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

Rational Transductions

A rational transducer MM is a 6-tuple (S,Σ,Δ,τ,s0,F)(S, \Sigma, \Delta, \tau, s_0, F), where SS is a finite set of states, Σ\Sigma and Δ\Delta are input and output alphabets respectively, the transition relation τ\tau is a finite relation between S×ΣS \times \Sigma^* and S×ΔS \times \Delta^*, s0Ss_0 \in S is the initial state and FSF\subseteq S is the set of final states.

For alphabets Σ,Δ\Sigma, \Delta, a transduction is a subset of Σ×Δ\Sigma^*\times\Delta^*. If AA is a transducer as above, then (A)\top (A) denotes its generated transduction, namely the set of all pairs (u,v)X×Y(u, v)\in X^*\times Y^* such that (s0,u,f,v)τ(s_0, u, f, v)\in \tau for some fFf \in F.

For a transduction TX×YT\subseteq X^*\times Y^* and a language LXL\subseteq X^*, we write TL={vYuL:(u,v)T}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 RXR\subseteq X^* and a rational transduction TX×YT\subseteq X^*\times Y^*, TRTR is regular.

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

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