alessandra palmigiano | Vrije Universiteit Amsterdam (original) (raw)

Papers by alessandra palmigiano

Research paper thumbnail of Algorithmic correspondence and canonicity for non-distributive logics

arXiv (Cornell University), Mar 28, 2016

We extend the theory of unified correspondence to a very broad class of logics with algebraic sem... more We extend the theory of unified correspondence to a very broad class of logics with algebraic semantics given by varieties of normal lattice expansions (LEs), also known as 'lattices with operators'. Specifically, we introduce a very general syntactic definition of the class of Sahlqvist formulas and inequalities, which applies uniformly to each LE-signature and is given purely in terms of the order-theoretic properties of the algebraic interpretations of the logical connectives. Together with this, we introduce a variant of the algorithm ALBA, specific to the setting of LEs, which effectively computes first-order correspondents of LE-inequalities, and is guaranteed to succeed on a wide class of inequalities (the so-called inductive inequalities) which significantly extend the Sahlqvist class. Further, we show that every inequality on which ALBA succeeds is canonical. The projection of these results yields state-of-the-art correspondence theory for many well known substructural logics, such as the Lambek calculus and its extensions, the Lambek-Grishin calculus, the logic of (not necessarily distributive) de Morgan lattices, and the multiplicative-additive fragment of linear logic.

Research paper thumbnail of Jónsson-style canonicity for ALBA-inequalities

Journal of Logic and Computation, 2015

The theory of canonical extensions typically considers extensions of maps A ! B to maps A ! B. In... more The theory of canonical extensions typically considers extensions of maps A ! B to maps A ! B. In the present paper, the theory of canonical extensions of maps A ! B to maps A ! B is developed, and is applied to obtain a new canonicity proof for those inequalities in the language of Distributive Modal Logic (DML) on which the algorithm ALBA [9] is successful.

Research paper thumbnail of Algorithmic correspondence for intuitionistic modal mu-calculus, Part 2

EPiC series in computing, Jan 23, 2018

Research paper thumbnail of Linear Logic Properly Displayed

ACM Transactions on Computational Logic, Jan 28, 2023

We introduce proper display calculi for intuitionistic, bi-intuitionistic and classical linear lo... more We introduce proper display calculi for intuitionistic, bi-intuitionistic and classical linear logics with exponentials, which are sound, complete, conservative, and enjoy cut elimination and subformula property. Based on the same design, we introduce a variant of Lambek calculus with exponentials, aimed at capturing the controlled application of exchange and associativity. Properness (i.e. closure under uniform substitution of all parametric parts in rules) is the main technical novelty of the present proposal, allowing both for the smoothest proof of cut elimination, and for the development of an overarching and modular treatment for a vast class of axiomatic extensions and expansions of intuitionistic, bi-intuitionistic and classical linear logics with exponentials. Our proposal builds on an algebraic and order-theoretic analysis of linear logic, and applies the guidelines of the multi-type methodology in the design of display calculi.

Research paper thumbnail of Multi-type Calculi and Structural Control

Electronic proceedings in theoretical computer science, Aug 7, 2023

Research paper thumbnail of Verwijtbaar handelen of nalaten, een categorisering

Tijdschrift voor ontslagrecht, Nov 1, 2022

Research paper thumbnail of Modelling competing theories

arXiv (Cornell University), May 28, 2019

We introduce a complete many-valued semantics for two normal lattice-based modal logics. This sem... more We introduce a complete many-valued semantics for two normal lattice-based modal logics. This semantics is based on reflexive many-valued graphs. We discuss an interpretation and possible applications of this logical framework in the context of the formal analysis of the interaction between (competing) scientific theories.

Research paper thumbnail of Modelling informational entropy

arXiv (Cornell University), Mar 29, 2019

By 'informational entropy', we understand an inherent boundary to knowability, due e.g. to percep... more By 'informational entropy', we understand an inherent boundary to knowability, due e.g. to perceptual, theoretical, evidential or linguistic limits. In this paper, we discuss a logical framework in which this boundary is incorporated into the semantic and deductive machinery, and outline how this framework can be used to model various situations in which informational entropy arises.

