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Gaplanguages and logtime complexity classes
 Theoretical Computer Science
, 1997
"... This paper shows that classical results about complexity classes involving “delayed diagonalization ” and “gap languages, ” such as Ladner’s Theorem and Schöning’s Theorem and independence results of a kind noted by Schöning and Hartmanis, apply at very low levels of complexity, indeed all the way d ..."
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Cited by 9 (6 self)
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This paper shows that classical results about complexity classes involving “delayed diagonalization ” and “gap languages, ” such as Ladner’s Theorem and Schöning’s Theorem and independence results of a kind noted by Schöning and Hartmanis, apply at very low levels of complexity, indeed all the way down in Sipser’s logtime hierarchy. This paper also investigates refinements of Sipser’s classes and notions of logtime reductions, following on from recent work by Cai, Chen, and others. 1
Is P versus NP formally independent
 Bulletin of the European Association for Theoretical Computer Science
, 2003
"... I have moved back to the University of Chicago and so has the web page for this column. See above for new URL and contact informaion. This issue Scott Aaronson writes quite an interesting (and opinionated) column on whether the P = NP question is independent of the usual axiom systems. Enjoy! ..."
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Cited by 8 (0 self)
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I have moved back to the University of Chicago and so has the web page for this column. See above for new URL and contact informaion. This issue Scott Aaronson writes quite an interesting (and opinionated) column on whether the P = NP question is independent of the usual axiom systems. Enjoy!
How to Prove RepresentationIndependent Independence Results
 Information Proc. Lett. 24
, 1987
"... A true assertion about the inputoutput behavior of a Turing Machine M may be independent of (i.e., impossible to prove in) a theory T because the computational behavior of M is particularly opaque, or because the function or set computed by M is inherently subtle. The latter sorts of representation ..."
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Cited by 6 (0 self)
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A true assertion about the inputoutput behavior of a Turing Machine M may be independent of (i.e., impossible to prove in) a theory T because the computational behavior of M is particularly opaque, or because the function or set computed by M is inherently subtle. The latter sorts of representationindependent independence results are more satisfying. For \Pi 2 assertions, the bestknown techniques for proving independence yield representationindependent results as a matter of course. This paper illustrates current understanding of unprovability for \Pi 2 assertions by demonstrating that very weak conditions on classses of sets S and R guarantee that there exists a set L 0 2 R \Gamma S such that L 0 is not provably infinite (hence, not provably nonregular, nondeterministic, noncontextfree, not in P, etc.). Under slightly stronger conditions, such L 0 s may be found within every L 2 R \Gamma S. 1 Introduction In a recent paper, Hartmanis shows how to use diagonalization techniques...
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...
Optimal Proof Systems and Complete Languages
 ELECTRONIC COLLOQUIUM ON COMPUTATIONAL COMPLEXITY, REPORT NO. 107
, 2008
"... We investigate the connection between optimal propositional proof systems and complete languages for promise classes. We prove that an optimal propositional proof system exists if and only if there exists a propositional proof system in which every promise class with the test set in coNP is repres ..."
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We investigate the connection between optimal propositional proof systems and complete languages for promise classes. We prove that an optimal propositional proof system exists if and only if there exists a propositional proof system in which every promise class with the test set in coNP is representable. Additionally, we prove that there exists a complete language for UP if and only if there exists a propositional proof system such that UP is representable in it. UP is the standard promise class with the test set in coNP.