List of speakers

Given a commutative distributive ℓ-monoid (M; ∨, ∧, +, 0) and an invertible element u ≥ 0 in M, we equip the set of elements of M between 0 and u with the MV-flavored operations ∨, ∧, ⊕, ⊙, 0, 1. For the algebras arising in this manner, we provide an axiomatization that is both equational and finite, and we name these algebras MV-monoidal algebras.

From a categorical perspective, we establish an adjunction that restricts to an equivalence between commutative distributive l-monoids with strong order-unit and MV-monoidal algebras. The equivalence can be further restricted to the celebrated equivalence between Abelian ℓ-groups with strong order-unit and MV-algebras.
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In this talk I will discuss a new method for proving Blok-Esakia theorems for varieties and universal classes of Heyting algebras and bi-Heyting algebras on the one hand and varieties and universal classes of Grz-algebras and tense Grz-algebras on the other via Esakia duality and stable canonical formulas and rules.

This is joint work with Antonio Maria Cleani.

When it comes to information, its potential incompleteness, uncertainty, and contradictoriness needs to be dealt with adequately. Separately, these characteristics have been taken into account by various appropriate logical formalisms and (classical) probability theory. While incompleteness and uncertainty are typically accommodated within one formalism, e.g. within various models of imprecise probability, contradictoriness and uncertainty less so — conflict or contradictoriness of information is rather chosen to be resolved than to be reasoned with. To reason with conflicting information, positive and negative support—evidence in favour and evidence against—a statement are quantified separately in the semantics. This two-dimensionality gives rise to logics interpreted over twisted-product algebras or bi-lattices, e.g. the well known Belnap-Dunn logic of First Degree Entailment.

In this talk, we introduce two-dimensional many-valued logics for uncertainty which are interpreted over twisted-product algebras based on the [0,1] real interval. They can be seen to account for the two-dimensionality of positive and negative component of (the degree of) belief based on potentially contradictory information. The logics include extensions of Łukasiewicz or Gödel logic with a de-Morgan negation which swaps between the positive and negative component. The logics inherit completeness and decidability properties of Łukasiewicz or Gödel logic respectively. Extensions of Gödel logic [2] turn out to include extensions of Nelson's paraconsistent logic N4, or Wansing's paraconsistent logic I₄C₄, with the prelinearity axiom.

Such logics can be applied to reason about belief based on evidence: In [1], a logical framework in which belief is based on potentially contradictory information obtained from multiple, possibly conflicting, sources and is of a probabilistic nature, has been suggested, using a two-layer modal logical framework to account for evidence and belief separately. The logics we introduce can be used on the upper level in this framework. The lower level uses Belnap-Dunn logic to model evidence, and its probabilistic extension to give rise to a belief modality.

(Based on joint work with S. Frittella and D. Kozhemiachenko)

[1] M. Bílková, S. Frittella, O. Majer and S. Nazari: Belief based on inconsistent information, DaLi 2020, LNCS, vol. 12569, Springer, 2020, pp. 68–86.
[2] M. Bílková, D. Kozhemiachenko and S. Frittella, Constraint tableaux for two-dimensional fuzzy logics, accepted at TABLEAUX 2021.
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There are a number of dualities for the category KHaus of compact Hausdorff spaces and continuous maps. The oldest of these is Gelfand duality (or Gelfand-Naimark-Stone duality) which establishes that KHaus is dually equivalent to the category of uniformly complete bounded archimedean ℓ-algebras. This duality is obtained by associating with each compact Hausdorff space X the lattice-ordered ℝ-algebra of real-valued continuous functions on X. Other dualities include Isbell duality and de Vries duality. Isbell duality is obtained by associating with each compact Hausdorff space X the compact regular frame of open subsets of X, and de Vries duality is obtained by associating with each X the de Vries algebra of regular open subsets of X. In this talk we describe how these three dualities are connected to each other.

