On the NP-Isomorphism Problem with Respect to Random Instances

On complete one-way functions

“…We do not describe in detail the properties of semi-Thue systems; they are discussed, for example, in [26][27][28]. We only note a relevant property: in [29] it was shown that the distributional accessibility problem for semi-Thue systems is complete for the DistNP complexity class. The proof relies on the fact that semi-Thue systems can model Turing machines.…”

“…This can be done with a so-called dynamic binary coding scheme developed by Yu. Gurevich and used in [6,29,36]. Proof.…”

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“…In case µ ν and ν µ, the distributions µ and ν are called equivalent (in symbols: µ ≡ ν) [WB95]. A function f from Σ * to the non-negative integers is polynomial on µ-average [Lev86] if there exists a constant > 0 such that Distributional NP search problems.…”

“…In the proof of Theorem 6 we used the completeness of the bounded halting problem for the class DistNP consisting of all pairs (L, µ) such that L is in NP and µ is p-computable [Lev86,Gur91]. In fact, the proof also works for other DistNP-complete problems such as the tiling problem, Post's correspondence problem, the word problem for Thue systems and groups, and LR(k) testing of context-free grammars [Lev86,Gur91,WB95,Wan95,Kar97].…”

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“…Although this conjecture has not been solved, it has stimulated a host of papers in structural complexity theory. Wang and Belanger [WB95] observed that some standard NP-complete problems, while isomorphic on the worst case, are not isomorphic on the average case under commonly used distributions. On the other hand, all the average-case NP-complete problems may still be isomorphic.…”