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10
Robust reductions
 Theory of Computing Systems
, 1999
"... We continue the study of robust reductions initiated by Gavaldà and Balcázar. In particular, a 1991 paper of Gavaldà and Balcázar [7] claimed an optimal separation between the power of robust and nondeterministic strong reductions. Unfortunately, their proof is invalid. We reestablish their theorem ..."
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We continue the study of robust reductions initiated by Gavaldà and Balcázar. In particular, a 1991 paper of Gavaldà and Balcázar [7] claimed an optimal separation between the power of robust and nondeterministic strong reductions. Unfortunately, their proof is invalid. We reestablish their theorem. Generalizing robust reductions, we note that robustly strong reductions are built from two restrictions, robust underproductivity and robust overproductivity, both of which have been separately studied before in other contexts. By systematically analyzing the power of these reductions, we
Algebraic properties for selector functions
 SIAM JOURNAL ON COMPUTING
, 2005
"... The nondeterministic advice complexity of the Pselective sets is known to be exactly linear. Regarding the deterministic advice complexity of the Pselective sets—i.e., the amount of Karp– Lipton advice needed for polynomialtime machines to recognize them in general—the best current upper bound is ..."
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Cited by 5 (4 self)
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The nondeterministic advice complexity of the Pselective sets is known to be exactly linear. Regarding the deterministic advice complexity of the Pselective sets—i.e., the amount of Karp– Lipton advice needed for polynomialtime machines to recognize them in general—the best current upper bound is quadratic [Ko83] and the best current lower bound is linear [HT96]. We prove that every associatively Pselective set is commutatively, associatively Pselective. Using this, we establish an algebraic sufficient condition for the Pselective sets to have a linear upper bound (which thus would match the existing lower bound) on their deterministic advice complexity: If all Pselective sets are associatively Pselective then the deterministic advice complexity of the Pselective sets is linear. The weakest previously known sufficient condition was P = NP. We also establish related results for algebraic properties of, and advice complexity of, the nondeterministically selective sets.
The Complexity of Finding TopTodaEquivalenceClass Members
, 2003
"... We identify two properties that for Pselective sets are effectively computable. Namely we show that, for any Pselective set, finding a string that is in a given length's top Toda equivalence class (very informally put, a string from Σ^n that the set's Pselector function declar ..."
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Cited by 5 (3 self)
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We identify two properties that for Pselective sets are effectively computable. Namely we show that, for any Pselective set, finding a string that is in a given length's top Toda equivalence class (very informally put, a string from &Sigma;^n that the set's Pselector function declares to be most likely to belong to the set) is FP computable, and we show that each Pselective set contains a weaklyP rankable subset.
On the Reducibility of Sets Inside NP to Sets with Low Information Content
, 2002
"... We study whether sets inside NP can be reduced to sets with low information content but possibly still high computational complexity. Examples of sets with low information content are tally sets, sparse sets, Pselective sets and membership comparable sets. For the graph automorphism... ..."
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We study whether sets inside NP can be reduced to sets with low information content but possibly still high computational complexity. Examples of sets with low information content are tally sets, sparse sets, Pselective sets and membership comparable sets. For the graph automorphism...
Polynomialtime multiselectivity
, 1997
"... We introduce a generalization of Selman's Pselectivity that yields a more flexible notion of selectivity, called (polynomialtime) multiselectivity, in which the selector is allowed to operate on multiple input strings. Since our introduction of this class, it has been used [HJRW96] to prove ..."
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Cited by 2 (1 self)
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We introduce a generalization of Selman's Pselectivity that yields a more flexible notion of selectivity, called (polynomialtime) multiselectivity, in which the selector is allowed to operate on multiple input strings. Since our introduction of this class, it has been used [HJRW96] to prove the first known (and optimal) lower bounds for generalized selectivitylike classes in terms of EL2, the second level of the extended low hierarchy. We study the resulting selectivity hierarchy, denoted by SH, which we prove does not collapse. In particular, we study the internal structure and the properties of SH and completely establish, in terms of incomparability and strict inclusion, the relations between our generalized selectivity classes and Ogihara's Pmc (polynomialtime membershipcomparable) classes. Although SH is a strictly increasing infinite hierarchy, we show that the core results that hold for the Pselective sets and that prove them structurally simple also hold for SH. In particular, all sets in SH have small circuits; the NP sets in SH are in Low2, the second level of the low hierarchy within NP; and SAT cannot be in SH unless P = NP. Finally, it is known that PSel, the class of Pselective sets, is not closed under union or intersection. We provide an extended selectivity hierarchy that is based on SH and that is large enough to capture those closures of the Pselective sets, and yet, in contrast with the Pmc classes, is refined enough to distinguish them.
