The class \Theta p 2 of languages polynomial-time truth-table reducible to sets in NP has a wide range of different characterizations. We consider several functional versions of \Theta p 2 based on these characterizations. We show that in this way the three function classes FL NP log , FP NP log , and FP NP k are obtained. In contrast to the language case the function classes seem to all be different. We give evidence in support of this fact by showing that FL NP log coincides with any of the other classes then L = P, and that the equality of the classes FP NP log and FP NP k would imply that the number of nondeterministic bits needed for the computation of any problem in NP can be reduced by a polylogarithmic factor, and that the problem can be computed deterministically with a sub-exponential time bound of order 2 n O(1= log log n) . 1 Introduction The study of nondeterministic computation is a central topic in structural complexity theory. The acceptance mechanism of...

We prove that the Minimum Equivalent DNF problem is \Sigma p 2 -complete, resolving a conjecture due to Stockmeyer. The proof involves as an intermediate step a variant of a related problem in logic minimization, namely, that of finding the shortest implicant of a Boolean function. We also obtain certain results concerning the complexity of the Shortest Implicant problem that may be of independent interest. When the input is a formula, the Shortest Implicant problem is \Sigma p 2 - complete, and \Sigma p 2 -hard to approximate to within an n 1=2\Gammaffl factor. When the input is a circuit, approximation is \Sigma p 2 - hard to within an n 1\Gammaffl factor. However, when the input is a DNF formula, the Shortest Implicant problem cannot be \Sigma p 2 -complete unless \Sigma p 2 = NP[log 2 n] NP . 1. Introduction Two-level (DNF) logic minimization is a central practical problem in logic synthesis and also one of the more natural problems in the polynomial hierarchy....

. The relationship between nondeterminism and other computational resources is investigated based on the "guess-then-check" model GC. Systematic techniques are developed to construct natural complete languages for the classes defined by this model. This improves a number of previous results in the study of limited nondeterminism. Connections of the model GC to computational optimization problems are exhibited. Key words. computational complexity, nondeterminism, complete languages, computational optimization AMS subject classifications. 68Q05, 68Q10, 68Q15, 68Q25 PII. S0097539793258295 1. Introduction. The study of the power of nondeterminism is central to complexity theory. The relationship between nondeterminism and other computational resources still remains unclear. Two fundamental questions are those of how much computational resource we should pay in order to eliminate nondeterminism and how much computational resource we can save if we are granted nondeterminism. A computation ...

. The fi hierarchy consists of classes fi k = NP[log k n] ` NP. Unlike collapses in the polynomial hierarchy and the Boolean hierarchy, collapses in the fi hierarchy do not seem to translate up, nor does closure under complement seem to cause the hierarchy to collapse. For any consistent set of collapses and separations of levels of the hierarchy that respects P = fi 1 ` fi 2 ` \Delta \Delta \Delta ` NP, we can construct an oracle relative to which those collapses and separations hold; at the same time we can make distinct levels of the hierarchy closed under computation or not, as we wish. To give two relatively tame examples: For any k 1, we construct an oracle relative to which P = fi k 6= fi k+1 6= fi k+2 6= \Delta \Delta \Delta and another oracle relative to which P = fi k 6= fi k+1 = PSPACE: We also construct an oracle relative to which fi 2k = fi 2k+1 6= fi 2k+2 for all k. Key words. structural complexity theory, limited nondeterminism, hierarchies, oracles AMS subject clas...

The maximum number of strands used is an important measure of a molecular algorithm's complexity. This measure is also called the volume used by the algorithm. Every problem that can be solved by an NP Turing machine with b(n) binary nondeterministic choices can be solved by molecular computation in a polynomial number of steps, with four test tubes, in volume 2 b(n) . We identify a large class of recursive algorithms that can be implemented using bounded nondeterminism. This yields improved molecular algorithms for important problems like 3-SAT, independent set, and 3-colorability. 1. A model of molecular computing Molecular computation was first studied in [1, 20]. The models we define were inspired as well by the work of [3, 28]. A molecular sequence is a string over an alphabet \Sigma (we can use any alphabet we like, encoding characters of \Sigma by finite sequences of base pairs). A test tube is a multiset of molecular sequences. We describe the allowable operations below. Whe...

The Transversal Hypergraph Generation is the problem of generating, given a hypergraph, the set of its minimal transversals, i.e., the hypergraph whose hyperedges are the minimal hitting sets of the given one. The purpose of this paper is to present an efficient and practical algorithm for solving this problem. We show that the proposed algorithm operates in a way that rules out regeneration and, thus, its memory requirements are polynomially bounded to the size of the input hypergraph. Although no time bound for the algorithm is given, experimental evaluation and comparison with other approaches have shown that it behaves well in practice and it can successfully handle large problem instances.

. We investigate the complexity and approximability of a basic optimization problem in the second level of the Polynomial Hierarchy, that of finding shortest implicants. We show that the DNF variant of this problem is complete for a complexity class in the second level of the hierarchy utilizing log 2 n-limited nondeterminism. We obtain inapproximability results for the DNF and formula variants of the shortest implicant problem that show that trivial approximation algorithms are optimal for these problems, up to lower order terms. It is hoped that these results will be useful in studying the complexity and approximability of circuit minimization problems, which have close connections to implicant problems. 1 Introduction Circuit minimization problems have been extensively studied since the 1950's as important practical problems in logic synthesis, and, since the early 1970's, as natural problems in the second level of the Polynomial Hierarchy. These problems are widely be...

We review some of the known results about the complexity of computing solutions or proofs of membership for problems in NP. Trying to capture the complexity of this problem, we consider the classes of functions FP NP , FP NP [f ] (for certain bounded functions f ), NPSV, and FP NP tt and provide some examples of NP problems with search functions in these classes. We also consider whether NP-complete problems can have such proofs of membership. We use the problem of obtaining solutions to compare the relative powers of the function classes above . Finally, we consider the situation in the nondeterministic logarithmic space setting, showing how the complexity of obtaining solutions for NL sets compares with the NP case. 1 Introduction Problems in the class NP have traditionally been studied from a decisional point of view. This has been so mainly because in all natural cases an algorithm providing a yes/no answer to an NP problem can be used to obtain a solution for the problem, ...

this paper we are interested in absolute results and consider advice classes slightly smaller than P=poly and circuit classes smaller than polynomial-size. And we establish several new lower bounds for exponential time problems with respect to these classes. This is not a new tack to explore. In last years' Structures Bin Fu [Fu93] considered lower bounds for polynomial time reductions to sparse sets, where limits are placed on the number of queries to the sparse set. The main result of his paper was that there are sets in EXP which are not polynomial time Turing reducible to a sparse set when the reduction is restricted to querying the sparse set no more than n

We define the sharply bounded hierarchy, SBH (QL), a hierarchy of classes within P , using quasilinear-time computation and quantification over values of length log n. It generalizes the limited nondeterminism hierarchy introduced by Buss and Goldsmith, while retaining the invariance properties. The new hierarchy has several alternative characterizations. We define both SBH (QL) and its corresponding hierarchy of function classes, FSBH(QL),and present a variety of problems in these classes, including ql m -complete problems for each class in SBH (QL). We discuss the structure of the hierarchy, and show that certain simple structural conditions on it would imply P 6= PSPACE. We present characterizations of SBH (QL) relations based on alternating Turing machines and on first-order definability, as well as recursion-theoretic characterizations of function classes corresponding to SBH (QL).