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The Logic of Choice

Published online by Cambridge University Press:  12 March 2014

Andreas Blass  and
Yuri Gurevich
Andreas Blass
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, Mi 48109-1109, E-mail: ablass@math.lsa.umich.edu
Yuri Gurevich
Affiliation:
Microsoft Research, One Microsoft Way, Redmond, WA 98052 (on leave of absence from)Department of Eecs, University of Michigan, Ann Arbor, Michigan E-mail: gurevich@microsoft.com
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Abstract

The choice construct (choose x: φ(x)) is useful in software specifications. We study extensions of first-order logic with the choice construct. We prove some results about Hilbert's ε operator, but in the main part of the paper we consider the case when all choices are independent.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 65 , Issue 3 , September 2000 , pp. 1264 - 1310
Copyright
Copyright © Association for Symbolic Logic 2000

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References

REFERENCES

Abiteboul, Serge, Simon, Eric, and Vianu, Victor [1990], Non-deterministic languages to express deterministic transformations, Proceedings of ACM symposium on principles of database systems.Google Scholar
Abiteboul, Serge and Vianu, Victor [1991], Nondeterminism in logic-based languages, Annals of Mathematics and Artificial Intelligence, vol. 3, pp. 151–186.CrossRefGoogle Scholar
Asm [∞], ASM web site, the Michigan web site on abstract state machines, http://www.eecs.umich.edu/gasm/, maintained by James K. Huggins..Google Scholar
Börger, Egon, Grädel, Erich, and Gurevich, Yuri [1996], The classical decision problem, Perspectives in Mathematical Logic, Springer Verlag.Google Scholar
Caicedo, Xavier [1995], Hilbert's ε-symbol in the presence of generalized quantifiers, Quantifiers: Logics, models and computation, volume II (Krynicki, M.et al., editors), Kluwer Publishers, pp. 63–78.Google Scholar
Del Castillo, Giuseppe, Gurevich, Yuri, and Stroetmann, Karl [1998], Typed abstract state machines, ASM Web Site.Google Scholar
Church, Alonzo [1936], An unsohable problem of elementary number theory, American Journal of Mathematics, vol. 58, pp. 345–363, [Davis 1965, pages 88–107].CrossRefGoogle Scholar
Davis, Martin [1965], The undecidable: Basic papers on undecidable propositions, unsohable problems and computable functions, Raven Press, New York.Google Scholar
Ebbinghaus, Heinz-Dieter and Flum, Jörg [1995], Finite model theory, Springer.Google Scholar
Fagin, Ron [1974], Generalized first-order spectra and polynomial time recognizable sets, Complexity of computation (Karp, R.M., editor), SIAM-AMS Proceedings, vol. 7, pp. 43–73.Google Scholar
Garey, Michael R. and Johnson, David S. [1979], Computers and intractability: a guide to the theory ofNP completeness, Freeman.Google Scholar
Greenlaw, Raymond, Hoover, H. James, and Ruzzo, Walter L. [1995], Limits to parallel computation: P-completeness theory, Oxford University Press.CrossRefGoogle Scholar
Gurevich, Yuri [1984], Toward logic tailored for computational complexity, Computation and proof theory (Logic Colloquium 1983) (Richter, M.et al., editors), Lecture Notes in Mathematics, vol. 1104, Springer Verlag, pp. 175–216.Google Scholar
Gurevich, Yuri [1988], Logic and the challenge of computer science, Current trends in theoretical computer science (Börger, E., editor), Computer Science Press, pp. 1–57.Google Scholar
Gurevich, Yuri [1991], Evolving algebras: An attempt to discover semantics, Bulletin of European Association for Theoretical Computer Science, vol. 43, pp. 264–284, A slightly revised version appeared in G. Rozenberg and A. Salomaa, editors, “Current Trends in Theoretical Computer Science”, World Scientific, 1993, 266–292.Google Scholar
Gurevich, Yuri [1995], Evolving algebra 1993: Lipari guide, Specification and validation methods (Börger, E., editor), Oxford University Press, See also “May 1997 Draft of the ASM Guide”, Technical Report CSE-TR-336-97, EECS Department, University of Michigan, 1997. Found at http://www.eecs.umich.edu/gasm/, pp. 9–36.Google Scholar
Gurevich, Yuri [1997], A guide to abstract state machines, unpublished notes.Google Scholar
Hilbert, David and Bernays, Paul [1939], Grundlagen der Mathematik, Volume 2, Springer Verlag.Google Scholar
Kleene, Stephen Cole [1952], Introduction to metamathematics, D. Van Nostrand, New York, reprinted by Wolters-Noordhoff, Groningen and North-Holland, Amsterdam in 1971.Google Scholar
Leisenring, A. C. [1969], Mathematical logic and Hubert's ε-symbol, Macdonald Technical and Scientific, London.Google Scholar
Moschovakis, Yiannis N. [1974], Elementary induction on abstract structures, North-Holland, Amsterdam.Google Scholar
Moschovakis, Yiannis N. [1980], Descriptive set theory, North Holland, Amsterdam.Google Scholar
Otto, Martin [1998], Epsilon-logic is more expressive than first-order logic over finite structures, to appear.Google Scholar
Päppinghaus, Peter and Wirsing, Martin [1983], Nondeterministic three-valued logic: Isotonic and guarded truth-functions, Studia Logica, vol. XLII.1, pp. 1–22.Google Scholar
Rescher, Nicholas [1969], Many-valued logics, McGraw-Hill, New York.Google Scholar
Trakhtenbrot, Boris A. [1950], Impossibility of an algorithm for the decision problem on finite classes, Doklady Akad. Nauk, vol. 70, pp. 569–572.Google Scholar
Turing, Alan M. [1936], On computable numbers, with an application to the Entscheidungsproblem, Proceedings of London Mathematical Society, vol. 2, no. 42, pp. 230–236, and no. 43 (1936), 544–546. [Davis 1965, pages, 115–153.].Google Scholar