It is shown that the Euclidean traveling salesman problem lies in the counting hierarchy - the previous best upper bound for this important problem (in terms of classical complexity classes) being PSPACE.Expand

In this chapter, P, NP, and related complexity classes are defined, and the use of diagonalization and padding techniques to prove relationships between classes are illustrated, and it is shown how to prove that a problem is NP-complete.Expand

It is proved that in the commutative setting, uniform semi-unbounded arithmetic circuits of logarithmic depth are as powerful as uniform arithmetic circuit of polynomial degree (and unrestricted depth); earlier proofs did not work in the uniform setting.Expand

It is shown that the permanent cannot be computed by uniform constantdepth threshold circuits of size T (n), for any function T such that for all k, T ( n) = o(2), and any problem that is hard for the complexity class C=P requires circuits of this size.Expand

It is shown that division lies in the complexity class FOM + POW obtained by augmenting FOM with a predicate for powering modulo small primes, and that the predicate POW itself lies in FOM.Expand

The author presents a very simple proof of the fact that any language accepted by polynomial-size depth-k unbounded-fan-in circuits of AND and OR gates is accepted by depth-three threshold circuits… Expand

It is shown that this natural complexity class for which the problems of determining if a system of linear equations is feasible and computing the rank of an integer matrix are complete under logspace reductions is closed under NC1-reducibility.Expand

A notion of resource-bounded measure for P and other subexponential-time classes is presented, based on Lutz's notion of measure, but overcomes the limitations that cause Lutz’s definitions to apply only to classes at least as large as E.Expand

It is shown that in the context of nonuniform complexity, nondeterministic logarithmic space bounded computation can be made unambiguous in the class of problems reducible to context-free languages.Expand

A very simple proof of theorem of Jung is given, showing that probabilistic logspace-bounded (PL) machines lose none of their computational power if they are restricted to run in polynomial time.Expand