Graph Structure and Monadic Second-Order Logic

Courcelle, Bruno; Engelfriet, Joost

doi:10.1017/cbo9780511977619

Cited by 384 publications

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“…This approach has its roots in finite model theory (Enderton, 2001;Hedman, 2004;Courcelle & Engelfriet, 2012). (See Potts & Pullum (2002) and Rogers et al (2013) for model-theoretic approaches to phonology.)…”

Section: Representing Candidatesmentioning

confidence: 99%

A formal analysis of Correspondence Theory

Payne

Heinz

2017

AMP

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This paper provides a computational analysis of the complexity of GEN and Correspondence Theory in terms of the nature of the logic involved in their formulation. The first result of this analysis shows that the GEN function is not definable in Monadic Second Order (MSO) logic. Second, we show that the set of input-output Correspondence-theoretic candidates from a given underlying representation is definable in First Order (FO) logic, which is less complex than MSO-logic. Third, we present some case studies where the correct input-output Correspondence-theoretic candidate from a given underlying representation can be accomplished with FO-definable, language-specific, inviolable constraints without recourse to optimization.

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Section: Representing Candidatesmentioning

confidence: 99%

A formal analysis of Correspondence Theory

Payne

Heinz

2017

AMP

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“…Each regular tree language L of closed terms thus represents a family of finite graphs {[[t]] | t ∈ L}. For a concise treatment of graph grammars and finite graphs we refer to the surveys [69,59] and the forthcoming book [53].…”

Section: Graph Grammars and Graph Algebrasmentioning

confidence: 99%

“…Since no classical order-or metric-theoretic notion of limit seems to exist for graphs, we use the more general categorical notion of colimit [11]. We outline this framework in which an infinite term (over the graph operations Θ) yields a countable graph; details may be found in [55,11] and the monograph [53].…”

Section: Graph Grammars and Graph Algebrasmentioning

confidence: 99%

Automata-based presentations of infinite structures

Bárány¹,

Grädel²,

Rubin³

2011

Finite and Algorithmic Model Theory

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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 1

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“…With the results of [6] this would imply our expressivity result (not the succinctness result). However, the result from [1] relies on an earlier proof of how to define tree decompositions of bounded width in MSO whose correctness has been doubted by Courcelle and Engelfriet [5]. A proof of our expressivity result for ≤-inv-FO along this lines is nevertheless possible.…”

Section: Introductionmentioning

confidence: 96%

Expressivity and Succinctness of Order-Invariant Logics on Depth-Bounded Structures

Eickmeyer

Elberfeld

Harwath

2014

Mathematical Foundations of Computer Science 2014

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Abstract. We study the expressive power and succinctness of orderinvariant sentences of first-order (FO) and monadic second-order (MSO) logic on graphs of bounded tree-depth. Order-invariance is undecidable in general and, therefore, in finite model theory, one strives for logics with a decidable syntax that have the same expressive power as orderinvariant sentences. We show that on graphs of bounded tree-depth, order-invariant FO has the same expressive power as FO, and orderinvariant MSO has the same expressive power as the extension of FO with modulo-counting quantifiers. Our proof techniques allow for a finegrained analysis of the succinctness of these translations. We show that for every order-invariant FO sentence there exists an FO sentence whose size is elementary in the size of the original sentence, and whose number of quantifier alternations is linear in the tree-depth. Our techniques can be adapted to obtain a similar quantitative variant of a known result that the expressive power of MSO and FO coincides on graphs of bounded tree-depth.

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Graph Structure and Monadic Second-Order Logic

Cited by 384 publications

References 0 publications

A formal analysis of Correspondence Theory

A formal analysis of Correspondence Theory

Automata-based presentations of infinite structures

Expressivity and Succinctness of Order-Invariant Logics on Depth-Bounded Structures

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