Research paper thumbnail of Algebraic semantics and model completeness for Intuitionistic Public Announcement Logic

Annals of Pure and Applied Logic, Apr 1, 2014

In the present paper, we start studying epistemic updates using the standard toolkit of duality t... more In the present paper, we start studying epistemic updates using the standard toolkit of duality theory. We focus on public announcements, which are the simplest epistemic actions, and hence on Public Announcement Logic (PAL) without the common knowledge operator. As is well known, the epistemic action of publicly announcing a given proposition is semantically represented as a transformation of the model encoding the current epistemic setup of the given agents; the given current model being replaced with its submodel relativized to the announced proposition. We dually characterize the associated submodelinjection map as a certain pseudo-quotient map between the complex algebras respectively associated with the given model and with its relativized submodel. As is well known, these complex algebras are complete atomic BAOs (Boolean algebras with operators). The dual characterization we provide naturally generalizes to much wider classes of algebras, which include, but are not limited to, arbitrary BAOs and arbitrary modal expansions of Heyting algebras (HAOs). Thanks to this construction, the benefits and the wider scope of applications given by a point-free, intuitionistic theory of epistemic updates are made available. As an application of this dual characterization, we axiomatize the intuitionistic analogue of PAL, which we refer to as IPAL, prove soundness and completeness of IPAL w.r.t. both algebraic and relational models, and show that the well known Muddy Children Puzzle can be formalized in IPAL.

Research paper thumbnail of Probabilistic Epistemic Updates on Algebras

ACM Transactions on Computational Logic, Aug 21, 2019

General rights Copyright and moral rights for the publications made accessible in the public port... more General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal ? Take down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim.

Research paper thumbnail of Bilattice logic properly displayed

Fuzzy Sets and Systems, May 1, 2019

We introduce a proper multi-type display calculus for bilattice logic (with conflation) for which... more We introduce a proper multi-type display calculus for bilattice logic (with conflation) for which we prove soundness, completeness, conservativity, standard subformula property and cut-elimination. Our proposal builds on the product representation of bilattices and applies the guidelines of the multi-type methodology in the design of display calculi.

Research paper thumbnail of Algorithmic correspondence and canonicity for distributive modal logic

Annals of Pure and Applied Logic, Mar 1, 2012

We define the algorithm ALBA for the language of the same distributive modal logic (DML) for whic... more We define the algorithm ALBA for the language of the same distributive modal logic (DML) for which a Sahlqvist theorem was proved by Gehrke, Nagahashi, and Venema. Successful executions of ALBA compute the local first-order correspondents of input DML inequalities, and also guarantee their canonicity. The class of inequalities on which ALBA is successful is strictly larger than the newly introduced class of inductive inequalities, which in its turn properly extends the Sahlqvist inequalities of Gehrke et. al. As their name suggests, evidence is given to the effect that inductive inequalities are the distributive counterpart of the inductive formulas of Goranko and Vakarelov in the classical setting.

Research paper thumbnail of Non-distributive description logic

arXiv (Cornell University), Jul 18, 2023

We define LE-ALC, a generalization of the description logic ALC based on the propositional logic ... more We define LE-ALC, a generalization of the description logic ALC based on the propositional logic of general (i.e. not necessarily distributive) lattices, and semantically interpreted on relational structures based on formal contexts from Formal Concept Analysis (FCA). The description logic LE-ALC allows us to formally describe databases with objects, features, and formal concepts, represented according to FCA as Galois-stable sets of objects and features. We describe ABoxes and TBoxes in LE-ALC, provide a tableaux algorithm for checking the consistency of LE-ALC knowledge bases with acyclic TBoxes, and show its termination, soundness and completeness. Interestingly, consistency checking for LE-ALC with acyclic TBoxes is in PTIME, while the complexity of the consistency checking of classical ALC with acyclic TBoxes is PSPACE-complete.

