具有时间可达迭代的逻辑

摘要：

一阶逻辑在表达能力上的不足导致了不动点算子对一阶逻辑的大量扩展，如最小不动点（LFP）、膨胀不动点（IFP）、部分不动点（PFP）等。这些逻辑在有限模型论、数据库论、描述复杂性等方面得到了广泛的研究。本文介绍了一种统一的框架--带迭代算子的逻辑，在这种框架中，迭代步骤可以用时态逻辑公式来表示。我们表明，建议的逻辑FO+TAI包含所有提到的不动点扩展以及许多其他不动点逻辑的自然片段。另一方面，我们表明，在有限的结构FO+TAI是没有更多的表达比FO+PFP。此外，我们表明，增加相同的机器的逻辑单调归纳（FO+LFP）并没有增加其表达能力。

英文摘要：

Deficiency in expressive power of the first-order logic has led to developing its numerous extensions by fixed point operators, such as Least Fixed-Point (LFP), inflationary fixed-point (IFP), partial fixed-point (PFP), etc. These logics have been extensively studied in finite model theory, database theory, descriptive complexity. In this paper we introduce unifying framework, the logic with iteration operator, in which iteration steps may be accessed by temporal logic formulae. We show that proposed logic FO+TAI subsumes all mentioned fixed point extensions as well as many other fixed point logics as natural fragments. On the other hand we show that over finite structures FO+TAI is no more expressive than FO+PFP. Further we show that adding the same machinery to the logic of monotone inductions (FO+LFP) does not increase its expressive power either.

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