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

Trifonov, Trifon (2012): Analysis of methods for extraction of programs from non-constructive proofs. Dissertation, LMU München: Faculty of Mathematics, Computer Science and Statistics

Abstract

The present thesis compares two computational interpretations of non-constructive proofs: refined A-translation and Gödel's functional "Dialectica" interpretation. The behaviour of the extraction methods is evaluated in the light of several case studies, where the resulting programs are analysed and compared. It is argued that the two interpretations correspond to specific backtracking implementations and that programs obtained via the refined A-translation tend to be simpler, faster and more readable than programs obtained via Gödel's interpretation. Three layers of optimisation are suggested in order 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 abd thus obtaining a near linear size bound of extracted terms. The second improvement allows declaring syntactically computational parts of the proof as irrelevant and that this can be used to remove redundant parameters, possibly improving the efficiency of the program. Finally, a special case of induction is identified, for which a more efficient recursive extracted term can be defined. It is shown the outcome of case distinctions can be memoised, which can result in exponential improvement of the average time complexity of the extracted program.

• Item Type: Theses (Dissertation, LMU Munich)
• Keywords: program extraction, non-constructive proofs, Dialectica interpretation, refined A-translation, infinite pigeonhole principle
• Subjects: 000 Computers, Information and General Reference > 004 Data processing computer science; 000 Computers, Information and General Reference
• Faculties: Faculty of Mathematics, Computer Science and Statistics
• Language: English
• Date of oral examination: 15. February 2012
• 1. Referee: Schwichtenberg, Helmut
• Signature of the printed copy: 0001/UMC 20069