Research paper thumbnail of Non-normal logics: semantic analysis and proof theory (extended version)

arXiv (Cornell University), Sep 29, 2020

We introduce proper display calculi for basic monotonic modal logic, the conditional logic CK and... more We introduce proper display calculi for basic monotonic modal logic, the conditional logic CK and a number of their axiomatic extensions. These calculi are sound, complete, conservative and enjoy cut elimination and subformula property. Our proposal applies the multi-type methodology in the design of proper display calculi, starting from a semantic analysis which motivates syntactic translations from single-type non-normal modal logics to multi-type normal poly-modal logics.

Research paper thumbnail of SGF-quantales and their groupoids

Quantales. The term quantale was introduced by Mulvey as the 'quantum' counterpart of the term lo... more Quantales. The term quantale was introduced by Mulvey as the 'quantum' counterpart of the term locale. Locales can be thought of as pointfree topological spaces, and as such, in the locally compact Hausdorff case, they are dual to commutative C *-algebras via the Gelfand duality. Mulvey considered quantales in the context of a research program aimed at providing dual counterparts to general C *-algebras, and extending Gelfand duality to noncommutative C *algebras. In Gelfand duality, the algebra-to-space direction consists of associating any commutative C *-algebra with its maximal ideal space. This construction was extended to noncommutative C *-algebras by considering the spectrum Max A of any unital C *-algebra A, i.e. the unital involutive quantale of closed linear subspaces of A. This gives rise to a functor Max which was extensively studied for more than a decade as it was considered the best candidate for the C *-algebra-toquantale direction of a noncommutative Gelfand-Naimark duality. Remarkably, M axA is a complete invariant of A, i.e. if A and A are C *-algebras such that Max A and Max A are isomorphic, then A and A are isomorphic. However, there are several problems with Max: 1) it has no adjoints, which is a necessary condition for its providing one direction of a duality; 2) it is not full on isomorphisms, i.e. some isomorphisms of spectra of C *-algebras do not arise from C *-algebra morphisms [5]; 3) there is no purely algebraic characterization of the class of quantales isomorphic to quantales of type Max A; 4) there is no canonical way of constructing A from Max A. These difficulties motivate the quest for alternative ways of linking C *-algebras and quantales. Besides their interest in relation to C *-algebras, quantales have been extensively studied in logic and theoretical computer science: not only do they provide the standard algebraic semantics for various resource-sensitive logics such as linear logic [2,12], they have also been applied to the study of the semantics of concurrent systems and their observable behaviour, described in terms of finite observations. Finite observations are formalized as semidecidable properties, and can therefore be identified with open sets of a topological space [11]; however, this perspective does not account for those (quantum-theoretic) situations where performing finite observations on a systems produces changes in the system itself. In those cases, the set of the finite observations that can be performed on a system has a natural noncommutative structure of quantale. The basic view on quantales as generalized topologies can be retrieved also in this context. In [1], this perspective on quantales was applied to provide a uniform algebraic framework for process semantics and develop a systematic study of various notions of observational equivalence between processes. Merging perspectives: the case study of Penrose tilings. Recently, investigation has focused on ways to integrate the two perspectives on quantales as noncommutative topologies and as algebras of experimental observations on computational

