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The Structure of Complete Degrees
, 1990
"... This paper surveys investigations into how strong these commonalities are. More concretely, we are concerned with: What do NPcomplete sets look like? To what extent are the properties of particular NPcomplete sets, e.g., SAT, shared by all NPcomplete sets? If there are are structural differences ..."
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Cited by 29 (3 self)
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This paper surveys investigations into how strong these commonalities are. More concretely, we are concerned with: What do NPcomplete sets look like? To what extent are the properties of particular NPcomplete sets, e.g., SAT, shared by all NPcomplete sets? If there are are structural differences between NPcomplete sets, what are they and what explains the differences? We make these questions, and the analogous questions for other complexity classes, more precise below. We need first to formalize NPcompleteness. There are a number of competing definitions of NPcompleteness. (See [Har78a, p. 7] for a discussion.) The most common, and the one we use, is based on the notion of mreduction, also known as polynomialtime manyone reduction and Karp reduction. A set A is mreducible to B if and only if there is a (total) polynomialtime computable function f such that for all x, x 2 A () f(x) 2 B: (1) 1
Relativization: A revisionistic retrospective
 Bulletin of the EATCS
, 1992
"... In this column we examine the role of relativization in complexity theory in light of recent nonrelativizing results involving interactive protocols. We begin with the twicetold tale of the relativization principle and ponder upon its possible demise. Then, we discuss whether usual assumptions are ..."
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Cited by 7 (0 self)
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In this column we examine the role of relativization in complexity theory in light of recent nonrelativizing results involving interactive protocols. We begin with the twicetold tale of the relativization principle and ponder upon its possible demise. Then, we discuss whether usual assumptions are historically accurate. 1 The TwiceTold Tale For almost two decades, contradictory relativization has been a central theme in complexity theory. This concept was first introduced by Baker, Gill and Solovay [BGS75] in their ground breaking paper where they exhibited oracles A and B such that P A = NP A and P B � = NP B. This result was startling because almost all results in recursion theory remain true in the presence of oracles and most techniques used in complexity theory had been resource bounded versions of those used in recursion theory. Baker, Gill and Solovay offered the following explanation of their results [BGS75]: We feel that this is further evidence of the difficulty of the P? = NP question....It seems unlikely that ordinary diagonalization methods are adequate for producing an example of a language in NP but not in P; such diagonalizations, we would expect, would apply equally well to the relativized classes....On the other hand, we do not feel that one can give a general method for simulating nondeterministic machines by deterministic machines in polynomial time, since such a method should apply as well to relativized machines.
The Intensional Content of Rice’s Theorem
"... The proofs of major results of Computability Theory like Rice, RiceShapiro or Kleene’s fixed point theorem hide more information of what is usually expressed in their respective statements. We make this information explicit, allowing to state stronger, complexity theoreticversions of all these the ..."
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Cited by 5 (1 self)
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The proofs of major results of Computability Theory like Rice, RiceShapiro or Kleene’s fixed point theorem hide more information of what is usually expressed in their respective statements. We make this information explicit, allowing to state stronger, complexity theoreticversions of all these theorems. In particular, we replace the notion of extensional set of indices of programs, by a set of indices of programs having not only the same extensional behavior but also similar complexity (Complexity Clique). We prove, under very weak complexity assumptions, that any recursive Complexity Clique is trivial, and any r.e. Complexity Clique is an extensional set (and thus satisfies RiceShapiro conditions). This allows, for instance, to use Rice’s argument to prove that the property of having polynomial complexity is not decidable, and to use RiceShapiro to conclude that it is not even semidecidable. We conclude the paper with a discussion of “complexitytheoretic ” versions of Kleene’s
Looking for an analogue of Rice's Theorem in circuit complexity theory
 Mathematical Logic Quarterly
, 1989
"... Abstract. Rice’s Theorem says that every nontrivial semantic property of programs is undecidable. In this spirit we show the following: Every nontrivial absolute (gap, relative) counting property of circuits is UPhard with respect to polynomialtime Turing reductions. For generators [31] we show a ..."
