A problem is a situation (difficult/easy) experienced by an agent. The problem is solved by a sequence of actions that reduce the difference between the initial situation and the goal. A problem when defined with its components is called a well-defined problem.

Criteria for defining the problem:

**Initial State**: It is the starting state or initial step of the agent towards its goal.**Actions**: It is the description of the possible actions available to the agent.**Transition Model**: It describes what each action does.**Goal Test**: It determines if the given state is a goal state.**Path cost**: It assigns a numeric cost to each path that follows the goal. The problem-solving agent selects a cost function, which reflects its performance measure.

**Constraint Satisfaction Problem:**

The problem is intended to illustrate or exercise various problem-solving methods. It can be given a concise, exact description and hence is usable by different researchers to compare the performance of algorithms.

A constraint satisfaction problem (CSP) consists of

➢ a set of variables,

➢ a domain for each variable, and

➢ a set of constraints.

The aim is to choose a value for each variable so that the resulting possible world satisfies the constraints.

Example: **Crypto Arithmetic Problem**

States: Arrangement of any numbers on the given arithmetic.

Initial State: Unsolved digits.

Actions: Replace the digits with unique numbers.

Transition model: Returns the arithmetic with the added numbers.

Goal test: Checks whether all the bits are replaced.

Path cost: There is no need for path cost because only final states are counted.

Solution:

A B C

+D E F

=G H I

=>

1. A ≠ B ≠ C ≠ D ≠ E ≠ F ≠ G ≠ H ≠ I

2. C + F = I

C + F = 10 + I (I as carry)

3. B + E = H

B + E = 10 + H

B + E + 1 = H

B + E + 1 =10 + H (H as carry)

4. A + D = G

A + D + 1 = G

Step 1:

Domain of C = {1, 2, 3, 4, 5, 6, 7, 8, 9}

Domain of F = {1, 2, 3, 4, 5, 6, 7, 8, 9}

So,

Domain of I = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}

Select C = 4 & F = 9 Then I = 3 (Carry = 1)

So,

A B 4

+D E 9

=G H 3

Step 2:

Domain of B = {1, 2, 5, 6, 7, 8}

Domain of E = {1, 2, 5, 6, 7, 8}

So,

Domain of H = {0, 1, 2, 5, 6, 7, 8}

Select B = 2 & E = 8

Then H = 10+1 {previous carry = 1 + (Carry = 1)}

So,

A 2 4

+D 8 9

G 1 3

Step 3:

Domain of A = {0, 5, 6, 7}

Domain of D = {0, 5, 6, 7}

So, Domain of G = {5, 6}

Select A = 0 & D = 5 Then G = 6 (with addition of Carry)

So,

Hence, the required solutions are:

A = 0, B = 2, C = 4, D = 5, E = 8, F = 9, G = 6, H = 1, I = 3.

0 2 4

+5 8 9

6 1 3

**Real-World Problem:**

Those problem whose solution people actually care about. Such problems tend not to have a single agreed upon description, but we can give the general flavour of their formulations.

Example:

Consider the airline travel problems that must be solved by a travel-planning website:

States: Each state obviously includes a location (like an airport) and the current time. Furthermore, because the cost of an action ( a flight segment) may depend on previous segments, their fare bases, and their status of domestic or international, the state must record extra information about these “historical” aspects.

Initial state: Specified by user’s query.

Actions: Take any flight from current location, in any seat class, leaving after the current time, leaving enough time for within-airport transfer if needed.

Transition model: The state resulting from taking a flight will have the flight’s destination as the current location and the flight’s arrival time as the current time.

Goal test: Are we at the final destination specified by the user?

Path cost: This depends on monetary cost, waiting time, flight time, customs, immigration procedure, seat quality, time of day, type of airplane, frequent-flyer mileage awards and so on.

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