Looking for an analogue of Rice's Theorem in circuit complexity theory

Borchert, B.; Stephan, Frank

doi:10.1007/3-540-63385-5_37

Cited by 5 publications

(14 citation statements)

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“…A bold and exciting paper of Borchert and Stephan [4] proposes and initiates the search for complexity-theoretic analogs of Rice's Theorem. Borchert and Stephan note that Rice's Theorem deals with properties of programs, and they suggest as a promising complexity-theoretic analog properties of boolean circuits.…”

Section: Corollary 12 (Rice's Theorem Version Ii)mentioning

confidence: 99%

“…Proof of Theorem 2.4. Let A be a nonempty proper subset of N. The paper of Borchert and Stephan [4] (see Theorem 1.4 above) and-using different nomenclatureearlier papers [15,5] have shown that (a) if A is finite and nonempty, then Counting(A) is ≤ p m -hard for coNP, and (b) if A is cofinite and a proper subset of N, then Counting(A) is ≤ p m -hard for NP.…”

Section: ([2]mentioning

confidence: 99%

“…Rice's Theorem ( [28,29], see [4]) states that every nontrivial language property of the recursively enumerable sets is either RE-hard or coRE-hard-and thus is certainly undecidable (a corollary that itself is often referred to as Rice's Theorem).…”

Section: Introductionmentioning

confidence: 99%

“…[4], see also the comments at the start of the proof of Theorem 2.4) Let A be a nonempty proper subset of N. Then one of the following three classes is ≤ p m -reducible to Counting(A): NP, coNP, or UP coUP.…”

mentioning

confidence: 99%

“…Passing on a comment from an anonymous referee, we mention that the reader may want to also compare the UP coUP occurrence in Borchert and Stephan[4] with the so-called Rice-Shapiro Theorem (see, e.g.,[30,27]). We mention that in making such a comparison one should keep in mind that the Rice-Shapiro Theorem deals with showing non-membership in RE and coRE, rather than with showing many-one hardness for those classes.…”

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confidence: 99%

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A second step towards complexity-theoretic analogs of Rice's Theorem

Hemaspaandra

Rothe

2000

Theoretical Computer Science

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Rice's Theorem states that every nontrivial language property of the recursively enumerable sets is undecidable. Borchert and Stephan [4] initiated the search for complexity-theoretic analogs of Rice's Theorem. In particular, they proved that every nontrivial counting property of circuits is UP-hard, and that a number of closely related problems are SPP-hard.The present paper studies whether their UP-hardness result itself can be improved to SPP-hardness. We show that their UP-hardness result cannot be strengthened to SPP-hardness unless unlikely complexity class containments hold. Nonetheless, we prove that every P-constructibly bi-infinite counting property of circuits is SPP-hard. We also raise their general lower bound from unambiguous nondeterminism to constant-ambiguity nondeterminism. *

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Section: Corollary 12 (Rice's Theorem Version Ii)mentioning

confidence: 99%

Section: ([2]mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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A second step towards complexity-theoretic analogs of Rice's Theorem

Hemaspaandra

Rothe

2000

Theoretical Computer Science

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Restrictive acceptance suffices for equivalence problems

Borchert

Hemaspaandra

Rothe

1999

Fundamentals of Computation Theory

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One way of suggesting that an NP problem may not be NP-complete is to show that it is in the class UP. We suggest an analogous new approach-weaker in strength of evidence but more broadly applicable-to suggesting that concrete NP problems are not NP-complete. In particular we introduce the class EP, the subclass of NP consisting of those languages accepted by NP machines that when they accept always have a number of accepting paths that is a power of two. Since if any NP-complete set is in EP then all NP sets are in EP, it follows-with whatever degree of strength one believes that EP differs from NP-that membership in EP can be viewed as evidence that a problem is not NP-complete. We show that the negation equivalence problem for OBDDs (ordered binary decision diagrams [22, 13]) and the interchange equivalence problem for 2-dags are in EP. We also show that for boolean negation [25] the equivalence problem is in EP NP , thus tightening the existing NP NP upper bound. We show that FewP [2], bounded ambiguity polynomial time, is contained in EP, a result that is not known to follow from the previous SPP upper bound. For the three problems and classes just mentioned with regard to EP, no proof of membership/containment in UP is known, and for the problem just mentioned with regard to EP NP , no proof of membership in UP NP is known. Thus, EP is indeed a tool that gives evidence against NP-completeness in natural cases where UP cannot currently be applied.

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