我们的NP理论工作首重解读NP的二个流行定义及二个定义的关系，这篇英语文章是此工作的总结和深化。

文章追本溯源分析用于定义NP的NDTM（不确定性图灵机），揭示NDTM指称二个完全不同的概念：一个是Oracle（神喻机），另一个是Turing Machine（图灵机）。由于同一术语NDTM指称两个内涵完全不同的概念，故发生了“概念偷换”的逻辑错误，导致“可验证”成为现在NP的标准定义，自此NP的本质“不确定性”消失了，遂有“P versus NP”的千禧年难题。

由于NP的原始定义与Oracle有关，而Oracle又来自图灵关于“可计算性”的工作，故追本溯源到图灵的工作，应该是澄清NP定义的正途，也即我们的NP理论基于可计算性的深刻背景。

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In this paper, we interpret NDTM (NonDeterministicTuringMachine) used to define NP by tracing to the source of NP. Originally NP was defined as the class of problems solvable in polynomial time by a NDTM in Cook’s theorem, where the NDTM was represented as Query Machine of essence Oracle. Later a model consisting of a guessing module and a checking module was proposed to replace the NDTM. This model of essence TM has a fundamental difference from the NDTM of essence Oracle, but people still use the term NDTM to designate this model, which leads to the disguised displacement of NDTM and produces out the verifier-based definition of NP as the class of problems verifiable in polynomialtime by a TM (Turing Machine). This verifier-based one has been then accepted as the standard definition of NP where comes from the famous equivalence ofthe two definitions of NP. Since then the notion of nondeterminism is lost from NP, which causes ambiguities in understanding NP and then great difficulties in solving the P versus NP problem.