Question 7.5
The book and slides are a little confusing about the definition of a model.

Here’s what the book says on page 201:
“When we need to be precise, we will use the term model in place of “possible world.”
(We will also use the phrase “m is a model of a” to mean that sentence a is true in model
m.) Whereas possible worlds might be thought of as (potentially) real environments that the
agent might or might not be in, models are mathematical abstractions, each of which simply
fixes the truth or falsehood of every relevant sentence.”

Question 7.5 asks:
“Consider a vocabulary with only four propositions, A, B, C, and D. How many models
are there for the following sentences?

a. (A ^ B) V (B ^ C)
b. A V B
c. A B C ”

So does model in this case mean a world where the sentence is true, or does it mean any possible combination of truth values for the inputs?

I think the question is asking for us to tell the number of worlds where the setence is true. Can someone please clarify?

I’m assuming that when doing the truth tables, for ease of use we can use 0 and 1 to represent false and true respectively. Is this a valid assumption?

Quoting from the specification on the project 3…
“For exercise 7.5, you can consider A B C to be
equivalent to (A B) ^ (B C).”

If you have A B C, this turns out to be
(A B) C because everything has the same
priority, and the sentence will be evaluated in the order
from left to right. (just like 3 + 4 + 5 –> (3 + 4) + 5).

Let me first clarify this a bit, because for some reason the WordPress formatting system actually seems to remove the logical symbols, which would be confusing to another person reading the comment (this is probably because they look like some unknown HTML tag to it, since it starts and ends with a greater-than and less-than sign).

For this, then, let me use IFF to represent the biconditional (you can also refer to the original homework page):

What I said earlier is that A IFF B IFF C should be considered as (A IFF B) ^ (B IFF C).

The reason that I said that is that precisely because the interpretation that is created by AIMA’s logical sentence converter renders (A IFF B IFF C) as ((A IFF B) IFF C). That is to say, it says that C is true if and only iff (A IFF B) is also true. It struck me that that was not the intent of the authors, but a parsing similar to A = B = C would be correct.

Just as we can render A = B = C as (A = B) ^ (B = C), we can render A IFF B IFF C as (A IFF B) ^ (B IFF C).

I’m a bit confused by the truth table functions in the AIMA Code.
(truth-table “(A B) & (B C)”)
(truth-table “A B C”)
(truth-table ‘( A B C))
The above 3 lines all render different truth tables. I guess from the professor’s previous comment #6, I now understand why there are differences between the first 2. However the third representation seems to generate an even additional variation. Am I just to assume this is another fault of the authors?

I’m not sure why there is the additional variation for the third example. It shouldn’t be syntactically correct, because IFF is a binary relation. The table that is produced seems to use (IFF A B) and ignore the C entirely, if I read it correctly.

on September 26, 2007 at 1:13 amDonnieQuestion 7.5

The book and slides are a little confusing about the definition of a model.

Here’s what the book says on page 201:

“When we need to be precise, we will use the term model in place of “possible world.”

(We will also use the phrase “m is a model of a” to mean that sentence a is true in model

m.) Whereas possible worlds might be thought of as (potentially) real environments that the

agent might or might not be in, models are mathematical abstractions, each of which simply

fixes the truth or falsehood of every relevant sentence.”

Question 7.5 asks:

“Consider a vocabulary with only four propositions, A, B, C, and D. How many models

are there for the following sentences?

a. (A ^ B) V (B ^ C)

b. A V B

c. A B C ”

So does model in this case mean a world where the sentence is true, or does it mean any possible combination of truth values for the inputs?

I think the question is asking for us to tell the number of worlds where the setence is true. Can someone please clarify?

Thanks

on September 26, 2007 at 1:59 amRonA model is any set of truth values for the inputs for which the sentence is true.

on September 26, 2007 at 9:59 pmRyan CinoI’m assuming that when doing the truth tables, for ease of use we can use 0 and 1 to represent false and true respectively. Is this a valid assumption?

on September 26, 2007 at 11:39 pmRonOr T and F, or high and low, or š and š¦ … anything is fine as long as the semantics are clear.

on October 1, 2007 at 1:44 amts2883Quoting from the specification on the project 3…

“For exercise 7.5, you can consider A B C to be

equivalent to (A B) ^ (B C).”

If you have A B C, this turns out to be

(A B) C because everything has the same

priority, and the sentence will be evaluated in the order

from left to right. (just like 3 + 4 + 5 –> (3 + 4) + 5).

Then, it won’t be equivalent to (A B) ^ (B C).

What do I do?

on October 1, 2007 at 1:49 amRonLet me first clarify this a bit, because for some reason the WordPress formatting system actually seems to remove the logical symbols, which would be confusing to another person reading the comment (this is probably because they look like some unknown HTML tag to it, since it starts and ends with a greater-than and less-than sign).

For this, then, let me use IFF to represent the biconditional (you can also refer to the original homework page):

What I said earlier is that A IFF B IFF C should be considered as (A IFF B) ^ (B IFF C).

The reason that I said that is that precisely because the interpretation that is created by AIMA’s logical sentence converter renders (A IFF B IFF C) as ((A IFF B) IFF C). That is to say, it says that C is true if and only iff (A IFF B) is also true. It struck me that that was not the intent of the authors, but a parsing similar to A = B = C would be correct.

Just as we can render A = B = C as (A = B) ^ (B = C), we can render A IFF B IFF C as (A IFF B) ^ (B IFF C).

Does this help?

on October 1, 2007 at 2:32 amJackieI’m a bit confused by the truth table functions in the AIMA Code.

(truth-table “(A B) & (B C)”)

(truth-table “A B C”)

(truth-table ‘( A B C))

The above 3 lines all render different truth tables. I guess from the professor’s previous comment #6, I now understand why there are differences between the first 2. However the third representation seems to generate an even additional variation. Am I just to assume this is another fault of the authors?

on October 1, 2007 at 2:33 amJackieEdit to post #7:

(truth-table ā(A IFF B) & (B IFF C)ā)

(truth-table āA IFF B IFF Cā)

(truth-table ā(IFF A B C))

the above post took out my arrow IFF symbols.

on October 1, 2007 at 2:07 pmRonI’m not sure why there is the additional variation for the third example. It shouldn’t be syntactically correct, because IFF is a binary relation. The table that is produced seems to use (IFF A B) and ignore the C entirely, if I read it correctly.

on October 1, 2007 at 4:59 pmRyan CinoIs there any turn-in specifications such as filenames like there have been in the past?

on October 1, 2007 at 11:53 pmJohannesWhat does the extra credit entail? For 7.5, for instance, do we just give the code used to generate the truth tables?

on October 2, 2007 at 12:02 amRonComment #10: Use your best judgment. The code should be a separate attachment.

Comment #11: You should show some output, too, just like for Project 2.