Suppose that a CS student wants to encode events taking place in an ice$-$hockey game to a string

using alphabet $\backslash$Gamma $=\{$a$,$ b$,$ c$,1,2,3,$ x$,$ y$,|\},$ where

a$,$ b$,$ c $1$st$,2$nd and $3$rd wing of the home team enters the ice

$1,2,31$st$,2$nd and $3$rd wing of the visiting team enters the ice

x home team scores

y visitor scores

$|$ end of period

For simplicity, we assume that only full wings can be changed, which takes place when the game

is paused $($i$.$e$.,$ individual players cannot change at anytime$).$ Valid strings include exactly two $|$

delimiters to indicate the end of the $1$st and $2$nd period. The game ends when the string ends, i$.$e$.,$

there is no final $|$ symbol. Moreover, at the start of each period both teams must put some wing

on ice, i$.$e$.,$ each period starts with two symbols of which one is from $\{$a$,$ b$,$ c$\}$ and the other from

$\{1,2,3\}($in either order$).$ After that wings can change and both teams can score. Hence, e$.$g$.,$

a$2$b$1|$a$1$x$|3$c is a valid $($yet short$)$ string corresponding to a game that home team won $10.$

a$)$ Let A denote the language corresponding to all valid strings, i$.$e$.$ entries that are correctly

recorded. Show that A is a regular language. Hint: give first a RE for one period.

b$)$ Let B denote those interesting games where at no point in time either team leads by more than

$1$ goal. Is B regular or non$-$regular? $($Argue why!$)$

c$)$ Plus$-$minus statistic is an often used performance metric: a goal means $+1$ for the wing, and

when opponent scores the wing on ice receives $-1.$ Let C denote the games where home team$$s

$1$st wing has a positive plus$-$minus statistic. Show that C is context$-$free.