Pietro Codara

Pietro Codara
University of Milan | UNIMI · Department of Computer Science

Ph.D.

About

33
Publications
1,274
Reads
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85
Citations
Additional affiliations
January 2012 - present
University of Milan
Position
  • PostDoc Position
November 2008 - December 2011
University of Milan
Position
  • PostDoc Position
November 2004 - November 2008
University of Milan
Position
  • Ph.D. student in Mathematics and Statistics for Computational Sciences
Education
November 2004 - November 2008
University of Milan
Field of study
  • Mathematics and Statistics for Computational Sciences
September 1997 - February 2004
University of Milan
Field of study
  • Computer Science

Publications

Publications (33)
Article
We introduce a family of Eulerian digraphs, E, associated with Dyck words. We provide the algorithms implementing the bijection between E and W, the set of Dyck words. To do so, we exploit a binary matrix, that we call Dyck matrix, representing the cycles of an Eulerian digraph.
Conference Paper
Beside algebraic and proof-theoretical studies, a number of different approaches have been pursued in order to provide a complete intuitive semantics for many-valued logics. Our intention is to use the powerful tools offered by formal concept analysis (FCA) to obtain further intuition about the intended semantics of a prominent many-valued logic, n...
Article
Full-text available
In this paper we present, by way of case studies, a proof of concept, based on a prototype working on a automotive data set, aimed at showing the potential usefulness of using formulas of {\L}ukasiewicz propositional logic to query databases in a fuzzy way. Our approach distinguishes itself for its stress on the purely linguistic, contraposed with...
Article
The Euler characteristic can be defined as a special kind of valuation on finite distributive lattices. This work begins with some brief consideration on the role of the Euler characteristic on NM algebras, the algebraic counterpart of Nilpotent Minimum logic. Then, we introduce a new valuation, a modified version of the Euler characteristic we cal...
Article
The paper deals with some generalizations of Fibonacci and Lucas sequences, arising from powers of paths, and cycles, respectively. In the first part of the work we provide a formula for the number of edges of the Hasse diagram of the independent subsets of the h-th power of a path ordered by inclusion. For h=1 such a diagram is called a Fibonacci...
Article
Full-text available
The main goal of this work is to establish a bijection between Dyck words and a family of Eulerian digraphs. We do so by providing two algorithms implementing such bijection in both directions. The connection between Dyck words and Eulerian digraphs exploits a novel combinatorial structure: a binary matrix, we call Dyck matrix, representing the cyc...
Article
Using the lattice-theoretic version of the Euler characteristic introduced by V. Klee and G.-C. Rota in the Sixties, we define the Euler characteristic of a formula in G\"{o}del logic (over finitely or infinitely many truth-values). We then prove that the information encoded by the Euler characteristic is classical, i.e. coincides with the analogou...
Article
In this paper, we investigate the notion of partition of a finite partially ordered set (poset, for short). We will define three different notions of partition of a poset, namely, monotone, regular, and open partition. For each of these notions we will find three equivalent definitions, that will be shown to be equivalent. We start by defining part...
Article
With this work we aim to show how Mathematica can be a useful tool to investigate properties of combinatorial structures. Specifically, we will face enumeration problems on independent subsets of powers of paths and cycles, trying to highlight the correspondence with other combinatorial objects with the same cardinality. Then we will study the stru...
Article
In this work we solve a special case of the problem of building an n-dimensional parallelepiped using a given set of n-dimensional parallelepipeds. Consider the identity x^3 = x(x-1)(x-2)+3x(x-1+x). For sufficiently large x, we associate with x^3 a cube with edges of size x, with x(x-1)(x-2) a parallelepiped with edges x, x-1, x-2, with 3x(x-1+x) t...
Article
Full-text available
An open partition \pi{} [Cod09a, Cod09b] of a tree T is a partition of the vertices of T with the property that, for each block B of \pi, the upset of B is a union of blocks of \pi. This paper deals with the number, NP(n), of open partitions of the tree, V_n, made of two chains with n points each, that share the root.
Article
Mathematica offers, by way of the package Combinatorics, many useful functions to work on graphs and ordered structures, but none of these functions was specific enough to meet the needs of our research group. Moreover, the existing functions are not always helpful when one has to work on new concepts. In this paper we present a package of features...
Article
