# A survey of some recent results on computational complexity in weak theories of arithmetic

Invited Lectures

First Online:

## Keywords

Polynomial Time Conjunctive Normal Form Peano Arithmetic Weak Theory Nonstandard Model
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Preview

Unable to display preview. Download preview PDF.

## References

- 1.Adleman, L. and Manders, K., "Computational complexity of decision procedures for polynomials," Proceedings of the 17th Annual IEEE Symp on Switching and Automata Theory, pp. 169–177 (1976).Google Scholar
- 2.Baker, T. P., "On ‘provable’ analogs of P and NP," Math Systems Theory Vol. 12 pp. 213–218 (1979).Google Scholar
- 3.Berman, L. and Hartmanis, J., "On the isomorphism and density of NP and other complete sets," SIAM J Comput Vol. 6 pp. 305–322 (Dec 1977).Google Scholar
- 4.Blum, M., "A machine independent theory of the complexity of the recursive functions," J Association Comput Mach Vol. 14(2) pp. 322–336 (1967).Google Scholar
- 5.Brainerd, W. and Landweber, L., Theory of Computation, John Wiley & Sons, New York (1974).Google Scholar
- 6.Csirmaz, L., "Structure of program runs of non-standard time," Acta Cybernetica Vol. 4(4) pp. 325–331 (1980).Google Scholar
- 7.Csirmaz, L., "Programs and program verification in a general setting," Theor Comput Sci, (to appear).Google Scholar
- 8.DeMillo, R. A. and Lipton, R. J., "Some connections between mathematical logic and complexity theory," Proceedings of the 11th Symposium on the Theory of Computation, pp. 153–159 (1979).Google Scholar
- 9.DeMillo, R. A. and Lipton, R. J., "The consistency of ‘P = NP’ and related problems with fragments of number theory," Proceedings of the 12th Symposium on the Theory of Computing, pp. 45-57 (May 1980).Google Scholar
- 10.Dimitracopolous, D. and Paris, J., "Truth definitions for delta-zero formulas," Preprint, (1980).Google Scholar
- 11.Fischer, P. C., "Theory of provably recursive functions," Trans Amer Math Soc Vol. 117(5) pp. 494–520 (May 1965).Google Scholar
- 12.Fortune, S., Topics in Computational Complexity, Cornell University (1979). (PhD Dissertation)Google Scholar
- 13.Fortune, S., Leivant, D., and O'Donnell, M., "The expressiveness of simple and second order type structures," IBM Research Report #RC 8542, (1980).Google Scholar
- 14.Goldrei, D. C., Macintyre, A., and Simmons, H., "The forcing companions of number theories," Israel J Math Vol. 14 pp. 317–337 (1973).Google Scholar
- 15.Gordon, D., "Complexity classes of provably recursive functions," J Comput and System Sci Vol. 18(3) pp. 294–303 (June 1979).Google Scholar
- 16.Grant, P. W., "Some more independence results in complexity theory," Theor Comput Sci Vol. 12 pp. 119–126 (1980).Google Scholar
- 17.Hajek, P., "Arithmetical complexity of some problems in computer science," pp. 282–287 in Mathematical Foundations of Computer Science, ed. J. Gruska,Springer-Verlag (1977).Google Scholar
- 18.Hajek, P., "Arithmetical hierarchy and complexity of computation," Theor Comput Sci Vol. 8 pp. 227–237 (1979).Google Scholar
- 19.Harrow, K., "The bounded arithmetic hierarchy," Inf and Control Vol. 36 pp. 102–117 (1978).Google Scholar
- 20.Hartmanis, J., "Relation between diagonalization, proof systems and complexity gaps," Proceedings of the 9th Symposium on the Theory of Computing, pp. 223–227 (1977).Google Scholar
- 21.Hartmanis, J. and Hopcroft, J. E., "Independence results in computer science," SIGACT News Vol. 8(4) pp. 13–24 (1976).Google Scholar
- 22.Hirschfeld, J. and Wheeler, W. H., Forcing, arithmetic, and division rings, Springer-Verlag Lecture Notes in Mathematics #454, Berlin & New York (1975).Google Scholar
- 23.Homer, S. and Reif, J. H., "The complexity of provable properties of first order theories," Preprint, (May 1980).Google Scholar
- 24.Joseph, D., On the power of formal systems for analyzing linear and polynomial time program behavior, Purdue University (1981). (Ph.D. Dissertation)Google Scholar
- 25.Joseph, D. and Young, P., "Fast programs for initial segments and polynomial time computation in weak models of arithmetic (preliminary abstract)," Proceedings of the 13th Symposium on the Theory of Computing, pp. 55–61 (May 1981).Google Scholar
