On the Constructive Content of Proofs

This thesis aims at exploring the scopes and limits of techniques for extracting programs from proofs. We focus on constructive theories of inductive definitions and classical systems allowing choice principles. Special emphasis is put on optimizations that allow for the extraction of realistic programs. Our main field of application is infinitary combinatorics. Higman's Lemma, having an elegant non-constructive proof due to Nash-Williams, constitutes an interesting case for the problem of discovering the constructive content behind a classical proof. We give two distinct solutions to this problem. First, we present a proof of Higman's Lemma for an arbitrary alphabet in a theory of inductive definitions. This proof may be considered as a constructive counterpart to Nash-Williams' minimal-bad-sequence proof. Secondly, using a refined $A$-translation method, we directly transform the classical proof into a constructive one and extract a program. The crucial point in the latter is that we do not need to avoid the axiom of classical dependent choice but directly assign a realizer to its translation. A generalization of Higman's Lemma is Kruskal's Theorem. We present a constructive proof of Kruskal's Theorem that is completely formalized in a theory of inductive definitions. As a practical part, we show that these methods can be carried out in an interactive theorem prover. Both approaches to Higman's Lemma have been implemented in Minlog.Ziel der vorliegenden Arbeit ist es, die Reichweiten und Grenzen von Techniken zur Extraktion von Programmen aus Beweisen zu erforschen. Wir konzentrieren uns dabei auf konstruktive Theorien Induktiver Definitionen und klassische Systeme mit Auswahlprinzipien. Besonderes Gewicht liegt auf Optimierungen, die zur Extraktion von realisischen Programmen f"uhren. Unser Hauptanwendungsgebiet ist die unendliche Kombinatorik. Higmans Lemma, ein Satz mit einem eleganten klassischen, auf Nash-Williams zur"uckgehenden Beweis, ist ein interessantes Fallbeispiel f"ur die Suche nach dem konstruktiven Gehalt in einem klassischen Beweis. Wir zeigen zwei unterschiedliche L"osungen zu dieser Problemstellung auf. Zun"achst pr"asentieren wir einen induktiven Beweis von Higmans Lemma f"ur ein beliebiges Alphabet, der als konstruktives Pendant zum klassischen Beweis angesehen werden kann. Als zweiten Ansatz verwandeln wir mit Hilfe der verfeinerten $A$-"Ubersetzungs-methode den klassischen Beweis in einen konstruktiven und extrahieren ein Programm. Der entscheidende Punkt ist hierbei, dass wir einen direkten Realisierer f"ur das "ubersetzte Auswahlaxiom verwenden. Die Verallgemeinerung von Higmans Lemma f"uhrt zu Kruskals Satz. Wir geben einen konstruktiven Beweis von Kruskals Theorem, der vollst"andig auf den Induktiven Definitionen basiert. Der praktische Teil der Arbeit befasst sich mit der Ausf"uhrbarkeit dieser Methoden und Beweise in dem Beweissystem Minlog

Extracting Imperative Programs from Proofs: In-place Quicksort

The process of program extraction is primarily associated with functional programs with less focus on imperative program extraction. In this paper we consider a standard problem for imperative programming: In-place Quicksort. We formalize a proof that every array of natural numbers can be sorted and apply a realizability interpretation to extract a program from the proof. Using monads we are able to exhibit the inherent imperative nature of the extracted program. We see this as a first step towards an automated extraction of imperative programs. The case study is carried out in the interactive proof assistant Minlog

Extracting verified decision procedures: DPLL and Resolution

This article is concerned with the application of the program extraction technique to a new class of problems: the synthesis of decision procedures for the classical satisfiability problem that are correct by construction. To this end, we formalize a completeness proof for the DPLL proof system and extract a SAT solver from it. When applied to a propositional formula in conjunctive normal form the program produces either a satisfying assignment or a DPLL derivation showing its unsatisfiability. We use non-computational quantifiers to remove redundant computational content from the extracted program and translate it into Haskell to improve performance. We also prove the equivalence between the resolution proof system and the DPLL proof system with a bound on the size of the resulting resolution proof. This demonstrates that it is possible to capture quantitative information about the extracted program on the proof level. The formalization is carried out in the interactive proof assistant Minlog