Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-27T20:07:52.014Z Has data issue: false hasContentIssue false
  • English
  • Français

Complexity, decidability and completeness*

Published online by Cambridge University Press:  12 March 2014

Douglas Cenzer  and
Jeffrey B. Remmel
Douglas Cenzer
Affiliation:
Department of Mathematics, University of Florida, Gainesville, Fl 32611
Jeffrey B. Remmel
Affiliation:
Department of Mathematics, University of Californiaat San Diego, La Jolla, CA 92093
Get access Rights & Permissions [Opens in a new window]

Abstract

We give resource bounded versions of the Completeness Theorem for propositional and predicate logic. For example, it is well known that every computable consistent propositional theory has a computable complete consistent extension. We show that, when length is measured relative to the binary representation of natural numbers and formulas, every polynomial time decidable propositional theory has an exponential time (EXPTIME) complete consistent extension whereas there is a nondeterministic polynomial time (NP) decidable theory which has no polynomial time complete consistent extension when length is measured relative to the binary representation of natural numbers and formulas. It is well known that a propositional theory is axiomatizable (respectively decidable) if and only if it may be represented as the set of infinite paths through a computable tree (respectively a computable tree with no dead ends). We show that any polynomial time decidable theory may be represented as the set of paths through a polynomial time decidable tree. On the other hand, the statement that every polynomial time decidable tree represents the set of complete consistent extensions of some theory which is polynomial time decidable, relative to the tally representation of natural numbers and formulas, is equivalent to P = NP. For predicate logic, we develop a complexity theoretic version of the Henkin construction to prove a complexity theoretic version of the Completeness Theorem. Our results imply that that any polynomial space decidable theory Δ possesses a polynomial space computable model which is exponential space decidable and thus Δ has an exponential space complete consistent extension. Similar results are obtained for other notions of complexity.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 71 , Issue 2 , June 2006 , pp. 399 - 424
Copyright
Copyright © Association for Symbolic Logic 2006

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

*

Department of Commerce Agreement 70-NANB5H1164 and NSF grant DMS-9306427.

