The modal intuitionistic epistemic logic \(\mathbf{IEL}^{-}\) was proposed by Artemov and Protopo... more The modal intuitionistic epistemic logic \(\mathbf{IEL}^{-}\) was proposed by Artemov and Protopopescu as the intuitionistic version of belief logic. We construct the modal lambda calculus which is Curry-Howard isomorphic to \(\mathbf{IEL}^{-}\) as the type-theoretical representation of applicative computation widely known in functional programming.
In this paper, we show that the class of representable residuated semigroups has the finite repre... more In this paper, we show that the class of representable residuated semigroups has the finite representation property. That is, every finite representable residuated semigroup is representable over a finite base. This result gives a positive solution to Hirsch and Hodkinson (2002, Relation Algebras by Games). The finite representation property for residuated semigroups also implies that the Lambek calculus has the finite model property with respect to relational models, the so-called $R$-models. We also show that the class of representable join semilattice-ordered semigroups is pseudo-universal and it has a recursively enumerable axiomatization. For this purpose, we introduce representability games for join semilattice-ordered semigroups.
The system of intuitionistic modal logic IEL´was proposed by S. Artemov and T. Protopopescu as th... more The system of intuitionistic modal logic IEL´was proposed by S. Artemov and T. Protopopescu as the intuitionistic version of belief logic [3]. We construct the modal lambda calculus which is Curry-Howard isomorphic to IEL´as the type-theoretical representation of applicative computation widely known in functional programming. We also provide a categorical interpretation of this modal lambda calculus considering coalgebras associated with a monoidal functor on a cartesian closed category. Finally, we study Heyting algebras and locales with corresponding operators. Such operators are used in point-free topology as well. We study compelete Kripke-Joyal-style semantics for predicate extensions of IEL´and related logics using Dedekind-MacNeille completions and modal cover systems introduced by Goldblatt [26]. The paper extends the conference paper published in the LFCS'20 volume [55].
In this paper, we show that the class of representable residuated semigroups has the finite repre... more In this paper, we show that the class of representable residuated semigroups has the finite representation property. That is, every finite representable residuated semigroup is isomorphic to some algebra over a finite base. This result gives a positive solution to Problem 19.17 from the monograph by Hirsch and Hodkinson \cite{hirsch2002relation}.
In this paper, we show that the class of representable residuated semigroups has the finite repre... more In this paper, we show that the class of representable residuated semigroups has the finite representation property. That is, every finite representable residuated semigroup is isomorphic to some algebra over a finite base. This result gives a positive solution to Problem 19.17 from the monograph by Hirsch and Hodkinson [13].
In this paper, we consider the polymodal version of Lambek calculus with subexponential modalitie... more In this paper, we consider the polymodal version of Lambek calculus with subexponential modalities initially introduced by Kanovich, Kuznetsov, Nigam, and Scedrov [10] and its quantale semantics. In our approach, subexponential modalities have an interpretation in terms of quantic conuclei. We show that this extension of Lambek calculus is complete w.r.t quantales with quantic conuclei. Also, we prove a representation theorem for quantales with quantic conuclei and show that Lambek calculus with subexponentials is relationally complete. Finally, we extend this representation theorem to the category of quantales with quantic conuclei. Some of these results were presented here [20]. Here we assign the special syntactic categories to the words of this sentence. np denotes "noun phrase", n-noun, ad-adjective, p-phrase, s-sentence. This sequent denotes that this Oscar Wilde's quote is a well-formed sentence. The verb "changed" has type "np under (s over p)". In other words, one needs to apply some noun phrase ("The Thames nocturne of blue and gold") from the left, apply some phrase from the right ("Changed to Harmony in grey") and obtain sentence after that. The other syntactic categories might be considered similarly. The general case of such derivations in categorial grammars is axiomatised via Lambek calculus [13], non-commutative linear logic:
In this paper, we show that the class of representable residuated semigroups has the finite repre... more In this paper, we show that the class of representable residuated semigroups has the finite representation property. That is, every finite representable residuated semigroup is representable over a finite base. This result gives a positive solution to Problem 19.17 from the monograph by Hirsch and Hodkinson [13]. We also show that the class of representable join semilattice-ordered semigroups is pseudo-universal and it has a recursively enumerable axiomatisation. For this purpose, we introduce representability games for join semilattice-ordered semigroups.
