If all NP complete sets are isomorphic under deterministic polynomial time mappings (p-isomorphic) then P @@@@ NP and if all PTAPE complete sets are p-isomorphic then P @@@@ PTAPE. We show that all NP complete sets known (in the literature) are indeed p-isomorphic and so are the known PTAPE complete sets. Thus showing that, inspite of the radically different origins and attempted simplification of these sets, all the known NP complete sets are identical but for polynomially time bounded permutations.
Furthermore, if all NP complete sets are p-isomorphic then they all must have similar densities and, for example, no language over a single letter alphabet can be NP complete, nor can any sparse language over an arbitrary alphabet be NP complete. We show that complete sets in EXPTIME and EXPTAPE cannot be sparse and therefore they cannot be over a single letter alphabet. Similarly, we show that the hardest context-sensitive languages cannot be sparse. We also relate the existence of sparse complete sets to the existence of simple combinatorial circuits for the corresponding truncated recognition problem of these languages.
如果在确定性多项式时间映射下所有NP成套都是同构的(p同构),则P @@@@ NP;如果所有PTAPE成套都是p同构的,则P @@@@ PTAPE。我们证明(文献中)所有已知的NP完全确实是p同构的,已知的PTAPE完全也是如此。因此表明,尽管起源完全不同,并且试图简化这些集合,但所有已知的NP完全集合都是相同的,只是在多项式时间上有界。 P>
此外,如果所有NP完整集都是p同构的,那么它们都必须具有相似的密度,例如,单个字母字母表上的语言都不能是NP完整的,任意字母表上的稀疏语言也不能是NP完整的。我们表明,EXPTIME和EXPTAPE中的完整集合不能稀疏,因此它们不能超过单个字母。类似地,我们表明最困难的上下文相关语言不能稀疏。对于这些语言的相应截断识别问题,我们还将稀疏成套的存在与简单组合电路的存在联系起来。 P>
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机译:NP和其他成套的同构和密度
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