Adams · A Dual Approach to Many-Valued Semantics
FP7 — People (Marie Curie Actions)
- Duration
- 2013-08-01 → 2015-07-31
- EU contribution
- €183,806
- Participants
- 1
- Scheme
- MC-IEF
Lines connect the coordinator with its partners.
Results in brief
A Dual Approach to Many-Valued Semantics
Research activities One of the main aims of research has been to generalise the known dualities in many-valued logic. This has been accomplished in a particularly satisfactory way, as the fellow (together with O. Caramello and V. Marra) provided a general framework for dualities that encompasses not only some of the most important duality in many-valued logics, like the one for finitely presented MV-algebras [1,2] and the one for Goedel algebra, but it also includes several important dualities in Mathematics such as Gelfand duality, Baker-Beynon duality, Pontryagin duality, and more. The outcomes of this research have also highlighted unexpected connections between the theory of duality, Birkhoff’s Subdirect Representation Theorem, and the theory of Transforms. The general framework and some of its applications are presented in [3,4]. Sometimes dualities are best understood for finitely presented/generated algebras e.g., for the class of MV-algebras. A possible way of passing from the finitely presented algebras to the full variety is to use ind-completions. With the aim of gaining a better understanding of this abstract construction, the fellow together with V. Marra have discovered a criterion for isomorphism in the ind-completion, when the diagrams have linearly ordered. This result is presented in [4]. Another fellow’s achievement during the fellowship is the discovery of “canonical formulas” for substructural logics. Indeed, in [5] the applicant (together with N. Bezhanishvili and N. Galatos) presented a uniform axiomatisation of all subvarieties of k-potent residuated lattices using canonical formulas. Canonical formulas are a pivotal tool for the study of superintuitionistic logics and were developed using the relational semantics of those logics. In [5], it is shown how such a tool can be exported to the wider realm of substructural logics, using locally finite subreducts and the Finite Embeddability Property (FEP). Finally, the fellow (together with J. Gil-Férez, L. Spada, C. Tsinakis, and H. Zhou) studied the relation between the FEP and join-completions for residuated lattices. This theme of research connects the two aforementioned topics of dualities and canonical formulas. Indeed, on the one hand the FEP is a crucial property for the machinery of canonical formulas to work. On the other hand, to provide relational semantics for substructural and many-valued logics one needs, a way to build a residuated lattice starting from a residuated frame and join-completions and nuclei operators provide a universal method for doing so. The investigations on the relation between join completions and FEP, together with applications to the Word Problem are contained in [6,7]. [1] Vincenzo Marra and Luca Spada.Duality, projectivity, and unification inŁukasiewiczlogic and MV-algebras.Annals of Pure and Applied Logic 164:192-210. 2013.(Preprint page) [2] Vincenzo Marra and Luca Spada.The dual adjunction between MV-algebras and Tychonoff spaces,Studia Logica100(1-2):253-278, 2012. Special issue of Studia Logica in memoriam Leo Esakia (L. Beklemishev, G. Bezhanishvili, D. Mundici and Y. Venema Editors). [3] Olivia Caramello, Vincenzo Marra and Luca Spada. General affine adjunctions, Nullstellensätze, and dualities. Preprint available on ArXiv.org 2014. [4] Vincenzo Marra and Luca Spada. A concrete categorical dual for finitely generated MV-algebras. In preparation. [5] Nick Bezhanishvili, Nick Galatos, and Luca Spada. Canonical formulas for k-potent commutative, integral, residuated lattices. Preprint 2015. [6] José Gil-Férez, Luca Spada, Constantine Tsinakis, and Hongjun Zhou. Join completions of residuated lattices. Preprint 2015. [7] José Gil-Férez, Luca Spada, Constantine Tsinakis, and Hongjun Zhou. The Finite Embeddability Property for Ordered Structures. Preprint 2015. Other activities: The fellow started his activities at the ILLC by integrating into the Institute. He initially delivered three seminars a different levels: he delivered a introductory seminar on Many-valued and substructural logics, designed for a wide audience; a second general talk was given at the Amsterdam Logic Colloquium, about a possible abstract framework for dualities; finally, he delivered a more technical seminar, describing a criterion for isomorphism for limits of finitely presented/generated objects. During the two year scholarship, the fellow has organised a permanent biweekly seminar at the ILLC, called Algebra|Coalgebra on topics surrounding dualities. The fellow has taught two courses: “Algebra and Coalgebra” and “Many-valued Logics” and on project-course: “Duality Theory”. Finally the fellow has participated to several conferences both as invited speaker and with contributed talks, disseminating his research activity.
Data: CORDIS, © European Union
Project objective
Mathematical logic plays a crucial role in the 21st century society, indeed, in an era where information has become a key aspect of our lives, it provides a powerful mathematical framework for the study of the structure and the dynamics of information. A particularly interesting aspect of mathematical logic is that it allows to take in account uncertainty, vagueness or subjectivity, yet in the rigorous setting proper to mathematics.Drawn from the applications, disciplines like many-valued logic, modal logic, reasoning under uncertainty, etc. have quickly developed a weaponry of sophisticated investigation tools that enabled the disclosure of a number of substantial facts about their expressiveness, functionality and scope. Nonetheless the investigations in these topics are far from being concluded, rather they are ready to be shifted to another level which could take into account inter-relations between those topics and offer an exchange of knowledge and methodology among their practitioners.This project fits into this tendency shift, proposing an investigation of new semantics for many-valued logic with tools traditionally developed in modal and intuitionistic logic. Indeed, while the algebraic approach has enabled a fine comprehension of many-valued logic, leading in few years to a very reasonable framework of study, the time is mature for seeking new interpretations and concrete representations of these logical systems. As recent results put in evidence, the instruments developed in fifty years of studies on modal and intuitionistic logic, ranging from dualities to correspondence theory, are certainly the most qualified to help addressing this issues.The applicant has a strong background in many-valued logic and has already shown independency and vocation to research. The host institution, widely considered a pivotal research centre in modal and intuitionistic logic, is probably the best place in Europe to carry out such a research.
Original text from CORDIS.
Participants
- UNIVERSITEIT VAN AMSTERDAM · AmsterdamCoordinatorNetherlands
Links
Data: CORDIS, © European Union