This paper is dedicated to the memory of Ronald V. Book, 1937-1997.

We prove that if P ≠ NP, then there exists a set in NP that is not polynomial time bounded truth-table reducible (in short, p btt -reducible) to any sparse set. In other words, we prove that no sparse p btt -hard set exists for NP unless P = NP. By using the technique proving this result, we investigate intractability of several number theoretic decision problems, i.e., decision problems defined naturally from number theoretic problems. We show that for these number theoretic decision problems, if it is not in P, then it is not p btt -reducible to any sparse set.

We study the polynomial time counting hierarchy, a hierarchy of complexity classes related to the notion of counting. We investigate some of their structural properties, settling many open questions dealing with oracle characterizations, closure under boolean operations, and relations with other complexity classes. We develop a new combinatorial technique to obtain relativized separations for some of the studied classes, which imply absolute separations for some logarithmic time bounded complexity classes.

Chang and Kadin have shown that if the difference hierarchy over NP collapses to level k, then the polynomial hierarchy (PH) is equal the kth level of the difference hierarchy over \Sigma p 2 . We simplify their proof and obtain a slightly stronger conclusion: If the difference hierarchy over NP collapses to level k, then PH collapses to i P NP (k\Gamma1)-tt j NP , the class of sets recognized in polynomial time with k \Gamma 1 nonadaptive queries to a set in NP NP and an unlimited number of queries to a set in NP. We also extend the result to classes other than NP: For any class C that has p m -complete sets and is closed under p conj - and NP m -reductions (alternatively, closed under p disj - and co-NP m -reductions), if the difference hierarchy over C collapses to level k, then PH C = i P NP (k\Gamma1)-tt j C . Then we show that the exact counting class C=P is closed under p disj - and co-NP m - reductions. Consequently, if the difference hiera...

this paper. The low hierarchy, as defined in [Sc-83], can only be used to classify the complexity of sets in NP. In order to talk about related sets that are not in NP, the extended low hierarchy was introduced in [BBS-86]. (The levels of this hierarchy are labeled EL 2 ,EL 2 , A preliminary version of this paper was presented at the 16th International Colloquium on Automata, Languages, and Programming [AH-89a]

This paper studies a notion called polynomial-time membership comparable sets. For a function g, a set A is polynomial-time g-membership comparable if there is a polynomialtime computable function f such that for any x 1 ; \Delta \Delta \Delta ; xm with m g(maxfjx 1 j; \Delta \Delta \Delta ; jx m jg), outputs b 2 f0; 1g m such that (A(x 1 ); \Delta \Delta \Delta ; A(xm )) 6= b. The following is a list of major results proven in the paper. 1. Polynomial-time membership comparable sets construct a proper hierarchy according to the bound on the number of arguments. 2. Polynomial-time membership comparable sets have polynomial-size circuits. 3. For any function f and for any constant c ? 0, if a set is p f(n)-tt -reducible to a Pselective set, then the set is polynomial-time (1 + c) log f(n)-membership comparable. 4. For any C chosen from fPSPACE;UP;FewP;NP;C=P;PP;MOD 2 P; MOD 3 P; \Delta \Delta \Deltag, if C ` P-mc(c log n) for some c ! 1, then C = P. As a corollary of the last tw...

this paper we study two different ways to restrict the power of NP: We consider languages accepted by nondeterministic polynomial time machines with a small number of accepting paths in case of acceptance, and also investigate subclasses of NP that are low for complexity classes not known to be in the polynomial time hierarchy. The first complexity class defined following the idea of bounding the number of accepting paths was Valiant's class UP (unique P) [Va76] of languages accepted by nondeterministic Turing machines that have exactly one accepting computation path for strings in the language, and none for strings not in the language. This class plays an important role in the areas of one-way functions and cryptography, for example in [GrSe84] it is shown that P6=UP if and only if one-way functions exist. The class UP can be generalized in a natural way by allowing a polynomial number of accepting paths. This gives rise to the class FewP defined by Allender [Al85] in connection with the notion of P-printable sets. We study complexity classes defined by such path-restricted nondeterministic polynomial time machines, and show results that exploit the fact that the machines for these classes have a bounded number of accepting computation paths. We will not only consider these subclasses of NP, namely UP and FewP, but also the class Few, an extension of FewP defined by Cai and Hemachandra [CaHe89], in which the accepting mechanism of the machine is more flexible. 1 The three classes UP, FewP and Few are all defined in terms of nondeterministic machines with a bounded number of accepting paths for every input string, but for the last two classes this number is not known beforehand, and can range over a space of polynomial size. We show in Section 3 that a polynomial numb...

The process of reconstructing evolutionary trees can be viewed formally as an optimization problem. Recently, decision problems associated with the most commonly used approaches to reconstructing such trees have been shown to be NP-complete [Day87, DJS86, DS86, DS87, GF82, Kri88, KM86]. In this thesis, a framework is established that incorporates all such problems studied to date. Within this framework, the NP-completeness results for decision problems are extended by applying theorems from [CT91, Gas86, GKR92, JVV86, KST89, Kre88, Sel91] to derive bounds on the computational complexity of several functions associated with each of these problems, namely ffl evaluation functions, which return the cost of the optimal tree(s), ffl solution functions, which return an optimal tree, ffl spanning functions, which return the number of optimal trees, ffl enumeration functions, which systematically enumerate all optimal trees, and ffl random-selection functions, which return a random...

this paper was presented at the 16th International Colloquium on Automata, Languages, and Programming [3]

Circuit-size complexity is compared with deterministic and nondeterministic time complexity in the presence of pseudorandom oracles. The following separations are shown to hold relative to every pspace-random oracle A, and relative to almost every oracle A 2 ESPACE.