This is based on joint work with G. Bezhanishvili and P. Morandi.
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In this talk we will present a relatively novel semantics for modal logic which interprets modal formulas as polyhedra in an Euclidean space. This is a variation of topological semantics, since the models are based on topological spaces and the modal operators are interpreted topologically, however the valuations are restricted to geometrically well-behaved (piecewise linear) subsets, polyhedra. The resulting logical systems - which we call polyhedral logics - turn out to be well-behaved as well, e. g. all of them possess the finite model property. In the talk we will present some results on identifying and axiomatizing polyhedral logics as well as applications of polyhedral semantics to the field of spatial model checking.

The research program of investigating polyhedral semantics for modal and intuitionistic logics is conducted jointly by research groups at Amsterdam (N. Bezhanishvili, S. Adam-Day), Milan (V. Marra), Pisa (V. Ciancia, G. Grilletti, D. Latella, M. Massink) and Tbilisi (D. Gabelaia, M. Jibladze, L. Uridia, E. Kuznetsov, K. Gogoladze, K. Razmadze).
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[This is joint work with Diego Calvanese, Alessandro Gianola, Marco Montali, Andrey Rivkin]

Uniform interpolants were largely studied in non-classical propositional logics since the nineties, and their connection to model completeness was pointed out in the literature. A successive parallel research line inside the automated reasoning community investigated uniform quantifier-free interpolants (sometimes referred to as “covers”) in first-order theories. In this paper, we investigate cover transfer to first-order theory combinations in the disjoint signatures case. We prove that, for convex theories, cover algorithms can be transferred to theory combinations under the same hypothesis needed to transfer quantifier-free interpolation (i.e., the equality interpolating property, aka strong amalgamation property). The key feature of our algorithm relies on the extensive usage of the Beth definability property for primitive fragments to convert implicitly defined variables into their explicitly defining terms. In the non-convex case, we show by a counterexample that covers may not exist in the combined theories, even in case combined quantifier-free interpolants do exist. However, we exhibit a cover transfer algorithm operating also in the non-convex case for special kinds of theory combinations; these combinations (called ‘tame combinations’) concern multi-sorted theories arising in many model-checking applications (in particular, the ones oriented to verification of data-aware processes).
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We make the observation that the canonical extension of a bounded distributive lattice D is the free completely distributive extension of D. From this it follows that the category of completely distributive lattices and complete homomorphisms is a (non-full) reflective subcategory of the category of bounded distributive lattices. We also discuss several additional mapping theorems and relate them to the literature.

Joint work with G. Bezhanishvili and M. Jibladze.
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The recent PhD thesis [1] presents a very detailed study of various aspects of compact ordered spaces and their duality theory. Among the many interesting results, it is shown there that the dual of the category of compact ordered spaces and homomorphisms is a variety, generalising this way the corresponding well-known result for compact Hausdorff spaces.

Our interest in these structures stems from our recent study of Stone-type dualities [2], where we extended the context from order structures to quantale-enriched (in particular metric) structures and therefore went from ordered compact Hausdorff spaces to quantale-enriched compact Hausdorff spaces. In particular, this step led naturally to the notion of quantale-enriched Priestley space. In this talk we investigate the category of quantale-enriched Priestley spaces and morphisms, with emphasis on those properties which identify the dual of this category as some kind of algebraic category. In particular, for certain quantales, we characterise the ℵ₁-copresentable objects in and show that this category is locally ℵ₁-copresentable.

This work is based on joint work with Pedro Nora.

[1] M. Abbadini, On the axiomatisability of the dual of compact ordered spaces, PhD thesis, Università degli Studi di Milano, 2021.
[2] D. Hofmann and P. Nora, Enriched Stone-type dualities, Advances in Mathematics, 330 (2018), pp. 307-360.
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One of the most convenient ways to describe the canonical extension of a boolean algebra is as the power set of the Stone space of the algebra. However, this requires the axiom of choice. Wes Holliday and Nick Bezhanishvili gave a choice-free description of the canonical extension of a boolean algebra. In this talk we give point-free (and thus also choice-free) descriptions of the canonical extension of a boolean algebra and of a bounded archimedean lattice-ordered algebra.
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A logic is said to admit an equational completeness theorem when it can be interpreted into the equational consequence of some class of algebras. Even if most completeness theorems in the literature take this form, logics lacking any equational completeness theorem are known since the work of Blok and Köhler's work on quasi-classical modal logics [1].

