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15.07.2020 | Focus

A constructive sequence algebra for the calculus of indications

verfasst von: Rocco Gangle, Gianluca Caterina, Fernando Tohmé

Erschienen in: Soft Computing | Ausgabe 23/2020

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Abstract

In this paper, we investigate some aspects of Spencer–Brown’s Calculus of Indications. Drawing from earlier work by Kauffman and Varela, we present a new categorical framework that allows to characterize the construction of infinite arithmetic expressions as sequences taking values in grossone.

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This is akin to Peirce’s \(\alpha \) graphs (Zeman 1974; Gangle et al. 2020), although with a basic but immaterial difference between the representations of True and False in the two formalisms.

The introduction of the numeral ① may require, in some contexts, the augmentation of the set of natural \({\mathbb {N}}\) to a larger set \(\hat{{\mathbb {N}}}\), where

https://static-content.springer.com/image/art%3A10.1007%2Fs00500-020-05121-1/MediaObjects/500_2020_5121_Equ5_HTML.png

For the purpose of this paper we will not need to make use of \(\hat{{\mathbb {N}}}\).

Evidence of the efficacy of the grossone approach is highlighted by its successful application to many fields of applied mathematics, including optimization (see Cococcioni et al. 2018, 2020; De Cosmis and De Leone 2012; Sergeyev et al. 2018; Iavernaro et al. 2020; Iudin et al. 2012; Sergeyev 2007, 2013b; Zhigljavsky 2012), fractals and cellular automata (see Caldarola 2018; D’Alotto 2012, 2015, 2013) as well as infinite decision-making processes, game theory, and probability (see Fiaschi and Cococcioni 2018; Rizza 2018, 2019), whereas the formal logical foundation of grossone has been investigated in Lolli (2012); Margenstern (2011); Montagna et al. (2015). The approach presented in this paper bears similarities to the application of grossone to Turing machines as described in Sergeyev and Garro (2010). Further investigation of these analogies remains open.

The two systems are slightly different in terms of their rules of deduction (or rewriting). We do not examine this point any further in this paper.

The relevance of grossone to this construction will appear in the subsequent sections.

Our approach is similar to that of Varela and Goguen (1978) in that it involves limits of sequences. But by calculating values of the sequences themselves, and not only their limits, the (complex algebra) waveform interpretation of Kauffman and Varela (1980) is partially maintained as well.

Here the objects are the natural numbers, and there is a morphism between two natural numbers \(m\rightarrow n\) if and only if \(m\ge n\). The choice of reversing the natural order is made in order for the functor F to be contravariant.

The condition given here guarantees that all nested cuts are of at most depth \(n-1\).

A monic arrow in any category is defined as follows. Given any morphism \(f: A\longrightarrow B\), f is said to be monic if, for any two morphisms \(g,h: Z\longrightarrow A\) such that \(f\circ g=f\circ h\), it is the case that \(g=h\).

\(\coprod \) is well defined in \({\mathcal {CI}}\) since it is a topos.

A formal categorical characterization of a variant of this procedure, applied to cuts-only \(\alpha \) graphs can found in Gangle et al. (2020).

Note that it is important to distinguish this operator

https://static-content.springer.com/image/art%3A10.1007%2Fs00500-020-05121-1/MediaObjects/500_2020_5121_Figc_HTML.gif

(as an operation on \(\alpha \)) from the circle drawn around 1 in the grossone numeral ① .

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Cococcioni M, Cudazzo A, Pappalardo M, Sergeyev YD (2020) Solving the lexicographic multi-objective mixed-integer linear programming problem using branch-and-bound and grossone methodology. Commun Nonlinear Sci Numer Simul 84:1–20MathSciNetCrossRef

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Titel: A constructive sequence algebra for the calculus of indications
verfasst von: Rocco Gangle
Gianluca Caterina
Fernando Tohmé
Publikationsdatum: 15.07.2020
Verlag: Springer Berlin Heidelberg
Erschienen in: Soft Computing / Ausgabe 23/2020
Print ISSN: 1432-7643
Elektronische ISSN: 1433-7479
DOI: https://doi.org/10.1007/s00500-020-05121-1

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