The modal intuitionistic epistemic logic IEL´was proposed by S. Artemov and T. Protopopescu as th... more The modal intuitionistic epistemic logic IEL´was proposed by S. Artemov and T. Protopopescu as the intuitionistic version of belief logic [3]. We construct the modal lambda calculus which is Curry-Howard isomorphic to IEL´as the type-theoretical representation of applicative computation widely known in functional programming. We also provide a categorical interpretation of this modal lambda calculus considering coalgebras associated with a monoidal functor on a cartesian closed category. Finally, we study Heyting algebras and locale with corresponding operators. Such operators are used in point-free topology as well. We study compelete semanticsà la Kripke-Joyal for predicate extensions of IELá nd IEL using Dedekind-MacNeille completions and cover systems introduced by Goldblatt [31]. The paper extends the conference paper published in the LFCS'20 volume [59].
We describe a family of decidable propositional dynamic logics, where atomic modalities satisfy s... more We describe a family of decidable propositional dynamic logics, where atomic modalities satisfy some extra conditions (for example, given by axioms of the logics K5, S5, or K45 for different atomic modalities). It follows from recent results [KSZ14], [KSZ20] that if a modal logic L admits a special type of filtration (so-called definable filtration), then its enrichments with modalities for the transitive closure and converse relations also admit definable filtration. We use these results to show that if logics L1,. .. , Ln admit definable filtration, then the propositional dynamic logic with converse extended by the fusion L1 *. .. * Ln has the finite model property.
In this paper, we show that the class of representable residuated semigroups has the finite repre... more In this paper, we show that the class of representable residuated semigroups has the finite representation property. That is, every finite representable residuated semigroup is representable over a finite base. This result gives a positive solution to Hirsch and Hodkinson (2002, Relation Algebras by Games). The finite representation property for residuated semigroups also implies that the Lambek calculus has the finite model property with respect to relational models, the so-called $R$-models. We also show that the class of representable join semilattice-ordered semigroups is pseudo-universal and it has a recursively enumerable axiomatization. For this purpose, we introduce representability games for join semilattice-ordered semigroups.
In this paper, we consider the polymodal version of Lambek calculus with subexponential modalitie... more In this paper, we consider the polymodal version of Lambek calculus with subexponential modalities initially introduced by Kanovich, Kuznetsov, Nigam, and Scedrov [10] and its quantale semantics. In our approach, subexponential modalities have an interpretation in terms of quantic conuclei. We show that this extension of Lambek calculus is complete w.r.t quantales with quantic conuclei. Also, we prove a representation theorem for quantales with quantic conuclei and show that Lambek calculus with subexponentials is relationally complete. Finally, we extend this representation theorem to the category of quantales with quantic conuclei. Some of these results were presented here [20]. Here we assign the special syntactic categories to the words of this sentence. np denotes "noun phrase", n-noun, ad-adjective, p-phrase, s-sentence. This sequent denotes that this Oscar Wilde's quote is a well-formed sentence. The verb "changed" has type "np under (s over p)". In other words, one needs to apply some noun phrase ("The Thames nocturne of blue and gold") from the left, apply some phrase from the right ("Changed to Harmony in grey") and obtain sentence after that. The other syntactic categories might be considered similarly. The general case of such derivations in categorial grammars is axiomatised via Lambek calculus [13], non-commutative linear logic:
In this paper, we study logics of bounded distributive residuated lattices with modal operators c... more In this paper, we study logics of bounded distributive residuated lattices with modal operators considering ✷ and ✸ in a noncommutative setting. We introduce relational semantics for such substructural modal logics. We prove that any canonical logic is Kripke complete via discrete duality and canonical extensions. That is, we show that a modal extension of the distributive full Lambek calculus is the logic of its frames if its variety is closed under canonical extensions. After that, we establish a Priestleystyle duality between residuated distributive modal algebras and topological Kripke structures based on Priestley spaces.
In this paper, we show that the class of representable residuated semigroups has the finite repre... more In this paper, we show that the class of representable residuated semigroups has the finite representation property. That is, every finite representable residuated semigroup is isomorphic to some algebra over a finite base. This result gives a positive solution to Problem 19.17 from the monograph by Hirsch and Hodkinson [11]. We also show that the class of representable join semilattice-ordered semigroups has a recursively enumerable axiomatisation using back-andforth games.