Despite the simplicity of this concept, intrinsic characterizations of logics with admitting an equational completeness theorem have proved elusive [5]. This is partly because equational completeness theorems can take unexpected forms, e.g., in view of Glivenko's Theorem, classical propositional logic is related to the variety of Heyting algebras by a (nonstandard) equational completeness theorem. Indeed, Blok and Rebagliato proved that nonstandard equational completeness theorems of this form are ubiquitous [3].

In this talk, we present a characterization of logics admitting an equational completeness theorem among the families of locally tabular logics and of logics with at least one tautology. As a consequence, we obtain that a nontrivial protoalgebraic logic lacks an equational completeness theorem iff syntactic equality and logical equivalence coincide. This settles the problem of understanding which logics admit an equational completeness theorem for most concrete logics and confirms the weakness of this notion, as opposed to algebraizability [2]. While the problem of determining whether a logic admits an algebraic semantics will be shown to be decidable for logics (resp. locally tabular logics) presented by a finite set of finite matrices (resp. by a finite Hilbert calculus), we shall see that it becomes undecidable for arbitrary logics presented by finite Hilbert calculi.

These observations have been collected in [4].

[1] W. J. Blok and P. Köhler. Algebraic semantics for quasi-classical modal logics. The Journal of Symbolic Logic, 48:941–964, 1983.
[2] W. J. Blok and D. Pigozzi. Algebraizable logics, volume 396 of Memoirs of the American Mathematical Society, Providence, January 1989.
[3] W. J. Blok and J. Rebagliato. Algebraic semantics for deductive systems.Studia Logica, Special Issue on Abstract Algebraic Logic, Part II, 74(5):153–180, 2003.
[4] T. Moraschini. On equational completeness theorems. To appear in the Journal of Symbolic Logic, 2021. Available online.
[5] J. G. Raftery. A perspective on the algebra of logic. Quaestiones Mathematicae, 34:275–325, 2011.
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Categorical semantics and duality theory play an important role in the study of propositional logics. In the first-order setting, a number of categorical tools are available, such as hyperdoctrines and ultracategories, but applications are few and far between. I shall discuss a different approach, recently introduced by Samson Abramsky, Anuj Dawar et al., whereby several notions of logical resources (e.g., quantifier-rank, number of variables, or modal-depth) are captured by coalgebras for appropriate comonads. Interestingly enough, this approach leads to novel connections between categorical semantics and (finite) model theory and combinatorics.
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Filtrations are known to be a powerful tool for proving the finite model property of modal logics. Moreover, filtrations provide a strong sufficient condition for the FMP/decidability of many derived modal logics: if a logic admits filtration, then its expansions with the inverse (temporal) and transitive closure modalities preserve the finite model property (Kikot, Zolin, Sh, 2014-2020). One of interesting corollaries of this is Kripke completeness of many non-canonical modal logics. An important family of logics that admit filtration is locally finite (locally tabular) modal logics. In the second part of my talk I will discuss how filtrations relate to local finiteness of modal logics and modal algebras.
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Regular projective algebras in varieties that are the equivalent algebraic semantics of a logic are interesting objects from several points of view: algebraic (as retracts of free algebras), categorical (they are preserved under categorical equivalence), and logical (they are connected to unification problems). Using dualities and categorical equivalences with known structures, such as lattice-ordered groups and rational polyhedra, we characterize the finitely generated projective algebras in some varieties that are the equivalent algebraic semantics of interesting many-valued logics. In particular: product algebras, the variety of perfect MV-algebras, and Wajsberg hoops. Moreover, we obtain some new results about projective algebras in the general setting of (bounded) commutative integral residuated lattices. This is partly joint work with Paolo Aglianò.