ERRATUM: the polynomial time hierarchy collapses if the boolean hierarchy collapses
01 Dec 1988-SIAM Journal on Computing (Society for Industrial and Applied Mathematics)-Vol. 17, Iss: 2, pp 1263-1282
TL;DR: It is shown that if the Boolean hierarchy (BH) collapses, then there exists a sparse set S such that ${\text{co-NP}} \subseteq {\text{ NP}}^S $, and therefore the polynomial time hierarchy (PH) col...
Abstract: It is shown that if the Boolean hierarchy (BH) collapses, then there exists a sparse set S such that ${\text{co-NP}} \subseteq {\text{ NP}}^S $, and therefore the polynomial time hierarchy (PH) col...
Citations
More filters
02 Jan 1991
TL;DR: This chapter discusses the concepts needed for defining the complexity classes, a set of problems of related resource-based complexity that can be solved by an abstract machine M using O(f(n) of resource R, where n is the size of the input.
Abstract: Publisher Summary This chapter discusses the concepts needed for defining the complexity classes. A complexity class is a set of problems of related resource-based complexity. A typical complexity class has a definition of the form—the set of problems that can be solved by an abstract machine M using O(f(n)) of resource R , where n is the size of the input. The simpler complexity classes are defined by various factors. The type of computational problem in which the most commonly used problems are decision problems. However, complexity classes can be defined based on function problems, counting problems, optimization problems, promise problems, etc. The most common model of computation is the deterministic Turing machine, but many complexity classes are based on nondeterministic Turing machines, etc.
618 citations
TL;DR: The Boolean hierarchy is generalized in such a way that it is possible to characterize P and O in terms of the generalization, and the class $P^{\text{NP}}[O(\log n)]$ can be characterized in very different ways.
Abstract: Polynomial time machines having restricted access to an NP oracle are investigated. Restricted access means that the number of queries to the oracle is restricted or the way in which the queries are made is restricted (e.g., queries made during truth-table reductions). Very different kinds of such restrictions result in the same or comparable complexity classes. In particular, the class $P^{\text{NP}}[O(\log n)]$ can be characterized in very different ways. Furthermore, the Boolean hierarchy is generalized in such a way that it is possible to characterize $P^{\text{NP}}$ and $P^{\text{NP}}[O(\log n)]$ in terms of the generalization.
328 citations
TL;DR: This work proves that the deduction problem for arbitrary propositional theories under the extended closed-world assumption or under circumscription is Π P 2 -complete, i.e., complete for a class of the second level of the polynomial hierarchy.
185 citations
TL;DR: In this article, the collapse of Buss' bounded arithmetic is characterized in terms of the provable collapse of the polynomial time hierarchy, and a general model-theoretical investigation on fragments of bounded arithmetic are presented.
Abstract: We characterize the collapse of Buss' bounded arithmetic in terms of the provable collapse of the polynomial time hierarchy. We include also some general model-theoretical investigations on fragments of bounded arithmetic.
126 citations
Proceedings Article•
01 Jan 1987
TL;DR: In this article, it was shown that if there exists a sparse set S ϵ NP such that co-NP ( NPs) is contained in PNP[O(log n), then the polynomial hierarchy (PH) of PNP(n) = DP.
Abstract: PNP[O(log n)] is the class of languages recognizable by deterministic polynomial time machines that make O(log n) queries to an oracle for NP. Our main result is that if there exists a sparse set S ϵ NP such that co-NP ( NPs, then the polynomial hierarchy (PH) is contained in PNP[O(log n)]. Thus if there exists a sparse ⩾PT-complete set for NP, PH ( PNP[O(log n)]. We show that this collapse is optimal by showing for any function f (n) that is o(log n), there exists a relativized world where NP has a sparse ⩾PT-complete set and yet PNP[O(log n)] (/ PNP[f(n)]. We also discuss complete problems for PNP[O(log n)] and show languages related to the optimum solution size of Clique and K-SAT are ⩾Pm-complete. In related research, we investigate when the class of languages ⩾Pm-reducible to a set C equals PC[O(log n)]. We obtain results that allow us to prove that if DP is closed under complementation, then PNP[O(log n)] = DP.
