Exercise $2(60$ points$).$ Consider the following world and agent:

$-$ We have discrete locations $\backslash (1,2,3\backslash$ldots $\backslash )$ and times $\backslash (0,1,2\backslash$ldots $\backslash ).$

$-$ The agent can only be in one location at any given time.

$-$ The agent can only take one of two actions at any time: either move $($always to the next location, i$.$e$.,$ from $\backslash ($ i $\backslash )$ to $\backslash ($ i$+1\backslash )),$ or grab $($but grabbing can only be done once during the game$).$ Executing the grab action results in the agent catching the gold only if it is at a location that contains gold.

$-$ At the start of the game, the agent is at location $1,$ and there is gold in exactly one location.

$-$ If at some time $\backslash ($ t $\backslash )$ the agent is at a location containing the gold, it perceives it$,$ as a symbol gold $\backslash (\{\}^\{$t$\}\backslash ).$ Otherwise, it does not perceive it: $\backslash ($

eg $\backslash )$ gold $\backslash (\{\}^\{$t$\}\backslash ).$

$-$ The goal is for the agent to catch the gold.

Design a propositional logic system that could be used to model this:

$1.(10$ points$)$ Create the relevant symbols and explain their meaning.

$2.(20$ points$)$ List relevant axioms and explain their meaning.

$3.(5$ points$)$ List relevant sentences that encode the start of the game.

$4.(5$ points$)$ Formulate an assertion $\backslash (\backslash$alpha $\backslash )$ to indicate success $($which we would use, together with the knowledge base KB composed of your axioms and sentences above, to formulate a plan, by calling Ask$($KB$,\backslash (\backslash$alpha $\backslash ))).$

Apply to the particular case where the gold is at location $3.$ Running SATplan on your KB with a suitable goal would produce a plan $($sequence of actions$)$ that achieves the goal, if such a plan exists $($and assuming you correctly formulated your KB$),$ but this would involve many symbols and sentences and it would be too complicated. Instead, just play out two scenarios where the agent acts in two ways:

$5.(10$ points$)$ Give, for times $0$ to $4,$ the truth values of all symbols $($including the percepts$)$ for a successful agent behavior $($the agent takes a sequence of actions that lead to getting the gold$).$ You must obtain the values by applying your axioms to your initial state formulation $($not by guessing what the values should be$).$ Tabulate them like this $($where $0$ and $1$ stand for false and true, respectively$)$:

$6.(10$ points$)$ Repeat but for an unsuccessful agent behavior $($the agent takes a sequence of actions that lead to not getting the gold$).$