Change Actions: Models of Generalised Differentiation

Cai et al. have recently proposed change structures as a semantic framework for incremental computation. We generalise change structures to arbitrary cartesian categories and propose the notion of change action model as a categorical model for (higher-order) generalised differentiation. Change action models naturally arise from many geometric and computational settings, such as (generalised) cartesian differential categories, group models of discrete calculus, and Kleene algebra of regular expressions. We show how to build canonical change action models on arbitrary cartesian categories, reminiscent of the F\`aa di Bruno construction

Fixed-point elimination in the intuitionistic propositional calculus

It is a consequence of existing literature that least and greatest fixed-points of monotone polynomials on Heyting algebras-that is, the algebraic models of the Intuitionistic Propositional Calculus-always exist, even when these algebras are not complete as lattices. The reason is that these extremal fixed-points are definable by formulas of the IPC. Consequently, the $\mu$-calculus based on intuitionistic logic is trivial, every $\mu$-formula being equivalent to a fixed-point free formula. We give in this paper an axiomatization of least and greatest fixed-points of formulas, and an algorithm to compute a fixed-point free formula equivalent to a given $\mu$-formula. The axiomatization of the greatest fixed-point is simple. The axiomatization of the least fixed-point is more complex, in particular every monotone formula converges to its least fixed-point by Kleene's iteration in a finite number of steps, but there is no uniform upper bound on the number of iterations. We extract, out of the algorithm, upper bounds for such n, depending on the size of the formula. For some formulas, we show that these upper bounds are polynomial and optimal

Trace as an alternative decategorification functor

Categorification is a process of lifting structures to a higher categorical level. The original structure can then be recovered by means of the so-called "decategorification" functor. Algebras are typically categorified to additive categories with additional structure and decategorification is usually given by the (split) Grothendieck group. In this expository article we study an alternative decategorification functor given by the trace or the zeroth Hochschild--Mitchell homology. We show that this form of decategorification endows any 2-representation of the categorified quantum sl(n) with an action of the current algebra U(sl(n)[t]) on its center.Comment: 47 pages with tikz figures. arXiv admin note: text overlap with arXiv:1405.5920 by other author

SProc Categorically

. We provide a systematic reconstruction of Abramsky's category SProc of synchronous processes [Abr93]: SProc is isomorphic to a span category on a category of traces. The significance of the work is twofold: It shows that the original presentation of SProc in mixed formulations is unnecessary --- a simple categorical description exists. Furthermore, the techniques employed in the reconstruction suggest a general method of obtaining process categories with structure similar to SProc. In particular, the method of obtaining bisimulation equivalence in our setting, which represents an extension of the work of Joyal, Nielsen and Winskel [JNW93], has natural application in many settings. 1 Introduction In [Abr93], Abramsky proposed a new paradigm for the semantics of computation, interaction categories, where the following substitutions are made: Denotational semantics Categories Interaction categories Domains objects Interface specifications Continuous functions maps Commun..

Change actions: models of generalised differentiation

Change structures, introduced by Cai et al., have recently been proposed as a semantic framework for incremental computation. We generalise change actions, an alternative to change structures, to arbitrary cartesian categories and propose the notion of change action model as a categorical model for (higher-order) generalised differentiation. Change action models naturally arise from many geometric and computational settings, such as (generalised) cartesian differential categories, group models of discrete calculus, and Kleene algebra of regular expressions. We show how to build canonical change action models on arbitrary cartesian categories, reminiscent of the Fàa di Bruno construction

Modele mathematique d'une usine de pate a papier au sulfate. Bilans matiere, enthalpie, entropie et exergie

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