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An algebraic generalization of Kripke structures
The Kripke semantics of classical propositional normal modal logic is made
algebraic via an embedding of Kripke structures into the larger class of
pointed stably supported quantales. This algebraic semantics subsumes the
traditional algebraic semantics based on lattices with unary operators, and it
suggests natural interpretations of modal logic, of possible interest in the
applications, in structures that arise in geometry and analysis, such as
foliated manifolds and operator algebras, via topological groupoids and inverse
semigroups. We study completeness properties of the quantale based semantics
for the systems K, T, K4, S4, and S5, in particular obtaining an axiomatization
for S5 which does not use negation or the modal necessity operator. As
additional examples we describe intuitionistic propositional modal logic, the
logic of programs PDL, and the ramified temporal logic CTL.Comment: 39 page
Decidability of the Clark's Completion Semantics for Monadic Programs and Queries
There are many different semantics for general logic programs (i.e. programs
that use negation in the bodies of clauses). Most of these semantics are Turing
complete (in a sense that can be made precise), implying that they are
undecidable. To obtain decidability one needs to put additional restrictions on
programs and queries. In logic programming it is natural to put restrictions on
the underlying first-order language. In this note we show the decidability of
the Clark's completion semantics for monadic general programs and queries.
To appear in Theory and Practice of Logic Programming (TPLP
Indicative Conditionals and Dynamic Epistemic Logic
Recent ideas about epistemic modals and indicative conditionals in formal
semantics have significant overlap with ideas in modal logic and dynamic
epistemic logic. The purpose of this paper is to show how greater interaction
between formal semantics and dynamic epistemic logic in this area can be of
mutual benefit. In one direction, we show how concepts and tools from modal
logic and dynamic epistemic logic can be used to give a simple, complete
axiomatization of Yalcin's [16] semantic consequence relation for a language
with epistemic modals and indicative conditionals. In the other direction, the
formal semantics for indicative conditionals due to Kolodny and MacFarlane [9]
gives rise to a new dynamic operator that is very natural from the point of
view of dynamic epistemic logic, allowing succinct expression of dependence (as
in dependence logic) or supervenience statements. We prove decidability for the
logic with epistemic modals and Kolodny and MacFarlane's indicative conditional
via a full and faithful computable translation from their logic to the modal
logic K45.Comment: In Proceedings TARK 2017, arXiv:1707.0825
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