Succinct Inputs, Lindstr om Quantifiers, and a General Complexity Theoretic Operator Concept

We address the question of the power of several logics with Lindströ m quantifiers over finite ordered structures. We will see that in the first-order case this nicely fits into the framewo rk of Barrington, Immerman, and Straubing’s examination of constant depth circu it classes. In the second-order case we get a strong relationship to succinct encodings f languages via circuits. Some of these logics can be characterized as closures of succinct encodings under appropriate reducibilities, others by certain hierarch ies of circuit classes. We will see that in a special case second-order Lindström quant ifiers can equivalently be expressed in first-order logic, while in the general case t his quivalence seems unlikely.

[1]  Steven Lindell.  A purely logical characterization of circuit uniformity , 1992, [1992] Proceedings of the Seventh Annual Structure in Complexity Theory Conference.

[2]  Helmut Veith,et al.  Succinct representation, leaf languages, and projection reductions , 1996, Proceedings of Computational Complexity (Formerly Structure in Complexity Theory).

[3]  Neil Immerman,et al.  The Complexity of Iterated Multiplication , 1995, Inf. Comput..

[4]  Georg Gottlob.  Relativized logspace and generalized quantifiers over finite structures , 1995, Proceedings of Tenth Annual IEEE Symposium on Logic in Computer Science.

[5]  Pierre McKenzie,et al.  Extensions to Barrington's M-Program Model , 1993, Theor. Comput. Sci..

[6]  Hans-Jörg Burtschick.  Berechnungs- und Beschreibungskomplexität von Zählfunktionen und Lindströmquantoren , 1997 .

[7]  José L. Balcázar,et al.  The complexity of algorithmic problems on succinct instances , 1992 .

[8]  Andreas Blass,et al.  Henkin quantifiers and complete problems , 1986, Ann. Pure Appl. Log..

[9]  Neil Immerman,et al.  Time, hardware, and uniformity , 1994, Proceedings of IEEE 9th Annual Conference on Structure in Complexity Theory.

[10]  Georg Gottlob.  NP trees and Carnap's modal logic , 1995, JACM.

[11]  Heribert Vollmer,et al.  Gap-Languages and Log-Time Complexity Classes , 1997, Theor. Comput. Sci..

[12]  Seinosuke Toda,et al.  PP is as Hard as the Polynomial-Time Hierarchy , 1991, SIAM J. Comput..

[13]  Pierluigi Crescenzi,et al.  A Uniform Approach to Define Complexity Classes , 1992, Theor. Comput. Sci..

[14]  Iain A. Stewart,et al.  Comparing the Expressibility of Languages Formed using NP-Complete Operators , 1991, J. Log. Comput..

[15]  Heribert Vollmer,et al.  Lindström Quantifiers and Leaf Language Definability , 1996, Int. J. Found. Comput. Sci..

[16]  N. Vereshchagin.  RELATIVIZABLE AND NONRELATIVIZABLE THEOREMS IN THE POLYNOMIAL THEORY OF ALGORITHMS , 1994 .

[17]  N. Immerman,et al.  On uniformity within NC 1 . , 1988 .

[18]  Neil Immerman,et al.  Expressibility and Parallel Complexity , 1989, SIAM J. Comput..

[19]  Thomas Schwentick,et al.  On the power of polynomial time bit-reductions , 1993, [1993] Proceedings of the Eigth Annual Structure in Complexity Theory Conference.

[20]  Perlindström.  First Order Predicate Logic with Generalized Quantifiers , 1966 .

[21]  Antoni Lozano,et al.  Succinct Circuit Representations and Leaf Language Classes are Basically the Same Concept , 1996, Inf. Process. Lett..

[22]  Iain A. Stewart,et al.  Using the Hamiltonian Path Operator to Capture NP , 1990, J. Comput. Syst. Sci..

[23]  Steven Lindell.  A Purely Logical Characterization of Circuit Uniformity (Extended Draft) , 1992 .

[24]  Michael Sipser,et al.  Borel sets and circuit complexity , 1983, STOC.

[25]  Ulrich Hertrampf.  Acceptance by transformation monoids (with an application to local self reductions) , 1997, Proceedings of Computational Complexity. Twelfth Annual IEEE Conference.

[26]  Denis Thérien,et al.  Finite monoids and the fine structure of NC1 , 1987, STOC.

[27]  Neil Immerman,et al.  Languages that Capture Complexity Classes , 1987, SIAM J. Comput..

[28]  Y. Gurevich.  On Finite Model Theory , 1990 .

[29]  Denis Thérien,et al.  Logspace and Logtime Leaf Languages , 1996, Inf. Comput..