Title of the
project: Bilattices meet d-frames |

Bilattices and frames are
mathematical structures widely applied in Theoretical
Computer Science, although in quite different areas.

Bilattices were introduced within the context of
non-monotonic and paraconsistent reasoning in Artificial
Intelligence, while frames have a key role in Domain Theory,
the mathematical theory of computation introduced by Dana
Scott as a foundation for denotational semantics of
programs.

The introduction of d-frames, a generalization of frames
designed to handle partial and conflicting information, has
recently opened a way to combine these two formalisms.

The aim of the present proposal is to explore this
possibility in order to obtain a mathematically rigorous and
versatile formalism that unifies the approach of bilattices
and the one of d-frames, thus having a primary impact on
Domain Theory but also on bilattice-based formalisms.

This is to be accomplished, on the one hand, by applying the
algebraic and logical methods that proved to be successful
in the study of bilattices to the theory of d-frames, with
the main aim to develop and achieve a deeper understanding
of the logic underlying d-frames; on the other hand, by
extending the scope of bilattices to the setting of Domain
Theory, focusing on issues such as completeness and
topological duality.

This is to be accomplished by (1) applying the algebraic and
logical methods that proved to be successful in the study of
bilattices to the theory of d-frames, with the main aim of
developing and achieving a deeper understanding of the logic
underlying d-frames; and by (2) extending the scope of
bilattices to the setting of Domain Theory, focusing on
issues such as completeness and topological duality.

- 2011

November - December. As a first step toward a (bi)topological understanding of bilattices, we have developed a Priestley-style topological duality theory for different classes of bilattices (with and without negation, implication, conflation operators). UR has given a presentation based on this work at the TACL 2011 conference (Topology, Algebra, and Categories in Logic) and an invited seminar at Analytic Topology in Mathematics and Computer Science Seminar of the University of Oxford (UK). This research has also resulted in a paper (1) to appear on Studia Logica.

- 2012

January - February: During this period Prof Ramon Jansana from the University of Barcelona (Spain), one of the collaborators mentioned in the project, hes been visiting the University of Birmingham. Together with Prof Jansana we have started the study of modal operators on bilattices. This line of research is related to our previous investigation of topological spaces corresponding to bilattices, as there is a natural way of associating topological spaces to the structures used in Kripke-style semantics for modal logics. This seems to be a promising line of research, as the applications of modal operators within computer science are constantly growing. Some preliminary results obtained in this respect have been presented by UR in an invited presentation at the CLOG Seminar of the University of Leicester (UK).

March - April: UR has extended his investigation of modal operators on bilattices to related algebras, called twist-structures, which correspond to subreducts (i.e., subalgebras relative to a fragment of the algebraic language) of certain classes of bilattices. This work, partly carried out in collaboration with Prof Hiroakira Ono (Japan Advanced Institute of Science and Technology) resulted in a joint paper, submitted to a special issue of Studia Logica devoted to Non-classical Modal and Predicate Logics.

May - June: We have continued our exploratory study of modal operations on bilattices. Some results on this have been presented by UR in an invited talk at Algebra|Coalgebra Seminar of the Institute for Logic, Language & Computation (ILLC) in Amsterdam. This line of research has been further pursued by UR during a research stay at the University of Barcelona, in collaboration with Prof Ramon Jansana and Dr F??lix Bou, within the more general context of many-valued modal logics. UR presented related work on twist-structures at the Duality Theory in Algebra, Logic and Computer Science Workshop 1 (University of Oxford) and at the Trends in Logic XI international conference in Germany.

July - August: We have been studying bilattice-valued modal logics from two parallel perspectives: (1) from the point of many-valued modal logics, continuing along the line of the research started in collaboration with Prof Jansana and Dr Bou in Barcelona, succeeding in aximatizing the least modal logic over the four-element Belnap bilattice; (2) from a more general coalgebraic point of view, aiming at extending the framework of coalgebraic modal logic from the classical to the bilattice setting. This last line of research has been pursued in collaboratio with Prof Jung's doctoral student Liang-Ting Chen. UR presented some general results on the application of bilattice logics in computer science at the 15th Wessex Theory Seminar (University of Birmingham).

September - October: UR continued research on implicative twist-structures (i.e., twist-structures corresponding to somehow minimal subreducts of bilattices) during a research stay at the Japan Advanced Institute for Science and Technology (JAIST, Japan) in collaboration with Prof Hiroakira Ono. Some results from this work have been presented by UR at the Logic, Algebra and Truth Degrees 2012 international conference (Kanazawa, Japan). Parallel to this line of research, we have continued the investigation of modal operators on bilattices, achieving some first results on the undertanding of modal bilattice logics from a coalgebraic point of view and obtaining an algebraic completeness theorem and a topological representation for modal bilattice logics and the associated algebraic semantics.

- A. Jung, U. Rivieccio, ???Priestley Duality for Bilattices???, Studia Logica, 100, 1-2 (2012), p. 223-252.PDF
- U. Rivieccio, "An infinity of super-Belnap logics",
*Journal of Applied Non-Classical Logics*, 22, 4 (2012), p. 319-335. PDF - U. Rivieccio, "Representation of interlaced
trilattices",
*Journal of Applied Logic*, 11, 2 (2013), p. 174-189. PDF - A. Pietz, U. Rivieccio, ???Nothing but the truth???,
*Journal of Philosophical Logic*, 42, 1 (2013), p. 125-135. PDF - F. Bou, U. Rivieccio, ???Bilattices with implications???,
*Studia Logica*, 101, 4 (2013), p. 651-675. PDF - A. Jung, U. Rivieccio, ???Kripke semantics for modal
bilattice logic??? (extended abstract),
*Proceedings of the 28th Annual ACM/IEEE Symposium on Logic in Computer Science,*IEEE Computer Society Press, p. 438-447, 2013. PDF - H. Ono, U. Rivieccio, ???Modal twist-structures over residuated lattices???, Logic Journal of the IGPL, DOI 10.1093/jigpal/jzt043. PDF
- R. Jansana, U. Rivieccio, ???Priestley duality for
N4-lattices???, to appear in
*Proceedings of the 8th conference of the European Society for Fuzzy Logic and Technology (EUSFLAT-2013).*PDF - U. Rivieccio, ???Implicative twist-structures???, to appear on Algebra Universalis. PDF
- R. Jansana, U. Rivieccio, "Dualities for modal N4-lattices", submitted to the Logic Journal of the IGPL.
- A. Jung, U. Rivieccio,"Four-valued modal logic: Kripke
semantics and duality", submitted to the
*Journal of Logic and Computation*.

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last update: december 2010