Q$\mathrm{.}1$ Temporal Planning $/$ STN

$($a$)$ This question is about temporal planning. For reference, here is the evaluator domain.

$($define $($domain elevators$)$

$($:requirements :typing$)($:types elevator passenger num $-$ object$)$

$($:predicates

$($passenger$-$at $?$person $-$ passenger $?$floor $-$ num$)$

$($boarded $?$person $-$ passenger $?$lift $-$ elevator$)$

$($lift$-$at $?$lift $-$ elevator $?$floor $-$ num$)$

$($next $?$n$1-$ num $?$n$2-$ num$)$

$)$

$($:action move$-$up

:parameters $(?$lift $-$ elevator $?$cur $?$nxt $-$ num$)$

:precondition $($and $($lift$-$at $?$lift $?$cur$)($next $?$cur $?$nxt$))$

:effect $($and $($not $($lift$-$at $?$lift $?$cur$))($lift$-$at $?$lift $?$nxt$))$

$)$

$($:action move$-$down

:parameters $(?$lift $-$ elevator $?$cur $?$nxt $-$ num$)$

:precondition $($and $($lift$-$at $?$lift $?$cur$)($next $?$nxt $?$cur$))$

:effect $($and $($not $($lift$-$at $?$lift $?$cur$))($lift$-$at $?$lift $?$nxt$))$

$)$

$($:action board

:parameters $(?$person $-$ passenger $?$floor $-$ num $?$lift $-$ elevator$)$

:precondition $($and $($lift$-$at $?$lift $?$floor$)($passenger$-$at $?$person $?$floor$))$

:effect $($and $($not $($passenger$-$at $?$person $?$floor$))($boarded $?$person $?$lift$))$

$)$

$($:action leave

:parameters $(?$person $-$ passenger $?$floor $-$ num $?$lift $-$ elevator$)$

:precondition $($and $($lift$-$at $?$lift $?$floor$)($boarded $?$person $?$lift$))$

:effect $($and $($passenger$-$at $?$person $?$floor$)($not $($boarded $?$person $?$lift$)))$

$)$

$)$

Consider the following problem and plan:

$($define $($problem elevators$-$problem$)$

$($:domain elevators$)$

$($:objects n$1$ n$2$ n$3-$ num p$1$ p$2-$ passenger e$1-$ elevator$)$

$($:init

$($next n$1$ n$2)$

$($next n$2$ n$3)$

$($lift$-$at e$1$ n$1)$

$($passenger$-$at p$1$ n$2)$

$($passenger$-$at p$2$ n$2)$

$)$

$($:goal $($and $($passenger$-$at p$1$ n$1)$

$($passenger$-$at p$2$ n$3)$

$)$

$)$

$)$

P$1)0\mathrm{.}000$: $($move$-$up e$1$ n$1$ n$2)[0\mathrm{.}001]$

P$2)0\mathrm{.}001$: $($board p$2$ n$2$ e$1)[0\mathrm{.}001]$

P$3)0\mathrm{.}002$: $($move$-$up e$1$ n$2$ n$3)[0\mathrm{.}001]$

P$4)0\mathrm{.}004$: $($move$-$down e$1$ n$3$ n$2)[0\mathrm{.}001]$

P$5)0\mathrm{.}005$: $($move$-$down e$1$ n$2$ n$1)[0\mathrm{.}001]$

P$6)0\mathrm{.}006$: $($leave p$1$ n$1$ e$1)[0\mathrm{.}001]$

Is this plan valid? If not, list all of the reasons that make it invalid.

$($b$)$ Using the above example, consider converting the actions into the durative actions as following:

$-$ Passenger p$1$ takes $2$ seconds for each action: board evaluator, leave evaluator

$-$ Passenger p$2$ takes $3$ seconds for each action: board evaluator, leave evaluator

$-$ Evaluator e$1$ takes $1$ seconds to move up or down $3$ meters

$-$ Distance between floor numbers n$1$ and n$2$ is $3$ meters

$-$ Distance between floor numbers n$2$ and n$3$ is $4$ meters

List all the actions of a valid plan while considering to reduce the total time taken to reach the same goal. $($c$)$ Using the following STN$,$ convert upper$-$bound$/$lower$-$bound format to single$-$bound format.

$($d$)$ Using floyd$-$warshall algorithm, convert the following graph of STN to all$-$pairs shortest path graph.