To reformulate a state$-$space search problem as a Constraint Satisfaction Problem $($CSP$),$ we need to map each component of the state$-$space search problem to the corresponding components of a CSP$.$ Here's how we can do it:

Variables: Define variables representing each step in the plan up to the maximum number of steps,

$$

T$.$ Let's denote these variables as

$$

$0$

$,$

$$

$1$

$,$

$.$

$.$

$.$

$,$

$$

$$

X

$0$

$$

$,$X

$1$

$$

$,...,$X

T

$$

$,$ where each variable

$$

$$

X

t

$$

represents the action to be taken at step

$$

t$.$ The domain of each variable

$$

$$

X

t

$$

includes all actions in

$$

A$,$ including the NoOp action.

Initial State: Define a constraint that enforces the initial state. This can be represented by an equality constraint where

$$

$0$

X

$0$

$$

is assigned the action that corresponds to the initial state

$$

$0$

s

$0$

$$

$.$

Transition Model: Define constraints that ensure that the transition from one state to another is valid according to the transition model. For each step

$$

t from $0$ to

$$

$$

$1$

T$1,$ we have a constraint that enforces that the action

$$

$$

X

t

$$

results in the state

$$

$$

$+$

$1$

s

t$+1$

$$

according to the transition model.

Goal States: Define constraints that enforce the goal states. For each possible goal state

$$

g in

$$

G$,$ we have a constraint that checks if the last action in the plan leads to

$$

g$.$

With this formulation, finding a solution to the CSP will yield a plan represented by the sequence of actions

$$

$0$

$,$

$$

$1$

$,$

$.$

$.$

$.$

$,$

$$

$$

X

$0$

$$

$,$X

$1$

$$

$,...,$X

T

$$

that transforms the initial state

$$

$0$

s

$0$

$$

into a goal state

$$

$$

s

T

$$

$.$ The plan can be easily extracted by looking at the values assigned to the

$$

$$

X

t

$$

variables.

This reformulation allows us to use a CSP solver to find a valid plan for the state$-$space search problem, even if we don't have access to traditional search algorithms. It provides a systematic way to represent and solve planning problems using constraints.