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For some text algorithms, the real measure for the complexity analysis is not the string itself but its structure stored in its prefix table or equivalently border table. In this paper, we define the combinatorial class of prefix lists, namely a sequence of integers together with their size, and an injection ψ from the class of prefix tables to the class of prefix lists. We call a valid prefix list the image by ψ of a prefix table. In particular, we describe algorithms converting a prefix/border table to a prefix list and inverse linear algorithms from computing from a prefix list L = ψ(P) two words respectively in a minimal size alphabet and on a maximal size alphabet with P as prefix table. We then give a new upper bound on the number of prefix tables for strings of length n (on any alphabet) which is of order (1 + ϕ)n (with $\varphi=\frac{1+\sqrt{5}}{2}$ the golden mean) and also present a corresponding lower bound.

For some text algorithms, the real measure for the complexity analysis is not the string itself but its structure stored in its prefix table (or border table, as border and prefix tables can be proved to be equivalent). We give a new upper bound on the number of prefix tables for strings of length n (on any alphabet) which is of order (1 + φ) n (with \(\varphi={{1+\sqrt{5}}\over{2}}\) the golden mean) and present also a lower bound.

In this paper we generalize an encoding method due to Girod (cf. [6]) using prefix codes, that allows a bidirectional decoding of the encoded messages. In particular we generalize it to any finite alphabet A, to any operation defined on A, to any code with finite deciphering delay and to any key x ∈ A+ , on a length depending on the deciphering delay. We moreover define, as in [4], a deterministic transducer for such generalized method. We prove that, fixed a code X ∈ A* with finite deciphering delay and a key x ∈ A *, the transducers associated to different operations are isomorphic as unlabelled graphs. We also prove that, for a fixed code X with finite deciphering delay, transducers associated to different keys have an isomorphic non trivial strongly connected component.

In this paper we describe a "light" algorithm for the on-line construction of a small automaton recognising a finite set of words. The algorithm runs in linear time. We carried out good experimental results on real dictionaries, on biological sequences and on the sets of suffixes (resp. factors) of a set of words that shows how our automaton is near to the minimal one. For the suffixes of a text, we propose a modified construction that leads to an even smaller automaton. We moreover construct linear algorithms for the insertion and deletion of a word in a finite set, directly from the constructed automaton.

We construct a transducer for the bidirectional decoding of words encoded by the method introduced by Girod (1999) in [5] and we prove that it is bideterministic and that it can be used both for the left-to-right and the right-to-left decoding.We also give a similar construction for a transducer that decodes in both directions words encoded by a generalization of Girod’s encoding method. We prove that it has the same properties as those of the previous transducer. In addition we show that it has a single initial/final state and that it is minimal.

We investigate the intersection of two finitely generated submonoids of the free monoid on a finite alphabet. To this purpose, we consider automata that recognize such submonoids and we study the product automata recognizing their intersection. By using automata methods we obtain a new proof of a result of Karhumaki on the cha- racterization of the intersection of two submonoids of rank two, in the case of prefix (or suffix) generators. In a more general setting, for an arbitrary number of generators, we prove that if H and K are two finitely generated submonoids generated by prefix sets such that the product automaton associated to H ∩ K has a given special property then �(H ∩ K) ≤ �(H)�(K )w here�(L )=m ax(0,rk(L) − 1) for any submonoid L. Mathematics Subject Classification. 68Q70, 68Q45, 20M35.

Submitted, 21 pages.; International audience; Considering the uniform distribution on sets of m non-empty words whose sum of lengths is n, we establish that the average state complexities of the rational operations are asymptotically linear.