Final Report Summary - BMDF (Bilattices meet d-Frames)
Bilattices and frames are mathematical structures that have been widely applied in theoretical computer science, although in quite different areas.
Bilattices were introduced in the context of non-monotonic and inconsistency-tolerant reasoning in artificial intelligence, while frames play a key role in domain theory, the mathematical theory of computation introduced by Dana Scott as a foundation for denotational semantics of programs.
The recent introduction of d-frames, a generalization of frames designed to handle partial and potentially conflicting information, has opened a way to combine these two formalisms.
We realized that the two approaches, bilattices and d-frames, which had been studied independently and by non-overlapping communities, were in fact very close both on a formal level and from the point of view of their motivation.
This project has explored the potentialities offered by the interplay between the two formalisms, combining a mathematically rigorous approach with one mainly driven by applications. We have focused in particular on investigating and developing new logical formalisms that allow for an effective treatment of partial and conflicting information, with potential applications in different areas of computer science context.
The methodology that allowed us to carry out this research resulted from a fruitful combination of the well-developed methods of algebraic logic together with the semantical insights provided by Stone-type topological dualities for various classes of structures.
Our study initially focused on extensions of Belnap-Dunn four-valued logic, which is the core of bilattice-based logics and one of the best known and established logical frameworks for dealing with partial and inconsistent information. We achieved a partial classification of the extensions of Belnap-Dunn logic which can be obtained by adding new axioms or finitary rules. Thanks to an algebraic construction known as twist-structure, we have also reached a deep understanding of the algebraic counterparts of these logics and of the behaviour of implication connectives within these systems. Three papers resulting from this line of research have already been published on international peer-reviewed journals, and one more has been accepted for publication.
A further step in our research, in which we have extensively used the logical and algebraic insights gained during the first stage of the project, involved working towards achieving a thorough understanding of bilattices and related algebraic structures from a topological point of view. This led to the development of a duality theory for distributive bilattices and also for other algebras related to inconsistency-tolerant logic (e.g. so-called N4-lattices, which are the algebraic semantics of paraconsistent Nelson logic).
During the last phase of the project we focused on the problem of introducing and axiomatizing modal expansions of Belnap-Dunn and bilattice logics. This line of research relates to recent work done by several authors on modal many-valued logics, and is particularly relevant to computer science because of the connection of modal logic with state-based systems and verification. Modal languages are in fact used to devise verification techniques that analyze the behaviour of software and hardware systems over time in order to check their correctness, reliability, or identify flaws in their design. This area would also benefit from the use of inconsistency-tolerant logics. For example, performing partial analyses dramatically reduces complexity, but when different parts of a complex system are analyzed separately, inconsistencies are likely to arise. Such analyses may still provide us with valuable information, but we need to shift from a classical to a paraconsistent setting. Our work on developing bilattice modal logics may thus lead to devising new languages for describing and verifying properties of state-based systems using an underlying logic that allows one to deal with partial information and inconsistencies in a more effective way than classical logic.
Two papers resulting from this line of research have already been published; another one has been accepted and is currently in press, and two more have recently been submitted.
DISSEMINATION ACTIVITIES
The project has resulted in seven publications in international peer-reviewed journals, plus two publications in conference proceedings. Two more papers have been submitted to peer-reviewed journals.
The fellow presented results at several international conferences (in Spain, Germany, Japan, Brazil, USA, and Italy), as well as giving seminars at five universities, three in the UK (Oxford, Leicester, Birmingham), one in Spain (Barcelona) and one in the Netherlands (Amsterdam). Several short talks have also been given within the University of Birmingham.
The scientist in charge presented at one international conference (in the USA) and gave seminars in the UK (Oxford), Spain (Barcelona) and France (Paris).
A course based on the project has been taught jointly by the fellow and the scientist in charge at the international logic summer school ESSLLI 2013 (Dusseldorf, Germany).
As part of the project, an international collaboration with members in the UK and Spain has been set up. The fellow travelled to work with team members in 2012, and again in 2013 together with the scientist in charge.
A project website was brought online in 2011, describing the project goals and providing up to date news on progress.