Research paper thumbnail of Sahlqvist via Translation

arXiv (Cornell University), Mar 27, 2016

In recent years, unified correspondence has been developed as a generalized Sahlqvist theory whic... more In recent years, unified correspondence has been developed as a generalized Sahlqvist theory which applies uniformly to all signatures of normal and regular (distributive) lattice expansions. A fundamental tool for attaining this level of generality and uniformity is a principled way, based on order theory, to define the Sahlqvist and inductive formulas and inequalities in every such signature. This definition covers in particular all (bi-)intuitionistic modal logics. The theory of these logics has been intensively studied over the past seventy years in connection with classical polyadic modal logics, using versions of Gödel-McKinsey-Tarski translations, suitably defined in each signature, as main tools. In view of this state-of-the-art, it is natural to ask (1) whether a general perspective on Gödel-McKinsey-Tarski translations can be attained, also based on order-theoretic principles like those underlying the general definition of Sahlqvist and inductive formulas and inequalities, which accounts for the known Gödel-McKinsey-Tarski translations and applies uniformly to all signatures of normal (distributive) lattice expansions; (2) whether this general perspective can be used to transfer correspondence and canonicity theorems for Sahlqvist and inductive formulas and inequalities in all signatures described above under Gödel-McKinsey-Tarski translations. In the present paper, we set out to answer these questions. We answer (1) in the affirmative; as to (2), we prove the transfer of the correspondence theorem for inductive inequalities of arbitrary signatures of normal distributive lattice expansions. We also prove the transfer of canonicity for inductive inequalities, but only restricted to arbitrary normal modal expansions of bi-intuitionistic logic. We also analyze the difficulties involved in obtaining the transfer of canonicity outside this setting, and indicate a route to extend the transfer of canonicity to all signatures of normal distributive lattice expansions.

Research paper thumbnail of Algorithmic correspondence and canonicity for non-distributive logics

arXiv (Cornell University), Mar 28, 2016

We extend the theory of unified correspondence to a very broad class of logics with algebraic sem... more We extend the theory of unified correspondence to a very broad class of logics with algebraic semantics given by varieties of normal lattice expansions (LEs), also known as 'lattices with operators'. Specifically, we introduce a very general syntactic definition of the class of Sahlqvist formulas and inequalities, which applies uniformly to each LE-signature and is given purely in terms of the order-theoretic properties of the algebraic interpretations of the logical connectives. Together with this, we introduce a variant of the algorithm ALBA, specific to the setting of LEs, which effectively computes first-order correspondents of LE-inequalities, and is guaranteed to succeed on a wide class of inequalities (the so-called inductive inequalities) which significantly extend the Sahlqvist class. Further, we show that every inequality on which ALBA succeeds is canonical. The projection of these results yields state-of-the-art correspondence theory for many well known substructural logics, such as the Lambek calculus and its extensions, the Lambek-Grishin calculus, the logic of (not necessarily distributive) de Morgan lattices, and the multiplicative-additive fragment of linear logic.

Research paper thumbnail of Rough concepts

HAL (Le Centre pour la Communication Scientifique Directe), Mar 10, 2020

Research paper thumbnail of Logics for Rough Concept Analysis

arXiv (Cornell University), Nov 17, 2018

Taking an algebraic perspective on the basic structures of Rough Concept Analysis as the starting... more Taking an algebraic perspective on the basic structures of Rough Concept Analysis as the starting point, in this paper we introduce some varieties of lattices expanded with normal modal operators which can be regarded as the natural rough algebra counterparts of certain subclasses of rough formal contexts, and introduce proper display calculi for the logics associated with these varieties which are sound, complete, conservative and with uniform cut elimination and subformula property. These calculi modularly extend the multi-type calculi for rough algebras to a 'nondistributive' (i.e. general lattice-based) setting.

Research paper thumbnail of Slanted canonicity of analytic inductive inequalities

arXiv (Cornell University), Mar 27, 2020

We prove an algebraic canonicity theorem for normal LE-logics of arbitrary signature, in a genera... more We prove an algebraic canonicity theorem for normal LE-logics of arbitrary signature, in a generalized setting in which the non-lattice connectives are interpreted as operations mapping tuples of elements of the given lattice to closed or open elements of its canonical extension. Interestingly, the syntactic shape of LE-inequalities which guarantees their canonicity in this generalized setting turns out to coincide with the syntactic shape of analytic inductive inequalities, which guarantees LE-inequalities to be equivalently captured by analytic structural rules of a proper display calculus. We show that this canonicity result connects and strengthens a number of recent canonicity results in two different areas: subordination algebras, and transfer results via Gödel-McKinsey-Tarski translations. CCS Concepts: • Theory of computation → Modal and temporal logics.

