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11
The complexity of decision versus search
- SIAM Journal on Computing
, 1994
"... A basic question about NP is whether or not search reduces in polynomial time to decision. We indicate that the answer is negative: under a complexity assumption (that deterministic and non-deterministic double-exponential time are unequal) we construct a language in NP for which search does not red ..."
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Cited by 30 (1 self)
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A basic question about NP is whether or not search reduces in polynomial time to decision. We indicate that the answer is negative: under a complexity assumption (that deterministic and non-deterministic double-exponential time are unequal) we construct a language in NP for which search does not reduce to decision. These ideas extend in a natural way to interactive proofs and program checking. Under similar assumptions we present languages in NP for which it is harder to prove membership interactively than it is to decide this membership, and languages in NP which are not checkable. Keywords: NP-completeness, self-reducibility, interactive proofs, program checking, sparse sets,
A Downward Collapse Within The Polynomial Hierarchy
, 1998
"... . Downward collapse (also known as upward separation) refers to cases where the equality of two larger classes implies the equality of two smaller classes. We provide an unqualified downward collapse result completely within the polynomial hierarchy. In particular, we prove that, for
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Cited by 22 (8 self)
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.<F3.803e+05> Downward collapse (also known as upward separation) refers to cases where the equality of two larger classes implies the equality of two smaller classes. We provide an unqualified downward collapse result completely within the polynomial hierarchy. In particular, we prove that, for<F3.319e+05> k ><F3.803e+05> 2, if P<F2.821e+05> #<F2.795e+05> p k<F2.821e+05> [1]<F3.803e+05> = P<F2.821e+05> #<F2.795e+05> p k<F2.821e+05> [2]<F3.803e+05> then #<F2.562e+05> p k<F3.803e+05> = #<F2.562e+05> p k<F3.803e+05> = PH. We extend this to obtain a more general downward collapse result.<F4.005e+05> Key words.<F3.803e+05> computational complexity theory, easy-hard arguments, downward collapse, polynomial hierarchy<F4.005e+05> AMS subject classifications.<F3.803e+05> 68Q15, 68Q10, 03D15, 03D10<F4.005e+05> PII.<F3.803e+05> S0097539796306474<F5.353e+05> 1. Introduction.<F4.529e+05> The theory of NP-completeness does not resolve the issue of whether P and NP are equal. However, it do...
Languages that are Easier than their Proofs
, 1991
"... A basic question about NP is whether or not search reduces in polynomial time to decision. We indicate that the answer is negative: under a complexity assumption (that deterministic and nondeterministic doubleexponential time are unequal) we construct a language in NP for which search does not reduc ..."
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Cited by 13 (7 self)
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A basic question about NP is whether or not search reduces in polynomial time to decision. We indicate that the answer is negative: under a complexity assumption (that deterministic and nondeterministic doubleexponential time are unequal) we construct a language in NP for which search does not reduce to decision. These ideas extend in a natural way to interactive proofs and program checking. Under similar assumptions we present languages in NP for which it is harder to prove membership interactively than it is to decide this membership. Similarly we present languages where checking is harder than computing membership. Each of the following properties --- checkability, random-self-reducibility, reduction from search to decision, and interactive proofs in which the prover's power is limited to deciding membership in the language itself --- implies coherence, one of the weakest forms of self-reducibility. Under assumptions about triple-exponential time, we construct incoherent sets in NP....
Upward Separation for FewP and Related Classes
, 1994
"... This paper studies the range of application of the upward separation technique that has been introduced by Hartmanis to relate certain structural properties of polynomial-time complexity classes to their exponential-time analogs and was first applied to NP [Har83]. Later work revealed the limitation ..."
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Cited by 13 (3 self)
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This paper studies the range of application of the upward separation technique that has been introduced by Hartmanis to relate certain structural properties of polynomial-time complexity classes to their exponential-time analogs and was first applied to NP [Har83]. Later work revealed the limitations of the technique and identified classes defying upward separation. In particular, it is known that coNP as well as certain promise classes such as BPP, R, and ZPP do not possess upward separation in all relativized worlds [HIS85; HJ93], and it had been suspected that this was also the case for other promise classes such as UP and FewP [All91]. In this paper, we refute this conjecture by proving that, in particular, FewP does display upward separation, thus providing the first upward separation result for a promise class. In fact, this follows from a more general result the proof of which heavily draws on Buhrman, Longpr'e, and Spaan's recently discovered tally encoding of sparse sets. As ...
On the limitations of locally robust positive reductions
- International Journal of Foundations of Computer Science
, 1991
"... Polynomial-time positive reductions, as introduced by Selman, are by definition globally robust — they are positive with respect to all oracles. This paper studies the extent to which the theory of positive reductions remains intact when their global robustness assumption is removed. We note that tw ..."
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Cited by 8 (1 self)
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Polynomial-time positive reductions, as introduced by Selman, are by definition globally robust — they are positive with respect to all oracles. This paper studies the extent to which the theory of positive reductions remains intact when their global robustness assumption is removed. We note that two-sided locally robust positive reductions — reductions that are positive with respect to the oracle to which the reduction is made — are sufficient to retain all crucial properties of globally robust positive reductions. In contrast, we prove absolute and relativized results showing that one-sided local robustness fails to preserve fundamental properties of positive reductions, such as the downward closure of NP. Keywords: Structural complexity theory; Polynomial-time reductions; Complexity classes.
