Sheaves, Games, and Model Completions

Ghilardi, Silvio; Zawadowski, Marek

doi:10.1007/978-94-015-9936-8

Cited by 39 publications

(30 citation statements)

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“…His technique builds the free Heyting algebra on a distributive lattice step by step by freely adding to the original lattice the implications of degree n, for each n ∈ ω. Ghilardi [10] used this technique to show that every finitely generated free Heyting algebra is a bi-Heyting algebra. A more detailed account of Ghilardi's construction can be found in [7] and [12]. Ghilardi and Zawadowski [12], based on this method, derive a model-theoretic proof of Pitts' uniform interpolation theorem.…”

Section: Introductionmentioning

confidence: 99%

“…A more detailed account of Ghilardi's construction can be found in [7] and [12]. Ghilardi and Zawadowski [12], based on this method, derive a model-theoretic proof of Pitts' uniform interpolation theorem. In [3] a similar construction is used to describe free linear Heyting algebras over a finite distributive lattice and [16] uses the same method to construct high order cylindric Heyting algebras.…”

Section: Introductionmentioning

confidence: 99%

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Free Heyting Algebras: Revisited

Bezhanishvili

Gehrke

2009

Algebra and Coalgebra in Computer Science

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Abstract. We use coalgebraic methods to describe finitely generated free Heyting algebras. Heyting algebras are axiomatized by rank 0-1 axioms. In the process of constructing free Heyting algebras we first apply existing methods to weak Heyting algebras-the rank 1 reducts of Heyting algebras-and then adjust them to the mixed rank 0-1 axioms. On the negative side, our work shows that one cannot use arbitrary axiomatizations in this approach. Also, the adjustments made for the mixed rank axioms are not just purely equational, but rely on properties of implication as a residual. On the other hand, the duality and coalgebra perspective does allow us, in the case of Heyting algebras, to derive Ghilardi's (Ghilardi, 1992) powerful representation of finitely generated free Heyting algebras in a simple, transparent, and modular way using Birkhoff duality for finite distributive lattices.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Free Heyting Algebras: Revisited

Bezhanishvili

Gehrke

2009

Algebra and Coalgebra in Computer Science

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“…The theory T * 0 is included in the theory BA * of atomless Boolean algebras (recall that a Boolean algebra is said to be atomless iff it does not have non-zero minimal elements): the axioms of T * 0 are in fact provable in BA * , as it is evident from the quantifier elimination procedure for BA * (see, e.g., [17]). Since every join semilattice with a greatest element embeds into an atomless Boolean algebra, 13 this shows both that T * 0 is the positive-universal model completion of T 0 , and that the theory of Boolean algebras is positive-universally compatible with the theory of join semilattices with a greatest element.…”

Section: Surjective Connectionsmentioning

confidence: 99%

Connecting Many-Sorted Theories

Baader

Ghilardi

2005

Automated Deduction – CADE-20

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Abstract. Basically, the connection of two many-sorted theories is obtained by taking their disjoint union, and then connecting the two parts through connection functions that must behave like homomorphisms on the shared signature. We determine conditions under which decidability of the validity of universal formulae in the component theories transfers to their connection. In addition, we consider variants of the basic connection scheme. Our results can be seen as a generalization of the so-called E-connection approach for combining modal logics to an algebraic setting. §1. Introduction. The combination of decision procedures for logical theories arises in many areas of logic in computer science, such as constraint solving, automated deduction, term rewriting, modal logics, and description logics. In general, one has two first-order theories T 1 and T 2 over signatures Σ 1 and Σ 2 , for which validity of a certain type of formulae (e.g., universal, existential positive, etc.) is decidable. These theories are then combined into a new theory T over a combination Σ of the signatures Σ 1 and Σ 2 . The question is whether decidability transfers from T 1 , T 2 to their combination T .One way of combining the theories T 1 , T 2 is to build their union T 1 ∪ T 2 . Both the Nelson-Oppen combination procedure [23,22] and combination procedures for the word problem [26,28,24,7] address this type of combination, but for different types of formulae to be decided. Whereas the original combination procedures were restricted to the case of theories over disjoint signatures, there are now also solutions for the non-disjoint case [12,31,8,13, 16,4,5], but they always require some additional restrictions since it is easy to see that in the unrestricted case decidability does not transfer. Similar combination problems have also been investigated in modal logic, where one asks whether decidability of (relativized) validity transfers from two modal logics to their fusion [19,29,32,6]. The approaches in [16,4,5] actually generalize these results from equational theories induced by modal logics to more general first-order theories satisfying certain model-theoretic restrictions: the theories T 1 , T 2 must be compatible with their shared theory T 0 , and this shared theory must be locally finite (a condition ensuring that finitely generated models are finite). The theory T i is compatible with the shared theory T 0 iff (i) T 0 ⊆ T i ; (ii) T 0 has a model completion T * 0 ; and (iii) every model of T i embeds into a model of T i ∪ T * 0 .

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“…Recall (e.g., from [27,30]) that BA admits as a model completion the theory of atomless Boolean algebras. 19 A Boolean algebra B is said to be atomless iff it does not have atoms, where an atom is a nonzero element a ∈ B such that for all b ∈ B either a ≤ b or a ≤ b.…”

Section: Boolean Algebrasmentioning

confidence: 99%

“…From the closure of L under uniform substitution, we obtain for arbitrary formulae t, u that A L |= t BA ≈ u BA iff t ⇔ u ∈ L; for u = 1, we also get (by modus ponens 29 Readers familiar with this construction will notice that the closure conditions required by Definition 6.9 are precisely the closure conditions that make the construction work. 30 That ≡ is in fact an equivalence relation follows from modus ponens and tautologies.…”

Section: Equational Theories Induced By Modal Logicsmentioning

confidence: 99%

A New Combination Procedure for the Word Problem That Generalizes Fusion Decidability Results in Modal Logics

Baader

Ghilardi

Tinelli

2004

Automated Reasoning

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Previous results for combining decision procedures for the word problem in the nondisjoint case do not apply to equational theories induced by modal logics-which are not disjoint for sharing the theory of Boolean algebras. Conversely, decidability results for the fusion of modal logics are strongly tailored towards the special theories at hand, and thus do not generalize to other types of equational theories.In this paper, we present a new approach for combining decision procedures for the word problem in the non-disjoint case that applies to equational theories induced by modal logics, but is not restricted to them. The known fusion decidability results for modal logics are instances of our approach. However, even for equational theories induced by modal logics our results are more general since they are not restricted to so-called normal modal logics.

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Sheaves, Games, and Model Completions

Cited by 39 publications

References 0 publications

Free Heyting Algebras: Revisited

Free Heyting Algebras: Revisited

Connecting Many-Sorted Theories

A New Combination Procedure for the Word Problem That Generalizes Fusion Decidability Results in Modal Logics

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