António Machiavelo

Chebyshev polynomials over finite fields and reversibility of σ-automata on square grids

Using number theory on function fields and algebraic number fields, we prove results about Chebyshev polynomials over finite prime fields to investigate reversibility of two-dimensional additive cellular automata on finite square grids. For example, we show that there are infinitely many primitive irreversible additive cellular automata on square grids when the base field has order two or three.

ON THE AVERAGE SIZE OF GLUSHKOV AND PARTIAL DERIVATIVE AUTOMATA

In this paper, the relation between the Glushkov automaton and the partial derivative automaton of a given regular expression, in terms of transition complexity, is studied. The average transition complexity of was proved by Nicaud to be linear in the size of the corresponding expression. This result was obtained using an upper bound of the number of transitions of . Here we present a new quadratic construction of that leads to a more elegant and straightforward implementation, and that allows the exact counting of the number of transitions. Based on that, a better estimation of the average size is presented. Asymptotically, and as the alphabet size grows, the number of transitions per state is on average 2. Broda et al. computed an upper bound for the ratio of the number of states of to the number of states of which is about ½ for large alphabet sizes. Here we show how to obtain an upper bound for the number of transitions in , which we then use to get an average case approximation. In conclusion, assymptotically, and for large alphabets, the size of is half the size of the . This is corroborated by some experiments, even for small alphabets and small regular expressions.

A Hitchhiker's Guide to descriptional complexity through analytic combinatorics

Nowadays, increasing attention is being given to the study of the descriptional complexity in the average case. Although the underlying theory for such a study seems intimidating, one can obtain interesting results in this area without too much effort. In this gentle introduction we take the reader on a journey through the basic analytical tools of that theory, giving some illustrative examples using regular expressions. Additionally, new asymptotic average-case results for several @e-NFA constructions are presented, in a unified framework. It turns out that, asymptotically, and in the average case, the complexity gap between the several constructions is significantly larger than in the worst case. Furthermore, one of the @e-NFA constructions approaches the corresponding @e-free NFA construction, asymptotically and on average.

ON THE AVERAGE STATE COMPLEXITY OF PARTIAL DERIVATIVE AUTOMATA: AN ANALYTIC COMBINATORICS APPROACH

The partial derivative automaton () is usually smaller than other nondeterministic finite automata constructed from a regular expression, and it can be seen as a quotient of the Glushkov automaton (). By estimating the number of regular expressions that have e as a partial derivative, we compute a lower bound of the average number of mergings of states in and describe its asymptotic behaviour. This depends on the alphabet size, k, and for growing ks its limit approaches half the number of states in . The lower bound corresponds to consider the automaton for the marked version of the regular expression, i.e. where all its letters are made different. Experimental results suggest that the average number of states of this automaton, and of the automaton for the unmarked regular expression, are very close to each other.

On the average size of glushkov and equation automata for KAT expressions

Kleene algebra with tests (KAT) is an equational system that extends Kleene algebra, the algebra of regular expressions, and that is specially suited to capture and verify properties of simple imperative programs. In this paper we study two constructions of automata from KAT expressions: the Glushkov automaton (

The average transition complexity of Glushkov and partial derivative automata

\mathcal{A}_{\mathsf{pos}}

On the invertibility of finite linear transducers

), and a new construction based on the notion of prebase (equation automata,

On the average number of states of partial derivative automata

\mathcal{A}_{\mathsf{eq}}

On the State Complexity of Partial Derivative Automata For Regular Expressions with Intersection

). Contrary to other automata constructions from KAT expressions, these two constructions enjoy the same descriptional complexity behaviour as their counterparts for regular expressions, both in the worst-case as well as in the average-case. In particular, our main result is to show that, asymptotically and on average the number of transitions of the

On the Equivalence of Automata for KAT -expressions

\mathcal{A}_{{\mathsf{pos}}}