Under the assumption that NP does not have p-measure 0, we investigate reductions to NP-complete sets and prove the following: 1. Adaptive reductions are more powerful than nonadaptive reductions: there is a problem that is Turing-complete for NP but not truth-table-complete. 2. Strong nondeterministic reductions are more powerful than deterministic reductions: there is a problem that is SNP-co...

Cook's Theorem [Cormen, T.H., Leiserson, C.E., Rivest, R.L., 2001. Introduction to Algorithms, second ed., The MIT Press; Garey, M.R., Johnson, D.S., 1979. Computer and Intractability, Freeman, San Fransico, CA] is that if one algorithm for an NP-complete or an NP-hard problem will be developed, then other problems will be solved by means of reduction to that problem. Cook's Theorem has been de...

A computer which has access to a closed timelike curve, and can thereby send the results of calculations into its own past, can exploit this to solve difficult computational problems efficiently. I give a specific demonstration of this for the problem of factoring large numbers, and argue that a similar approach can solve NP-complete and PSPACE-complete problems. I discuss the potential impact ...

We show that if an NP-complete problem has a non-adaptive self-corrector with respect to a samplable distribution then coNP is contained in NP/poly and the polynomial hierarchy collapses to the third level. Feigenbaum and Fortnow (SICOMP 22:994-1005, 1993) show the same conclusion under the stronger assumption that an NP-complete problem has a non-adaptive random self-reduction. Our result show...

Conditional logics, introduced by Lewis and Stalnaker, have been utilized in artiicial intelligence to capture a broad range of phenomena. In this paper we examine the complexity of several variants discussed in the literature. We show that, in general, deciding satis-ability is PSPACE-complete for formulas with arbitrary conditional nesting and NP-complete for formulas with bounded nesting of ...

Many NP-complete problems have counting versions which are #P-complete. On the other hand, #Perfect Matchings is also Cook-complete for #P, which is surprising as Perfect Matching is actually in P (which implies that #Perfect Matchings cannot be Karp-complete for #P). Here, we study the complexity class #PE (functions of #P with easy decision version). The inclusion #PE ⊆ #P is proper unless P ...

The problem of determining the size of a finite concept lattice is shown to be #P-complete. Since any finite lattice can be represented as a concept lattice, the problem of determining the size of a lattice given by the ordered sets of its irreducibles is also #P-complete. Some results about NP-completeness or polynomial tractability of decision problems related to concepts with bounded extent,...

A model of learning from positive and negative examples in concept lattices is considered. Latticeand graph-theoretic interpretations of learning concept-based classi0cation rules (called hypotheses) and classi0cation in this model are given. The problems of counting all formal concepts, all hypotheses, and all minimal hypotheses are shown to be #P-complete. NP-completeness of some decision pro...