Problems:

A problem can be defined formally by four components:

1] Initial state

2] Actions

3] Goal test

4] Path cost

1] Initial state.

The Initial state that the agent start in example: the Initial state for our agent in the Romania might be described as In(Arad).

2] Actions.

A description of the possible actions available to the agent. The most common formulation uses a successor function.

Given a particular state X , SUCCESSOR – FN(x) returns a set of <action, successor> ordered pairs, where each action is one of the legal actions in state X and each successor is a state that can be reached from X. by applying the action. Example: From the state In(Arad), the successor function for the Romania problem would return,

{ < Gocsibiu.), In (sibiu) >, < Go (Timisoara ),

In(Timisoara)>, <Go(Zerin), In(Zerind))}</p>

Together, the initial state and success or function implicitly define the state space of the problem – the set of all states reachable from the initial state. The state space forms a graph in which nodes are states and the ares between nodes art actions. A path in the state space is a sequence of states connected by a sequence of actions.

3] The goal test, which determines whether a given state is a goal state.

4] A path cost function that assigns a numeric cost to each path. The problem solving agent chooses a cost function that reflects its own performance measure. For the agent trying to get Bucharest, time is of the essence, so the cost of a path might be its length in kilometer.

The step cost of taking action a to go from state X to state Y is denoted by c(x,a,y).

A solution to a problem is a path from the initial state to a goal state. Solution quality is measured by the path cost function and an optimal solution has the lowest path cost among all solutions.

Figure (2) show the step cost for Romania as route this.

Example – problems.

Toy and real world problems.

A toy problem is intended to illustrate or exercise various problem solving methods. It can be given a concise, exact description. This means that it can be used easily by different researchers to compare the performance of algorithms.

A real world problem is one whose solutions people actually care about.they tend not to have a single agreed-upon description, but we will attempt to give the general flavor of their formulations.

Toy problems.

1] Vacuum world. This can be formulated as a problem as follows:

- State: The agent is in one of the two locations, each of which might or might not contain dirt. Thus, therefore, $2 \times 2^2 = 8$ possible world std.

- Initial state: Any state can be designated as the initial state.

- Successor function : This generates the legal states that results from trying the three action (left, right and suck). The complete state space is shown in figure (3).

- Goal test: This checks whether all the are clean.

- Path cost : Each step costs 1, so the path cost is the number of steps in the path. Figure (3) The state space or the vacuum world, Ares denote actions: L = Left, R = Right, S = Suck.

Compared with the real world, this toy problem has discrete locations, discrete dirt, reliable cleaning, and it never gets messed up once cleaned. One important thing to note is that the state is determined by both the agent location and the dirt locations. A larger environment with n locations has $h2^n$ states.