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Cited by 5 (1 self)
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Abstract. Rice’s Theorem says that every nontrivial semantic property of programs is undecidable. In this spirit we show the following: Every nontrivial absolute (gap, relative) counting property of circuits is UPhard with respect to polynomialtime Turing reductions. For generators [31] we show a perfect analogue of Rice’s Theorem. Mathematics Subject Classification: 03D15, 68Q15. Keywords: Rice’s Theorem, Counting problems, Promise classes, UPhard, NPhard, generators.
An example of a theorem that has contradictory relativizations and a diagonalization proof
 Bulletin of the European Association for Theoretical Computer Science
, 1990
"... The central questions in complexity theory (e.g. the P =?NP question) can only be solved ..."
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Cited by 3 (2 self)
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The central questions in complexity theory (e.g. the P =?NP question) can only be solved
Effectivizing Inseparability
, 1991
"... Smullyan's notion of effectively inseparable pairs of sets is not the best effective /constructive analog of Kleene's notion of pairs of sets inseparable by a recursive set. We present a corrected notion of effectively inseparable pairs of sets, prove a characterization of our notion, and show that ..."
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Cited by 2 (0 self)
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Smullyan's notion of effectively inseparable pairs of sets is not the best effective /constructive analog of Kleene's notion of pairs of sets inseparable by a recursive set. We present a corrected notion of effectively inseparable pairs of sets, prove a characterization of our notion, and show that the pairs of index sets effectively inseparable in Smullyan's sense are the same as those effectively inseparable in ours. In fact we characterize the pairs of index sets effectively inseparable in either sense thereby generalizing Rice's Theorem. For subrecursive index sets we have sufficient conditions for various inseparabilities to hold. For inseparability by sets in the same subrecursive class we have a characterization. The latter essentially generalizes Kozen's (and Royer's later) Subrecursive Rice Theorem, and the proof of each result about subrecursive index sets is presented "Rogers style" with care to observe subrecursive restrictions. There are pairs of sets effectively inseparab...
Diagonalization
, 2000
"... We give a modern historical and philosophical discussion of diagonalization as a tool to prove lower bounds in computational complexity. We will give several examples and discuss four possible approaches to use diagonalization for separating logarithmicspace from nondeterministic polynomialtime ..."
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We give a modern historical and philosophical discussion of diagonalization as a tool to prove lower bounds in computational complexity. We will give several examples and discuss four possible approaches to use diagonalization for separating logarithmicspace from nondeterministic polynomialtime. 1 Introduction The greatest embarrassment in computational complexity theory comes from our inability to achieve signicant complexity class separations. In recent years we have seen many interesting results come from an old techniquediagonalization. Deceptively simple, diagonalization, combined with techniques for collapsing classes, can yield quite interesting lower bounds on computation. In 1874, Cantor [Can74] rst used diagonalization for showing the set of reals is not countable. The proof worked by assuming an enumeration of the reals and designing a set that onebyone is dierent from every set in the enumeration. Drawn as a table this process considers the diagonal set and re...
Research Reports on
, 2005
"... Employing the variational approach having the twobody reduced density matrix (RDM) as variables to compute the ground state energies of atomicmolecular systems has been a long time dream in electronic structure theory in chemical physics/physical chemistry. Realization of the RDM approach has b ..."
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Employing the variational approach having the twobody reduced density matrix (RDM) as variables to compute the ground state energies of atomicmolecular systems has been a long time dream in electronic structure theory in chemical physics/physical chemistry. Realization of the RDM approach has benefited greatly from recent developments in semidefinite programming (SDP). We present the actual state of this new application of SDP as well as the formulation of these SDPs, which can be arbitrarily large. Numerical results using parallel computation on high performance computers are given. The RDM method has several advantages including robustness and provision of high accuracy compared to traditional electronic structure methods, although its computational time and memory consumption are still extremely large.