Complete Axiomatizations of Fragments of Monadic Second-Order Logic on Finite Trees

Gheerbrant, Amélie; Cate, Balder ten

doi:10.2168/lmcs-8(4:12)2012

Cited by 5 publications

(6 citation statements)

References 22 publications

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“…Therefore, we suggest that an alternative axiomatization of S1S, and completeness proof for it, could be sought by axiomatizing existentially closed (♦, X, I )-algebras. In a similar direction, we note that Gheerbrant and Ten Cate [8] used modal logic techniques to axiomatize monadic second order logic on finite trees, and fragments. An extension of our results in this paper to finite trees could also be connected to the results in [8].…”

Section: 4mentioning

confidence: 96%

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A Model-Theoretic Characterization of Monadic Second Order Logic on Infinite Words

Ghilardi¹,

Gool²

2017

J. symb. log.

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Monadic second order logic and linear temporal logic are two logical formalisms that can be used to describe classes of infinite words, i.e., first-order models based on the natural numbers with order, successor, and finitely many unary predicate symbols.Monadic second order logic over infinite words (S1S) can alternatively be described as a first-order logic interpreted in P(ω), the power set Boolean algebra of the natural numbers, equipped with modal operators for 'initial', 'next' and 'future' states. We prove that the first-order theory of this structure is the model companion of a class of algebras corresponding to a version of linear temporal logic (LTL) without until.The proof makes crucial use of two classical, non-trivial results from the literature, namely the completeness of LTL with respect to the natural numbers, and the correspondence between S1S-formulas and Büchi automata.

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Section: 4mentioning

confidence: 96%

“…In a similar direction, we note that Gheerbrant and Ten Cate [8] used modal logic techniques to axiomatize monadic second order logic on finite trees, and fragments. An extension of our results in this paper to finite trees could also be connected to the results in [8].…”

mentioning

confidence: 96%

A Model-Theoretic Characterization of Monadic Second Order Logic on Infinite Words

Ghilardi¹,

Gool²

2017

J. symb. log.

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“…This means that the theory of MSO over finite strings has a recursive and complete axiomatization. One such axiomatization is given in [12], and therefore for a set Γ ∪ {ϕ} of MSO formulas, Γ ⊢ ϕ means that ϕ is derivable from these axioms and Γ (Γ may be omitted when empty). Since FO over finite strings also has a decidable validity problem, it likewise has a recursive and complete axiomatization.…”

Section: A Msomentioning

confidence: 99%

Axiomatizations and Computability of Weighted Monadic Second-Order Logic

Achilleos¹,

Pedersen²

2021

Preprint

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Weighted monadic second-order logic is a weighted extension of monadic second-order logic that captures exactly the behaviour of weighted automata. Its semantics is parameterized with respect to a semiring on which the values that weighted formulas output are evaluated. Gastin and Monmege (2018) gave abstract semantics for a version of weighted monadic secondorder logic to give a more general and modular proof of the equivalence of the logic with weighted automata. We focus on the abstract semantics of the logic and we give a complete axiomatization both for the full logic and for a fragment without general sum, thus giving a more fine-grained understanding of the logic. We discuss how common decision problems for logical languages can be adapted to the weighted setting, and show that many of these are decidable, though they inherit bad complexity from the underlying first-and second-order logics. However, we show that a weighted adaptation of satisfiability is undecidable for the logic when one uses the abstract interpretation.

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“…Another trend relies on model-theoretic techniques. For instance [tCF10,GtC12] give complete axiomatizations of MSO and the modal µ-calculus over finite trees; a reworking of the completeness of MSO on ωwords [Sie70] is proposed in [Rib12]; and [SV10] gives a model-theoretic completeness proof for a fragment of the modal µ-calculus. An attractive feature of model-theoretic completeness proofs for the aforementioned logics is that they allow elegant reformulations of algebraic approaches to these logics.…”

Section: Introductionmentioning

confidence: 99%

A Functional (Monadic) Second-Order Theory of Infinite Trees

Das,

Riba

2019

Preprint

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This paper presents a complete axiomatization of Monadic Second-Order Logic (MSO) over infinite trees. MSO on infinite trees is a rich system, and its decidability ("Rabin's Tree Theorem") is one of the most powerful known results concerning the decidability of logics.By a complete axiomatization we mean a complete deduction system with a polynomial-time recognizable set of axioms. By naive enumeration of formal derivations, this formally gives a proof of Rabin's Tree Theorem. The deduction system consists of the usual rules for second-order logic seen as two-sorted first-order logic, together with the natural adaptation In addition, it contains an axiom scheme expressing the (positional) determinacy of certain parity games.The main difficulty resides in the limited expressive power of the language of MSO. We actually devise an extension of MSO, called Functional (Monadic) Second-Order Logic (FSO), which allows us to uniformly manipulate (hereditarily) finite sets and corresponding labeled trees, and whose language allows for higher abstraction than that of MSO.

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Complete Axiomatizations of Fragments of Monadic Second-Order Logic on Finite Trees

Cited by 5 publications

References 22 publications

A Model-Theoretic Characterization of Monadic Second Order Logic on Infinite Words

A Model-Theoretic Characterization of Monadic Second Order Logic on Infinite Words

Axiomatizations and Computability of Weighted Monadic Second-Order Logic

A Functional (Monadic) Second-Order Theory of Infinite Trees

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