" Ella in realtà è più radiosa del sole e supera ogni costellazione, paragonata alla luce risulta più luminosa "

Pietro Codara

Artificial Intelligence Research Institute (IIIA-CSIC)
Campus de la UAB, E-08193 Bellaterra, Barcelona, Spain



  • The research activity of P. Codara concerned mainly three areas.


    MANY-VALUED LOGIC. Codara's interest for many-valued logics dates back to his master thesis [T1]. Since then this interest has never diminished. Research has mainly involved two considerable many-valued logics belonging to the Hájek's framework of logics with a continuous t-norm, namely Łukasiewicz and Gödel logic, but it has also involved other non-classical logic worthy of attention for the specific scope. Many of the works apply a categorical duality to the algebraic counterpart of the logic. This is the case, for instance, of [C1], and of the talks held at the conference on Topology, Algebra and Categories in Logic, TACL 2015 and TACL 2009, where the notions of free object, coproduct, and subalgebra, respectively, are investigated in a dual category involving, as objects, order combinatorial structures. An important rôle in the research on many-valued logic has been played by the investigation of the notion of probability of non-classical events. The research devoted to a logical approach to subjective probability à la de Finetti, by using Łukasiewicz and other many-valued logics to describe non-classical events has been carried on, mainly, within the research projects Probability theory of non-classical events, led by Prof. Marra, and Probability of non-classical events: logical foundations and applications, a post-doc project of P. Codara. The paper [C3] deals with this topic. A special attention has been devoted, recently, to alternative (intended) semantics for many-valued logics. A first attempt to apply this research is [C11]. A substantial part of the research in non-classical logic has been devoted to applications, especially to the possibility of exploiting theoretical insights in order to explain or justify the use of some tools in engineering applications (e.g., fuzzy control). This is the case, for instance, of the research works [J5], [J1], [C7], and [C6].


    COMBINATORICS. The research activity in the field of combinatorics followed, in almost every step, that concerning many-valued logic. In fact, in Codara's research, combinatorics has always been a key tool for obtaining significant results concerning many-valued logics. Sometimes, the reverse path has been followed, obtaining results of interest to researchers in combinatorics thanks to techniques derived from the logic, and in particular from category theory. In fact, the use of categorical duality in Codara's research for obtaining such kind of results began with the doctoral thesis [T2]. Here, it is provided an intrinsic characterization of the concept of partition (the concepts, since more than one notion of partition is eligible) of a finite partially ordered set. The use of a categorical duality, the Birkhoff duality, which allows switching between finite lattices and finite partially ordered sets, is crucial, along with the study of the classes of epimorphisms in the dual category of finite partially ordered sets and order-preserving maps. On the subject, besides the doctoral thesis, see [B1], which extends the results to the category of finite partially ordered sets and open maps (or p-morphisms). Analogous techniques has been used in several other papers, allowing, via a combinatorial investigation of ordered structures, to obtain major results about some many-valued logic. We cite for instance [J1], [C3], and [C1]. Some other results are interesting for the combinatorics itself. This is the case of [J2], and [C10], where the notion of Euler characteristic on finite distributive lattices, introduced by Rota, is investigating in a logical framework. Other results are purely combinatorial results on ordered structures and graphs, without evident connections with logic. This is the case of [J6], [J4], and [P2]. While [J4] is connected to the so-called Boson normal ordering problem, [P2] is strongly related with classic results on context-free languages (the Chomsky–Schützenberger representation theorem). Finally, a particular focus has always been paid to the exploitation of symbolic computation platforms and programming languages in order to treat combinatorial problems (see [N5], [N4], [N3], [N2], and [N1]).


    REASONING UNDER UNCERTAINTY. A special attention has always been devoted to the applications of the theoretical results described in the previous paragraphs. A major field of application is that of the treatment of uncertain information. In [J5], and [J1], we investigate the main tools used by engineers when dealing with fuzzy control techniques: fuzzy sets. We show that, once elected the logical framework to work within, the logic itself entails information about the fuzzy sets. In the case of Łukasiewicz logic and triangular fuzzy sets, the logic is capable to describe almost exactly the shape of the fuzzy sets, i.e., there exists a logical theory encoding almost all information. Other papers dealing with fuzzy control are [C7] and [C6], where, besides analyzing the rôle of implication in applications of this kind, we implement and examine an expert system based on many-valued logics capable to make previsions on financial markets using technical analysis. We argue that the performance of the logic-based system (both with Łukasiewicz, and with Gödel logic) is comparable to that of a Mamdani-type system. Finally, the paper [C9] and [C8] are a first attempt to deal with indiscernibility relations in an ordered universe, via the notion of partition of partially ordered sets introduced in the doctoral thesis, while [P4], and [P3] aim to be a preliminary logical approach to FCA (formal context analysis).



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