Counting the number of solutions: a survey of recent inclusion results in the area of counting classes

Preprin

Structural properties of the counting hierarchies

We study three different hierarchies related to the notion of counting: the polynomial time counting hierarchy, the hierarchy of counting functions, and the logarithmic time counting hierarchy. We investigate the connections between these hierarchies and study some of their structural properties, settling many open questions dealing with oracle characterizations, closure under boolean operations, lowness, complete problems, succint representations, and relations with other complexity classes. We develop a new combinatorial technique to obtain relativized separations, and we obtain also absolute separations for some of the studied classes.Postprint (published version

Structural properties of the counting hierarchies

We study three different hierarchies related to the notion of counting: the polynomial time counting hierarchy, the hierarchy of counting functions, and the logarithmic time counting hierarchy. We investigate the connections between these hierarchies and study some of their structural properties, settling many open questions dealing with oracle characterizations, closure under boolean operations, lowness, complete problems, succint representations, and relations with other complexity classes. We develop a new combinatorial technique to obtain relativized separations, and we obtain also absolute separations for some of the studied classes

A combinatorial technique for separating counting complexity classes

We introduce a new combinatorial technique to obtain relativized separations of certain complexity classes related to the idea of counting, like PP, G (exact counting), and ¿P (parity). To demonstrate its usefulness we present three relativizations separating NP from G, NP from ¿P and ¿P from PP. Other separations follow from these results, and as a consequence we obtain an oracle separating PP from PSPACE, thus solving an open problem proposed by Angluin in [An,80]. From the relativized separations to obtain absolute separations for counting complexity classes with log-time bounded computation time.Postprint (published version

A combinatorial technique for separating counting complexity classes

We introduce a new combinatorial technique to obtain relativized separations of certain complexity classes related to the idea of counting, like PP, G (exact counting), and ¿P (parity). To demonstrate its usefulness we present three relativizations separating NP from G, NP from ¿P and ¿P from PP. Other separations follow from these results, and as a consequence we obtain an oracle separating PP from PSPACE, thus solving an open problem proposed by Angluin in [An,80]. From the relativized separations to obtain absolute separations for counting complexity classes with log-time bounded computation time

Turing machines with few accepting computations

In this paper we study complexity classes defined by path-restricted nondeterministic machines. We prove that for every language L in the class Few a polynomial time nondeterministic machine can be constructed which has f(x)+1 accepting paths for strings x ¿ L, and f(x) accepting paths for strings that are not in L, being f a function in PF. From this result we obtain lowness properties of the class Few, and positive relativizations of different counting classes.Postprint (published version

Kolmogorov complexity of #P functions

Are the outputs of #P functions "random"? We way phrase this question more precisely: are there #P functions whose outputs cannot in general be compressed into a string of small length, and recovered quickly from that string, also given the input as additional information? We prove that the answer to this question is "yes" if and only if P ¿ PP.Postprint (published version

Classes of bounded nondeterminism

We study certain language classes located between P and N P that are defined by polynomial time machines with a bounded amount of nondeterminism. We observe that these classes have complete problems and find a characterization of the classes using robust machines with bounded access to the oracle, obtaining some other results in this direction. We also study questions related with the existence of complete tally sets in these classes and closure of the classes under different types of polynomial time reducibilities

Computing functions with parallel queries to NP

The class "Theta_2^p" of languages polynomial time truth-table reducible to sets in NP has a wide range of different characterizations. We survey some of them studying the classes obtained when the characterizations are used to define functions instead of languages. We show that in this way the three function classes FP_parallel^NP, FP_log^NP and FL_log^NP are obtained. We give an overview about the known relationships between these classes, including some original results.Postprint (published version