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

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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Markdown	Appears as
`italics` or `_italics_`	italics
`bold` or `__bold__`	bold
- bulleted - list	bulleted list
1. numbered 2. list	numbered list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1 paragraph 2	paragraph 1 paragraph 2
`[example link](https://brilliant.org)`	example link
`> This is a quote`	This is a quote
# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"	# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"

Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$