Chapter 7 or 8

John plays tennis if sunny and weekend day.
If John plays tennis, Mary goes shopping.
It is Saturday.
It is sunny.
•
Specific: Does John play tennis?
•
All: what may one conclude?

• What are the States?
• What are the legal operators?
• What is an appropriate search?
• What do we want?

• Collection of boolean formula in boolean variables.
•
Proposition variables stand for a statement that may be either true or false.
• Ex. It is the weekend. Q
• Ex. It is Saturday. P
• Ex. It is Saturday implies is weekend:
P =>Q
Initial State: what you know
{ P, P=>Q} meaning clauses are true.

• Operators take a previous state (collection of formula) and add new formula.
• Modus Ponens: If A is true, and A implies
B, then B is true.
• Model:
A = it is Saturday, B = it is weekend and A is true, and A=>B is true, then B is true.

• If some A are B, and some B are C, then some A are C.
• If A implies B, and B is false, then A is false.

• Models are particular instantiations of the variables.
• If some A are B, and some B are C, then some A are C.
• A = women, B= students, C = men
• If some women are students, and some students are men, then ….
• Bad Rule.

• What does it mean to say a statement is true?
• What are a good set of operators?
• What can we say in propositional logic?
• What is the efficiency?
• Can we guarantee to infer all true conclusions?

• Model = possible world
• x+y = 4 is true in the world x=3, y=1.
• x+y = 4 is false in the world x=3, y = 2.
•
Entailment S1,S2,..Sn |= S means in every world where S1…Sn are true, S is true.
• Careful: No mention of proof – just checking all the worlds.
• Some cognitive scientists argue that this is the way people reason.

• Proof is a syntactic property.
• Rules for deriving new sentences from old ones.
•
Sound : any derived sentence is true.
•
Complete : any true sentence is derivable.
• NOTE: Logical Inference is monotonic.
Can’t change your mind.

• See text for complete rules
• Atomic Sentence: true, false, variable
• Complex Sentence: connective applied to atomic or complex sentence.
• Connectives: not, and, or, implies, equivalence, etc.
• Defined by tables.

• Truth tables: p =>q |= ~p or q t t t t p t f t t q t f t t p =>q t f t t
~p or q

• If 2+2 = 5 then monkeys are cows. TRUE
• If 2+2 = 5 then cows are animals. TRUE
• Indicates a difference with natural reasoning. Single incorrect or false belief will destroy reasoning. No weight of evidence.

• Does s1,..sk entail s?
• Say variables (symbols) v1…vn.
• Check all 2^n possible worlds.
• In each world, check if s1..sk is true, that s is true.
• Complexity: approximately O(2^n).
• Complete: possible worlds finite for propositional logic, unlike for arithmetic.

• If it rains, then the game will be cancelled.
• If the game is cancelled, then we clean house.
• Can we conclude?
– If it rains, then we clean house.
• p = it rains, q = game cancelled r = we clean house.
• If p then q. not p or q
• If q then r. not q or r
• if p then r. not p or r (resolution)

•
Equivalence : two sentences are equivalent if they are true in same models or worlds.
•
Validity : a sentence is valid if it is true in all models. (tautology) e.g. P or not P.
– Sign: Members or not Members only.
– Berra: It’s not over till its over.
•
Satisfiability : a sentence is satisfied if it true in some model.

• Goldbach’s conjecture: Every even number
(>2) is the sum of 2 primes.
• This is either valid or not.
• It may not be provable.
• Godel: No axiomization of arithmetic will be complete, i.e. always valid statements that are not provable.

• Modus Ponens: p, p=>q |-- q.
– Sound
• Resolution example (sound)
– p or q, not p or r |-- q or r
• Abduction (unsound, but common)
– q, p=>q |-- p
– ground wet, rained => ground wet |-- rained
– medical diagnosis

• Typically have dozen of rules.
• Difficult for people to use.
• Expensive for computation.
– e.g. a |-- a or b
– a and b |-- a
• All known systems take exponential time in worse case. (co-np complete)

• clause 1: x1 +x2+..xn+y (+ = or)
• clause 2: -y + z1 + z2 +… zm
• clauses contain complementary literals.
• x1 +.. xn +z1 +… zm
• y and not y are complementary literals.
• Theorem: If s1,…sn |= s then s1,…sn |-- s by resolution.
Refutation Completeness.
Factoring: (simplifying: x or x goes to x)

• Horn clauses have 1 positive literal.
• They have the form a,b,c,…=> d
• Modus Ponens is “Horn Clause” complete.
• Means: If KB is a set of horn clauses, and
KB => horn clause c, then KB -> c by modus ponens.
• Resolution is also “horn clause” complete since it yields modus ponens.

• To apply resolution we need to write what we know as a conjunct of disjuncts.
• Pg 215 contains the rules for doing this transformation.
• Basically you remove all  and => and move “not’s” inwards. Then you may need to apply distributive laws.

• P
• (P&Q) =>R
• (S or T) => Q
• T
• Distributive laws:
• (-s&-t) or q 
(-s or q)&(-t or q).
• P
• -P or –Q or R
• -S or Q
• -T or Q
• T
• Remember: implicit adding.

• P (1)
• -P or –Q or R (2)
• -S or Q (3)
• -T or Q (4)
• T (5)
• ~R (6)
• -P or –Q : 7 by 2 & 6
• -Q : 8 by 7 & 1.
• -T : 9 by 8 & 4
• empty: by 9 and 5.
• Done: order only effects efficiency.

To prove s1, s2..sn |-- s
1. Put s1,s2,..sn & not s into cnf.
2. Resolve any 2 clauses that have complementary literals
3. If you get empty, done
4. Continue until set of clauses doesn’t grow.
Search can be expensive (exponential).

Prolog only allows Horn clauses.
• if a, b, c then d => not a or not b or not c or d
• Prolog writes this:
– d :- a, b, c.
• Prolog thinks: to prove d, set up subgoals a, b, c and prove/verify each subgoal.

• From facts to conclusions
• Given s1: p, s2: q, s3: p&q=>r
• Rewrite in clausal form: s3 = (-p+-q+r)
• s1 resolve with s3 = -q+r (s4)
• s2 resolve with s4 = r
• Generally used for processing sensory information.

• From Negative of Goal to data
• Given s1: p, s2: q, s3: p&q=>r
• Goal: s4 = r
• Rewrite in clausal form: s3 = (-p+-q+r)
• Resolve s4 with s3 = -p +-q (s5)
• Resolve s5 with s2 = -p (s6)
• Resolve s6 with s1 = empty. Eureka r is true.

• Quantification: every student has a father.
• Relations: If X is married to Y, then Y is married to X.
• Probability: There is an 80% chance of rain.
• Combine Evidence: This car is better than that one because…
• Uncertainty: Maybe John is playing golf.
• Changing world: actions