Abstract
We begin by reviewing the natural deduction rules for the →∧∀-fragment of minimal logic. It is shown how intuitionistic and classical logic can be embedded. Recursion and induction is added to obtain a more realistic proof system. Simple types are added in order to make the language more expressive. We also consider two alternative methods to deal with the strong or constructive existential quantifier ∃*. Finally we discuss the well-known notion of an extracted program of a derivation involving ∃*, in order to set up a relation between the two alternatives.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Ulrich Berger. Program extraction from normalization proofs. In M. Bezem and J.F. Groote, editors, Typed Lambda Calculi and Applications, pages 91–106. Springer Lecture Notes in Computer Science Vol. 664, 1993.
Roy Dyckhoff. Contraction-free sequent calculi for intuitionistic logic. Journal of Symbolic Logic, 57: 793–807, 1992.
Harvey Friedman. Classically and intuitionistically provably recursive functions. In Dana S. Scott and Gert H. Müller, editors, Higher Set Theory, pages 21–28. Springer Lecture Notes in Mathematics, Volume 699, 1978.
Gerhard Gentzen. Untersuchungen über das logische Schließen. Mathematische Zeitschrift, 39:176–210, 405–431, 1934.
Christopher Alan Goad. Computational uses of the manipulation of formal proofs. PhD thesis, Stanford University, August 1980. Stanford Department of Computer Science Report No. STAN-CS-80-819.
Jörg Hudelmaier. Bounds for cut elimination in intuitionistic propositional logic. Archive for Mathematical Logic, 31: 331–354, 1992.
Dale Miller. A logic programming language with lambda-abstraction, function variables and simple unification. Journal of Logic and Computation, 2 (4): 497–536, 1991.
G.E. Mints. On e-theorems (in russian). Zapiski, 40: 110–118, 1974.
Chetan Murthy. Extracting constructive content from classical proofs. Technical Report 90-1151, Dep. of Comp.Science, Cornell Univ., Ithaca, New York, 1990. PhD thesis.
Tobias Nipkow. Functional unification of higher-order patterns. In Proc. 8th IEEE Symp. Logic in Computer Science, pages 64–74, 1993.
Dag Prawitz. Natural Deduction, volume 3 of Acta Universitatis Stockholmiensis. Stockholm Studies in Philosophy. Almqvist amp; Wiksell, Stockholm, 1965.
Dan Sahlin, Torkel Franzen, and Seif Haridi. An intuitionistic predicate logic theorem prover. Journal of Logic and Computation, 2 (6): 619–656, 1992.
Helmut Schwichtenberg. Proofs as programs. In Peter Aczel, Harold Simmons, and Stanley S. Wainer, editors, Proof Theory. A selection of papers from the Leeds Proof Theory Programme 1990, pages 81–113. Cambridge University Press, 1992.
N. Shankar. Proof search in intuitionistic sequent calculus, 1991.
Wilfried Sieg and Richard Scheines. Searching for proofs (in sentential logic). In Leslie Burkholder, editor, Philosophy and the computer, pages 137–159, Boulder, San Francisco, Oxford, 1992. Westview Press.
Martin Stein. Interpretationen der Heyting-Arithmetik endlicher Typen. PhD thesis, Universität Münster, Fachbereich Mathematik, 1976.
Anne S. Troelstra, editor. Metamathematical Investigations of Intuitionistic Arithmetic and Analysis, volume 344 of Lecture Notes in Mathematics. Springer, Berlin, 1973.
Anne S. Troelstra and Dirk van Dalen. Constructivism in Mathematics. An Introduction, volume 121,123 of Studies in Logic and the Foundations of Mathematics. North-Holland, Amsterdam, 1988.
Anton Wallner. Komplexe Existenzbeweise in der Arithmetik. Master’s thesis, Mathematisches Institut der Universität München, 1993.
Hermann Weyl. Uber die neue Grundlagenkrise der Mathematik. Mathematische Zeitschrift, 10, 1921.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1995 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Berger, U., Schwichtenberg, H. (1995). Program Development by Proof Transformation. In: Schwichtenberg, H. (eds) Proof and Computation. NATO ASI Series, vol 139. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-79361-5_1
Download citation
DOI: https://doi.org/10.1007/978-3-642-79361-5_1
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-79363-9
Online ISBN: 978-3-642-79361-5
eBook Packages: Springer Book Archive