References

REFERENCES

[1]Brent, R., Fast multiple-precision evaluation of elementary functions, Journal of the ACM, vol. 23 (1976). pp. 242–251.CrossRefGoogle Scholar
[2]Cenzer, D., Downey, R., and Remmel, J. B., Feasible torsion-free Abelian groups, To appear.Google Scholar
[3]Cenzer, D. and Remmel, J. B.. Feasible graphs with standard universe, Computability theory, Oberwolfach 1996. Special issue of Annals of Pure and Applied Logic, vol. 94 (1998), pp. 21–35.Google Scholar
[4]Cenzer, D. and Remmel, J. B.. Polynomial-time versus recursive models, Annals of Pure and Applied Logic, vol. 54 (1991). pp. 17–58.CrossRefGoogle Scholar
[5]Cenzer, D. and Remmel, J. B., Polynomial-time Abelian groups, Annals of Pure and Applied Logic, vol. 56(1992). pp. 313–363.CrossRefGoogle Scholar
[6]Cenzer, D. and Remmel, J. B., Recursively presented games and strategies, Mathematical Social Sciences, vol. 24 (1992), pp. 117–139.CrossRefGoogle Scholar
[7]Cenzer, D. and Remmel, J. B., Feasible graphs and colorings, Mathematical Logic Quarterly, vol. 41 (1995), pp. 327–352.CrossRefGoogle Scholar
[8]Cenzer, D. and Remmel, J. B., Feasibly categorical abelian groups, Feasible mathematics II (Clote, P. and Remmel, J., editors). Progess in Computer Science and Applied Logic, vol. 13. Birkhäuser, 1995, pp. 91–154.CrossRefGoogle Scholar
[9]Cenzer, D. and Remmel, J. B., Feasibly categorical models, Logic and computational complexity (Leivant, D., editor). Lecture Notes in Computer Science, vol. 960. Springer-Verlag, 1995, pp. 300–312.CrossRefGoogle Scholar
[10]Cenzer, D. and Remmel, J. B., Complexity-theoretic model theory and algebra, Handbook of recursive mathematics, Vol. I: Recursive model theory (Ersov, Y., Goncharov, S., Marek, V., Nerode, A., and Remmel, J., editors), Elsevier Studies in Logic and the Foundations of Mathematics, vol. 138, 1998, pp. 381–513.CrossRefGoogle Scholar
[11], Π10 classes. Handbook of recursive mathematics, Vol. 2: Recursive algebra, analysis and combinatorics (Ersov, Y., Goncharov, S.. Marek, V., Nerode, A., and Remmel, J., editors), Elsevier Studies in Logic and the Foundations of Mathematics, vol. 139, 1998, pp. 623–821.Google Scholar
[12]Ebbinghaus, H. and Flum, J., Finite model theory, Perspectives in Mathematical Logic, Springer-Verlag, 1995.Google Scholar
[13]Ehrenfeucht, A., Separable theories, Bulletin de l'Academie Polonaise des Sciences. Série des Sciences Mathématiques, Astronomiques et Physiques, vol. 9 (1961), pp. 17–19.Google Scholar
[14]Ferrante, J. and Rackoff, C., Complexity of logical theories, Lecture Notes in Mathematics, vol. 718, Springer-Verlag, 1979.CrossRefGoogle Scholar
[15]Grigorieff, S., Every recursive linear ordering has a copy in DTIME(n). this Journal, vol. 55 (1990), pp. 260–276.Google Scholar
[16]Harizanov, V.. Pure computable model theory, Handbook of recursive mathematics (Ershov, Yu. et al., editors), vol. 1, Elsevier, 1998, pp. 3–114.Google Scholar
[17]Hopcroft, J. and Ullman, J., Formal languages and their relations to automata, Addison Wesley, 1969.Google Scholar
[18]Jockusch, C. G. and Soare, R. I., Π10 classes and degrees of theories, Transactions of the American Mathematical Society, vol. 173 (1972), pp. 33–56.Google Scholar
[19]Ko, Ker-I, Complexity theory of real functions, Progress in Theoretical Computer Science, Birkhäuser, 1991.CrossRefGoogle Scholar
[20]Nerode, A. and Remmel, J. B., Complexity theoretic algebra I. vector spaces over finite fields, Proceedings of structure in complexity, second annual conference, IEEE Computer Society, 1987, pp. 218–239.Google Scholar
[21]Nerode, A. and Remmel, J. B., Complexity theoretic algebra II, the free Boolean algebra, Annals of Pure and Applied Logic, vol. 44 (1989). pp. 71–99.CrossRefGoogle Scholar
[22]Nerode, A. and Remmel, J. B.Complexity theoretic algebra: vector space bases. Feasible mathematics (Buss, S. and Scott, P., editors), Progress in Computer Science and Applied Logic, vol. 9. Birkhäuser, 1990, pp. 293–319.CrossRefGoogle Scholar
[23]Odifreddi, P., Classical recursion theory, North–Holland. Amsterdam, 1989.Google Scholar
[24]Papadimitriou, C., Computational complexity, Addison-Wesley, 1994.Google Scholar
[25]Remmel, J. B., When is every recursive linear ordering of type μ recursively Isomorphic to a p-time linear order over the binary representation of the natural numbers?, Feasible mathematics (Buss, S. and Scott, P., editors), Progress in Computer Science and Applied Logic, vol. 9. Birkhäuser, 1990, pp. 321–341.CrossRefGoogle Scholar
[26]Polynomial-time categoricity and linear orderings, Logical methods (Crossley, J.. Remmel, J., Shore, R., and Sweedler, M.. editors), Progress in Computer Science and Applied Logic, vol. 12. Birkhäuser, 1993. pp. 321–341.CrossRefGoogle Scholar
[27]Shoenfield, J. R.. Degrees of models, this Journal, vol. 25 (1960). pp. 233–237.Google Scholar
[28]Soare, R.. Recursively enumerable sets and degrees, Springer-Verlag, Berlin, 1987.CrossRefGoogle Scholar