A crash course in subadditivity, part 1

Today, the 1st of March 2018, I gave what ended up being the first of a series of Theory Lunch talks about subadditive functions. The idea is to give an introduction to the subject, following Hille’s and Lind and Marcus’s textbooks, and stating an important theorem by the Hungarian mathematician Mihály Fekete; then, discuss some extensions to the case of many variables and their implications in the theory of cellular automata, referring to two of my papers from 2008, one of them with Tommaso Toffoli and Patrizia Mentrasti.

Let’s start from the beginning: Continue reading

Nonuniversality in computation: A proof by semantic shift?

Today, the 8th of September 2016, we had a very interesting discussion about a theorem, due to Selim G. Akl, pointed to me in a tweet by Andy Adamatzky. Such theorem has, according to Akl, the consequence that the Church-Turing thesis, a basic tenet of theoretical computer science, is false. Of course, surprising statements require solid arguments: is Akl’s solid enough?

First of all, let us recall what the Church-Turing thesis is, and what it is not. Its statement, as reported by the Stanford Encyclopedia of Philosophy, goes as follows: Continue reading

Second-order theories should not be taken lightly

First-order formal logic is a standard topic in computer science. Not so for second-order logic: which, though used the default in fields of mathematics such as topology and analysis, is usually not treated in standard courses in mathematical logic. For today’s Theory Lunch I discussed some classical theorems that hold for first-order logic, but not for second-order logic: Continue reading

A concrete piece of evidence for incompleteness

On Thursday, the 25th of March 2015, Venanzio Capretta gave a Theory Lunch talk about Goodstein’s theorem. Later, on the 9th of March, Wolfgang Jeltsch talked about ordinal numbers, which are at the base of Goodstein’s proof. Here, I am writing down a small recollection of their arguments.

Given a base $b \geq 2$, consider the base- $b$ writing of the nonnegative integer $n = b^m \cdot a_m + b^{m-1} \cdot a_{m-1} + \ldots + b \cdot a_1 + a_0$

where each $a_i$ is an integer between $0$ and $b-1$. The Cantor base- $b$ writing of $n$ is obtained by iteratively applying the base- $b$ writing to the exponents as well, until the only values appearing are integers between $0$ and $b$. For example, for $b = 2$ and $n = 49$, we have $49 = 32 + 16 + 1 = 2^{2^2 + 1} + 2^{2^2} + 1$

and also $49 = 27 + 9 \cdot 2 + 3 + 1 = 3^3 + 3^2 \cdot 2 + 3 + 1$

Given a nonnegative integer $n$, consider the Goodstein sequence defined for $i \geq 2$ by putting $x_2 = n$, and by constructing $x_{i+1}$ from $x_i$ as follows: Continue reading

Playing with a beautiful mind

Today’s talk’s topic is an idea so important in game theory, and with so many applications in different fields including computer science, that it earned its discoverer, together with Reinhard Selten and John Harsanyi, the 1994 Nobel Memorial Prize in Economic Sciences.

To introduce this idea, together with other basic game-theoretic notions, we resort to some examples. Here goes the first one: Continue reading