光谱问题五十年：综述和新结果
Fifty Years of the Spectrum Problem: Survey and New Results
摘要：
1952年，海因里希·肖尔茨（Heinrich Scholz）在《符号逻辑杂志》（Journal of Symbolic Logic）上发表了一个问题，要求对谱进行表征，即：作为一阶句子的有限模型的基数的自然数的集合。Günter Asser问光谱的补集是否总是光谱。这些天真的问题对有限模型论和描述复杂性的发展产生了深远的影响。在本文中，我们调查的发展，在过去的50多年来有关的频谱问题。我们的介绍遵循概念的发展，而不是时间顺序。最初是一个数论问题，它已接近的递归论，资源有限的复杂性理论，分类的复杂性定义的句子，最后在结构图论。虽然Scholz的问题有各种各样的答案，但Asser的问题仍然是开放的。一个附录是对G. Asser，A. Mostowski，J. Bennett和S.密苏里州另一个附录载有一个问题和解答的简编，这些问题和解答仍然是开放的。
英文摘要：
In 1952, Heinrich Scholz published a question in the Journal of Symbolic Logic asking for a characterization of spectra, i.e., sets of natural numbers that are the cardinalities of finite models of first order sentences. Günter Asser asked whether the complement of a spectrum is always a spectrum. These innocent questions turned out to be seminal for the development of finite model theory and descriptive complexity. In this paper we survey developments over the last 50-odd years pertaining to the spectrum problem. Our presentation follows conceptual developments rather than the chronological order. Originally a number theoretic problem, it has been approached in terms of recursion theory, resource bounded complexity theory, classification by complexity of the defining sentences, and finally in terms of structural graph theory. Although Scholz' question was answered in various ways, Asser's question remains open. One appendix paraphrases the contents of several early and not easily accesible papers by G. Asser, A. Mostowski, J. Bennett and S. Mo. Another appendix contains a compendium of questions and conjectures which remain open.
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