On the elementary theory of linear order

Läuchli, H.; Leonard, Jack E.

doi:10.4064/fm-59-1-109-116

Cited by 96 publications

(53 citation statements)

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“…• In a sequence of papers by Ehrenfeucht, L auchli, Shelah and Gurevich, cf. [44,76,77,98,149] it emerged that Theorems 1.3 and 1.4 remain true for MSOL rather than FOL in the case of the sum Th MSOL i∈I A i and the multiple disjoint union Th MSOL i∈I A but not for products.…”

Section: Theorem 13 (Feferman and Vaughtmentioning

confidence: 91%

Algorithmic uses of the Feferman–Vaught Theorem

Makowsky

2004

Annals of Pure and Applied Logic

161

222

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The classical Feferman-Vaught Theorem for First Order Logic explains how to compute the truth value of a ÿrst order sentence in a generalized product of ÿrst order structures by reducing this computation to the computation of truth values of other ÿrst order sentences in the factors and evaluation of a monadic second order sentence in the index structure. This technique was later extended by L auchli, Shelah and Gurevich to monadic second order logic. The technique has wide applications in decidability and deÿnability theory.Here we give a uniÿed presentation, including some new results, of how to use the FefermanVaught Theorem, and some new variations thereof, algorithmically in the case of Monadic Second Order Logic MSOL.We then extend the technique to graph polynomials where the range of the summation of the monomials is deÿnable in MSOL. Here the Feferman-Vaught Theorem for these polynomials generalizes well known splitting theorems for graph polynomials. Again, these can be used algorithmically.Finally, we discuss extensions of MSOL for which the Feferman-Vaught Theorem holds as well. (J.A. Makowsky).1 At the occasion of his 75th birthday. It is also a tribute to Feferman's work in model theory, which went seemingly unnoticed at the celebrations of his 70th birthday. Feferman returned several times to topics related to the preservation of truth under various model theoretic constructions, so in [54][55][56][57][58]60]. The FefermanVaught Theorem can also be viewed as falling into this category.

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Section: Theorem 13 (Feferman and Vaughtmentioning

confidence: 91%

Algorithmic uses of the Feferman–Vaught Theorem

Makowsky

2004

Annals of Pure and Applied Logic

161

222

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“…As the constructions are effective, this gives in particular a decision procedure that relies on automata theory for the theory of MSO[<] interpreted over boundedwidth posets of SP ⋄ . This decision result can also be obtained without automata theory, using for example model-theoretical techniques as in [20], or the methods of [13]. As a corollary, the inclusion problem of rational languages is also decidable.…”

Section: Introductionmentioning

confidence: 89%

Logic and Bounded-Width Rational Languages of Posets over Countable Scattered Linear Orderings

Bedon

2008

Logical Foundations of Computer Science

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Abstract. In this paper we consider languages of labelled N -free posets over countable and scattered linear orderings. We prove that a language of such posets is series-rational if and only if it is recognizable by a finite depth-nilpotent algebra if and only if it is bounded-width and monadic second-order definable. This extends previous results on languages of labelled N -free finite and ω-posets and on languages of labelled countable and scattered linear orderings.

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“…Towards this end, we first present a new suitable notation for describing models in a concrete way. The compositional approach presented here was hinted at in [25], and traces back to pioneering work in [18] and [3]. It uses a small number of distinct operations for putting together a larger model from one or more smaller ones, or copies thereof.…”

Section: Introductionmentioning

confidence: 98%

Synthesis for continuous time

French

McCabe-Dansted

Reynolds

2015

Theoretical Computer Science

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When considering applications of reasoning in formal temporal logics, and particularly applications that need reasoning about distributed concurrent systems, multi-agent systems, or understanding natural language, it is convenient to use a logic based on a real-number flow of time rather than the usual discrete flow. Such a logic may be employed for several distinct reasoning tasks and one that has potential to be one of the most useful is building a model from a specification, a simple form of synthesis. In this paper, we consider the task of model-building for temporal logic specifications over the real-number linear flow of time. We present a new notation for giving a detailed description of the compositional construction of such a model and an efficient procedure for finding such a description from the temporal specification.

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On the elementary theory of linear order

Cited by 96 publications

References 0 publications

Algorithmic uses of the Feferman–Vaught Theorem

Algorithmic uses of the Feferman–Vaught Theorem

Logic and Bounded-Width Rational Languages of Posets over Countable Scattered Linear Orderings

Synthesis for continuous time

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