Counting classes consist of languages defined in terms of the number of accepting computations of nondeterministic polynomial-time Turing machines. Well known examples of counting classes are NP, co-NP, \PhiP, and PP. Every counting class is a subset of P #P[1] , the class of languages computable in polynomial time using a single call to an oracle capable of determining the number of accepting paths of an NP machine. Using closure properties of #P, we systematically develop a complexity theory for counting classes defined in terms of thresholds and moduli. An unexpected result is that MOD k iP = MOD k P for prime k. Finally, we improve a result of Cai and Hemachandra by showing that recognizing languages in the class Few is as easy as distinguishing uniquely satisfiable formulas from unsatisfiable formulas (or detecting unique solutions, as in [28]). 1. Introduction Valiant [27] defined the class #P of functions whose values equal the number of accepting paths of polynomial-time bo...

We show how to use various notions of genericity as tools in oracle creation. In particular, 1. we give an abstract definition of genericity that encompasses a large collection of different generic notions; 2. we consider a new complexity class AWPP, which contains BQP (quantum polynomial time), and infer several strong collapses relative to SP-generics; 3. we show that under additional assumptions these collapses also occur relative to Cohen generics; 4. we show that relative to SP-generics, ULIN ∩ co-ULIN ̸ ⊆ DTIME(n k) for any k, where ULIN is unambiguous linear time, despite the fact that UP ∪ (NP ∩ co-NP) ⊆ P relative to these generics; 5. we show that there is an oracle relative to which NP/1∩co-NP/1 ̸ ⊆ (NP∩co-NP)/poly; and 6. we use a specialized notion of genericity to create an oracle relative to which NP BPP ̸ ⊇ MA.

The study of the complexity of sets encompasses two complementary aims: (1) establishing -- usually via explicit construction of algorithms -- that sets are feasible, and (2) studying the relative complexity of sets that plausibly might be feasible but are not currently known to be feasible (such as the NP-complete sets and the PSPACE-complete sets). For the study of the complexity of closure properties, a recent urry of results [21, 33, 49, 6, 7, 16] has established an analog of (1); these papers explicitly demonstrate many closure properties possessed by PP and C=P (and the proofs implicitly give closure properties of the function class #P). The present paper presents and develops, for function classes such as #P, SpanP, OptP, and MidP, an analog of (2): a general theory of the complexity of closure properties. In particular, we show that subtraction is hard for the closure properties of each of these classes: each is closed under subtraction if and only if it is closed under every polynom...

It is shown that determining whether a quantum computation has a nonzero probability of accepting is at least as hard as the polynomial time hierarchy. This hardness

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...

We show that if every NP set is polynomial-time bounded truthtable reducible to some P-selective set, then NP is contained in DTIME(2 n O(1= p log n) ). In the proof, we implement a recursive procedure that reduces the number of nondeterministic steps of a given nondeterministic computation.

this article mention all of the amazing research in computational complexity theory. We survey various areas in complexity choosing papers more for their historical value than necessarily the importance of the results. We hope that this gives an insight into the richness and depth of this still quite young eld

this article mention all of the amazing research in computational complexity theory. We survey various areas in complexity choosing papers more for their historical value than necessarily the importance of the results. We hope that this gives an insight into the richness and depth of this still quite young eld

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