A Characterization Result for Non-Distributive Logics

Recent published work has addressed the Shalqvist correspondence problem for non-distributive logics. The natural question that arises is to identify the fragment of first-order logic that corresponds to logics without distribution, lifting van Benthem's characterization result for modal logic to this new setting. Carrying out this project is the contribution of the present article. The article is intended as a demonstration and application of a project of reduction of non-distributive logics to (sorted) residuated modal logics. The reduction is an application of recent representation results by this author for normal lattice expansions and a generalization of a canonical and fully abstract translation of the language of substructural logics into the language of their companion sorted, residuated modal logics. The reduction of non-distributive logics to sorted modal logics makes the proof of a van Benthem characterization of non-distributive logics nearly effortless, by adapting and reusing existing results, demonstrating the usefulness and suitability of this approach in studying logics that may lack distribution

Reconciliation of Approaches to the Semantics of Logics without Distribution

This article contributes in that it clarifies and indeed completes an approach (initiated by Dunn and this author several years ago and again pursued by the present author over the last three years or so) to the relational semantics of logics that may lack distribution (Dunn's non-distributive gaggles). The approach uses sorted frames with an incidence relation on sorts (polarities), equipped with additional sorted relations, but, in the spirit of Occam's razor principle, it drops the extra assumptions made in the generalized Kripke frames approach, initiated by Gehrke, that the frames be separated and reduced (RS-frames). We show in this article that, despite rejecting the additional frame restrictions, all the main ideas and results of the RS-frames approach relating to the semantics of non-distributive logics are captured in this simpler framework. This contributes in unifying the research field, and, in an important sense, it complements and completes Dunn's gaggle theory project for the particular case of logics that may drop distribution

Semantics for Finite Delay

We produce a fully abstract model for a notion of process equivalence taking into account issues of fairness, called by Milner fair bisimilarity. The model uses Aczel's anti-foundation axiom and it is constructed along the lines of the anti-founded model for SCCS given by Aczel. We revisit Aczel's semantics for SCCS where we prove a unique fixpoint theorem under the assumption of guarded recursion. Then we consider Milner's extension of SCCS to include a finite delay operator ". Working with fair bisimilarity we construct a fully abstract model, which is also fully abstract for fortification. We discuss the solution of recursive equations in the model. The paper is concluded with an investigation of the algebraic theory of fair bisimilarity. Keywords: fairness, anti-foundation, finite delay, parallelism, fair bisimilarity, fortification. This paper was composed while I was unemployed and an unofficial visitor at the Department of Mathematics, University of Ioannina, Greece. My than..