The modal intuitionistic epistemic logic \(\mathbf{IEL}^{-}\) was proposed by Artemov and Protopo... more The modal intuitionistic epistemic logic \(\mathbf{IEL}^{-}\) was proposed by Artemov and Protopopescu as the intuitionistic version of belief logic. We construct the modal lambda calculus which is Curry-Howard isomorphic to \(\mathbf{IEL}^{-}\) as the type-theoretical representation of applicative computation widely known in functional programming.
In this paper, we show that the class of representable residuated semigroups, residuated semigrou... more In this paper, we show that the class of representable residuated semigroups, residuated semigroups on binary relations, has the finite representation property. That is, every finite representable residuated semigroup is isomorphic to some algebra over a finite base. This result gives a positive solution to Problem 19.17 from the monograph by Hirsch and Hodkinson \cite{hirsch2002relation}.
In this paper, we show that the class of representable residuated semigroups has the finite repre... more In this paper, we show that the class of representable residuated semigroups has the finite representation property. That is, every finite representable residuated semigroup is isomorphic to some algebra over a finite base. This result gives a positive solution to Problem 19.17 from the monograph by Hirsch and Hodkinson \cite{hirsch2002relation}.
The system of intuitionistic modal logic $\textbf{IEL}^{-}$ was proposed by S. Artemov and T. Pro... more The system of intuitionistic modal logic $\textbf{IEL}^{-}$ was proposed by S. Artemov and T. Protopopescu as the intuitionistic version of belief logic (S. Artemov and T. Protopopescu. Intuitionistic epistemic logic. The Review of Symbolic Logic, 9, 266–298, 2016). We construct the modal lambda calculus, which is Curry–Howard isomorphic to $\textbf{IEL}^{-}$ as the type-theoretical representation of applicative computation widely known in functional programming.We also provide a categorical interpretation of this modal lambda calculus considering coalgebras associated with a monoidal functor on a Cartesian closed category. Finally, we study Heyting algebras and locales with corresponding operators. Such operators are used in point-free topology as well. We study complete Kripke–Joyal-style semantics for predicate extensions of $\textbf{IEL}^{-}$ and related logics using Dedekind–MacNeille completions and modal cover systems introduced by Goldblatt (R. Goldblatt. Cover semantics for...
The modal intuitionistic epistemic logic \(\mathbf{IEL}^{-}\) was proposed by Artemov and Protopo... more The modal intuitionistic epistemic logic \(\mathbf{IEL}^{-}\) was proposed by Artemov and Protopopescu as the intuitionistic version of belief logic. We construct the modal lambda calculus which is Curry-Howard isomorphic to \(\mathbf{IEL}^{-}\) as the type-theoretical representation of applicative computation widely known in functional programming.
In this paper, we show that the class of representable residuated semigroups has the finite repre... more In this paper, we show that the class of representable residuated semigroups has the finite representation property. That is, every finite representable residuated semigroup is representable over a finite base. This result gives a positive solution to Hirsch and Hodkinson (2002, Relation Algebras by Games). The finite representation property for residuated semigroups also implies that the Lambek calculus has the finite model property with respect to relational models, the so-called $R$-models. We also show that the class of representable join semilattice-ordered semigroups is pseudo-universal and it has a recursively enumerable axiomatization. For this purpose, we introduce representability games for join semilattice-ordered semigroups.
The system of intuitionistic modal logic IEL´was proposed by S. Artemov and T. Protopopescu as th... more The system of intuitionistic modal logic IEL´was proposed by S. Artemov and T. Protopopescu as the intuitionistic version of belief logic [3]. We construct the modal lambda calculus which is Curry-Howard isomorphic to IEL´as the type-theoretical representation of applicative computation widely known in functional programming. We also provide a categorical interpretation of this modal lambda calculus considering coalgebras associated with a monoidal functor on a cartesian closed category. Finally, we study Heyting algebras and locales with corresponding operators. Such operators are used in point-free topology as well. We study compelete Kripke-Joyal-style semantics for predicate extensions of IEL´and related logics using Dedekind-MacNeille completions and modal cover systems introduced by Goldblatt [26]. The paper extends the conference paper published in the LFCS'20 volume [55].
In this paper, we show that the class of representable residuated semigroups has the finite repre... more In this paper, we show that the class of representable residuated semigroups has the finite representation property. That is, every finite representable residuated semigroup is isomorphic to some algebra over a finite base. This result gives a positive solution to Problem 19.17 from the monograph by Hirsch and Hodkinson \cite{hirsch2002relation}.
