An analysis of the logic of Riesz Spaces with strong unit

We study \L ukasiewicz logic enriched with a scalar multiplication with scalars taken in $[0,1]$. Its algebraic models, called {\em Riesz MV-algebras}, are, up to isomorphism, unit intervals of Riesz spaces with a strong unit endowed with an appropriate structure. When only rational scalars are considered, one gets the class of {\em DMV-algebras} and a corresponding logical system. Our research follows two objectives. The first one is to deepen the connections between functional analysis and the logic of Riesz MV-algebras. The second one is to study the finitely presented MV-algebras, DMV-algebras and Riesz MV-algebras, connecting them from logical, algebraic and geometric perspective

Towards understanding the Pierce-Birkhoff conjecture via MV-algebras

Our main issue was to understand the connection between \L ukasiewicz logic with product and the Pierce-Birkhoff conjecture, and to express it in a mathematical way. To do this we define the class of \textit{f}MV-algebras, which are MV-algebras endowed with both an internal binary product and a scalar product with scalars from $[0,1]$. The proper quasi-variety generated by $[0,1]$, with both products interpreted as the real product, provides the desired framework: the normal form theorem of its corresponding logical system can be seen as a local version of the Pierce-Birkhoff conjecture

On theories of random variables

We study theories of spaces of random variables: first, we consider random variables with values in the interval $[0,1]$, then with values in an arbitrary metric structure, generalising Keisler's randomisation of classical structures. We prove preservation and non-preservation results for model theoretic properties under this construction: i) The randomisation of a stable structure is stable. ii) The randomisation of a simple unstable structure is not simple. We also prove that in the randomised structure, every type is a Lascar type

Many valued logics: interpretations, representations and applications

2015 - 2016This thesis, as the research activity of the author, is devoted to establish new connections and to strengthen well-established relations between different branches of mathematics, via logic tools. Two main many valued logics, logic of balance and L ukasiewicz logic, are considered; their associated algebraic structures will be studied with different tools and these techniques will be applied in social choice theory and artificial neural networks. The thesis is structured in three parts. Part I The logic of balance, for short Bal(H), is introduced. It is showed: the relation with `-Groups, i.e. lattice ordered abelian groups (Chapter 2); a functional representation (Chapter 3); the algebraic geometry of the variety of `-Groups with constants (Chapter 4). Part II A brief historical introduction of L ukasiewicz logic and its extensions is provided. It is showed: a functional representation via generalized states (Chapter 5); a non-linear model for MV-algebras and a detailed study of it, culminating in a categorical theorem (Chapter 6). Part III Applications to social choice theory and artificial neural network are presented. In particular: preferences will be related to vector lattices and their cones, recalling the relation between polynomials and cones studied in Chapter 4; multilayer perceptrons will be elements of non-linear models introduced in Chapter 6 and networks will take advantages from polynomial completeness, which is studied in Chapter 2. We are going to present: in Sections 1.2 and 1.3 all the considered structures, our approach to them and their (possible) applications; in Section 1.4 a focus on the representation theory for `-Groups and MV-algebras. Note that: algebraic geometry for `-Groups provides a modus operandi which turns out to be useful not only in theoretical field, but also in applications, opening (we hope) new perspectives and intuitions, as we made in this first approach to social theory; non-linear models here presented and their relation to neural networks seem to be very promising, giving both intuitive and formal approach to many concrete problems, for instance degenerative diseases or distorted signals. All these interesting topics will be studied in future works of the author. [edited by author]Questa tesi, come l’attivit`a di ricerca dell’autore, `e dedicata a stabilire nuove connessioni e a rafforzare le relazioni ben consolidate tra diversi settori della matematica, attraverso strumenti logici. Sono considerate due principali logiche a piu` valori, logic of balance e L ukasiewicz logic; le loro strutture algebriche associate verranno studiate con strumenti diversi e queste tecniche saranno applicate nella teoria della scelta sociale e nelle reti neurali artificiali. La tesi `e strutturata in tre parti. Part I Viene introdotta la Logic of balance. Viene mostrato: la relazione con `-Groups, gruppi abeliani ordinati reticolarmente (Chapter 2); una rappresentazione funzionale (Chapter 3); geometria algebrica della variet`a degli `-Groups con costanti (Chapter 4). Part II Viene fornita una breve introduzione storica della logica di L ukasiewicz e delle sue estensioni. Viene mostrato: una rappresentazione funzionale tramite stati generalizzati (Chapter 5); Un modello non lineare per le MV-algebre e uno studio dettagliato di esso, culminando in un teorema categoriale (Chapter 6). Part III Sono presentate applicazioni alla teoria delle scelte sociali e delle rete neurali artificiali. In particolare: le preferenze saranno correlate ai reticoli vettoriali e ai loro coni, richiamando la relazione tra polinomi e coni studiati nel Capitolo 4; I multilayer perceptrons saranno elementi di modelli non lineari introdotti nel Capitolo 6 e le reti prenderanno vantaggi dalla completezza polinomiale, studiata nel Capitolo 2. La geometria algebrica per gli `-Groups fornisce un modus operandi che risulta utile non solo nel campo teorico, ma anche nelle applicazioni, aprendo (speriamo) nuove prospettive e intuizioni, come abbiamo fatto in questo primo approccio alla teoria sociale; I modelli non lineari qui presentati e la loro relazione con le reti neurali sembrano molto promettenti, offrendo un approccio intuitivo e formale a molti problemi concreti, ad esempio malattie degenerative o segnali distorti. Tutti questi argomenti saranno oggetto di studio in opere future dell’autore. [a cura dell'autore]XV n.s. (XXIX