A bimachine (Definition B.2.2 of Transducers) is a deterministic model for the rational functions. It consists of input and output alphabets $A$ and $B$, two deterministic automata over $A$ without accepting states — the prefix automaton and the suffix automaton — and an output function

On an input $a_1 \cdots a_n$ it considers, for every $i \in \{0, \ldots, n\}$, the factorisation into the prefix $a_1 \cdots a_i$ and the suffix $a_{i+1} \cdots a_n$, runs the prefix automaton on the prefix and the suffix automaton on the reverse of the suffix, applies the output function to the resulting pair of states, and concatenates the $n + 1$ pieces in increasing order of $i$. Bimachines compute exactly the rational functions (Theorem B.2.3). A bimachine is aperiodic if both its automata are aperiodic, i.e. satisfy the stabilisation condition on state transformations; aperiodic bimachines compute exactly the first-order relabellings (Theorem C.4.16).