- 26.Joseph, D. and Young, P., "Independence results in computer science?," J Comput Systems Sci, (to appear). (preliminary version in Proc. 12th SIGACT Symp Theor Comput, 1980)Google Scholar
- 27.Karp, R., "Reducibility among combinatorial problems," in Complexity of Computer Computations, ed. Miller and Thatcher,Plenum Press (1972).Google Scholar
- 28.Kent, C. and Hodgson, B., "An arithmetical characterization of NP," Lakehead Univ Technical Report, #6-80, (1980). (Dept of Mathematics, Thunder Bay, Ontario, Can.)Google Scholar
- 29.Kirby, L., Private communicationGoogle Scholar
- 30.Kirby, L. and Paris, J., "Initial segments of models of Peano's axioms," in Proceedings Conference Set Theory & Hierarchy Theory V; Bieutowice, Poland, Springer-Verlag Lecture Notes in Mathematics #619, Berlin & New York (1976).Google Scholar
- 31.Leivant, D., "The complexity of parameter passing in polymorphic procedures," Proceedings of the 13th Symposium on the Theory of Computing, pp. 38–45 (May 1981).Google Scholar
- 32.Leivant, D., "Unprovability of theorems of complexity theory in weak number theories," Theor Comput Sci, (to appear).Google Scholar
- 33.Lipton, R. J., "Model theoretic aspects of complexity theory," Proceedings of the 19th Symposium of the Foundations of Computer Science, pp. 193–200 (1978).Google Scholar
- 34.Macintyre, A. and Simmons, H., "Algebraic properties of number theories," Israel J Math Vol. 22 pp. 7–27 (1975).Google Scholar
- 35.Mahaney, S., "Sparse complete sets for NP: Solution of a conjecture of Berman and Hartmanis," Proc. of the 21st Symposium on the Foundations of Computer Science, pp. 54–60 (1980).Google Scholar
- 36.Manders, K., "Computational complexity of decision problems in elementary number theory," in Proceedings of the Set Theory and Hierarchy Theory Conference, Springer-Verlag, Karpacz (1979). (Lecture Notes in Mathematics Series.)Google Scholar
- 37.Matijesevic, Y., "Enumerable sets are diophantine," Dokl Akad Nauk SSSR Vol. 191 pp. 279–282 (1970).Google Scholar
- 38.McAloon, K.(ed.), Modeles de l'Arithmetique, Societe Mathematique de France, Paris (1980).Google Scholar
- 39.O'Donnell, M., "A programming language theorem which is independent of Peano Arithmetic," Proceedings of the 11th Symposium on the Theory of Computation, pp. 176–188 (1979).Google Scholar
- 40.Paris, J. and Wilkie, A., "Models of arithmetic and rudimentary sets," Preprint, (1980).Google Scholar
- 41.Paris, J. B., "Some independence results for Peano Arithmetic," J Symbolic Logic Vol. 43 pp. 725–730 (Dec. 1978).Google Scholar
- 42.Paris, J. B. and Harrington, L., "A mathematical incompleteness in Peano arithmetic," in Handbook of Mathematical Logic, ed. J. Barwise,North-Holland, New York (1978).Google Scholar
- 43.Phillips, N. C. K., "The P = NP question and recent independence results," Quaestiones Informaticae Vol. 1 pp. 26–29 (June 1979).Google Scholar
- 44.Rabin, M., "Degree of difficulty of computing a function and a partial ordering of recursive sets," Technical Report Hebrew Univ, (1960).Google Scholar
- 45.Rogers, H., The Theory of Recursive Functions and Effective Computability, McGraw-Hill, New York (1967).Google Scholar
- 46.Simmons, H., "Existentially closed models of basic number theory," pp. 325–369 in Logic Colloquium 76, ed. R. Gandy and J. M. R. Hyland,North-Holland (1977).Google Scholar
- 47.Smullyan, R., "Theory of formal systems," in Annals of Mathematics no. 47, Princeton Univ Press (1961).Google Scholar
- 48.Stockmeyer, L., "The polynomial-time hierarchy," Theo Comput Sci Vol. 3 pp. 1–22 (1977).Google Scholar
- 49.Wilkie, A., "to appear," in Proceedings of the Set Theory and Hierarchy Theory Conference, Springer-Verlag, Karpacz (1979). (Lecture Notes in Mathematics Series.)Google Scholar
- 50.Wilkie, A., "Applications of complexity theory to sigma-zero definability problems in arithmetic," Preprint, (1980).Google Scholar
- 51.Wrathall, C., "Rudimentary predicates and relative computability," Siam J Comput Vol. 7 pp. 194–209 (1978).Google Scholar
- 52.Young, P., "Optimization among provably equivalent programs," J Assoc Comput Mach Vol. 24(4) pp. 693–700 (1977). (Prelim abstract in IEEE SWAT Proc 1973).Google Scholar

## Copyright information

© Springer-Verlag Berlin Heidelberg 1981