Expanding Belnap: dualities for a new class of default bilattices
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Algebra Universalis
Expanding Belnap: dualities for a new class of default bilattices Andrew P. K. Craig, Brian A. Davey and Miroslav Haviar Dedicated to the memory of Prof. Beloslav Rieˇcan. Abstract. Bilattices provide an algebraic tool with which to model simultaneously knowledge and truth. They were introduced by Belnap in 1977 in a paper entitled How a computer should think. Belnap argued that instead of using a logic with two values, for ‘true’ (t) and ‘false’ (f ), a computer should use a logic with two further values, for ‘contradiction’ () and ‘no information’ (⊥). The resulting structure is equipped with two lattice orders, a knowledge order and a truth order, and hence is called a bilattice. Prioritised default bilattices include not only values for ‘true’ (t 0 ), ‘false’ (f 0 ), ‘contradiction’ and ‘no information’, but also indexed families of default values, t 1 , . . . , t n and f 1 , . . . , f n , for simultaneous modelling of degrees of knowledge and truth. We focus on a new family of prioritised default bilattices: Jn , for n ∈ ω. The bilattice J0 is precisely Belnap’s seminal example. We obtain a multisorted duality for the variety Vn generated by Jn , and separately a singlesorted duality for the quasivariety Jn generated by Jn . The main tool for both dualities is a unified approach that enables us to identify the meet-irreducible elements of the appropriate subuniverse lattices. Our results provide an interesting example where the multi-sorted duality for the variety has a simpler structure than the single-sorted duality for the quasivariety. Mathematics Subject Classification. 06D50, 08C20, 03G25. Keywords. Bilattice, Default bilattice, (Multi-sorted)Natural duality.
Presented by R. Willard. The first author acknowledges the support of the NRF South Africa (grant 127266) and the third author acknowledges the support of Slovak grant VEGA 1/0337/16. 0123456789().: V,-vol
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A. P. K. Craig et al.
Algebra Univers.
1. Introduction We describe a new class of default bilattices { Jn | n ∈ ω } for use in prioritised default logic. While the first of these bilattices (n = 0) is Belnap’s original four-element bilattice FOUR [1], for n 1 these bilattices provide new algebraic structures for dealing with inconsistent and incomplete information. In particular, the structure of the knowledge order gives a new method for interpreting contradictory responses from amongst a hierarchy of ‘default true’ and ‘default false’ responses. The first two bilattices from our new class, drawn in their knowledge order (k ) and their truth order (t ), are shown in Figure 1, while Jn is shown in Figure 2. To place both our family of bilattices, and our results concerning them, in an appropriate context, we recall that bilattices were investigated in the late 1980’s by Ginsberg [14,15] as a method for inference with incomplete and contradictory information. These investigations built on the simple example introduced by Belnap [1] about a decade earlier. Belnap’s idea is represented by the four-elemen
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