Flavio D'Alessandro

A combinatorial problem on Trapezoidal words

In this paper, we investigate some combinatorial properties concerning the family of the so-called Trapezoidal words. Trapezoidal words, considered in de Luca (Theoret. Comput. Sci. 218 (1999) 13-39 are finite words over the two-letter alphabet A={a,b} whose subword complexity has the same behaviour as that of finite Sturmian words. In de Luca (Theoret. Comput. Sci. 218 (1999) 13-39 it has been proved that the family of Finite Sturmian words is properly contained in that one of Trapezoidal words. We carry on with the studying of the family of Trapezoidal words and, in particular, of its relation with that one of finite Sturmian words.

The Synchronization Problem for Strongly Transitive Automata

The synchronization problem is investigated for a new class of deterministic automata called strongly transitive. An extension to unambiguous automata is also considered.

The Synchronization Problem for Locally Strongly Transitive Automata

The synchronization problem is investigated for a new class of deterministic automata called locally strongly transitive. An application to synchronizing colorings of aperiodic graphs with a cycle of prime length is also considered.

Well quasi-orders and context-free grammars

Let G be a context-free grammar and let L be the language of all the words derived from any variable of G. We prove the following generalization of Higmans theorem: any division order on L is a well quasi-order on L. We also give applications of this result to some quasi-orders associated with unitary grammars.

Commutativity in free inverse monoids

This article characterizes the solutions of the commutativity equation xy = yx in free inverse monoids. The main result implies the following interesting property that is the natural generalization to free inverse monoids of the solutions of the same equation in free monoids. Let x and y be non-idempotent elements of a free inverse monoid such that xy = yx. Then there exist some elements χ and z such that x and y are conjugate by χ to some positive powers of z, namely xχ = χzn and yχ = χzm, with n, m⩾ 1. We also show that the centralizer of a given non-idempotent element is a rational, non-recognizable subset of the free inverse monoid.

Quasi-polynomials, linear Diophantine equations and semi-linear sets

We investigate the family of semi-linear sets of N^t and Z^t. We study the growth function of semi-linear sets and we prove that such a function is a piecewise quasi-polynomial on a polyhedral partition of N^t. Moreover, we give a new proof of combinatorial character of a famous theorem by Dahmen and Micchelli on the partition function of a system of Diophantine linear equations.

FREE GROUPS OF QUATERNIONS

This paper deals with the study of the free noncommutative group in the multiplicative group of the skewfield of the real Hamilton quaternions. The main results proved in this paper allows us to obtain the following interesting corollary: let G be a subgroup of rational quaternions. Then G is either solvable or contains the free noncommutative group.

On the separability of sparse context-free languages and of bounded rational relations

This paper proves two results. (1) Given two bounded context-free languages, it is recursively decidable whether or not there exists a regular language which includes the first and is disjoint with the second and (2) given two rational k-ary bounded relations it is recursively decidable whether or not there exists a recognizable relation which includes the first and is disjoint with the second.

Well quasi-orders, unavoidable sets, and derivation systems

Let I be a finite set of words and ⇒* I be the derivation relation generated by the set of productions {e → u |u ∈ I}. Let L ∈ I be the set of words u such that e ⇒* I u. We prove that the set I is unavoidable if and only if the relation ⇒* I is a well quasi-order on the set L ∈ I . This result generalizes a theorem of [Ehrenfeucht et al., Theor. Comput. Sci. 27(1983) 311-332]. Further generalizations are investigated.

On the commutative equivalence of bounded context-free and regular languages: The semi-linear case

This is the third paper of a group of three where we prove the following result. Let A be an alphabet of t letters and let ψ:A⁎⟶Nt be the corresponding Parikh morphism. Given two languages L1,L2⊆A⁎, we say that L1 is commutatively equivalent to L2 if there exists a bijection f:L1⟶L2 from L1 onto L2 such that, for every u∈L1, ψ(u)=ψ(f(u)). Then every bounded context-free language is commutatively equivalent to a regular language.