By Thomas Forster

Philosophical issues, that are frequently neglected or handled casually, are given cautious attention during this creation. Thomas Forster areas the thought of inductively outlined units (recursive datatypes) on the middle of his exposition leading to an unique research of good verified subject matters. The presentation illustrates tough issues and comprises many routines. Little earlier wisdom of good judgment is needed and just a wisdom of normal undergraduate arithmetic is believed.

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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.