Translating Equality Downwards
, 1997
"... Downward translation of equality refers to cases where a collapse of some pair of complexity classes would induce a collapse of some other pair of complexity classes that (a priori) one expects are smaller. Recently, the first downward translation of equality was obtained that applied to the polynom ..."
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Cited by 7 (5 self)
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Downward translation of equality refers to cases where a collapse of some pair of complexity classes would induce a collapse of some other pair of complexity classes that (a priori) one expects are smaller. Recently, the first downward translation of equality was obtained that applied to the polynomial hierarchy---in particular, to bounded access to its levels [HHH97a]. In this paper, we provide a much broader downward translation that subsumes not only that downward translation but also that translation's elegant enhancement by Buhrman and Fortnow [BF96]. Our work also sheds light on previous research on the structure of refined polynomial hierarchies [Sel95, Sel94]. 1 Introduction Does the collapse of low-complexity classes imply the collapse of higher-complexity classes? Does the collapse of high-complexity classes imply the collapse of lower-complexity classes? These questions---known respectively as downward and upward translation of equality---have long been central topics in co...
Space-Efficient Recognition Of Sparse Self-Reducible Languages
, 1994
"... . Mahaney and others have shown that sparse self-reducible sets have time-ecient algorithms, and have concluded that it is unlikely that NP has sparse complete sets. Mahaney's work, intuition, and a 1978 conjecture of Hartmanis notwithstanding, nothing has been known about the density of complet ..."
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Cited by 5 (3 self)
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. Mahaney and others have shown that sparse self-reducible sets have time-ecient algorithms, and have concluded that it is unlikely that NP has sparse complete sets. Mahaney's work, intuition, and a 1978 conjecture of Hartmanis notwithstanding, nothing has been known about the density of complete sets for feasible classes until now. This paper shows that sparse self-reducible sets have space-ecient algorithms, and in many cases, even have time-space-ecient algorithms. We conclude that NL, NC k , AC k , LOG(DCFL), LOG(CFL), and P lack complete (or even Turing-hard) sets of low density unless implausible complexity class inclusions hold. In particular, if NL (respectively P, k , or NP) has a polylog-sparse logspace-hard set, then NL SC (respectively P SC, k SC, or PH SC), and if P has subpolynomially sparse logspace-hard sets, then P 6= PSPACE. Subject classications. 68Q15, 03D15. 1. Introduction Complete sets are the quintessences of their complexity cla...
Tally NP Sets and Easy Census Functions
, 1998
"... We study the question of whether every P set has an easy (i.e., polynomialtime computable) census function. We characterize this question in terms of unlikely collapses of language and function classes such as #P 1 ` FP, where #P 1 is the class of functions that count the witnesses for tally NP sets ..."
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Cited by 2 (1 self)
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We study the question of whether every P set has an easy (i.e., polynomialtime computable) census function. We characterize this question in terms of unlikely collapses of language and function classes such as #P 1 ` FP, where #P 1 is the class of functions that count the witnesses for tally NP sets. We prove that every #P PH 1 function can be computed in FP #P #P 1 1 . Consequently, every P set has an easy census function if and only if every set in the polynomial hierarchy does. We show that the assumption #P 1 ` FP implies P = BPP and PH ` MOD k P for each k 2, which provides further evidence that not all sets in P have an easy census function. We also relate a set's property of having an easy census function to other well-studied properties of sets, such as rankability and scalability (the closure of the rankable sets under P-isomorphisms). Finally, we prove that it is no more likely that the census function of any set in P can be approximated (more precisely, can be n ff -e...
A Moment of Perfect Clarity I: The Parallel Census Technique
, 2000
"... We discuss the history and uses of the parallel census technique|an elegant tool in the study of certain computational objects having polynomially bounded census functions. A sequel [GH] will discuss advances (including [CNS95] and Glaer [Gla00]), some related to the parallel census technique and ..."
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Cited by 2 (2 self)
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We discuss the history and uses of the parallel census technique|an elegant tool in the study of certain computational objects having polynomially bounded census functions. A sequel [GH] will discuss advances (including [CNS95] and Glaer [Gla00]), some related to the parallel census technique and some due to other approaches, in the complexity-class collapses that follow if NP has sparse hard sets under reductions weaker than (full) truth-table reductions. 1 Introduction Have you ever used a pair of binoculars? Then you know the process one goes through to initially set the distance between the two eyepieces|sometimes the view may black out, yet if one goes too far the other way one has two circles of view that don't coincide. However, there is a point where things are just right: All is crisply aligned and one can enjoy the view of that pileated woodpecker, at least if it has been so polite as to wait while one was playing with the interocular adjustment. Supported in part...
Types of Separability
, 2000
"... In this paper we demonstrate that the studies of structural properties of the boolean hierarchy of NP-partitions are not only worthwhile in their own, e.g., as a framework for capturing the complexity of classication problems but have interesting ties with other research in computational complexity: ..."
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Cited by 2 (0 self)
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In this paper we demonstrate that the studies of structural properties of the boolean hierarchy of NP-partitions are not only worthwhile in their own, e.g., as a framework for capturing the complexity of classication problems but have interesting ties with other research in computational complexity: We discuss the relationships to the study of separable NP sets.
