we achieved. For a complete and precise description of program extraction we refer to [25].…”

A constructive notion of codimension

“…In concrete terms, each statement can be understood as a specification of a program, and its proof as a program realizing this specification together with its proof of correctness [16]. Recent work has already shown the feasibility of such a program for a considerable part of commutative algebra [14], confirming "the feeling that commutative algebra can be seen computationally as a machine that produces algebraic identities" [17].…”