$1.[15$ points$]$ Formulate the $\backslash (5\backslash$times $5\backslash )$ Futoshiki puzzle as described below as a Constraint Satisfaction Problem $($CSP$.)$ Define the set of variables, their domains values, and the set of constraints according to the rules of the puzzle and the initial game board configuration in the figure below. $($You do not have to apply the Backtracking Algorithm to generate a search tree and find solution to the problem. Just formulate the problem.$)$

$\backslash (5\backslash$times $5\backslash )$ Futoshiki puzzle

$-$ The game board consists of $\backslash (5\backslash$times $5\backslash )$ cells. Some of the cells already have numbers assigned to them and there are inequality signs $(\backslash (<mtext></mtext><mi>\backslash </mi><mi>)</mi>$ or $\backslash (>\backslash ))$ between some of the adjacent cells as shown in the figure below.

$-$ The goal is to find assignments $(1$ to $5)$ for the empty cells such that the inequality relationships between all pairs of adjacent cells with an inequality sign between them are satisfied. In addition, a digit can only appear once in every row and column; that is$,$ the digits in every row and column must be different from each other. $-$ Backtracking search algorithm for CSP:

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function BACKTRACKING$-$SEARCH$($CSp$)$ returns solution or failure

return BACKTRACK$(\{\},$ CSP$)$

function BACKTRACK$($assignment$,$ csp$)$ returns a solution or failure

if assignment is complete then return assignment

var$\}\backslash$leftarrow$\backslash$mathrm$\{$ SELECT$-$UNASSIGNED$-$VARIABLE$($csp$)$

for each value in ORDER$-$DOMAIN$-$VALUES$($var$,$ assignment, csp$)$ do

if value is consistent with assignment then

add $\{$var $=$ value$\}$ to assignment

inferences $\backslash$leftarrow INFERENCE$($csp$,$ var, value$)$

if inferences $!=$ failure then

add inferences to assignment

result & BACKTRACK$($assignment$,$ csp$)$

if result $/=$ failure then

return result

remove $\{$var$=$value$\}$ and inferences from assignment

return failure

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