An Abstract Approach to Consequence Relations

We generalise the Blok-J\'onsson account of structural consequence relations, later developed by Galatos, Tsinakis and other authors, in such a way as to naturally accommodate multiset consequence. While Blok and J\'onsson admit, in place of sheer formulas, a wider range of syntactic units to be manipulated in deductions (including sequents or equations), these objects are invariably aggregated via set-theoretical union. Our approach is more general in that non-idempotent forms of premiss and conclusion aggregation, including multiset sum and fuzzy set union, are considered. In their abstract form, thus, deductive relations are defined as additional compatible preorderings over certain partially ordered monoids. We investigate these relations using categorical methods, and provide analogues of the main results obtained in the general theory of consequence relations. Then we focus on the driving example of multiset deductive relations, providing variations of the methods of matrix semantics and Hilbert systems in Abstract Algebraic Logic

Sequent and Hypersequent Calculi for Abelian and Lukasiewicz Logics

We present two embeddings of infinite-valued Lukasiewicz logic L into Meyer and Slaney's abelian logic A, the logic of lattice-ordered abelian groups. We give new analytic proof systems for A and use the embeddings to derive corresponding systems for L. These include: hypersequent calculi for A and L and terminating versions of these calculi; labelled single sequent calculi for A and L of complexity co-NP; unlabelled single sequent calculi for A and L.Comment: 35 pages, 1 figur

Convolution, Separation and Concurrency

A notion of convolution is presented in the context of formal power series together with lifting constructions characterising algebras of such series, which usually are quantales. A number of examples underpin the universality of these constructions, the most prominent ones being separation logics, where convolution is separating conjunction in an assertion quantale; interval logics, where convolution is the chop operation; and stream interval functions, where convolution is used for analysing the trajectories of dynamical or real-time systems. A Hoare logic is constructed in a generic fashion on the power series quantale, which applies to each of these examples. In many cases, commutative notions of convolution have natural interpretations as concurrency operations.Comment: 39 page

NIDUS IDEARUM. Scilogs, V: joining the dots

My lab[oratory] is a virtual facility with noncontrolled conditions in which I mostly perform scientific meditation and chats: a nest of ideas (nidus idearum, in Latin). I called the jottings herein scilogs (truncations of the words scientific, and gr. Λόγος – appealing rather to its original meanings ground , opinion , expectation ), combining the welly of both science and informal (via internet) talks (in English, French, and Romanian). * In this fifth book of scilogs collected from my nest of ideas, one may find new and old questions and solutions, mostly referring to topics on NEUTROSOPHY – email messages to research colleagues, or replies, notes about authors, articles, or books, so on. Feel free to budge in or just use the scilogs as open source for your own ideas

Generalized Multisets: From ZF to FSM

We study generalized multisets (multisets that allow possible negative multiplicities) both in the Zermelo-Fraenkel framework and in the finitely supported mathematics. We extend the notion of generalized multiset over a finite alphabet, and we replace it by the notion of algebraically finitely supported generalized multiset over a possibly infinite alphabet. We analyze the correspondence between some properties of generalized multisets obtained in finitely supported mathematics where only finitely supported objects are allowed, and those obtained in the classical Zermelo-Fraenkel framework