In this paper, we show that the class of representable residuated semigroups has the finite repre... more In this paper, we show that the class of representable residuated semigroups has the finite representation property. That is, every finite representable residuated semigroup is isomorphic to some algebra over a finite base. This result gives a positive solution to Problem 19.17 from the monograph by Hirsch and Hodkinson [13].
In this paper, we consider the polymodal version of Lambek calculus with subexponential modalitie... more In this paper, we consider the polymodal version of Lambek calculus with subexponential modalities initially introduced by Kanovich, Kuznetsov, Nigam, and Scedrov [10] and its quantale semantics. In our approach, subexponential modalities have an interpretation in terms of quantic conuclei. We show that this extension of Lambek calculus is complete w.r.t quantales with quantic conuclei. Also, we prove a representation theorem for quantales with quantic conuclei and show that Lambek calculus with subexponentials is relationally complete. Finally, we extend this representation theorem to the category of quantales with quantic conuclei. Some of these results were presented here [20]. Here we assign the special syntactic categories to the words of this sentence. np denotes "noun phrase", n-noun, ad-adjective, p-phrase, s-sentence. This sequent denotes that this Oscar Wilde's quote is a well-formed sentence. The verb "changed" has type "np under (s over p)". In other words, one needs to apply some noun phrase ("The Thames nocturne of blue and gold") from the left, apply some phrase from the right ("Changed to Harmony in grey") and obtain sentence after that. The other syntactic categories might be considered similarly. The general case of such derivations in categorial grammars is axiomatised via Lambek calculus [13], non-commutative linear logic:
In this paper, we show that the class of representable residuated semigroups has the finite repre... more In this paper, we show that the class of representable residuated semigroups has the finite representation property. That is, every finite representable residuated semigroup is representable over a finite base. This result gives a positive solution to Problem 19.17 from the monograph by Hirsch and Hodkinson [13]. We also show that the class of representable join semilattice-ordered semigroups is pseudo-universal and it has a recursively enumerable axiomatisation. For this purpose, we introduce representability games for join semilattice-ordered semigroups.
The modal intuitionistic epistemic logic IEL´was proposed by S. Artemov and T. Protopopescu as th... more The modal intuitionistic epistemic logic IEL´was proposed by S. Artemov and T. Protopopescu as the intuitionistic version of belief logic [3]. We construct the modal lambda calculus which is Curry-Howard isomorphic to IEL´as the type-theoretical representation of applicative computation widely known in functional programming. We also provide a categorical interpretation of this modal lambda calculus considering coalgebras associated with a monoidal functor on a cartesian closed category. Finally, we study Heyting algebras and locale with corresponding operators. Such operators are used in point-free topology as well. We study compelete semanticsà la Kripke-Joyal for predicate extensions of IELá nd IEL using Dedekind-MacNeille completions and cover systems introduced by Goldblatt [31]. The paper extends the conference paper published in the LFCS'20 volume [59].
We describe a family of decidable propositional dynamic logics, where atomic modalities satisfy s... more We describe a family of decidable propositional dynamic logics, where atomic modalities satisfy some extra conditions (for example, given by axioms of the logics K5, S5, or K45 for different atomic modalities). It follows from recent results [KSZ14], [KSZ20] that if a modal logic L admits a special type of filtration (so-called definable filtration), then its enrichments with modalities for the transitive closure and converse relations also admit definable filtration. We use these results to show that if logics L1,. .. , Ln admit definable filtration, then the propositional dynamic logic with converse extended by the fusion L1 *. .. * Ln has the finite model property.
In this paper, we show that the class of representable residuated semigroups has the finite repre... more In this paper, we show that the class of representable residuated semigroups has the finite representation property. That is, every finite representable residuated semigroup is representable over a finite base. This result gives a positive solution to Hirsch and Hodkinson (2002, Relation Algebras by Games). The finite representation property for residuated semigroups also implies that the Lambek calculus has the finite model property with respect to relational models, the so-called $R$-models. We also show that the class of representable join semilattice-ordered semigroups is pseudo-universal and it has a recursively enumerable axiomatization. For this purpose, we introduce representability games for join semilattice-ordered semigroups.
