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...

We characterize the complexity of some natural and important problems in linear algebra. In particular, we identify natural complexity classes for which the problems of (a) determining if a system of linear equations is feasible and (b) computing the rank of an integer matrix, (as well as other problems), are complete under logspace reductions. As an important

The Random Oracle Hypothesis, attributed to Bennett and Gill, essentially states that the relationships between complexity classes which hold for almost all relativized worlds must also hold in the unrelativized case. Although this paper is not the first to provide a counterexample to the Random Oracle Hypothesis, it does provide a most compelling counterexample by showing that for almost all oracles A, IP A 6= PSPACE A . If the Random Oracle Hypothesis were true, it would contradict Shamir's result that IP = PSPACE. In fact, it is shown that for almost all oracles A, co-NP A 6` IP A . These results extend to the multi-prover proof systems of Ben-Or, Goldwasser, Kilian and Wigderson. In addition, this paper shows that the Random Oracle Hypothesis is sensitive to small changes in the definition. A class IPP, similar to IP, is defined. Surprisingly, the IPP = PSPACE result holds for all oracle worlds. 1 Department of Computer Science, Cornell University, Ithaca, NY 14853, U.S.A...

this paper to give simple proofs, in a uniform format, of the major known (pre-1992) results relating how polynomial-time reductions of SAT to sparse sets collapse the polynomial-time hierarchy. To help the reader familiar with basic facts of complexity theory follow the main flow of ideas, while keeping the exposition self-contained, straight forward proofs from elementary complexity theory are relegated to footnotes. We treat polynomial-time Turing reductions (i.e., Cook reductions) in Section 2. Bounded truth-table reductions (and many-one reductions) are treated in Section 3. Sections 2 and 3 may be read independently of each other. Section 4 uses the definitions of Section 3 to give simple proofs of results on conjunctive and disjunctive reductions. A comprehensive discussion of early work on how reductions to sparse sets collapse the polynomial-time hierarchy may be found in [Ma-89]. Additional discussions of this topic, as well as extensive bibliographies, may be found in [JY-90] and in [Yo-90]. 2 Polynomial-Time Turing Reductions 2.1 Introduction In [Lo-82], Long explicitly used the result (KL-1) to prove:

. This paper focuses on complexity classes of partial functions that are computed in polynomial time with oracles in NPMV, the class of all multivalued partial functions that are computable nondeterministically in polynomial time. Concerning deterministic polynomial-time reducibilities, it is shown that 1. A multivalued partial function is polynomial-time computable with k adaptive queries to NPMV if and only if it is polynomial-time computable via 2 k \Gamma 1 nonadaptive queries to NPMV. 2. A characteristic function is polynomial-time computable with k adaptive queries to NPMV if and only if it is polynomial-time computable with k adaptive queries to NP. 3. Unless the Boolean hierarchy collapses, for every k, k adaptive (nonadaptive) queries to NPMV is different than k + 1 adaptive (nonadaptive) queries to NPMV. Nondeterministic reducibilities, lowness and the difference hierarchy over NPMV are also studied. The difference hierarchy for partial functions does not collapse unless the Boolean hierarchy collapses, but, surprisingly, the levels of the difference and bounded query hierarchies do not interleave (as is the case for sets) unless the polynomial hierarchy collapses. Key words. computational complexity, complexity classes, relativized computation, bounded query classes, Boolean hierarchy, multivalued functions, NPMV AMS subject classifications. 68Q05, 68Q10, 68Q15, 03D10, 03D15 1.

This paper derives a new and surprisingly low complexity result for inference in a new form of Reiter's propositional default logic [18]. The problem studied here is the default inference problem whose fundamental importance was pointed out by Kraus, Lehmann, and Magidor [12]. We prove that "normal" default inference, in propositional logic, is a problem complete for co-NP(3), the third level of the Boolean hierarchy [4]. Our result (by changing the underlying semantics) contrasts favorably with a similar result of Gottlob [7], who proves that standard default inference is \Pi P 2 -complete. Our inference relation also obeys all of the laws for preferential consequence relations set forth by Kraus, Lehmann, and Magidor. In particular, we get the property of being able to reason by cases and the law of cautious monotony. Both of these laws fail for standard propositional default logic. The key technique for our results is the use of Scott's domain theory to integrate defaults into pa...

Witness reduction has played a crucial role in several recent results in complexity theory. These include Toda's result that PH ` BP \Delta \PhiP, the "collapsing" of PH into \PhiP with a high probability; Toda and Ogiwara's results which "collapses" PH into various counting classes with a high probability; and hard functions for various function classes studied by Ogiwara and Hemachandra. Ogiwara and Hemachandra's results establish a connection between functions being hard for #P and functions interacting with the class to effect witness reduction. In fact, we believe that the ability to achieve some form of witness reduction is what makes a function hard for a class of functions. To support our thesis we define new function classes and obtain results analogous to those of Ogiwara and Hemachandra. We also introduce the notion of randomly hard functions and obtain similar results. 1 Introduction Consider any problem in NP, for example, the CLIQUE problem. Let #Cliques(G; k) denote th...

This paper investigates nondeterministic bounded query classes in relation to the complexity of NP-hard approximation problems and the Boolean Hierarchy. Nondeterministic bounded query classes turn out be rather suitable for describing the complexity of NP-hard approximation problems. The results in this paper take advantage of this machine-based model to prove that in many cases, NP-approximation problems have the upward collapse property. That is, a reduction between NP-approximation problems of apparently different complexity at a lower level results in a similar reduction at a higher level. For example, if MaxClique reduces to (log n)-approximating MaxClique using many-one reductions, then the Traveling Salesman Problem (TSP) is equivalent to MaxClique under many-one reductions. Several upward collapse theorems are presented in this paper. The proofs of these theorems rely heavily on the machinery provided by the nondeterministic bounded query classes. In fact, these results depend on a surprising connection between the Boolean Hierarchy and nondeterministic bounded query classes.

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7 We consider polynomial-time Turing machines that have access to two oracles and investigate when the order of oracle queries is significant. The oracles used here are complete languages for the Polynomial Hierarchy (PH). We prove that, for solving decision problems, the order of oracle queries does not matter. This improves upon the previous result of Hemaspaandra, Hemaspaandra and Hempel, who showed that the order of the queries does not matter if the base machine asks only one query to each oracle. On the other hand, we prove that, for computing functions, the order of oracle queries does matter, unless PH collapses. 4 Address: Department of Electrical Engineering and Computer Science, University of Illinois at Chicago, 851 South Morgan Street (M/C 154), Chicago, IL 60607-7053. Supported in part by the National Science Foundation under grants CCR-9415410 & CCR-9700417 and by NASA under grant NAG 52895. Research performed while this author was at the University of Maryland Human-...