CIS587: The Wumpus World

• Rules of the Wumpus World
• The Situation Calculus
• The Frame Problem
• Representing Knowledge about the Wumpus World
• Model-Based and Diagnostic Reasoning
• Some Questions
• A variety of "worlds" are being used as examples for Knowledge Representation, Reasoning, and Planning. Among them the Vacuum World, the Block World, and the Wumpus World. We will examine the Wumpus World and in this context introduce the Situation Calculus, the Frame Problem, and a variety of axioms.

The Wumpus World was introduced by Genesereth, and is discussed in Russell-Norvig. The Wumpus World is a simple world (as is the Block World) for which to represent knowledge and to reason.

It is a cave with a number of rooms, represented as a 4x4 square.

```	+---------+---------+---------+---------+
4	| stench  |         | breeze  |  pit    |
|         |         |	      |         |
|         |         |	      |         |
+---------+---------+---------+---------+
3	| wumpus  | stench  | pit     | breeze  |
|         | breeze  |	      |         |
|         | gold    |	      |         |
+---------+---------+---------+---------+
2	| stench  |         | breeze  |         |
|         |         |	      |         |
|         |         |	      |         |
+---------+---------+---------+---------+
1	| start   | breeze  | pit     | breeze  |
|         |         |	      |         |
|   ==>   |         |	      |         |
+---------+---------+---------+---------+
1         2         3         4
```

Rules of the Wumpus World

The neighborhood of a node consists of the four squares north, south, east, west of the given square.

In a square the agent gets a vector of percepts, with components

Stench,Breeze,Glitter,Bump,Scream
For example [Stench,None,Glitter,None,None]

• Stench is perceived at a square iff the wumpus is at this square or in its neighborhood.
• Breeze is perceived at a square iff a pit is in the neighborhood of this square.
• Glitter is perceived at a square iff gold is in this square
• Bump is perceived at a square iff the agent goes Forward into a wall
• Scream is perceived at a square iff the wumpus is killed anywhere in the cave

An agent can do the following actions (one at a time):

Turn(Right), Turn(Left), Forward, Shoot, Grab, Release, Climb
• The agent can go Forward in the direction it is currently facing, or Turn Right, or Turn Left. Going Forward into a wall will generate a Bump percept.
• The agent has a single arrow that it can Shoot. It will go straight in the direction faced by the agent until it hits (and kills) the wumpus, or hits (and is absorbed by) a wall.
• The agent can Grab a portable object at the current square or it can Release an object that it is holding.
• The agent can Climb out of the cave if at the Start square.
The Start square is (1,1) and initially the agent is facing east. The agent dies if it is in the same square as the wumpus.
The objective of the game is to kill the wumpus, to pick up the gold, and to climb out with it.

The Situation Calculus

The Situation Calculus is used in cases as the one that we are now considering. With the name Situation we mean a state of the world. The Situation Calculus is used to reason about actions and their effect on the possible states of the world.
We will limit our discussion of the Situation Calculus to the case of the Wumpus World.

The Frame Problem

We are reasoning about possible states of the world, where the states are identified by the actions by which we got to that state from the initial state. The Frame Problem is concerned with the question of what happens to the truth-value of the statements that describe the world as we go from one world to the world resulting by application of an action.

We deal with the frame problem by introducing a certain number of axioms, that go by names such as Effect Axioms, Frame Axioms, and Successor-State Axioms.

Representing our Knowledge about the Wumpus World

Percept(x,y)
where x must be a percept vector and y must be a situation. It means that at situation y the agent perceives x.
For convenience we introduce the following definitions:
• Percept([Stench,y,z,w,v],t) = > Stench(t)
• Percept([x,Breeze,z,w,v],t) = > Breeze(t)
• Percept([x,y,Glitter,w,v],t) = > AtGold(t)
Holding(x,y)
where x is an object and y is a situation. It means that the agent is holding the object x in situation y.
Action(x,y)
where x must be an action (i.e. Turn(Right), Turn(Left), Forward, ..) and y must be a situation. It means that at situation y the agent takes action x.
At(x,y,z)
where x is an object, y is a Location, i.e. a pair [u,v] with u and v in {1,2,3,4}, and z is a situation. It means that the agent x in situation z is at location y.
Present(x,s)
means that object x is in the current room in the situation s.
Result(x,y)
It means that the result of applying action x to the situation y is the situation Result(x,y). Note that Result(x,y) is a term, not a statement.
For example we can say
• Result(Forward, S0) = S1
• Result(Turn(Right),S1) = S2