We provide a formula for the number of edges of the Hasse diagram of the independent subsets of the h-th power of a path ordered by inclusion. For h=1 such a value is the number of edges of a Fibonacci cube. We show that, in general, the number of edges of the diagram is obtained by convolution of a Fibonacci-like sequence with itself.
Article
Continuing to pursue a research direction that we already explored in connection with G\"odel-Dummett logic and Ruspini partitions, we show here that {\L}ukasiewicz logic is able to express the notion of pseudo-triangular basis of fuzzy sets, a mild weakening of the standard notion of triangular basis. En route to our main result we obtain an eleme...
Article
In the first part of this work we provide a formula for the number of edges of the Hasse diagram of the independent subsets of the h-th power of a path ordered by inclusion. For h=1 such a value is the number of edges of a Fibonacci cube. We show that, in general, the number of edges of the diagram is obtained by convolution of a Fibonacci-like seq...
Conference Paper
In a recently published work the author investigates indiscernibility relations on information systems with a partially ordered universe. Specifically, he introduces a notion of compatibility between the (partially ordered) universe and an indiscernibility relation on its support, and establishes a criterion for compatibility. In this paper we make...
Conference Paper
Let the pair (U, A) be an information system, where U is a collection of objects, the universe, and A is a finite set of attributes. If we consider a subset B of the set of attributes A, we can associate with B an indiscernibility relation on U, and thus a partition of the set U. Endow U with a partial order, obtaining a partially ordered set P, an...
Conference Paper
This paper is divided into two parts. In the present Part I, our main objective is to analyse Mamdani-type fuzzy control systems in logical terms, with special emphasis on the fuzzy inference process. To that end, we provide our own inference procedure, cast in the language of standard many-valued logics. We give an ample discussion of the logical...
Conference Paper
This paper is divided into two parts. In Part I, our main objective was to analyse Mamdani-type fuzzy control systems in logical terms, with special emphasis on the fuzzy inference process. To that end, we provided our own inference procedure, cast in the language of standard many-valued logics. We gave an ample discussion of the logical meaning of...
Conference Paper
Full-text available
Using the lattice-theoretic version of the Euler characteristic introduced by V. Klee and G.-C. Rota, we define the Euler characteristic of a formula in Gödel logic (over finitely or infinitely many truth-values). We then prove that the information encoded by the Euler characteristic is classical, i.e., coincides with the analogous notion defined o...
Conference Paper
Full-text available
Fuzzy sets featuring in applications to fuzzy control systems are often required to satisfy specific conditions such as, e.g., convexity or normality. In the same connection, a widespread choice is to work with fuzzy sets whose graphs have triangular shape. The purpose of this paper is to show that the former conditions may be regarded as attempts...
Conference Paper
Full-text available
In the elementary case of finitely many events, we generalise to Gödel (propositional infinite-valued) logic — one of the fundamental fuzzy logics in the sense of Hájek — the classical correspondence between partitions, quotient measure spaces, and push-forward measures. To achieve this end, appropriate Gödelian analogues of the Boolean notions of...
Article
By a Ruspini partition we mean a finite family of fuzzy sets {f1,…,fn},fi:[0,1]→[0,1], such that ∑i=1nfi(x)=1 for all x∈[0,1], where [0,1] denotes the real unit interval. We analyze such partitions in the language of Gödel logic. Our first main result identifies the precise degree to which the Ruspini condition is expressible in this language, and...
Conference Paper
A Ruspini partition is a finite family of fuzzy sets {f 1, ..., f n }, f i : [0, 1] →[0, 1], such that \(\sum^n_{i=1} f_i(x) = 1\) for all x ∈ [0, 1]. We analyze such partitions in the language of Gödel logic. Our main result identifies the precise degree to which the Ruspini condition is expressible in this language, and yields inter alia a constr...
Conference Paper
Godel propositional logic is the logic of the minimum triangular norm, and can be axiomatized as propositional intuitionistic logic augmented by the prelinearity axiom (alpha rarr beta) V (beta rarr alpha). Its algebraic counterpart is the subvariety of Heyting algebras satisfying prelinearity, known as Godel algebras. A Delannoy path is a lattice...
Article
Full-text available
Riassunto Discutiamo due sommatorie doppie che, al momento, pare non possano essere calcolate con sistemi di calcolo simbolico. 2 . Qui l'osservazione da fare è che il numeratore di questa frazione è il prodotto della somma dei coefficienti che compaiono nella prima sommatoria della H*L moltiplicata per la somma dei coefficienti che compaiono nella...

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