In this paper, we consider the polymodal version of Lambek calculus with subexponential modalitie... more In this paper, we consider the polymodal version of Lambek calculus with subexponential modalities initially introduced by Kanovich, Kuznetsov, Nigam, and Scedrov [10] and its quantale semantics. In our approach, subexponential modalities have an interpretation in terms of quantic conuclei. We show that this extension of Lambek calculus is complete w.r.t quantales with quantic conuclei. Also, we prove a representation theorem for quantales with quantic conuclei and show that Lambek calculus with subexponentials is relationally complete. Finally, we extend this representation theorem to the category of quantales with quantic conuclei. Some of these results were presented here [20]. Here we assign the special syntactic categories to the words of this sentence. np denotes "noun phrase", n-noun, ad-adjective, p-phrase, s-sentence. This sequent denotes that this Oscar Wilde's quote is a well-formed sentence. The verb "changed" has type "np under (s over p)". In other words, one needs to apply some noun phrase ("The Thames nocturne of blue and gold") from the left, apply some phrase from the right ("Changed to Harmony in grey") and obtain sentence after that. The other syntactic categories might be considered similarly. The general case of such derivations in categorial grammars is axiomatised via Lambek calculus [13], non-commutative linear logic:
In this paper, we study logics of bounded distributive residuated lattices with modal operators c... more In this paper, we study logics of bounded distributive residuated lattices with modal operators considering ✷ and ✸ in a noncommutative setting. We introduce relational semantics for such substructural modal logics. We prove that any canonical logic is Kripke complete via discrete duality and canonical extensions. That is, we show that a modal extension of the distributive full Lambek calculus is the logic of its frames if its variety is closed under canonical extensions. After that, we establish a Priestleystyle duality between residuated distributive modal algebras and topological Kripke structures based on Priestley spaces.
In this paper, we show that the class of representable residuated semigroups has the finite repre... more In this paper, we show that the class of representable residuated semigroups has the finite representation property. That is, every finite representable residuated semigroup is isomorphic to some algebra over a finite base. This result gives a positive solution to Problem 19.17 from the monograph by Hirsch and Hodkinson [11]. We also show that the class of representable join semilattice-ordered semigroups has a recursively enumerable axiomatisation using back-andforth games.
The modal intuitionistic epistemic logic \(\mathbf{IEL}^{-}\) was proposed by Artemov and Protopo... more The modal intuitionistic epistemic logic \(\mathbf{IEL}^{-}\) was proposed by Artemov and Protopopescu as the intuitionistic version of belief logic. We construct the modal lambda calculus which is Curry-Howard isomorphic to \(\mathbf{IEL}^{-}\) as the type-theoretical representation of applicative computation widely known in functional programming.
In this paper, we show that the class of representable residuated semigroups, residuated semigrou... more In this paper, we show that the class of representable residuated semigroups, residuated semigroups on binary relations, has the finite representation property. That is, every finite representable residuated semigroup is isomorphic to some algebra over a finite base. This result gives a positive solution to Problem 19.17 from the monograph by Hirsch and Hodkinson \cite{hirsch2002relation}.
In this paper, we show that the class of representable residuated semigroups has the finite repre... more In this paper, we show that the class of representable residuated semigroups has the finite representation property. That is, every finite representable residuated semigroup is isomorphic to some algebra over a finite base. This result gives a positive solution to Problem 19.17 from the monograph by Hirsch and Hodkinson \cite{hirsch2002relation}.
The system of intuitionistic modal logic $\textbf{IEL}^{-}$ was proposed by S. Artemov and T. Pro... more The system of intuitionistic modal logic $\textbf{IEL}^{-}$ was proposed by S. Artemov and T. Protopopescu as the intuitionistic version of belief logic (S. Artemov and T. Protopopescu. Intuitionistic epistemic logic. The Review of Symbolic Logic, 9, 266–298, 2016). We construct the modal lambda calculus, which is Curry–Howard isomorphic to $\textbf{IEL}^{-}$ as the type-theoretical representation of applicative computation widely known in functional programming.We also provide a categorical interpretation of this modal lambda calculus considering coalgebras associated with a monoidal functor on a Cartesian closed category. Finally, we study Heyting algebras and locales with corresponding operators. Such operators are used in point-free topology as well. We study complete Kripke–Joyal-style semantics for predicate extensions of $\textbf{IEL}^{-}$ and related logics using Dedekind–MacNeille completions and modal cover systems introduced by Goldblatt (R. Goldblatt. Cover semantics for...
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