Propositional Logic Reasoning correctly computationally

1
Propositional Logic Reasoning correctly
computationally

• Chapter 7 or 8

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Natural Reasoning

• 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?

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State-Space Model?

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

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States

• 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 gtQ
• Initial State what you know
• P, PgtQ meaning clauses are true.

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Operators

• 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 AgtB is true, then B is
  true.

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What are the right operators?

• 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.

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A model

• 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.

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Concerns

• 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?

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Semantic definition of Truth

• Model possible world
• xy 4 is true in the world x3, y1.
• xy 4 is false in the world x3, y 2.
• Entailment S1,S2,..Sn S means in every
  world where S1Sn 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. Cant
  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 gtq p or q

p q p gtq p or q
t t t t
t f f f
t t t t
t t t t
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Beware Implies gt

• If 22 5 then monkeys are cows. TRUE
• If 22 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) v1vn.
• Check all 2n possible worlds.
• In each world, check if s1..sk is true, that s is
  true.
• Complexity approximately O(2n).
• 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 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 Its not over till its over.
• Satisfiability a sentence is satisfied if it
  true in some model.

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Validity ! Provability

• Goldbachs conjecture Every even number (gt2) 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, pgtq -- q.
• Sound
• Resolution example (sound)
• p or q, not p or r -- q or r
• Abduction (unsound, but common)
• q, pgtq -- p
• ground wet, rained gt 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..xny ( 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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Horn Clauses Prolog program

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

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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 gt and move
  nots inwards. Then you may need to apply
  distributive laws.

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Proposition -gt CNFGoal Proving R

• P
• (PQ) gtR
• (S or T) gt 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.

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Resolution Proof

• 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.

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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 doesnt grow.
• Search can be expensive (exponential).

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Forward and Backward Reasoning

• Prolog only allows Horn clauses.
• if a, b, c then d gt 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 pqgtr
• Rewrite in clausal form s3 (-p-qr)
• s1 resolve with s3 -qr (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 pqgtr
• Goal s4 r
• Rewrite in clausal form s3 (-p-qr)
• 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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What cant 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.
• Changing world actions