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Propositional Logic or how to reason correctly. Chapter 8 (new edition) Chapter 7 (old edition). Goals. Feigenbaum: In the knowledge lies the power. Success with expert systems. 70’s. What can we represent? Logic(s): Prolog Mathematical knowledge: mathematica

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### Propositional Logic or how to reason correctly

Chapter 8 (new edition)

Chapter 7 (old edition)

• Feigenbaum: In the knowledge lies the power. Success with expert systems. 70’s.

• What can we represent?

• Logic(s): Prolog

• Mathematical knowledge: mathematica

• Common Sense Knowledge: Lenat’s Cyc has a million statement in various knowledge

• Probabilistic Knowledge: Bayesian networks

• Reasoning: via search

• 300 BC Aristotle: Syllogisms

• Late 1600’s Leibnitz’s goal: mechanization of inference

• 1847 Boole: Mathematical Analysis of Logic

• 1879: Complete Propositional Logic: Frege

• 1965: Resolution Complete (Robinson)

• 1971: Cook: satisfiability NP-complete

• 1992: GSAT Selman min-conflicts

• Proposition = Statement that may be either true or false.

• John is in the classroom.

• Mary is enrolled in 270A.

• If A is true, and A implies B, then B is true.

• If some A are B, and some B are C, then some A are C.

• If some women are students, and some students are men, then ….

• What does it mean to say a statement is true?

• What are sound rules for reasoning

• What can we represent in propositional logic?

• What is the efficiency?

• Can we guarantee to infer all true statements?

• 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 = 1.

• 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

• 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.

• 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.

• Validity: a sentence is valid if it 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)

• 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&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

Proposition -> CNFGoal: Proving R

-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.

Resolution Proof

To prove s1, s2..sn |-- s

• Put s1,s2,..sn & not s into cnf.

• Resolve any 2 clauses that have complementary literals

• If you get empty, done

• Continue until set of clauses doesn’t grow.

Search can be expensive (exponential).

• Horn clause has at most 1 positive literal.

• 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.

Backwards Reasoning: what prolog does

• 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.

• Effective, complete propositional algorithm

• Basically: recursive backtracking with tricks.

• early termination: short circuit evaluation

• pure symbol: variable is always + or – (eliminate the containing clauses)

• one literal clauses: one undefined variable, really special cases of MRV

• Propositional satisfication is a special case of Constraint satisfication.

• Heuristic algorithm, like min-conflicts

• Randomly assign values (t/f)

• For a while do

• randomly select a clause

• with probability p, flip a random variable in clause

• else flip a variable which maximizes number of satisfied clauses.

• Of course, variations exists.

• Critical point: ratio of clauses/variables = 4.24 (empirical).

• If above, problems usually unsatsifiable.

• If below, problems usually satisfiable.

• Theorem: Critical range is bounded by [3.0003, 4.598].

• 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.