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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
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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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...