110 citations
References
More filters
IBM1
TL;DR: The problem of deciding validity in the theory of equality is shown to be complete in polynomial-space, and close upper and lower bounds on the space complexity of this problem are established.
1,402 citations
"ERRATUM: the polynomial time hierar..." refers background in this paper
...We also assume the reader is familiar with the PH defined in [21]....
[...]
TL;DR: It immediately follows that the context-sensitive languages are closed under complementation, thus settling a question raised by Kuroda.
Abstract: In this paper we show that nondeterministic space $s(n)$ is closed under complementation for $s(n)$ greater than or equal to $\log n$. It immediately follows that the context-sensitive languages are closed under complementation, thus settling a question raised by Kuroda [Inform. and Control, 7 (1964), pp. 207–233].
732 citations
"ERRATUM: the polynomial time hierar..." refers background in this paper
...Then Immerman and Szelepcsnyi independently showed that nondeterministic space classes are closed under complementation which implies that the logarithmic and linear space hierarchies collapse all the way to their first levels, nondeterministic logarithmic and linear space, respectively [9], [22]....
[...]
28 Apr 1980
TL;DR: This work aims to understand when nonuniform upper bounds can be used to obtain uniform upper bounds, and how to relate it to more common notions.
Abstract: It is well known that every set in P has small circuits [13]. Adleman [1] has recently proved the stronger result that every set accepted in polynomial time by a randomized Turing machine has small circuits. Both these results are typical of the known relationships between uniform and nonuniform complexity bounds. They obtain a nonuniform upper bound as a consequence of a uniform upper bound. The central theme here is an attempt to explore the converse direction. That is, we wish to understand when nonuniform upper bounds can be used to obtain uniform upper bounds. In this section we will define our basic notion of nonuniform complexity. Then we will show how to relate it to more common notions.
625 citations
"ERRATUM: the polynomial time hierar..." refers background or methods in this paper
...The proof uses a technique called oracle replacement which is embedded in the proofs of the sparse oracle results of Karp and Lipton [12], Mahaney [17], Long [16], and Yap [26]....
[...]
...In fact, these two concepts are used in [11] to unify all the sparse oracle results of Karp and Lipton [12], Mahaney [17], Long [16], Yap [26], and Kadin [10]....
[...]
TL;DR: It is shown that complete sets in EXPTIME and EXPTAPE cannot be sparse and therefore they cannot be over a single letter alphabet, and the hardest context-sensitive languages can be sparse.
Abstract: If all $NP$ complete sets are isomorphic under deterministic polynomial time mappings (p-isomorphic) then $P
e NP$ and if all PTAPE complete sets are P-isomorphic then $P
e {\text{PTAPE}}$. We show that all $NP$ complete sets known (in the literature) are indeed p-isomorphic and so are the known PTAPE complete sets. This shows that, in spite of the radically different origins and attempted simplification of these sets, all the known $NP$ complete sets are identical but for simple isomorphic codings computable in deterministic polynomial time.Furthermore, if all $NP$ complete sets are p-isomorphic then they all must have similar densities and, for example, no language over a single letter alphabet can be $NP$ complete, nor can any sparse language over an arbitrary alphabet be $NP$ complete. We show that complete sets in EXPTIME and EXPTAPE cannot be sparse and therefore they cannot be over a single letter alphabet. Similarly, we show that the hardest context-sensitive languages cannot be sparse. We als...
482 citations
"ERRATUM: the polynomial time hierar..." refers background in this paper
...There is a well-known result of Meyer published in [3] that relates sparse oracles for NP and polynomial size circuits for NP:...
[...]
TL;DR: Various forms of polynomial time reducibility are compared.
428 citations
"ERRATUM: the polynomial time hierar..." refers background in this paper
...Polynomial time truth-table and bounded truth-table reducibilities were defined in [14]....
[...]
...Polynomial time bounded truth-table reducibilities are defined in [14]....
[...]