Re ned Program Extraction from Classical Proofs

It is well known that it is undecidable in general whether a given programmeets its speci cation In contrast it can be checked easily by a machine whether a formal proof is correct and from a constructive proof one can automatically extract a corresponding program which by its very construction is correct as well This at least in principle opens a way to produce correct software e g for safety critical applications Moreover programs obtained from proofs are commented in a rather extreme sense Therefore it is easy to maintain them and also to adapt them to particular situations We will concentrate on the question of classical versus constructive proofs It is known that any classical proof of a speci cation of the form x yB with B quanti er free can be transformed into a constructive proof of the same formula However when it comes to extraction of a program from a proof obtained in this way one easily ends up with a mess Therefore some re nements of the standard transformation are necessary In this paper we develop a re ned method of extracting reasonable and sometimes unexpected programs from classical proofs Other interesting examples of program extraction from classical proofs have been studied byMurthy Coquand s group see e g in a type theoretic context and by Kohlenbach using a Dialectica interpretation We now describe in more detail what the paper is about In section we x our version of intuitionistic arithmetic for functionals and recall how classical arithmetic can be seen as a subsystem Then our argument goes as follows It is well known that from a derivation of a classical existential formula yA y A one generally cannot read o an instance A simple example has been given by Kreisel Let R be a primitive recursive relation such that zR x z is undecidable Clearly we have even logically