# Tessellate S.T.E.M.S. (2019) - Computer Science - College - Set 4 - Subjective Problem 1

Rational Transductions

A rational transducer $M$ is a 6-tuple $(S, \Sigma, \Delta, \tau, s_0, F)$, where $S$ is a finite set of states, $\Sigma$ and $\Delta$ are input and output alphabets respectively, the transition relation $\tau$ is a finite relation between $S \times \Sigma^*$ and $S \times \Delta^*$, $s_0 \in S$ is the initial state and $F\subseteq S$ is the set of final states.

For alphabets $\Sigma, \Delta$, a transduction is a subset of $\Sigma^*\times\Delta^*$. If $A$ is a transducer as above, then $\top (A)$ denotes its generated transduction, namely the set of all pairs $(u, v)\in X^*\times Y^*$ such that $(s_0, u, f, v)\in \tau$ for some $f \in F$.

For a transduction $T\subseteq X^*\times Y^*$ and a language $L\subseteq X^*$, we write $TL=\{v\in Y^*|\exists u\in L:(u,v)\in T\}$.

Prove/disprove the following statements:

• Composition of two rational transductions is a rational transduction.
• Inverse of a rational transduction is a rational transduction.
• For a regular language $R\subseteq X^*$ and a rational transduction $T\subseteq X^*\times Y^*$, $TR$ is regular.

This problem is a part of Tessellate S.T.E.M.S (2019)

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