In a run through the wumpus world, we have to tell the search engine in our AI man what actions are safe:
0110 - could mean many things:
| "2 is prime" | P |
| "I ate breakfast today" | Q |
syntax = how a sentence looks
sentence -> AtomicSentence | ComplexSentence
AtomicSentence -> T(rue) | F(alse) | Symbols
Symbols -> P | Q | R
ComplexSentence -> (Sentence) | NOT Sentence | Sentence Connective Sentence
example: (P AND Q) IMPLIES R
semantics = what a sentence means
interpretation: assigns each symbol a truth value, T or F
truth tables ("compositional semantics"):
| A | B | NOT A | A and B | A or B | A implies B | A equiv B |
|---|---|---|---|---|---|---|
| t | t | f | t | t | t | t |
| t | f | f | f | t | f | f |
| f | t | t | f | t | t | f |
| f | f | t | f | f | t | t |
EXAMPLES OR PITFALLS when sentences -> logic
2 is odd implies that 3 is even
It says something about B if A is T. Otherwise it doesn't say anything.
valid = always true
example:
This second statement is the same as saying P OR (NOT P). Check out the truth table:
Therefore it's both valid and satisfiable.
To say that a knowledge base entails a sentence =
example:
symbols:
background knolwedge:
Is there an interpretation, or assignment of truth values to the symbols, that
is false? If so, it's no longer valid. Well, if you make the following assignment:
H is false
It doesn't follow. To check if something is valid, you can write down all the interpretations
in a truth table and check. In this case, there are 2^4 = 16 interpretations since there are
4 symbols and 2 values.
Inference Procedures: these can infer the sentence from the knowledge base.
inference procedures = repeated application of inference rules. We can use these
instead of constructing massive truth tables, which are exponential in the number of symbols.
Either I go to the movies or I go swimming (inclusive vs. exclusive or)
(Implication doesn't imply causality)
satisfiable = there is an interpretation (at least one is true)
unsatisfiable = always false
2 is prime. (Is this always true?)
2 is prime or 2 is not prime.
P NOT P P OR (NOT P) T F T F T T
When an interpretation makes the knowledge base true, it also makes the sentence
true as well. The advantage is that the computer does not need to know the
interpretation.
Does "head, I win; tails, you lose"
- entail -
"I win"?
(note: background knowledge is important!)
H = coin lands heads up
T = coin lands tails up
I = I win
U = You lose
(H implies I) AND (T implies U)
entails
I
I is false
T is true
U is true
The inference procedure is sound if the knowledge base entails the sentence.