Knowledge-base for Wumpus world

As in the previous topic we have learned about the wumpus world and how a knowledge-based agent evolves the world. Now in this topic, we will create a knowledge base for the wumpus world, and will derive some proves for the Wumpus-world using propositional logic.

The agent starts visiting from first square [1, 1], and we already know that this room is safe for the agent. To build a knowledge base for wumpus world, we will use some rules and atomic propositions. We need symbol [i, j] for each location in the wumpus world, where i is for the location of rows, and j for column location.

Atomic proposition variable for Wumpus world:

• Let P_i,j be true if there is a Pit in the room [i, j].
• Let B_i,j be true if agent perceives breeze in [i, j], (dead or alive).
• Let W_i,j be true if there is wumpus in the square[i, j].
• Let S_i,j be true if agent perceives stench in the square [i, j].
• Let V_i,j be true if that square[i, j] is visited.
• Let G_i,j be true if there is gold (and glitter) in the square [i, j].
• Let OK_i,j be true if the room is safe.

Note: For a 4 * 4 square board, there will be 7*4*4= 122 propositional variables.

Some Propositional Rules for the wumpus world:

Note: lack of variables gives us similar rules for each cell.

Representation of Knowledgebase for Wumpus world:

Following is the Simple KB for wumpus world when an agent moves from room [1, 1], to room [2,1]:

Here in the first row, we have mentioned propositional variables for room[1,1], which is showing that room does not have wumpus(¬ W₁₁), no stench (¬S₁₁), no Pit(¬P₁₁), no breeze(¬B₁₁), no gold (¬G₁₁), visited (V₁₁), and the room is Safe(OK₁₁).

In the second row, we have mentioned propositional variables for room [1,2], which is showing that there is no wumpus, stench and breeze are unknown as an agent has not visited room [1,2], no Pit, not visited yet, and the room is safe.

In the third row we have mentioned propositional variable for room[2,1], which is showing that there is no wumpus(¬ W21), no stench (¬S₂₁), no Pit (¬P₂₁), Perceives breeze(B₂₁), no glitter(¬G₂₁), visited (V₂₁), and room is safe (OK₂₁).

Prove that Wumpus is in the room (1, 3)

We can prove that wumpus is in the room (1, 3) using propositional rules which we have derived for the wumpus world and using inference rule.

• Apply Modus Ponens with ¬S11 and R1:

We will firstly apply MP rule with R1 which is ¬S₁₁ → ¬ W₁₁ ^ ¬ W₁₂ ^ ¬ W₂₁, and ¬S₁₁ which will give the output ¬ W₁₁ ^ W₁₂ ^ W₁₂.

• Apply And-Elimination Rule:

After applying And-elimination rule to ¬ W₁₁ ∧ ¬ W₁₂ ∧ ¬ W₂₁, we will get three statements:
¬ W₁₁, ¬ W₁₂, and ¬W₂₁.

• Apply Modus Ponens to ¬S₂₁, and R2:

Now we will apply Modus Ponens to ¬S₂₁ and R2 which is ¬S₂₁ → ¬ W₂₁ ∧¬ W₂₂ ∧ ¬ W₃₁, which will give the Output as ¬ W₂₁ ∧ ¬ W₂₂ ∧¬ W₃₁

• Apply And -Elimination rule:

Now again apply And-elimination rule to ¬ W₂₁ ∧ ¬ W₂₂ ∧¬ W₃₁, We will get three statements:
¬ W₂₁, ¬ W₂₂, and ¬ W₃₁.

• Apply MP to S₁₂ and R4:

Apply Modus Ponens to S₁₂ and R₄ which is S₁₂ → W₁₃ ∨. W₁₂ ∨. W₂₂ ∨.W₁₁, we will get the output as W₁₃∨ W₁₂ ∨ W₂₂ ∨.W₁₁.

• Apply Unit resolution on W₁₃ ∨ W₁₂ ∨ W₂₂ ∨W₁₁ and ¬ W₁₁ :

After applying Unit resolution formula on W₁₃ ∨ W₁₂ ∨ W₂₂ ∨W₁₁ and ¬ W₁₁ we will get W₁₃ ∨ W₁₂ ∨ W₂₂.

• Apply Unit resolution on W₁₃ ∨ W₁₂ ∨ W₂₂ and ¬ W₂₂ :

After applying Unit resolution on W₁₃ ∨ W₁₂ ∨ W₂₂, and ¬W₂₂, we will get W₁₃ ∨ W₁₂ as output.

• Apply Unit Resolution on W₁₃ ∨ W₁₂ and ¬ W₁₂ :

After Applying Unit resolution on W₁₃ ∨ W₁₂ and ¬ W₁₂, we will get W₁₃ as an output, hence it is proved that the Wumpus is in the room [1, 3].