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**1 - 2**of**2**### There Are No Sparse NP_W-Hard Sets

"... . In this paper we prove that, in the context of weak machines over IR, there are no sparse NP-hard sets. 1 Introduction In [1977] Berman and Hartmanis conjectured that all NP-complete sets are polynomially isomorphic. That is, that for all NP-complete sets A and B, there exists a bijection &apos ..."

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. In this paper we prove that, in the context of weak machines over IR, there are no sparse NP-hard sets. 1 Introduction In [1977] Berman and Hartmanis conjectured that all NP-complete sets are polynomially isomorphic. That is, that for all NP-complete sets A and B, there exists a bijection ' : \Sigma ! \Sigma such that x 2 A if and only if '(x) 2 B. In addition both ' and its inverse are computable in polynomial time. Here \Sigma denotes the set f0; 1g and \Sigma the set of all finite sequences of elements in \Sigma. Should this conjecture be proved, we would have as a consequence that no "small" NP-complete set exists in a precise sense of the word "small". A set S ` \Sigma is said to be sparse when there is a polynomial p such that for all n 2 IN the subset S n of all elements in S having size n has cardinality at most p(n). If the Berman-Hartmanis conjecture is true, then there are no sparse NP-complete sets. Partially supported by CERG grant 9040393. 1 In 1982 Mahaney (...