Wumpus World

Inspired by a warm conversation with Prof. R K Mishra & Prof. Devender Singh, IIT BHU

Game Rules (Section 7.2)

Objective

Find the gold ✨, grab it, and climb out at [1,1] alive. Maximize your score.

The Cave

Grid — 4×4 (Classic) or 6×6 (Large/Nightmare)
Agent starts at [1,1] (bottom-left), facing right
Wumpus 👹 — a beast in one cell. Kills you if you enter. Doesn't move.
Pits 🕳️ — bottomless pits. Each cell (except [1,1]) has 20% chance of a pit.
Gold ✨ — hidden in one cell.

Percepts (what you sense)

Percept	Means
🟢 Stench	Wumpus is in an adjacent cell (up/down/left/right)
💨 Breeze	A pit is in an adjacent cell
✨ Glitter	Gold is in this cell — use Grab!
Bump	You walked into a wall (no movement)
🎯 Scream	Your arrow killed the Wumpus

Scoring

Event	Points
Grab gold + climb out at [1,1]	+1000
Death (pit or Wumpus)	−1000
Each action	−1
Shooting the arrow	−10

Arrow Rules

You have exactly 1 arrow
The arrow flies in a straight line in the direction you're currently facing
It travels through cells until it hits a Wumpus (kill!) or a wall (miss)
Your facing direction = the direction of your last move

Controls

Movement (also changes facing)

Input	Action
↑ button / Arrow Up / W	Move up (face up)
↓ button / Arrow Down / S	Move down (face down)
← button / Arrow Left / A	Move left (face left)
→ button / Arrow Right / D	Move right (face right)
Swipe on touchscreen	Move in swipe direction

Turning (change facing without moving)

Input	Action
↩ Q button / Q key	Turn left (counter-clockwise)
E ↪ button / E key	Turn right (clockwise)

Turning costs −1 point (it's an action). Use this to aim your arrow before shooting!

Actions

Input	Action
✋ Grab / G key	Pick up gold (if in current cell)
🏹 Shoot / F key	Fire arrow in facing direction (straight line)
🧗 Climb / C key	Climb out — only works at [1,1]
🔄 New	Start a new game

Grid Symbols

Symbol	Meaning
🧑‍🚀➡ 🧑‍🚀⬆ 🧑‍🚀⬇ 🧑‍🚀⬅	You (explorer + arrow shows facing direction)
👹	Wumpus (revealed on death/game end)
🕳️	Pit (revealed on death/game end)
✨	Gold
✅	KB inferred: cell is safe
❓	Unknown — cannot determine safety
☠️	Dead Wumpus (killed by your arrow)

1. Environment Definition (Ch. 2)

Section 2.3 PEAS Description

PEAS	Wumpus World
Performance	+1000 gold & escape, −1000 death, −1/action, −10 arrow
Environment	4×4 (or N×N) grid cave with pits, wumpus, gold
Actuators	Forward, TurnLeft, TurnRight, Grab, Shoot, Climb
Sensors	Stench, Breeze, Glitter, Bump, Scream

Section 2.3.2 Environment Properties

Property	Value
Observable	Partially — only current cell percepts
Deterministic	Yes in Classic; No in Stochastic mode
Episodic / Sequential	Sequential — past actions affect future
Static / Dynamic	Static — world doesn't change while agent thinks
Discrete / Continuous	Discrete
Single / Multi-agent	Single agent (Wumpus is environment, not agent)

→ Play Classic mode to observe all these properties firsthand.

2. Knowledge-Based Agents (Ch. 7)

Section 7.1 Core Architecture

A knowledge-based agent maintains a Knowledge Base (KB) of sentences in a formal language. It reasons by:
1. TELL — add new percept to KB
2. ASK — query KB to decide action
3. Act — perform the chosen action

Section 7.2 Wumpus World Specification

Grid: 4×4, agent starts at [1,1] facing right
Squares adjacent to Wumpus → Stench
Squares adjacent to Pit → Breeze
Square with gold → Glitter
Pits: each square (except [1,1]) has P=0.2 of containing a pit
Agent has exactly 1 arrow

→ Open the "Knowledge Base" panel above the grid to see this in action!

Propositional Logic Inference

Section 7.3–7.5

Key idea: encode percepts as propositions and use inference rules.

¬B₁,₁ → ¬P₁,₂ ∧ ¬P₂,₁

"No breeze at [1,1] implies no pit at [1,2] and [2,1]"

S₁,₂ ∧ ¬S₁,₁ → W₁,₃ ∨ W₂,₂

"Stench at [1,2] but not [1,1] → Wumpus is at [1,3] or [2,2]"

Methods to determine entailment (KB ⊨ α):

Model checking — enumerate all possible worlds (exponential)
Resolution — prove by contradiction using CNF
Forward/Backward chaining — for Horn clauses

→ In Classic mode, the KB panel shows exactly these inferences as you explore.

