Propositional logic or how to reason correctly
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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 l.jpg

Propositional Logic or how to reason correctly

Chapter 8 (new edition)

Chapter 7 (old edition)


Goals l.jpg
Goals

  • 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


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History

  • 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


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Syllogisms

  • 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 ….


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Concerns

  • 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?


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Semantics

  • 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.


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Reasoning or Inference Systems

  • 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.


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Proposition Logic: Syntax

  • 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.


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Propositional Logic: Semantics

  • Truth tables: p =>q |= ~p or q


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Implies =>

  • 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.


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Inference

  • 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.


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Translation into Propositional Logic

  • 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)


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Concepts

  • 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.


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Validity != Provability

  • 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.


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Natural Inference Rules

  • 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


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Natural Inference Systems

  • 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)


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Full Propositional Resolution

  • 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)


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Conjunctive Normal Form

  • 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.


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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.

Proposition -> CNFGoal: Proving R


Resolution proof l.jpg

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.

Resolution Proof


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Resolution Algorithm

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).


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Forward and Backward Reasoning

  • 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.


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Forward Reasoning

  • 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.


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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.


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Davis-Putnam Algorithm

  • 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.


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WalkSat

  • 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.


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Hard Satisfiability Problems

  • 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].


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What can’t we say?

  • 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.


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