Research paper thumbnail of Algorithmic correspondence and canonicity for non-distributive logics

arXiv (Cornell University), Mar 28, 2016

We extend the theory of unified correspondence to a very broad class of logics with algebraic sem... more We extend the theory of unified correspondence to a very broad class of logics with algebraic semantics given by varieties of normal lattice expansions (LEs), also known as 'lattices with operators'. Specifically, we introduce a very general syntactic definition of the class of Sahlqvist formulas and inequalities, which applies uniformly to each LE-signature and is given purely in terms of the order-theoretic properties of the algebraic interpretations of the logical connectives. Together with this, we introduce a variant of the algorithm ALBA, specific to the setting of LEs, which effectively computes first-order correspondents of LE-inequalities, and is guaranteed to succeed on a wide class of inequalities (the so-called inductive inequalities) which significantly extend the Sahlqvist class. Further, we show that every inequality on which ALBA succeeds is canonical. The projection of these results yields state-of-the-art correspondence theory for many well known substructural logics, such as the Lambek calculus and its extensions, the Lambek-Grishin calculus, the logic of (not necessarily distributive) de Morgan lattices, and the multiplicative-additive fragment of linear logic.

Research paper thumbnail of Jónsson-style canonicity for ALBA-inequalities

Journal of Logic and Computation, 2015

The theory of canonical extensions typically considers extensions of maps A ! B to maps A ! B. In... more The theory of canonical extensions typically considers extensions of maps A ! B to maps A ! B. In the present paper, the theory of canonical extensions of maps A ! B to maps A ! B is developed, and is applied to obtain a new canonicity proof for those inequalities in the language of Distributive Modal Logic (DML) on which the algorithm ALBA [9] is successful.

Research paper thumbnail of Algorithmic correspondence for intuitionistic modal mu-calculus, Part 2

EPiC series in computing, Jan 23, 2018

Research paper thumbnail of Linear Logic Properly Displayed

ACM Transactions on Computational Logic, Jan 28, 2023

We introduce proper display calculi for intuitionistic, bi-intuitionistic and classical linear lo... more We introduce proper display calculi for intuitionistic, bi-intuitionistic and classical linear logics with exponentials, which are sound, complete, conservative, and enjoy cut elimination and subformula property. Based on the same design, we introduce a variant of Lambek calculus with exponentials, aimed at capturing the controlled application of exchange and associativity. Properness (i.e. closure under uniform substitution of all parametric parts in rules) is the main technical novelty of the present proposal, allowing both for the smoothest proof of cut elimination, and for the development of an overarching and modular treatment for a vast class of axiomatic extensions and expansions of intuitionistic, bi-intuitionistic and classical linear logics with exponentials. Our proposal builds on an algebraic and order-theoretic analysis of linear logic, and applies the guidelines of the multi-type methodology in the design of display calculi.

Research paper thumbnail of Multi-type Calculi and Structural Control

Electronic proceedings in theoretical computer science, Aug 7, 2023

Research paper thumbnail of Verwijtbaar handelen of nalaten, een categorisering

Tijdschrift voor ontslagrecht, Nov 1, 2022

Research paper thumbnail of Modelling competing theories

arXiv (Cornell University), May 28, 2019

We introduce a complete many-valued semantics for two normal lattice-based modal logics. This sem... more We introduce a complete many-valued semantics for two normal lattice-based modal logics. This semantics is based on reflexive many-valued graphs. We discuss an interpretation and possible applications of this logical framework in the context of the formal analysis of the interaction between (competing) scientific theories.

Research paper thumbnail of Modelling informational entropy

arXiv (Cornell University), Mar 29, 2019

By 'informational entropy', we understand an inherent boundary to knowability, due e.g. to percep... more By 'informational entropy', we understand an inherent boundary to knowability, due e.g. to perceptual, theoretical, evidential or linguistic limits. In this paper, we discuss a logical framework in which this boundary is incorporated into the semantic and deductive machinery, and outline how this framework can be used to model various situations in which informational entropy arises.

