Monadic second order logic as the model companion of temporal logic

Ghilardi, Silvio

doi:10.1145/2933575.2933609

Cited by 8 publications

(3 citation statements)

References 18 publications

(19 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…An essential insight in the proof of Büchi's theorem is the fact that every monadic second-order formula is equivalent on words to an existential monadic second-order formula, and thus the iterative approach is not relevant as the hierarchy collapses. See (Ghilardi and van Gool, 2016) for a duality and type-theoretic approach via model companions. However, for the first-order fragment the iterative approach is very powerful.…”

Section: Three Examples Of Dual Spaces In Logicmentioning

confidence: 99%

A Cook's tour of duality in logic: from quantifiers, through Vietoris, to measures

Gehrke,

Jakl,

Reggio

2020

Preprint

View full text Add to dashboard Cite

We identify and highlight certain landmark results in Samson Abramsky's work which we believe are fundamental to current developments and future trends. In particular, we focus on the use of • topological duality methods to solve problems in logic and computer science;• category theory and, more particularly, free (and co-free) constructions;• these tools to unify the 'power' and 'structure' strands in computer science.

show abstract

Section: Three Examples Of Dual Spaces In Logicmentioning

confidence: 99%

A Cook's tour of duality in logic: from quantifiers, through Vietoris, to measures

Gehrke,

Jakl,

Reggio

2020

Preprint

View full text Add to dashboard Cite

show abstract

“…Interestingly, model completeness has other well-known applications in computer science. It has been applied: (i) to reveal interesting connections between temporal logic and monadic second-order logic [29,30]; (ii) in automated reasoning to design complete algorithms for constraint satisfiability in combined theories over non-disjoint signatures [1,23,31,[49][50][51]; (iii) again in automated reasoning in relationship with interpolation and symbol elimination [59,60]; (iv) in modal logic and in software verification theories [24,25], to obtain combined interpolation results.…”

Section: Introductionmentioning

confidence: 99%

Model Completeness, Uniform Interpolants and Superposition Calculus

et al. 2021

Self Cite

View full text Add to dashboard Cite

Uniform interpolants have been largely studied in non-classical propositional logics since the nineties; a successive research line within the automated reasoning community investigated uniform quantifier-free interpolants (sometimes referred to as “covers”) in first-order theories. This further research line is motivated by the fact that uniform interpolants offer an effective solution to tackle quantifier elimination and symbol elimination problems, which are central in model checking infinite state systems. This was first pointed out in ESOP 2008 by Gulwani and Musuvathi, and then by the authors of the present contribution in the context of recent applications to the verification of data-aware processes. In this paper, we show how covers are strictly related to model completions, a well-known topic in model theory. We also investigate the computation of covers within the Superposition Calculus, by adopting a constrained version of the calculus and by defining appropriate settings and reduction strategies. In addition, we show that computing covers is computationally tractable for the fragment of the language used when tackling the verification of data-aware processes. This observation is confirmed by analyzing the preliminary results obtained using the mcmt tool to verify relevant examples of data-aware processes. These examples can be found in the last version of the tool distribution.

show abstract

“…In algebraic logic some attention has been paid to the class of existentially closed structures in varieties coming from the algebraization of common propositional logics. In fact, there are relevant cases where such classes are elementary: this includes, besides the easy case of Boolean algebras, also Heyting algebras [GZ97,GZ02], diagonalizable algebras [Sha93,GZ02] and some universal classes related to temporal logics [GvG01], [GvG16]. However, very little is known about the related axiomatizations, with the remarkable exception of the case of the locally finite amalgamable varieties of Heyting algebras recently investigated in [DJ10] and of the simpler cases of posets and semilattices studied in [AB86].…”

Section: Introductionmentioning

confidence: 99%

Existentially Closed Brouwerian Semilattices

Carai¹,

Ghilardi²

2019

J. symb. log.

Self Cite

View full text Add to dashboard Cite

The variety of Brouwerian semilattices is amalgamable and locally finite, hence by well-known results [Whe76], it has a model completion (whose models are the existentially closed structures). In this paper, we supply for such a model completion a finite and rather simple axiomatization.2010 Mathematics subject classification. Primary 03G25; Secondary 03C10, 06D20.A co-Brouwerian semilattice, CBS for short, is a structure obtained by reversing the order of a Brouwerian semilattice. We will work with CBSes instead of Brouwerian semilattices.Definition 2.1. A poset (P, ≤) is said to be a co-Brouwerian semilattice if it has a least element, which we denote with 0, and for every a, b ∈ P there exists the sup of {a, b}, which we call join of a and b and denote with a ∨ b, and the difference a − b satisfying for every c ∈ PClearly, there is also an alternative equational definition for co-Brouwerian semilattices (which we leave to the reader, because it is dual to the equational definition for Brouwerian semilattices given above).Moreover, we will call co-Heyting algebras the structures obtained reversing the order of Heyting algebras. Obviously any co-Heyting algebra is a CBS.Definition 2.2. Let A, B be co-Brouwerian semilattices. A map f : A → B is a morphism of co-Brouwerian semilattices if it preserves 0, the join and difference of any two elements of A.Notice that such a morphism f is an order preserving map because, for any a, b elements of a co-Brouwerian semilattice, we have a ≤ b iff a ∨ b = b.Definition 2.3. Let L be a CBS. We say that g ∈ L is join-irreducible iff for every n ≥ 0 and b 1 , . . . , b n ∈ L, we have that g ≤ b 1 ∨ . . . ∨ b n implies g ≤ b i for some i = 1, . . . , n.Notice that taking n = 0 we obtain that join-irreducibles are different from 0.Remark 2.4. Let L be a CBS and g ∈ L. Then the following conditions are equivalent:1. g is join-irreducible.2. g = 0 and for any b 1 , b 2 ∈ L we have that g3. For every n ≥ 0 and b 1 , . . . , b n ∈ L we have that g = b 1 ∨ . . . ∨ b n implies g = b i for some i = 1, . . . , n.4. g = 0 and for any b 1 , b 2 ∈ L we have that g = b 1 ∨ b 2 implies g = b 1 or g = b 2 .5. g = 0 and for any a ∈ L we have that g − a = 0 or g − a = g.Proof. The implications 1 ⇔ 2, 3 ⇔ 4 and 1 ⇒ 3 are straightforward. For the remaining ones see Lemma 2.1 in [Köh81].Definition 2.5. Let L be a CBS and a ∈ L.A join-irreducible component of a is a maximal element among the join-irreducibles of L that are smaller than or equal to a.

show abstract

Monadic second order logic as the model companion of temporal logic

Cited by 8 publications

References 18 publications

A Cook's tour of duality in logic: from quantifiers, through Vietoris, to measures

A Cook's tour of duality in logic: from quantifiers, through Vietoris, to measures

Model Completeness, Uniform Interpolants and Superposition Calculus

Existentially Closed Brouwerian Semilattices

Contact Info

Product

Resources

About