The logics of kernels and closures

Abstract: Each subset D of a complete Boolean algebra generates both a closure operator and a kernel operator on the algebra, respectively mapping each element to its lower approximation (the join of all D-elements below it) and its upper approximation (the meet of all D-elements above it). I will discuss the bimodal logics of such approximation operators on Boolean algebras. By varying the constraints imposed on the generating set D, we obtain a natural family of modal logics. The resulting algebraic approximation semantics offers a new perspective on several common modal systems: I will show how well-known modal logics can be recovered as logics of approximation for particular choices of constraints on the generating set, and one can trace the emergence of various modal laws to simple structural features of the generating set. The logic of approximation operators generated by arbitrary subsets D is the subnormal logic EMNT4+EMNT4. I will give a simple criterion that characterizes the corresponding class of algebras: that is, algebras with abstract closure and kernel operators that are representable as approximation operators. The complete logic of approximation operators generated by a sublattice is the fusion S4+S4: the completeness result relies on a correspondence between sublattice-generated approximation operators and pairwise zero-dimensional bitopological spaces. When D is a complete sublattice, we obtain exactly the temporal logic S4t. When D is a subalgebra, the two modalities collapse into one as they become each other’s duals, and we obtain monomodal S5 (for complete subalgebras) and S4 (for subalgebras in general).

logic in computer sciencelogic

Audience: researchers in the topic

Online logic seminar

Series comments: Description: Seminar on all areas of logic