Research paper thumbnail of Algebraic semantics and model completeness for Intuitionistic Public Announcement Logic

Annals of Pure and Applied Logic, Apr 1, 2014

In the present paper, we start studying epistemic updates using the standard toolkit of duality t... more In the present paper, we start studying epistemic updates using the standard toolkit of duality theory. We focus on public announcements, which are the simplest epistemic actions, and hence on Public Announcement Logic (PAL) without the common knowledge operator. As is well known, the epistemic action of publicly announcing a given proposition is semantically represented as a transformation of the model encoding the current epistemic setup of the given agents; the given current model being replaced with its submodel relativized to the announced proposition. We dually characterize the associated submodelinjection map as a certain pseudo-quotient map between the complex algebras respectively associated with the given model and with its relativized submodel. As is well known, these complex algebras are complete atomic BAOs (Boolean algebras with operators). The dual characterization we provide naturally generalizes to much wider classes of algebras, which include, but are not limited to, arbitrary BAOs and arbitrary modal expansions of Heyting algebras (HAOs). Thanks to this construction, the benefits and the wider scope of applications given by a point-free, intuitionistic theory of epistemic updates are made available. As an application of this dual characterization, we axiomatize the intuitionistic analogue of PAL, which we refer to as IPAL, prove soundness and completeness of IPAL w.r.t. both algebraic and relational models, and show that the well known Muddy Children Puzzle can be formalized in IPAL.

Research paper thumbnail of Probabilistic Epistemic Updates on Algebras

ACM Transactions on Computational Logic, Aug 21, 2019

General rights Copyright and moral rights for the publications made accessible in the public port... more General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal ? Take down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim.

Research paper thumbnail of Bilattice logic properly displayed

Fuzzy Sets and Systems, May 1, 2019

We introduce a proper multi-type display calculus for bilattice logic (with conflation) for which... more We introduce a proper multi-type display calculus for bilattice logic (with conflation) for which we prove soundness, completeness, conservativity, standard subformula property and cut-elimination. Our proposal builds on the product representation of bilattices and applies the guidelines of the multi-type methodology in the design of display calculi.

Research paper thumbnail of Algorithmic correspondence and canonicity for distributive modal logic

Annals of Pure and Applied Logic, Mar 1, 2012

We define the algorithm ALBA for the language of the same distributive modal logic (DML) for whic... more We define the algorithm ALBA for the language of the same distributive modal logic (DML) for which a Sahlqvist theorem was proved by Gehrke, Nagahashi, and Venema. Successful executions of ALBA compute the local first-order correspondents of input DML inequalities, and also guarantee their canonicity. The class of inequalities on which ALBA is successful is strictly larger than the newly introduced class of inductive inequalities, which in its turn properly extends the Sahlqvist inequalities of Gehrke et. al. As their name suggests, evidence is given to the effect that inductive inequalities are the distributive counterpart of the inductive formulas of Goranko and Vakarelov in the classical setting.

Research paper thumbnail of Non-distributive description logic

arXiv (Cornell University), Jul 18, 2023

We define LE-ALC, a generalization of the description logic ALC based on the propositional logic ... more We define LE-ALC, a generalization of the description logic ALC based on the propositional logic of general (i.e. not necessarily distributive) lattices, and semantically interpreted on relational structures based on formal contexts from Formal Concept Analysis (FCA). The description logic LE-ALC allows us to formally describe databases with objects, features, and formal concepts, represented according to FCA as Galois-stable sets of objects and features. We describe ABoxes and TBoxes in LE-ALC, provide a tableaux algorithm for checking the consistency of LE-ALC knowledge bases with acyclic TBoxes, and show its termination, soundness and completeness. Interestingly, consistency checking for LE-ALC with acyclic TBoxes is in PTIME, while the complexity of the consistency checking of classical ALC with acyclic TBoxes is PSPACE-complete.

