Title of the
project: Bilattices meet d-frames
Scheme:
Marie Curie Intra-European Fellowships (IEF)
Call:
FP7-PEOPLE-2010-IEF
Research fellow:
Umberto Rivieccio
Host researcher:
Achim Jung
Host institution:
School of Computer Science, University of Birmingham
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Abstract
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.
Timeline
- 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.
Publications resulting from the project
- 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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