Pietro Codara

A logical analysis of Mamdani-type fuzzy inference, I theoretical bases

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 meaning of our procedure. We eventually show how to fully recover Mamdani-type fuzzy inference from the latter. In this sense, then, our proposal may be regarded as a logical interpretation of Mamdani-type fuzzy inference. In Part II of this paper, we report on the results of an experiment on the technical analysis of the financial markets based on fuzzy techniques. The core algorithm implements the inference procedure described in this first part of the paper. In Part II, we will argue that the experimental results support the claim that our present theoretical analysis provides a sound interpretation of Mamdani-type fuzzy inference.

A simple combinatorial interpretation of certain generalized Bell and Stirling numbers

In a series of papers, P. Blasiak et al. developed a wide-ranging generalization of Bell numbers (and of Stirling numbers of the second kind) that is relevant to the so-called boson normal ordering problem. They provided a recurrence and, more recently, also offered a (fairly complex) combinatorial interpretation of these numbers. We show that by restricting the numbers somewhat (but still widely generalizing Bell and Stirling numbers), one can supply a much more natural combinatorial interpretation. In fact, we offer two different such interpretations, one in terms of graph colourings and another one in terms of certain labelled Eulerian digraphs.

Independent subsets of powers of paths, and Fibonacci cubes

Abstract We provide a formula for the number of edges of the Hasse diagram of the independent subsets of the hth 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.

A logical analysis of Mamdani-type fuzzy inference, II. An experiment on the technical analysis of financial markets

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 our procedure. We eventually showed how to fully recover Mamdani-type fuzzy inference from the latter. In this sense, then, our proposal in Part I may be regarded as a logical interpretation of Mamdani-type fuzzy inference. In the present Part II of this paper, we report on the results of an experiment on the technical analysis of the financial markets based on fuzzy techniques. The core algorithm implements the inference procedure described in the first part of the paper. The experimental results support the claim that our theoretical analysis in Part I provides a sound interpretation of Mamdani-type fuzzy inference.

Open Partitions and Probability Assignments in Gödel Logic

In the elementary case of finitely many events, we generalise to Godel (propositional infinite-valued) logic -- one of the fundamental fuzzy logics in the sense of Hajek -- the classical correspondence between partitions, quotient measure spaces, and push-forward measures. To achieve this end, appropriate Godelian analogues of the Boolean notions of probability assignment and partition are needed. Concerning the former, we use a notion of probability assignment introduced in the literature by the third-named author et al. Concerning the latter, we introduce and use open partitions , whose definition is justified by independent considerations on the relational semantics of Godel logic (or, more generally, of the finite slice of intuitionistic logic). Our main result yields a construction of finite quotient measure spaces in the Godelian setting that closely parallels its classical counterpart.

Partitions of a Finite Partially Ordered Set

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 partitions of a poset in terms of fibres of some surjection having the poset as domain. We then obtain combinatorial characterisations of such notions in terms of blocks, without reference to surjection. Finally, we give a further, equivalent definition of each kind of partition by means of analogues of equivalence relations.

Propositional Gödel Logic and Delannoy Paths

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 path in Z2 that only uses northward, eastward, and northeastward steps. We establish a representation theorem for free n-generated Godel algebras in terms of the Boolean n-cube {0,1}n, enriched by suitably generalized Delannoy paths.

The Euler Characteristic of a Formula in Godel Logic

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 over Boolean logic. Building on this, we define k-valued versions of the Euler characteristic of a formula φ, for each integer k ≥ 2, and prove that they indeed provide information about the logical status of φ in Gödel k-valued logic. Specifically, our main result shows that the k-valued Euler characteristic is an invariant that separates k-valued tautologies from non-tautologies.

Best Approximation of Ruspini Partitions in Gödel Logic

A Ruspini partition is a finite family of fuzzy sets {f 1 , ..., f n }, f i : [0, 1] i¾?[0, 1], such that

Indiscernibility relations on partially ordered sets

\sum^n_{i=1} f_i(x) = 1