Research paper thumbnail of Non-normal logics: semantic analysis and proof theory (extended version)

arXiv (Cornell University), Sep 29, 2020

We introduce proper display calculi for basic monotonic modal logic, the conditional logic CK and... more We introduce proper display calculi for basic monotonic modal logic, the conditional logic CK and a number of their axiomatic extensions. These calculi are sound, complete, conservative and enjoy cut elimination and subformula property. Our proposal applies the multi-type methodology in the design of proper display calculi, starting from a semantic analysis which motivates syntactic translations from single-type non-normal modal logics to multi-type normal poly-modal logics.

Research paper thumbnail of SGF-quantales and their groupoids

Quantales. The term quantale was introduced by Mulvey as the 'quantum' counterpart of the term lo... more Quantales. The term quantale was introduced by Mulvey as the 'quantum' counterpart of the term locale. Locales can be thought of as pointfree topological spaces, and as such, in the locally compact Hausdorff case, they are dual to commutative C *-algebras via the Gelfand duality. Mulvey considered quantales in the context of a research program aimed at providing dual counterparts to general C *-algebras, and extending Gelfand duality to noncommutative C *algebras. In Gelfand duality, the algebra-to-space direction consists of associating any commutative C *-algebra with its maximal ideal space. This construction was extended to noncommutative C *-algebras by considering the spectrum Max A of any unital C *-algebra A, i.e. the unital involutive quantale of closed linear subspaces of A. This gives rise to a functor Max which was extensively studied for more than a decade as it was considered the best candidate for the C *-algebra-toquantale direction of a noncommutative Gelfand-Naimark duality. Remarkably, M axA is a complete invariant of A, i.e. if A and A are C *-algebras such that Max A and Max A are isomorphic, then A and A are isomorphic. However, there are several problems with Max: 1) it has no adjoints, which is a necessary condition for its providing one direction of a duality; 2) it is not full on isomorphisms, i.e. some isomorphisms of spectra of C *-algebras do not arise from C *-algebra morphisms [5]; 3) there is no purely algebraic characterization of the class of quantales isomorphic to quantales of type Max A; 4) there is no canonical way of constructing A from Max A. These difficulties motivate the quest for alternative ways of linking C *-algebras and quantales. Besides their interest in relation to C *-algebras, quantales have been extensively studied in logic and theoretical computer science: not only do they provide the standard algebraic semantics for various resource-sensitive logics such as linear logic [2,12], they have also been applied to the study of the semantics of concurrent systems and their observable behaviour, described in terms of finite observations. Finite observations are formalized as semidecidable properties, and can therefore be identified with open sets of a topological space [11]; however, this perspective does not account for those (quantum-theoretic) situations where performing finite observations on a systems produces changes in the system itself. In those cases, the set of the finite observations that can be performed on a system has a natural noncommutative structure of quantale. The basic view on quantales as generalized topologies can be retrieved also in this context. In [1], this perspective on quantales was applied to provide a uniform algebraic framework for process semantics and develop a systematic study of various notions of observational equivalence between processes. Merging perspectives: the case study of Penrose tilings. Recently, investigation has focused on ways to integrate the two perspectives on quantales as noncommutative topologies and as algebras of experimental observations on computational

