Autumn Semester

These are the pages of the Logic Seminar at the Università degli Studi di Milano and the Università degli Studi dell'Insubria.

## The Logic Seminar

Welcome to the web pages of the Logic Seminar at the Università degli Studi di Milano and the Università degli Studi dell'Insubria.

The Seminar usually meets every other week during both the autumn and the spring semester, currently on Tuesdays. The exact date, time, and place of each meeting are annonunced on these pages.

There are two types of talks at the Seminar. First, we invite visiting researchers, colleagues, or graduate students to give talks on their own research activity. Second, during each semester we run a proper seminar on a given subject that is of interest to a group of participants. Usually, the participants are not experts on that subject, but wish to learn more about it. The participants then take turns in preparing one- or two-hour talks on a specific topic within the chosen subject.

## Next Talk

**18 October 2017**, 10:00, Auletta 4, Dipartimento di Informatica, via Comelico 39/41, Milano

Stefania Boffa,

Università degli Studi dell'Insubria

*Sequences of orthopairs, Kleene algebras and IUML-algebras*

We study sequences of approximation of sets given by refining relations on the universe, and we show that such sequences can be equipped with a structure of finite centered Kleene algebra satisfying the interpolation property or with a structure of finite IUML-algebras.

**18 October 2017**, 10:40, Auletta 4, Dipartimento di Informatica, via Comelico 39/41, Milano

Diego Valota,

Università degli Studi di Milano

*Counting Spectra via Dualities: from Goedel logic to WNM logics*

Given a class of structures C and a natural number k≥1,
we define the following counting problems:

- Spectrum: counting the number of k-element structures in C;

- Fine Spectrum: counting the number of non-isomorphic k-element structures
in C;

- Free Spectrum: counting the elements of the free k-generated algebra in C
(when C is variety of algebras).

In this talk we will see how to exploit the dual equivalence between
the category of finite Godel algebras and their homomorphisms,
and the category of finite forests and open maps,
to solve two of the above mentioned Spectra problems,
namely the Free Spectrum and the Fine Spectrum problem.
Solutions to the Free Spectrum problem, when C is the variety of Godel
algebras G, can be easily found in literature. Indeed, already in 1969 Horn has
obtained a recurrence formula to compute the cardinalities of free
k-generated Godel algebras. Another solution to this problem can be
achieved by restating the Horn's recurrence in terms of finite forests
[D'Antona and Marra, 2006].
Conversely, to the best of our knowledge, the Fine Spectrum problem for
the class of Godel algebras has never been considered before.
We introduce an algorithm that given a natural number k≥1,
it generates a set of forests S_k such that for every F ∈ S_k
the number of subforests of F is exactly k.
That is, given a finite cardinal k we can build the set of finite
Godel algebras with k elements, solving in this way
the Fine Spectrum problem for G.
Finally, we will discuss some possible generalizations of this approach
to solve Spectra Problems for other varieties related to many-valued logics
in the WNM hierarchy.