On proofs about threshold circuits and counting hierarchies

We define theories of Bounded Arithmetic characterizing classes of functions computable by constant-depth threshold circuits of polynomial and quasipolynomial size. Then we define certain second-order theories and show that they characterize the functions in the Counting Hierarchy. Finally we show that the former theories are isomorphic to the latter via the so-called RSUV-isomorphism.

[1]  Uzi Vishkin,et al.  Constant Depth Reducibility , 1984, SIAM J. Comput..

[2]  Georg Schnitger,et al.  Parallel Computation with Threshold Functions , 1986, J. Comput. Syst. Sci..

[3]  Georg Schnitger,et al.  Parallel Computation with Threshold Functions , 1988, J. Comput. Syst. Sci..

[4]  Eric Allender,et al.  A note on the power of threshold circuits , 1989, 30th Annual Symposium on Foundations of Computer Science.

[5]  Seinosuke Toda.  On the computational power of PP and (+)P , 1989, 30th Annual Symposium on Foundations of Computer Science.

[6]  P. Clote.  Sequential, machine-independent characterizations of the parallel complexity classes AlogTIME, AC k , NC k and NC , 1990 .

[7]  S. Buss.  Axiomatizations and Conservation Results for Fragments of Bounded Arithmetic ∗ , 1990 .

[8]  Gaisi Takeuti,et al.  S3i andV2i(BD) , 1990, Arch. Math. Log..

[9]  Jan Krajícek,et al.  Bounded Arithmetic and the Polynomial Hierarchy , 1991, Ann. Pure Appl. Log..

[10]  J. Krajícek.  Fragments of bounded arithmetic and bounded query classes , 1993 .

[11]  P. Clote,et al.  Arithmetic, proof theory, and computational complexity , 1993 .

[12]  S. Buss,et al.  An Application of Boolean Complexity to Separation Problems in Bounded Arithmetic , 1994 .

[13]  Peter Clote,et al.  Computational Models and Function Algebras , 1994, LCC.

[14]  P. Clote,et al.  First Order Bounded Arithmetic and Small Boolean Circuit Complexity Classes , 1995 .

[15]  Heribert Vollmer,et al.  Recursion Theoretic Characterizations of Complexity Classes of Counting Functions , 1996, Theor. Comput. Sci..

[16]  Jan Johannsen.  A bounded arithmetic theory for constant depth threshold circuits , 1996 .

[17]  外史 竹内.  Bounded Arithmetic と計算量の根本問題 , 1996 .

[18]  Peter Clote,et al.  Computation Models and Function Algebras , 1999, Handbook of Computability Theory.

[19]  Eric Allender,et al.  The Permanent Requires Large Uniform Threshold Circuits , 1999, Chic. J. Theor. Comput. Sci..