13 research outputs found
The algebra of adjacency patterns: Rees matrix semigroups with reversion
We establish a surprisingly close relationship between universal Horn classes
of directed graphs and varieties generated by so-called adjacency semigroups
which are Rees matrix semigroups over the trivial group with the unary
operation of reversion. In particular, the lattice of subvarieties of the
variety generated by adjacency semigroups that are regular unary semigroups is
essentially the same as the lattice of universal Horn classes of reflexive
directed graphs. A number of examples follow, including a limit variety of
regular unary semigroups and finite unary semigroups with NP-hard variety
membership problems.Comment: 30 pages, 9 figure
Compositional Reasoning for Multi-Modal Logics ⋆
Abstract. We provide decomposition and quotienting results for multimodal logic with respect to a composition operator, traditionally used for epistemic models, due to van Eijck et al. (Journal of Applied Non-Classical Logics 21(3–4):397–425, 2011), that involves sets of atomic propositions and valuation functions from Kripke models. While the composition operator was originally defined only for epistemic S5 n models, our results apply to the composition of any pair of Kripke models. In particular, our quotienting result extends a specific result in the above mentioned paper by van Eijck et al. for the composition of epistemic models with disjoint sets of atomic propositions to compositions of any two Kripke models regardless of their sets of atomic propositions. We also explore the complexity of the formulas we construct in our decomposition result.