A Theory of NP-completeness and Ill-conditioning for Approximate Real Computations

G. Malajovich, M. Shub
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引用次数: 4

Abstract

We develop a complexity theory for approximate real computations. We first produce a theory for exact computations but with condition numbers. The input size depends on a condition number, which is not assumed known by the machine. The theory admits deterministic and nondeterministic polynomial time recognizable problems. We prove that P is not NP in this theory if and only if P is not NP in the BSS theory over the reals. Then we develop a theory with weak and strong approximate computations. This theory is intended to model actual numerical computations that are usually performed in floating point arithmetic. It admits classes P and NP and also an NP-complete problem. We relate the P vs. NP question in this new theory to the classical P vs. NP problem.
近似实际计算的np完备性和病态条件理论
我们发展了一个近似实际计算的复杂性理论。我们首先提出了一个精确计算的理论,但有条件数。输入的大小取决于一个条件数,而这个条件数是机器不知道的。该理论承认确定性和非确定性多项式时间可识别问题。当且仅当P在实数上的BSS理论中不是NP时,我们证明了P在这个理论中不是NP。然后,我们发展了一个具有弱近似计算和强近似计算的理论。该理论旨在模拟通常在浮点运算中执行的实际数值计算。它承认P类和NP类,也是一个NP完全问题。我们将这个新理论中的P与NP问题与经典的P与NP问题联系起来。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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