无限结构的自动机表示

V. Bárány, E. Grädel, S. Rubin
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引用次数: 26

摘要

有限结构的模型理论与计算机科学的各个领域密切相关,包括复杂性理论、数据库和验证。特别是,复杂性类和逻辑语言的表达能力之间有着密切的关系,正如描述性复杂性理论的基本定理所证明的那样,如费金定理和ImmermanVardi定理(参见[78,第3章]的调查)。然而,对于许多应用来说,对有限结构的严格限制已经被证明是过于严格的,并且已经有相当多的努力将相关的逻辑和算法方法从有限结构扩展到合适的无限结构类。特别是对于无限结构至关重要的数据库和验证而言[130]。算法模型理论旨在以系统的方式扩展有限模型理论的方法和方法,以及它与计算机科学的相互作用,从有限结构到有限呈现的无限结构。用有限的方式表现无限的结构有很多可能性。模型理论中的一个经典方法涉及可计算结构类;它们是自然数域上的可数结构,具有有限的可计算函数和关系集合。这种结构可以通过一系列算法有限地呈现,自20世纪60年代以来,它们在模型理论中得到了深入研究。然而,从算法模型理论的角度来看,这类可计算结构是有问题的。事实上,算法模型理论的中心问题之一是逻辑公式的有效评估,从合适的逻辑,例如,一阶逻辑(FO),一元二阶逻辑(MSO),或不动点逻辑,如LFP或模态μ微积分。但在可计算结构上,一般只有无量词的公式允许有效求值,一阶逻辑的存在片段已经是不可判定的,例如在可计算结构(N,+,·)上。这就引出了我们的核心要求,即对于合适的逻辑L(取决于预期的应用程序),C类有限呈现结构的模型检查问题应该是可以通过算法解决的。完全地
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Automata-based presentations of infinite structures
The model theory of finite structures is intimately connected to various fields in computer science, including complexity theory, databases, and verification. In particular, there is a close relationship between complexity classes and the expressive power of logical languages, as witnessed by the fundamental theorems of descriptive complexity theory, such as Fagin’s Theorem and the ImmermanVardi Theorem (see [78, Chapter 3] for a survey). However, for many applications, the strict limitation to finite structures has turned out to be too restrictive, and there have been considerable efforts to extend the relevant logical and algorithmic methodologies from finite structures to suitable classes of infinite ones. In particular this is the case for databases and verification where infinite structures are of crucial importance [130]. Algorithmic model theory aims to extend in a systematic fashion the approach and methods of finite model theory, and its interactions with computer science, from finite structures to finitely-presentable infinite ones. There are many possibilities to present infinite structures in a finite manner. A classical approach in model theory concerns the class of computable structures; these are countable structures, on the domain of natural numbers, say, with a finite collection of computable functions and relations. Such structures can be finitely presented by a collection of algorithms, and they have been intensively studied in model theory since the 1960s. However, from the point of view of algorithmic model theory the class of computable structures is problematic. Indeed, one of the central issues in algorithmic model theory is the effective evaluation of logical formulae, from a suitable logic such as, for instance, first-order logic (FO), monadic second-order logic (MSO), or a fixed point logic like LFP or the modal μ-calculus. But on computable structures, only the quantifier-free formulae generally admit effective evaluation, and already the existential fragment of first-order logic is undecidable, for instance on the computable structure (N,+, · ). This leads us to the central requirement that for a suitable logic L (depending on the intended application) the model-checking problem for the class C of finitely presented structures should be algorithmically solvable. At the very
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