Links Between Complexity Theory and Constrained Block Coding
"... The goal of this paper is to establish links between computational complexity theory and the theory and practice of constrained block coding. The complexities of several fundamental problems in constrained block coding are shown to be complete in various classes of the existing complexity theoretic ..."
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The goal of this paper is to establish links between computational complexity theory and the theory and practice of constrained block coding. The complexities of several fundamental problems in constrained block coding are shown to be complete in various classes of the existing complexity theoretic structure. The results include (relatively rare) , Eva , and NPm'completeness results. Two t3'pes of prob lems are considered: (1) the problem of designing encoder and decoder circuits using minimum or approximately minimum hardware for a given constraint and a given rate; (2) computing the maximum rate of a block code for a given constraint and codeword length. In both cases, a constraint is specified by a deterministic finite state transition dia gram. Another question studied is whether maximumrate block codes can always be implemented by encoders and decoders of polynomial size. The answer to this question is shown to be closely' related to the complexit3, of PP.
Advice for semifeasible sets and the complexitytheoretic cost(lessness) of algebraic properties
 International Journal of Foundations of Computer Science
"... This paper provides a tutorial overview of the advice complexity of the semifeasible sets—informally put, the class of sets having a polynomialtime algorithm that, given as input any two strings of which at least one belongs to the set, will choose one that does belong to the set. No previous famil ..."
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This paper provides a tutorial overview of the advice complexity of the semifeasible sets—informally put, the class of sets having a polynomialtime algorithm that, given as input any two strings of which at least one belongs to the set, will choose one that does belong to the set. No previous familiarity with either the semifeasible sets or advice complexity will assumed, and when we include proofs we will try to make the material as accessible as possible via providing intuitive, informal presentations. Karp and Lipton introduced advice complexity about a quarter of a century ago [KL80]. Advice complexity asks, for a given power of interpreter, how many bits of “help ” suffice to accept a given set. Thus, this is a notion that contains aspects both of informational complexity and of computational complexity. We will see that for some powers of interpreter the (worstcase) complexity of the semifeasible sets is known right down to the bit (and beyond), but that for the most central power of interpreter—deterministic polynomial time—the complexity is currently known only to be at least linear and at most quadratic. While overviewing the advice complexity of the semifeasible sets, we will stress also the issue of whether the functions at the core of semifeasibility—socalled selector functions—can without cost be chosen to possess such algebraic properties as commutativity and associativity. We will see that this is relevant, in ways both potential and actual, to the study of the advice complexity of the semifeasible sets.
LANGUAGES TO DIAGONALIZE AGAINST ADVICE CLASSES
"... Abstract. Variants of Kannan’s Theorem are given where the circuits of the original theorem are replaced by arbitrary recursively presentable classes of languages that use advice strings and satisfy certain mild conditions. Let poly k denote those functions in O(n k). These variants imply that DTIM ..."
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Abstract. Variants of Kannan’s Theorem are given where the circuits of the original theorem are replaced by arbitrary recursively presentable classes of languages that use advice strings and satisfy certain mild conditions. Let poly k denote those functions in O(n k). These variants imply that DTIME(nk ′ ) NE /polyk does not contain PNE, DTIME(2nk ′)/polyk does not contain EXP, SPACE(nk ′)/polyk does not contain PSPACE, uniform TC 0 /polyk does not contain CH, and uniform ACC/polyk does not contain ModPH. Consequences for selective sets are also obtained. In particular, it is shown that R DTIME(nk) T (NPsel) does not contain PNE, (Lsel) does not contain PSPACE. Finally, a circuit size hierarchy theorem is established.