Research paper thumbnail of Sahlqvist via Translation

arXiv (Cornell University), Mar 27, 2016

In recent years, unified correspondence has been developed as a generalized Sahlqvist theory whic... more In recent years, unified correspondence has been developed as a generalized Sahlqvist theory which applies uniformly to all signatures of normal and regular (distributive) lattice expansions. A fundamental tool for attaining this level of generality and uniformity is a principled way, based on order theory, to define the Sahlqvist and inductive formulas and inequalities in every such signature. This definition covers in particular all (bi-)intuitionistic modal logics. The theory of these logics has been intensively studied over the past seventy years in connection with classical polyadic modal logics, using versions of Gödel-McKinsey-Tarski translations, suitably defined in each signature, as main tools. In view of this state-of-the-art, it is natural to ask (1) whether a general perspective on Gödel-McKinsey-Tarski translations can be attained, also based on order-theoretic principles like those underlying the general definition of Sahlqvist and inductive formulas and inequalities, which accounts for the known Gödel-McKinsey-Tarski translations and applies uniformly to all signatures of normal (distributive) lattice expansions; (2) whether this general perspective can be used to transfer correspondence and canonicity theorems for Sahlqvist and inductive formulas and inequalities in all signatures described above under Gödel-McKinsey-Tarski translations. In the present paper, we set out to answer these questions. We answer (1) in the affirmative; as to (2), we prove the transfer of the correspondence theorem for inductive inequalities of arbitrary signatures of normal distributive lattice expansions. We also prove the transfer of canonicity for inductive inequalities, but only restricted to arbitrary normal modal expansions of bi-intuitionistic logic. We also analyze the difficulties involved in obtaining the transfer of canonicity outside this setting, and indicate a route to extend the transfer of canonicity to all signatures of normal distributive lattice expansions.

Research paper thumbnail of Algorithmic correspondence and canonicity for non-distributive logics

arXiv (Cornell University), Mar 28, 2016

We extend the theory of unified correspondence to a very broad class of logics with algebraic sem... more We extend the theory of unified correspondence to a very broad class of logics with algebraic semantics given by varieties of normal lattice expansions (LEs), also known as 'lattices with operators'. Specifically, we introduce a very general syntactic definition of the class of Sahlqvist formulas and inequalities, which applies uniformly to each LE-signature and is given purely in terms of the order-theoretic properties of the algebraic interpretations of the logical connectives. Together with this, we introduce a variant of the algorithm ALBA, specific to the setting of LEs, which effectively computes first-order correspondents of LE-inequalities, and is guaranteed to succeed on a wide class of inequalities (the so-called inductive inequalities) which significantly extend the Sahlqvist class. Further, we show that every inequality on which ALBA succeeds is canonical. The projection of these results yields state-of-the-art correspondence theory for many well known substructural logics, such as the Lambek calculus and its extensions, the Lambek-Grishin calculus, the logic of (not necessarily distributive) de Morgan lattices, and the multiplicative-additive fragment of linear logic.

Research paper thumbnail of Rough concepts

HAL (Le Centre pour la Communication Scientifique Directe), Mar 10, 2020

Research paper thumbnail of Logics for Rough Concept Analysis

arXiv (Cornell University), Nov 17, 2018

Taking an algebraic perspective on the basic structures of Rough Concept Analysis as the starting... more Taking an algebraic perspective on the basic structures of Rough Concept Analysis as the starting point, in this paper we introduce some varieties of lattices expanded with normal modal operators which can be regarded as the natural rough algebra counterparts of certain subclasses of rough formal contexts, and introduce proper display calculi for the logics associated with these varieties which are sound, complete, conservative and with uniform cut elimination and subformula property. These calculi modularly extend the multi-type calculi for rough algebras to a 'nondistributive' (i.e. general lattice-based) setting.

Research paper thumbnail of Slanted canonicity of analytic inductive inequalities

arXiv (Cornell University), Mar 27, 2020

We prove an algebraic canonicity theorem for normal LE-logics of arbitrary signature, in a genera... more We prove an algebraic canonicity theorem for normal LE-logics of arbitrary signature, in a generalized setting in which the non-lattice connectives are interpreted as operations mapping tuples of elements of the given lattice to closed or open elements of its canonical extension. Interestingly, the syntactic shape of LE-inequalities which guarantees their canonicity in this generalized setting turns out to coincide with the syntactic shape of analytic inductive inequalities, which guarantees LE-inequalities to be equivalently captured by analytic structural rules of a proper display calculus. We show that this canonicity result connects and strengthens a number of recent canonicity results in two different areas: subordination algebras, and transfer results via Gödel-McKinsey-Tarski translations. CCS Concepts: • Theory of computation → Modal and temporal logics.