# On the complexity of small description and related topics

Invited Lectures

First Online:

## Abstract

The class P/poly is known to be the class of sets with small descriptions, more specifically, polynomial size circuits. In this paper, we discuss the problem of obtaining the polynomial size circuits for a given set in P/poly by using the set as an oracle. Recent results on upper and lower bounds of the relative complexity of this problem are presented. We also introduce two related research topics — query learning and identity mapping network — and explain how they are related to this problem.

## Keywords

Polynomial Time Internal Node Turing Machine Input Pattern Target Concept
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

- [Ang87]D. Angluin: Learning regular sets from queries and counterexamples.
*Information and Computation*75 (1987), 87–106.zbMATHMathSciNetCrossRefGoogle Scholar - [Ang88]D. Angluin: Queries and concept leaxning.
*Machine Learning*2 (1988), 319–342.Google Scholar - [Ang90]D. Angluin: Negative results for equivalence queries.
*Machine Learning*5 (1990), 121–150.Google Scholar - [AK91]D. Angluin and M. Kharitonov: When won't membership queries help? In
*Proc. 23rd ACM Sympos. on Theory of Computing*, ACM (1991), 444–454.Google Scholar - [BDG88]J. Balcázar, J. Díaz, and J. Gabarró:
*Structural Complexity I*, EATCS Monographs on Theoretical Computer Science, Springer-Verlag (1988).Google Scholar - [BR87]P. Berman and R. Roos: Learning one-counter languages in polynomial time. In
*Proc. 28th IEEE Foundations of Computer Science*, IEEE (1987), 61–67.Google Scholar - [Boo92]R. Book: On sets with small information content. In
*Kolmogorov Complexity: Theory and Relations to Computational Complexity*O. Watanabe (ed.), to appear.Google Scholar - [Gav92]R. Gavaldà: On the second level of the polynomial-time hierachy relative to sparse sets. In
*Proc. 7th Structure in Complexity Theory Conference*(1992), to appear.Google Scholar - [GW91]R. Gavaldà and O. Watanabe: On the computational complexity of small descriptions. In
*Proc. 6th Structure in Complexity Theory Conference*, IEEE (1991), 89–101; the final version will appear in SIAM J. Comput.Google Scholar - [GS88]J. Grollmann and A. Selman: Complexity measures for public-key cryptosystems.
*SIAM J. Comput.*17 (1988), 309–335.zbMATHMathSciNetCrossRefGoogle Scholar - [GGM86]O. Goldreigh, S. Goldwasser, and S. Micali: How to construct random functions.
*J. ACM*33 (1986), 792–807.CrossRefGoogle Scholar - [Has90]J. Håstad: Pseudo-random generators under uniform assumptions. In
*Proc. 22nd ACM Sympos. on Theory of Computing*, ACM (1990), 395–415.Google Scholar - [ILL89]R. Impagliazzo, L. Levin, and M. Luby: Pseudo-random generation from one-way functions. In
*Proc. 21st ACM Sympos. on Theory of Computing*, ACM (1989), 12–24.Google Scholar - [Ish89]H. Ishizuka: Learning simple deterministic languages. In
*Proc. 2nd Workshop on Computational Learning Theory*, Morgan Kaufmann (1989), 162–174.Google Scholar - [KL80]R. Karp and R. Lipton: Some connections between nonuniform and uniform complexity classes In
*Proc. 12th ACM Sympos. on Theory of Computing*, ACM (1980), 302–309.Google Scholar - [KV89]M. Kearns and L. Valiant: Cryptographic limitations on learning boolean formulae and finite automata. In
*Proc. 21st ACM Sympos. on Theory of Computing*, ACM (1989), 433–444.Google Scholar - [Ko85]K. Ko: Continuous optimization problems and a polynomial hierarchy of real functions.
*J. Complexity*1 (1985), 210–231.zbMATHMathSciNetCrossRefGoogle Scholar - [LOW92]How hard are sparse sets? In
*Proc. 7th Structure in Complexity Theory Conference*(1992), to appear.Google Scholar - [PW88]L. Pitt and M. Warmuth: Reductions among prediction problems: on the difficulty of prediction automata. In
*Proc. 3rd Structure in Complexity Theory Conference*, IEEE (1988), 60–69.Google Scholar - [Sak88]Y. Sakakibara: Learning context-free grammars from structural data in polynomial time. In
*Proc. 1st Workshop on Computational Learning Theory*, Morgan Kaufmann (1988), 330–344.Google Scholar - [Sch85]U. Schöning:
*Complexity and Structure*, Lecture Notes in Computer Science 211, Springer-Verlag (1985).Google Scholar - [Val84]L. Valiant: A theory of the learnable.
*C. ACM*27 (1984), 1134–1142.zbMATHCrossRefGoogle Scholar - [Wat90]O. Watanabe: A formal study of learning via queries. In
*Proc. 17th International Colloquium on Automata, Languages and Programming*, Lecture Notes in Computer Science 443, Springer-Verlag (1990), 137–152.Google Scholar - [Wat91]O. Watanabe: On the complexity of three layer networks for identity mapping, IEICE Technical Report COMP91-84 (1991).Google Scholar
- [Wat92]O. Watanabe: A framework for polynomial time query learnability. Technical Report 92TR-0003, Dept. of Computer Science, Tokyo Institute of Technology (1992).Google Scholar
- [WG92]O. Watanabe and R. Gavaldà: Structural analysis of polynomial time query learnability. Technical Report 92TR-0004, Dept. of Computer Science, Tokyo Institute of Technology (1992).Google Scholar

## Copyright information

© Springer-Verlag Berlin Heidelberg 1992