Expanding Belnap 2: the dual category in depth

Document Type : Research Paper

Authors

1 Department of Mathematics and Applied Mathematics University of Johannesburg PO Box 524, Auckland Park, 2006 South~Africa

2 La Trobe University

3 Department of Mathematics Faculty of Natural Sciences, M. Bel University Tajovskeho 40, 974~01 Banska Bystrica Slovakia

10.52547/cgasa.2022.102443

Abstract

Bilattices, which provide an algebraic tool for simultaneously modelling knowledge and truth, were introduced by N.D. Belnap in a 1977 paper entitled How a computer should think. Prioritised default bilattices include not only Belnap’s four values, for ‘true’ (t), ‘false’(f), ‘contradiction’(⊤) and ‘no information’ (⊥), but also indexed families of default values for simultaneously modelling degrees of knowledge and truth. Prioritised default bilattices have applications in a number of areas including artificial intelligence. In our companion paper, we introduced a new family of prioritised default bilattices, Jn, for n ⩾ 0, with J0 being Belnap’s seminal example. We gave a duality for the variety Vn generated by Jn, with the dual category Xn consisting of multi-sorted topological structures. Here we study the dual category in depth. We axiomatise the category Xn and show that it is isomorphic to a category Yn of single-sorted topological structures. The objects of Yn are ranked Priestley spaces endowed with a continuous retraction. We show how to construct the Priestley dual of the underlying bounded distributive lattice of an algebra in Vn via its dual in Yn; as an application we show that the size of the free algebra FVn(1) is given by a polynomial in n of degree 6.

Keywords


[1] Belnap, N. D., ‘How a computer should think’, In: Contemporary Aspects of Philosophy (Oriel Press Ltd., 1977), pp. 30–56.
[2] Cabrer, L. M. and Priestley, H. A., ‘Coproducts of distributive lattice-based algebras’, Algebra Universalis 72 (2014), 251–286.
[3] Clark, D. M. and Davey, B. A., Natural Dualities for the Working Algebraist (Cambridge University Press, Cambridge, 1998).
[4] Craig, A.P.K., Davey, B.A. and Haviar, M., ‘Expanding Belnap: dualities for a new class of default bilattices’, Algebra Universalis 81(50) (2020).
[5] Davey, B. A., Haviar, M. and Priestley, H. A., ‘Piggyback dualities revisited’, Algebra Universalis 76, 245–285 (2016).
[6] Davey, B. A. and Priestley, H. A., Introduction to Lattices and Order, 2nd edn. (Cambridge University Press, Cambridge, 2002).
[7] Davey, B. A. and Priestley, H. A., ‘Generalized piggyback dualities and applications to Ockham algebras’, Houston J. Math. 13 (1987), 151–198.
[8] Davey, B. A. and Talukder, M. R., ‘Functor category dualities for varieties of Heyting algebras’, J. Pure Appl. Algebra 178 (2003), 49–71.
[9] Davey, B. A. and Werner, H., ‘Piggyback-Dualit¨aten’, Bull. Austral. Math. Soc. 32 (1985), 1–32.
[10] Davey, B. A. and Werner, H., ‘Piggyback dualities’, In: Lectures in Universal Algebra, Szab´o, L., Szendrei, ´A (eds.), Colloq. Math. Soc. J´anos Bolyai 43 (North- Holland, Amsterdam, 1986), pp. 61–83.
[11] Encheva, S. and Tumin, S., ‘Application of default logic in an intelligent tutoring system’, In: Network-Based Information Systems, LNCS 4658 (Springer, 2007), pp.486–494.
[12] Ginsberg, M. L., ‘Multi-valued logics’, In: Proceedings of the 5th National Conference on Artificial Intelligence (Morgan Kaufmann, 1986), pp. 243–249.
[13] Ginsberg, M. L., ‘Multivalued logics: a uniform approach to reasoning in artificial intelligence’, Computational Intelligence 4 (1988), 265–316.
[14] Priestley, H. A., ‘Representation of distributive lattices by means of ordered Stone spaces’, Bull. London Math. Soc. 2 (1970), 186–190.
[15] Priestley, H. A., ‘Ordered topological spaces and the representation of distributive lattices’, Proc. London Math. Soc. 24 (1972), 507–530.
[16] Reiter, R., ‘A logic for default reasoning’, Artif. Intell. 13 (1980), 81–132.
[17] Sakama, C., ‘Ordering default theories and nonmonotonic logic programs’, Theor. Comput. Sci. 338 (2005), 127–152.
[18] Shet, V. D., Harwood, D. and Davis, L. S., ‘Multivalued default logic for identity maintenance in visual surveillance’, In: Proceedings of the 9th European conferenceon Computer Vision, Part IV (Springer, 2006), pp. 119–132.