By Heinrich Rickert

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First-order definable transformations of finite words have been considered in [26] and [32]. Some of these connections have been considered under a stronger semantics, the origin semantics, in [9]. This paper surveys some of these important results. All the transducer-logic connections presented in this paper are effective. The setting of functional transformations of finite words is presented in details, in contrast to the other results, which come nevertheless with the main bibliographic references.

N β1 . . βm . In particular, w = w = w for all words w ∈ Σ ∗ . For all w ∈ Σ ∗ and n ∈ N, we denote by wn the concatenation of w, n times. In particular, w0 = , w1 = w and w2 = ww. e. R ⊆ Σ ∗ × Σ ∗ . e. R(u) = {v ∈ Σ ∗ | (u, v) ∈ R}. The word u is usually called an input word while the words v such that (u, v) ∈ R are called output words. e. dom(R) = {u ∈ Σ ∗ | R(u) = ∅} and range(R) = {v ∈ Σ ∗ | ∃u ∈ Σ ∗ , v ∈ R(u)}. e. e. |R(u)| ≤ 1. Functional transformations are rather denoted by f, g, h .

The register update substitutions are represented on the edges by the the assignment operator :=, while the output function is represented by the vertical arrows leading to an expression, here U . The function f1/2 can also be defined with only one state, but using one additional register V , as depicted by Logic-Automata Connections for Transformations a U := V V := U a σ∈Σ U := U σ V := V σ 43 σ U := U σ σ U := σU q0 q0 a | U := U a q0 q1 q0 q0 a | U := U U U (a) f1/2 U UV UU U (b) f1/2 (c) fcopy (d) fcopy (e) frev Fig.

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