Analysis of methods for extraction of programs from non-constructive proofs

Proofs in constructive logic correspond to functional programs in a direct and natural way. Computational content can also be found in proofs which use non-constructive principles, but more advanced techniques are required to interpret such proofs. Various methods have been developed to harvest programs from derivations in classical logic and experiments have yielded surprising and counterintuitive, yet correct and e cient algorithms. Nevertheless, the use of non-constructive arguments generally leads to an indirect backtracking computation, which is slower and more di cult to understand as compared to a program extracted from a constructive proof. Constructive proofs can be transformed into programs in an unambiguous manner by projecting only their computational components. However, for proofs in classical logic there seems to be no canonical de nition of their computational meaning, since every extraction method introduces speci c computational infrastructure to support non-constructive reasoning. Applying several techniques to the same proof can result in di erent correct programs, however the relation between them with respect to e ciency, size and readability has not been thoroughly explored. The rst part of the present work compares two computational interpretations of non-constructive proofs: re ned A-translation [BBS02] and Gödel's functional Dialectica interpretation [Göd58]. An arithmetical system is de ned in which both techniques can be applied to the same proof object. The behaviour of the extraction methods is evaluated in the light of several case studies, and the resulting programs are compared. It is argued that the two interpretations correspond to speci c backtracking schemes and that programs obtained via re ned A-translation tend to be simpler, faster and more readable than programs obtained via Gödel's interpretation. The second part of the thesis introduces three layers of optimisation of Gödel's interpretation to produce faster and more readable programs. First, it is shown that syntactic repetition of subterms can be reduced by using let-constructions instead of meta substitutions. The practical e ects of the modi cation are nearly linear size of extracted terms and improved e ciency, achieved by avoiding repeated evaluations of equal terms. The second improvement is an extension of previous work [Ber05, Her07b], which allows declaring syntactically computational parts of the proof as computationally irrelevant. It is shown that Gödel's interpretation admits a wide variety of such annotations, which can be used to remove redundant parameters, possibly improving the e ciency of the program. An additional feature is the ability to embed Kreisel's modi ed realisability, and thus the re nedA-translation, inside the extended Dialectica interpretation. Finally, a special case of induction is identi ed, for which a more e cient recursive extracted term can be de ned. It is shown the outcome of case distinctions can be memoised by a boolean ag, which can result in exponential improvement of the average time complexity of the extracted program.