Question

Ouestion 4 [1.5 ptsl: Figure 2 shows the Wumpus world game, where the agent starts from location [4.l], and does not sense breeze or stench. Please use propositional logics to write sentences to describe the observations and rules & backgrounds (after agent enters location [4.1], and only consider Pit) (0.5 pt) Given above settings, please use model checking to validate that location [3,1] is safe (or KP)Please list all models as a table (only consider Pit), and mark model(s) of KB and model(s) of o. Explain why KBP (1 pt) Breeze PIT PIT Breece PIT Breeze START 2

Answer 1

ANSWER :

Propositional Logic
The smalles addresable unit of propositional logic is a whole sentence which can be either TRUE or FALSE
That is, any expression such that "Is it the case that ?" is a meaningful question with a YES/NO answer.
Eg. asking this about = Romeo loves Juliet. is meaningful, so it is sentence in this sense.
On the other hand, e.g. neither "Is it the case that loves?" nor "Is it the case that Juliet?" is a meaningful question. We shall later present another logic where we can address also such parts of sentences.
Note also that "Is it the case that Romeo loves?" is another meaningful question - but this is a different sentence
The syntax used in this course and its book Russell and Norving for propositional logic is:
Sentence Atomic | Complex

Atomic TRUE | FALSE | Symbol

Complex Sentence

| (Sentence Sentence)

| (Sentence Sentence)

| (Sentence Sentence)

| (Sentence Sentence)

1.The Predicate Symbols are variable names which stand for different sentences. E.g. we might choose the name X to stand, when we are wrting the background knowledge , and so on
2.We suppress witing nested parentheses by stipulating that the connnectives are in descending binding power.
3.We Also stipulate that are associative, but are not.

In the semantics, a possible world w is just a function
: Symbol call this B

which gives for each Symbol Y its truth value (Y)

This extends from single Symbol to whole Sentences:
(c) = c if c

If we thing that FALSE < TRUE
The part of the background knowledge for our Wumpus world which deals with pits can be written in propositional logic as follows
1.Let the Symbol Pi,j (or Bi,j) stand for "Ther is a Pit (or Breeze ) in square [i,j]".
2.There is no pit in [1, 1] is then simply P1,1
3.We have the general rule that
"This square has a Breeze exactly when at least one of its adjacent squares has a Pit" (Rule 25)

But now we encounter a shortcoming of propositional logic:
Rule (25 ) combines together two whole sentence with "exactly when" This we can do with ' '
But Rule (25) is trying to combine them so that "this" and "it" are the same square.
Alas, propostional logic does not allows us to access such parts of whole statements.
Then we add into KB also the Breeze percepts at the top right Wumpus world
B1,1

B2,1

Let us consider an example gives as Figure on how the agent might reason and act in this Wumpus world
Top Left: Agent A starts at [1, 1] facing right.

1.Agent A is actually in the example world given figure, but does not know it.
2.The background knowledge assures agent A that e is at [1, 1] and that it is OK = certainly not deadly.
3.Agent A gets the percept "Stench = Breeze = Glitter = Bump = Scream = FALSE"
4.Agent A infers from this percept and that its both neighbouring squares [1, 2] and [2, 1] are also OK: "If there was a Pit(Wumpus), then here would be Breeze (Smell) but isn't, so"
5.The KB enables agent A to discover certainties about parts of its environment - even without visiting those parts.
Top Right: Agent A is cautious, and will only move to OK squares.

1.Agent A walks into [2, 1], because it is OK , and in the direction where agent A is facing, so it is cheper than the other choice [1, 2], Agent A also marks [1, 1] Visited.
2.Agent A perceives a Breeze but nothing else.
3.Agent A infers: "At least one of the adjacent squeares [1, 1], [2, 2] and [3, 1] must contain a Pit. There is no Pit in [1, 1] by my background knowledge . Hence [2, 2] or [3, 1] or both must contain a Pit"
4.Hence agent A cannot be certain of either [2, 2] or [3, 1], so [2, 1] is a dead end for a cautious agent like A.
Bottom Left: Agent A has turned back from the dead end [2, 1] and walked to examine the other OK choice [1, 2] insted.

1.Agent A preceives a Stench but nothing else.
2.Agent A infers using also earlier percepts: "The Wumpus is in an adjacent square. It is not in [1, 1]. It is not in [2, 2] either, because then I would have sensed a Stench in [2, 1]. Hence it is in [1,3]"
3.Agent A infers using also earlier inferences: "There is no Breeze here, so ther is no Pit in any adjacent square. In particular, there is no Pit in [2, 2] after all. Hence there is a Pit in [3, 1]."
4.Agent A finally infers: "[2, 2] is OK after all - now it is certain that it has neither a Pit nor the Wumpus"
This reasoning is too complicated for many animals - but not for the logical agent A.

Bottom right:

1.Agent A walks to the only unvisited OK choice [2, 2]. There is no Breeze here, and since the the square of the Wumpus is now known too, [2, 3] and [3, 2] are OK too.
2.Agent A walks into [2, 3] and senses the Glitter there, so he grabs the gold and succeeds