These definitions could be made more general. Since in the Wumpus World there is a single agent, there is no reason for us to make predicates and functions relative to a specific agent. In other "worlds" we should change things appropriately.
Another generalization is possible, to write all assertions about the world as terms, and then add the predicates

T(situation assertion-term)
to mean that in the specified situation the specified assertion is true.
Belief(agent situation assertion-term)
to mean that the specified agent in the specified situation believes the specified assertion to be true.
Here are a series of statements about the Wumpus World.

Effect Axioms

Effect axioms characterize what is changed because of an action. For example:
• Present(x,s) & Portable(x) = > Holding(x, Result(Grab,s))
• Not Holding(x, Result(Release,s))

Frame Axioms

Frame axioms characterize what has remained the same because of an action. For example:
• Holding(x,s) & (a/=Release) = > Holding(x,Result(a,s))
• NOT Holding(x,s) & ((a/=Grab)) | NOT(Present(x,s) & Portable(x)) = > NOT Holding(x,Result(a,s))

Successor-State Axioms

For each predicate that can change over time they characterize the actions under which it changes and the actions under which it remains the same. For example:
• Holding(x,Result(a,s)) IFF [(a=Grab & Present(x,s) & Portable(x)) OR (Holding(x,s) & (a/=Release))

More Definitions and Axioms

• Orientation(Agent,s0) = 0
• At(Agent,[1,1],s0)

• Portable(Gold)
• AtGold(s) = > Present(Gold, s)

• LocationToward([x,y],0) = [x+1,y]
• LocationToward([x,y],90) = [x,y+1]
• LocationToward([x,y],180) = [x-1,y]
• LocationToward([x,y],270) = [x,y-1]

• At(p,l,s) = > LocationAhead(p,s) = LocationToward(l, Orientation(p,s))
• Adjacent(l1,l2) IFF EXISTS d (l1 = LocationToward(l2,d))
• Wall([x,y]) IFF (x=0 OR x=5 OR y=0 OR y=5)
• At(p,l,Result(a,s)) IFF [(a=Forward & l=LocationAhead(p,s) & NOT Wall(l)) OR (At(p,l,s) & a/=Forward)]
• Orientation(p,Result(a,s))=d IFF [
(a=Turn(Right) & d=Mod(Orientation(p,s)-90,360)) OR
(a=Turn(Left) & d=Mod(Orientation(p,s)+90,360)) OR
(Orientation(p,s)=d & NOT(a=Turn(Right) & a=Turn(Left)))]

• At(Wumpus.l1,s) & Adjacent(l1,l2) = > Smelly(l2)
• At(Pit,l1,s) & Adjacent(l1,l2) = > Breezy(l2)

Model-Based and Diagnostic Reasoning

Causal Rules specify how aspects of the state of the world determine our percepts. Model-Based Reasoning is what we do when we use causal rules. Here are some causal rules:
• At(Wumpus,l1,s) & Adjacent(l1,l2) = > Smelly(l2)
• At(Pit,l1,s) & Adjacent(l1,l2) = > Breezy(l2)
Diagnostic Rules specify how to go from percepts to assertions about the state of the world. Diagnostic Reasoning is what we do when we use diagnostic rules. Here are some diagnostic rules:
• At(Agent,l,s) & Breeze(s) = > Breezy(l)
• At(Agent,l,s) & Stench(s) = > Smelly(l)
• Smelly(l1) = > (EXISTS l2 At(Wumpus,l2,s) & (l2=l1 OR Adjacent(l1,l2)))
• At(Wumpus,x,t) & NOT Pit(x) IFF OK(x)

Some Questions

Many questions arise because of what has been done above. Here are some such questions:
1. Have we specified enough axioms about the Wumpus World for us to solve our stated problem?
2. How does one acquire the needed domain knowledge about the Wumpus World?
3. How much of the work done for this world can be used when studying other worlds?
4. Clearly we could write an algorithm that would solve the Wumpus problem in an ad-hoc fashion. Is the approach that we have chosen ultimately more general, more effective than the use ad-hoc prescriptive procedures?

ingargiola@cis.temple.edu