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Strong extension axioms and Shelah's zero-one law for choiceless polynomial time

Published online by Cambridge University Press:  12 March 2014

Andreas Blass  and
Yuri Gurevich
Andreas Blass
Affiliation:
Mathematics Department, University of Michigan, Ann Arbor, MI 48109-1109, USA, E-mail: ablass@umich.edu
Yuri Gurevich
Affiliation:
Microsoft Research, One Microsoft Way, Redmond, WA 98052, USA, E-mail: gurevich@microsoft.com
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Abstract

This paper developed from Shelah's proof of a zero-one law for the complexity class “choiceless polynomial time,” defined by Shelah and the authors. We present a detailed proof of Shelah's result for graphs, and describe the extent of its generalizability to other sorts of structures. The extension axioms, which form the basis for earlier zero-one laws (for first-order logic, fixed-point logic, and finite-variable infinitary logic) are inadequate in the case of choiceless polynomial time; they must be replaced by what we call the strong extension axioms. We present an extensive discussion of these axioms and their role both in the zero-one law and in general.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 68 , Issue 1 , March 2003 , pp. 65 - 131
Copyright
Copyright © Association for Symbolic Logic 2003

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References

REFERENCES

[1]Blass, Andreas and Gurevich, Yuri, Choiceless polynomial time computation and the zero-one law, Computer science logic 2000 (Clote, Peter and Schwichtenberg, Helmut, editors), Lecture Notes in Computer Science, vol. 1862, Springer, 2000, pp. 18–40.Google Scholar
[2]Blass, Andreas, A new zero-one law and strong extension axioms, Bulletin of the European Association for Theoretical Computer Science, vol. 72 (2000), pp. 103–122.Google Scholar
[3]Blass, Andreas, Gurevich, Yuri, and Kozen, Dexter, A zero-one law for logic with a fixed-point operator, Information and Control, vol. 67 (1985), pp. 70–90.CrossRefGoogle Scholar
[4]Blass, Andreas, Gurevich, Yuri, and Shelah, Saharon, Choiceless polynomial time, Annals of Pure and Applied Logic, vol. 100 (1999), pp. 141–187.CrossRefGoogle Scholar
[5]Blass, Andreas and Harary, Frank, Properties of almost all graphs and complexes, Journal of Graph Theory, vol. 3 (1979), pp. 225–240.CrossRefGoogle Scholar
[6]Bollobás, Béla, Random graphs, Academic Press, 1985.Google Scholar
[7]Chernoff, H., A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations, Annals of Mathematical Statistics, vol. 23 (1952), pp. 493–507.CrossRefGoogle Scholar
[8]Ebbinghaus, Heinz-Dieter and Flum, Jörg, Finite model theory, Springer-Verlag, 1995.Google Scholar
[9]Fagin, Ron, Probabilities on finite models, this Journal, vol. 41 (1976), pp. 50–58.Google Scholar
[10]Glebskii, Y. V., Kogan, D. I., Liogonkii, M. I., and Taianov, V. A., Range and degree of realizability of formulas in the restricted predicate calculus, Cybernetics, vol. 5 (1969), pp. 142–154.CrossRefGoogle Scholar
[11]Gurevich, Yuri, Evolving algebras 1993: Lipari guide, Specification and validation methods (Börger, E., editor), Oxford University Press, 1995, pp. 9–36. See also the May 1997 draft of the ASM guide, Tech Report CSE-TR-336-97, EECS Dept., Univ. of Michigan, 1997. Found at: http://www.eecs.umich.edu/gasm/.Google Scholar
[12]Hagerup, Torben and Rüb, Christine, A guided tour of Chernoff bounds, Information Processing Letters, vol. 33 (1989/1990), pp. 305–308.CrossRefGoogle Scholar
[13]Kolaitis, Phokion and Vardi, Moshe, 0-1 laws and decision problems for fragments of secondorder logic, Information and Computation, vol. 87 (1990), pp. 302-339, (special issue for the 3rd IEEE Symposium on Logic in Computer Science, 1988).CrossRefGoogle Scholar
[14]Kolaitis, Phokion and Vardi, Moshe, 0-1 laws for infinitary logics, Proceedings of the 5th IEEE symposium on logic in computer science, 1990, pp. 156-167.Google Scholar
[15]Loeve, M., Probability theory, Van Nostrand, 1955.Google Scholar
[16]Shelah, Saharon, Choiceless polynomial time logic: inability to express [paper number 634], Computer science logic 2000 (Peter Clote and Helmut Schwichtenberg, editors), Lecture Notes in Computer Science, vol. 1862, Springer, 2000, pp. 72-125.CrossRefGoogle Scholar
[17]Talanov, V. A. and Knyazev, V. V., The asymptotic truth of infinite formulas, Proceedings of all-union seminar on discrete mathematics and its applications (Moscow 1984), 1986, pp. 56-61.Google Scholar