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11
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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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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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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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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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
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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 ..."
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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...
Random access to advice strings and collapsing results
"... We propose a model of computation where a Turing machine is given random access to an advice string. With random access, an advice string of exponential length becomes meaningful for polynomially bounded complexity classes. We compare the power of complexity classes under this model. It gives a more ..."
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We propose a model of computation where a Turing machine is given random access to an advice string. With random access, an advice string of exponential length becomes meaningful for polynomially bounded complexity classes. We compare the power of complexity classes under this model. It gives a more stringent notion than the usual model of computation with relativization. Under this model of random access, we prove that there exist advice strings such that the Polynomialtime Hierarchy PH and Parity Polynomialtime ⊕P all collapse to P. Our main proof technique uses the decision tree lower bounds for constant depth circuits [Yao85, Cai86, H˚as86], and the algebraic machinery of Razborov and Smolensky [Raz87, Smo87].
Abstract 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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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 Complexity Theory
, 1996
"... 1 Introduction Theorem 1 (Rice 1953) Looking for an Analogue of Rice's Theorem in Complexity Theory Bernd Borchert Universit at Heidelberg Frank Stephan Universit at Heidelberg Let be any nonempty proper subset of the partial recursive functions. Then the halting problem or its complement is ..."
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1 Introduction Theorem 1 (Rice 1953) Looking for an Analogue of Rice's Theorem in Complexity Theory Bernd Borchert Universit at Heidelberg Frank Stephan Universit at Heidelberg Let be any nonempty proper subset of the partial recursive functions. Then the halting problem or its complement is manyone reducible to the following problem: Given a program , does it compute a function from ? Im Neuenheimer Feld 294, 69120 Heidelberg, bb@math.uniheidelberg.de Im Neuenheimer Feld 294, 69120 Heidelberg, fstephan@math.uniheidelberg.de; supported by the Deutsche Forschungsgemeinschaft (DFG) grant Am 60/91. Rice's Theorem says that every nontrivial semantic property of programs is undecidable. It this spirit we show the following: Every nontrivial absolute (gap, relative) counting property of circuits is UPhard with respect to polynomialtime Turing reductions. One of the nicest theorems in Recursion Theory is Rice's Theorem [17, 18]. Informally speaking it says: Any nontrivial semantic p...