3. First-Order Logic (Ch. 8–9)

Section 8.3

Propositional logic needs a separate symbol for each cell: P₁,₁, P₁,₂, ... P₄,₄ (16 symbols just for pits!).
FOL uses variables and quantifiers — one rule covers all cells:

∀x,y Breeze(x,y) ⟺ ∃a,b (Adjacent(x,y,a,b) ∧ Pit(a,b))

"A cell has breeze iff some adjacent cell has a pit" — covers any grid size with one sentence.

∀x,y ¬Breeze(x,y) → ∀a,b (Adjacent(x,y,a,b) → ¬Pit(a,b))

"No breeze means all neighbors are pit-free."

→ Try Large Cave (6×6) mode — notice how many propositions you'd need vs. one FOL rule.

4. Uncertainty & Probability (Ch. 13)

Section 13.1

When the agent faces unknown cells with conflicting information, logical entailment can't decide. Instead, compute P(Pit | percepts) using Bayes' theorem:

P(Pit₃,₁ | known) = α · P(known | Pit₃,₁) · P(Pit₃,₁)

Where:

P(Pit₃,₁) = 0.2 (prior — each cell has 20% pit chance)
P(known | Pit₃,₁) = likelihood of observed breezes given pit there
α = normalization constant

Section 13.2 Bayesian Network for Wumpus

The book builds a Bayesian network where:

Pit nodes are parents of Breeze nodes
Observed breezes are evidence
Inference computes posterior probability of pits in unvisited cells

→ In Stochastic mode, your moves slip 20% of the time — pure logic fails, you must reason probabilistically about which cells are safe to attempt to reach.

5. Sensor Models & Filtering (Ch. 15)

Section 15.1–15.2

What if your sensors are unreliable? A breeze might be a false positive, or you might miss a real stench.
The sensor model defines: P(percept | true_state)

P(Stench_observed | Wumpus_adjacent) = 0.85

P(Stench_observed | ¬Wumpus_adjacent) = 0.15

Now even "safe" inferences might be wrong. You need to maintain belief states — probability distributions over possible worlds, updated at each step.

This connects to:

Hidden Markov Models — state changes over time, observations are noisy
Filtering — P(state_t | observations₁:t)

→ In Noisy Sensors mode, percepts lie 15% of the time. Watch the KB make wrong inferences! Dashed orange borders = uncertain cells.

6. Decision Theory (Ch. 16)

Section 16.1–16.3

With uncertain actions AND uncertain sensors, the agent needs a full decision-theoretic framework:
MEU — Maximum Expected Utility: choose the action that maximizes expected utility over all possible outcomes.

action* = argmax_a Σ_s P(s|evidence) · U(s, a)

Where U(s,a) accounts for:

Probability of dying (pit/wumpus) if you move there
Probability of slipping to wrong cell
Expected future reward (reaching gold)
Cost of each action (−1 per step)

→ Nightmare mode combines everything: stochastic moves + noisy sensors + 6×6 + 2 Wumpi. Pure decision theory territory!

7. Quick Reference — Symbols in Game

In Game	Formal	Meaning
💨 Breeze	B_x,y	Adjacent cell has a pit
🟢 Stench	S_x,y	Adjacent cell has Wumpus
✨ Glitter	G_x,y	Gold is in this cell
✅ Safe	KB ⊨ ¬P ∧ ¬W	Inferred: no pit and no Wumpus
❓ Unknown	KB ⊭ safe	Cannot determine safety
⚠️ SLIPPED	P(s'\|s,a) ≠ 1	Non-deterministic transition
🚫 sensors unreliable	P(obs\|state) < 1	Sensor model is noisy

8. Mode → Chapter Mapping

Game Mode	Chapters	Concepts Practiced
📖 Classic	Ch. 2, 7	PEAS, propositional logic, KB agents, entailment
🎲 Stochastic	Ch. 13	Probability, Bayes' theorem, uncertain actions*
📡 Noisy	Ch. 15	Sensor models, belief states, filtering*
🗺️ Large	Ch. 8–9	FOL scalability, universal quantification*
💀 Nightmare	Ch. 16	Decision theory, MEU, decision networks*

* Stochastic, Noisy, Large, and Nightmare are extensions inspired by textbook concepts. Classic mode is the exact specification from Section 7.2.

Reference: Russell, S. & Norvig, P. — Artificial Intelligence: A Modern Approach (4th Edition)

🏰 Wumpus World