By M. Lothaire

A sequence of significant purposes of combinatorics on phrases has emerged with the advance of automated textual content and string processing. the purpose of this quantity, the 3rd in a trilogy, is to give a unified therapy of a few of the foremost fields of functions. After an advent that units the scene and gathers jointly the elemental proof, there keep on with chapters during which purposes are thought of intimately. The components coated contain center algorithms for textual content processing, ordinary language processing, speech processing, bioinformatics, and components of utilized arithmetic similar to combinatorial enumeration and fractal research. No specific must haves are wanted, and no familiarity with the applying components or with the fabric coated by way of the former volumes is needed. The breadth of software, mixed with the inclusion of difficulties and algorithms and an entire bibliography will make this publication perfect for graduate scholars and pros in arithmetic, laptop technological know-how, biology and linguistics.

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Extra resources for Applied Combinatorics on Words

Example text

20. The minimization procedure starts with the partition e = {1, 3, 4}{2}. Since a−1 e = e, the action of letter a does not reﬁne e. On the contrary, b−1 e = {1, 4}{2}{3} and thus e is reﬁned to f = {1, 4}{2}{3} which is found to be stable. 20 on the right. There is a more complicated but more eﬃcient algorithm, known as Hopcroft’s algorithm, which can be used to minimize deterministic automata. We assume that the automaton is complete. The idea is to replace the global operation of intersection of two partitions by the reﬁnement of a partition by a single block.

The relation between a word written in lower-case letters a, b, c, . . and the corresponding upper-case letters A, B, C, . . is rational. Indeed, it is described by the expression ((a, A) ∪ (b, B) ∪ . )∗ . 5. 27. A transducer reads the input and writes the output. 28. 28. From lower case to upper case. synchronous. 6. 29. 29. The Fibonacci morphism. 29 realizes the same morphism. It is literal. 7. 30 realizes the circular right shift on a word on the alphabet {a, b} ending with letter a. The transformation consists in shifting cyclically each symbol one place to the right.

Of course, for any edge (p, a, q) one has h(p) > h(q). Since the automaton is trim, its initial state is the unique Version June 23, 2004 32 Algorithms on Words state of maximal height. The heights satisfy the formula h(p) = 0 1 + max(p,a,q) h(q) if p has no outgoing edge, otherwise. In the second case, the maximum is taken over all edges starting in p. Observe that this formula leads to an eﬀective algorithm for computing heights because the automaton has no cycle. The parameters in the algorithm are the number n of states of A, the number m of transitions, and the size k of the underlying alphabet.