By M. Lothaire

A chain of vital purposes of combinatorics on phrases has emerged with the advance of automatic textual content and string processing. the purpose of this quantity, the 3rd in a trilogy, is to provide a unified remedy of a few of the main fields of purposes. After an creation that units the scene and gathers jointly the elemental proof, there stick with chapters within which functions are thought of intimately. The parts coated comprise middle algorithms for textual content processing, typical language processing, speech processing, bioinformatics, and parts of utilized arithmetic reminiscent of combinatorial enumeration and fractal research. No particular necessities are wanted, and no familiarity with the appliance parts or with the cloth coated by means 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 booklet excellent for graduate scholars and execs in arithmetic, desktop technological know-how, biology and linguistics.

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The function Erase(v, U ) returns the set of half-edges obtained by erasing the preﬁx v of the words u appearing in the half-edges (u, q) ∈ U . In a second step, we build the set of states and the next state function of the resulting sequential transducer B. As for automata, we use a function Explore() which operates on the ﬂy. Explore(T , S, B) 1 T is a collection of sets of half-edges 2 S is an element of T 3 for each letter a do 4 (v, U ) ← Lcp(Next(S, a)) 5 NextB (S, a) ← (v, U ) 6 if U = ∅ and U ∈ / T then 7 T ←T ∪U 8 (T , B) ← Explore(T , U, B) 9 return (T , B) We can ﬁnally write the function realizing the determinization of a transducer into a sequential one.

The product of ρ and σ ⊂ A∗ × B ∗ is the relation ρσ = {(ur, vs) | (u, v) ∈ ρ, (r, s) ∈ σ}. Version June 23, 2004 40 Algorithms on Words The star of σ ⊂ A∗ × B ∗ is the relation σ ∗ = {(u1 u2 · · · un , v1 v2 · · · vn ) | (ui , vi ) ∈ σ, n ≥ 0}. A relation from A∗ to B ∗ is rational if it can be obtained from subsets of (A ∪ {ε}) × (B ∪ {ε}) by a ﬁnite number of operations of union, product and star. A rational relation that is a (partial) function is called a rational function. 4. g. on the alphabet {a, b} as ((a, aa) ∪ (b, bb))∗ .

The algorithm computing the composition of two transducers is easy to write. 5. 31. The right 2-shift. ComposeTransducers(S, T) 1 S and T are literal transducers 2 U ← NewTransducer() 3 for each edge (p, a, b, q) of S do 4 for each edge (r, b, c, s) of T do 5 add ((p, r), a, c, (q, s)) to the edges of U 6 for each edge (p, a, ε, q) of S do 7 for each state r of T do 8 add ((p, r), a, ε, (q, r)) to the edges of U 9 for each edge (r, ε, c, s) of T do 10 for each state p of S do 11 add ((p, r), ε, c, (p, s)) to the edges of U 12 InitialU ← InitialS × InitialT 13 TerminalU ← TerminalS × TerminalT 14 return U The composition can be used to compute an automaton that recognizes the image of a word (and more generally of a regular set) by a rational relation.