Monadic second order logic on tree-like structures

Walukiewicz, Igor

doi:10.1007/3-540-60922-9_33

Cited by 63 publications

(112 citation statements)

References 11 publications

Supporting

Mentioning

107

Contrasting

Unclassified

Order By: Relevance

“…The more difficult result that the unfolding operation, mapping graphs (G, v) to trees T (G,v) , is also MSO-compatible appeared in [61] (see also [57] for an exposition and a treatment of the simpler case of deterministic graphs). We note that this result also follows from Muchnik's Theorem [126,138,17] and that it generalises Rabin's theorem.…”

Section: Higher-order Data Structuressupporting

confidence: 69%

Automata-based presentations of infinite structures

Bárány¹,

Grädel²,

Rubin³

2011

Finite and Algorithmic Model Theory

View full text Add to dashboard Cite

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

show abstract

Section: Higher-order Data Structuressupporting

confidence: 69%

Automata-based presentations of infinite structures

Bárány¹,

Grädel²,

Rubin³

2011

Finite and Algorithmic Model Theory

View full text Add to dashboard Cite

show abstract

“…This result is proved in [10]. It subsumes the case of unfolding and we think it may help to solve some open questions in [8,9].…”

Section: ))))supporting

confidence: 52%

Semantical Evaluations as Monadic Second-Order Compatible Structure Transformations

Courcelle

2002

Lecture Notes in Computer Science

View full text Add to dashboard Cite

Abstract.A transformation of structures τ is monadic second-order compatible (MS-compatible) if every monadic second-order property P can be effectively rewritten into a monadic second-order property Q such that, for every structure S, if T is the transformed structure τ (S), then P (T ) holds iff Q(S) holds. We will review Monadic Second-order definable transductions (MS-transductions): they are MS-compatible transformations of a particular form, i.e., defined by monadic second-order (MS) formulas. The unfolding of a directed graph into a tree is an MS-compatible transformation that is not an MS-transduction. The MS-compatibility of various transformations of semantical interest follows. We will present three main cases and discuss applications and open problems. Overview of the lectureOur working logical language is Monadic Second-Order Logic, i.e., the extension of First-Order Logic with variables denoting sets of elements of the considered structures. It enjoys a number of interesting properties regarding decidability and construction of polynomial algorithms [4].We consider certain semantical evaluations that can be formalized as transformations of discrete structures like graphs or trees (and not as mappings from terms to values belonging to semantical domains as this is usual in denotational semantics).Our main concern will be to identify transformations such that the verification of an MS property P, of a structure T = τ (S) reduces to the verification of an MS property Q of the input structure S, where Q depends only on τ and P and, of course, of the fixed relational signatures of S and T .In such a case, if the MS theory of an infinite structure S is decidable (which means that there exists an algorithm that decides whether a monadic secondorder formula is true or not in S), then so is that of T = τ (S). We say that τ is Monadic Second-order compatible (MS-compatible).

show abstract

“…As a corollary of this result and Proposition 3.5, MSO (see Example 2.9) is in fact a coalgebraic fixpoint logic. Theorem 3.10 below can be seen as a more precise formulation of Fact 1.3; as mentioned there, the two statements can be found in Walukiewicz [23] and Janin & Walukiewicz [12], respectively. Theorem 3.10 1.…”

Section: Qedmentioning

confidence: 94%

“…This link between logic and automata theory essentially goes back to the work of Rabin and Büchi on stream and tree automata. The two statements in Fact 1.3 below can be found in Walukiewicz [23] and Janin & Walukiewicz [12], respectively. Fact 1.3 1.…”

Section: Introductionmentioning

confidence: 94%

Expressiveness Modulo Bisimilarity: A Coalgebraic Perspective

Venema

2014

Outstanding Contributions to Logic

View full text Add to dashboard Cite

Monadic second order logic on tree-like structures

Cited by 63 publications

References 11 publications

Automata-based presentations of infinite structures

Automata-based presentations of infinite structures

Semantical Evaluations as Monadic Second-Order Compatible Structure Transformations

Expressiveness Modulo Bisimilarity: A Coalgebraic Perspective

Contact Info

Product

Resources

About