# On logical descriptions of some concepts in structural complexity theory

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## Abstract

A logical framework is introduced which captures the behaviour of oracle machines and gives logical descriptions of complexity classes that are defined by oracle machines. Using this technique the notion of first-order selfreducibility is investigated and applied to obtain a structural result about non-uniform complexity classes below P.

## Keywords

Complexity Theory Logical Description Order Logic Travel Salesperson Problem First Order
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## References

- [1]S. Buss and L. Hay,
*On truth-table reducibility to SAT and the difference hierarchy over NP*, Proceedings of 3rd Conference on Structure in Complexity Theory 1988, 224–233.Google Scholar - [2]J. Cai and L. Hemachandra,
*The Boolean hierarchy: hardware over NP*, Proceedings of 1st Conference on Structure in Complexity Theory 1986, Lecture Notes in Computer Science Nr. 223, Springer 1986, 105–124.Google Scholar - [3]R. Fagin,
*Generalized First-Order Spectra and Polynomial-Time Recognizable Sets*, SIAM-AMS Proc.**7**(1974), 43–73.Google Scholar - [4]E. Grandjean,
*The Spectra of First-Order Sentences and Computational Complexity*, SIAM J. Comp.**13**(1984), 367–373.Google Scholar - [5]E. Grandjean,
*Universal Quantifiers and Time Complexity of Random Access Machines*, Math. Syst. Theory (1985), 171–187.Google Scholar - [6]E. Grandjean,
*First-Order Spectra with One Variable*, in: E. Börger (Ed.), “Computation Theory and Logic”, Lecture Notes in Computer Science Nr. 270, Springer 1987, 166–180.Google Scholar - [7]Y. Gurevich,
*Toward logic tailord for computational complexity*, in: M. M. Richter et al. (Eds), Computation and Proof Theory, Springer Lecture Notes in Mathematics Nr. 1104 (1984), 175–216.Google Scholar - [8]Y. Gurevich,
*Logic and the Challenge of Computer Science*, in: E. Börger (Ed), Trends in Theoretical Computer Science, Computer Science Press (1988), 1–57.Google Scholar - [9]N. Immerman,
*Relational Queries Computable in Polynomial Time*, Inf. and Control**68**(1986), 86–104.Google Scholar - [10]N. Immerman,
*Languages that Capture Complexity Classes*, SIAM J. Comput.**16**(1987), 760–778.CrossRefGoogle Scholar - [11]N. Immerman,
*Expressibility and Parallel Complexity*, Tech. Report 546, Yale University, Department of Computer Science (1987).Google Scholar - [12]N. Immerman,
*Expressibility as a Complexity Measure: Results and Directions*, Proc. of 2nd Conf. on Structure in Complexity Theory (1987), 194–202.Google Scholar - [13]N. Immerman,
*Nondeterministic space is closed under complementation*, SIAM J. Comput.**17**(1988), 935–939.CrossRefGoogle Scholar - [14]N. Immerman,
*Descriptive and Computational Complexity*, in: J. Hartmanis (Ed.), Computational Complexity Theory, Proc. of AMS Symposia in Appl. Math.**38**(1989), 75–91.Google Scholar - [15]N. Jones and W. Laaser,
*Complete problems for deterministic polynomial time*, Theoret. Comp. Sci**3**(1977), 105–117.Google Scholar - [16]R. Karp and R. Lipton,
*Turing Machines that Take Advice*, in: Logic and Algorithmic, Monographie Nr. 30 de L'Enseignement Mathématique, Genève 1982, 255–273.Google Scholar - [17]K. Ko,
*On self-reducibility and weak P-selectivity*, J. of Comput. Syst. Sci.**26**(1983), 209–221.Google Scholar - [18]K. Ko and U. Schöning,
*On circuit-size complexity and the low hierarchy in NP*, SIAM J. Comput.**14**(1985), 41–51.CrossRefGoogle Scholar - [19]C. Papadimitriou and M. Yannakakis,
*The complexity of facets (and some facets of complexity)*, Proceedings of 14th STOC 1982, 255–260.Google Scholar - [20]R. Szelepcsényi,
*The Method of Forced Enumeration for Nondeterministic Automata*, Acta Informatica**26**, (1988), 279–284.Google Scholar - [21]U. Schöning,
*A Low and a High Hierarchy within NP*, J. Comput. Syst. Sci.**27**(1983), 14–28.Google Scholar - [22]U. Schöning,
*Complexity and Structure*, Springer Lecture Notes in Computer Science Nr. 211 (1986).Google Scholar - [23]M. Vardi,
*Complexity of Relational Query Languages*, Proc. of 14th STOC (1982), 137–146.Google Scholar - [24]K. Wagner,
*Bounded Query Computations*, Proceedings of 3rd Conference on Structure in Complexity Theory 1988, 260–277.Google Scholar - [25]G. Wechsung (and K. Wagner),
*On the Boolean cloure of NP*, Proceedings of FCT 85, Lecture Notes in Computer Science Nr. 199, Springer 1985, 485–493.Google Scholar

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