CS451CS551EE565 ARTIFICIAL INTELLIGENCE

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CS451/CS551/EE565ARTIFICIAL INTELLIGENCE

• First Order Logic
• 10-13-2006
• Prof. Janice T. Searleman
• jets_at_clarkson.edu, jetsza

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Outline

• Review of HW4
• First order logic
• syntax
• quantifiers ?, ?
• Reading Assignment AIMA Chapter 9
• Exam1 Wednesday, 10/18, 700 pm, SC356

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Syntax of FOL Basic elements

• Constants KingJohn, 2, book23, A,B,C
• Predicates Brother(X,Y), gt, diplomat(X),
• Functions Sqrt(X), father-of(X), max(X,Y),
• Variables t, w, v, x, y, z
• Connectives ?, ?, ?, ?, ?
• Equality
• Quantifiers ?, ?

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Truth in first-order logic

• Sentences are true with respect to a model and an
  interpretation
• Must interpret variables, constants, functions,
  and predicate symbols by associating them with
  objects, functions and relations in the world.
• Truth values are relative to an interpretation
  (model)
• Express Johns father is a diplomat. with
  diplomat(father-of(john))).
• Express All birds fly. with
• (? x) bird(x) gt fly(x).
• An atomic sentence predicate(term1,...,termn) is
  true
• iff the objects referred to by term1,...,termn
• are in the relation referred to by predicate

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Universal Quantification

• (? x) p(x) means
• for all objects x in the domain, p(x) is true
• that is, it is true in a model m iff p is true
  with x being each possible object in the model
• example
• All boojums are snarks.
• expressed by
• (?x) boojum(x) ? snark(x).
• where
• boojum(x) means x is a boojum
• snark(x) means x is a snark

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Existential Quantification

• (? x) p(x) means
• there exist one or more objects x in the domain
  s.t. p(x) is true
• that is, it is true in a model m iff P is true
  with x being some possible object in the model
• example
• Not all snarks are boojums.
• expressed by
• (? x) snark(x) ? boojum(x).
• where
• boojum(x) means x is a boojum
• snark(x) means x is a snark

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Universal quantification, revisited

• ?ltvariablesgt ltsentencegt
• Everyone at Clarkson is smart
• ?x At(x , Clarkson) ? Smart(x).
• Roughly speaking, equivalent to the conjunction
  of instantiations of P
• At(KingJohn, Clarkson) ? Smart(KingJohn)
• ?
• At(Richard, Clarkson) ? Smart(Richard)
• ?
• At(Clarkson, Clarkson) ? Smart(Clarkson)
• ? ...

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A common mistake to avoid

• Typically, ? is the main connective with ?
• Common mistake using ? as the main connective
  with ?
• ?x At(x , Clarkson) ? Smart(x)
• means Everyone is at Clarkson and everyone is
  smart

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Existential quantification

• ?ltvariablesgt ltsentencegt
• Someone at SLU is smart
• (?x) At(x, SLU) ? Smart(x).
• Roughly speaking, equivalent to the disjunction
  of instantiations of P
• At(KingJohn, SLU) ? Smart(KingJohn)
• ?
• At(Richard, SLU) ? Smart(Richard)
• ?
• At(SLU, SLU) ? Smart(SLU)
• ? ...

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Another common mistake to avoid

• Typically, ? is the main connective with ?
• Common mistake using ? as the main connective
  with ?
• ?x At(x , SLU) ? Smart(x)
• is true if there is anyone who is not at SLU!

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Properties of quantifiers

• ?x ?y is the same as ?y ?x
• ?x ?y is the same as ?y ?x
• ?x ?y is not the same as ?y ?x
• ?x ?y Loves(x,y)
• There is a person who loves everyone in the
  world (including him/herself)
• ?y ?x Loves(x,y)
• Everyone in the world is loved by at least one
  person

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Properties of ? and ? (cont)

• Quantifier duality each can be expressed using
  the other e.g. consider the predicate likes(X,Y)
  meaning X likes Y.
• Everyone dislikes parsnips.
• (?x) likes(x,parsnips).
• (?x) likes(x,parsnips).
• Everyone likes ice cream.
• (?x) likes(x,iceCream)
• (?x) likes(x,iceCream).
• Someone likes persimmons.
• (?x) likes(x,persimmons)
• (?x) likes(x,persimmons).

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Equality

• term1 term2 is true under a given
  interpretation if and only if term1 and term2
  refer to the same object
• E.g., definition of Sibling in terms of Parent
• ?x,y Sibling(x,y) ? ?(x y) ? ?m,f ? (m f) ?
  Parent(m,x) ? Parent(f,x) ? Parent(m,y) ?
  Parent(f,y)

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Summary

• First-order logic
• objects and relations are semantic primitives
• syntax constants, functions, predicates,
  equality, quantifiers
• Increased expressive power sufficient to define
  wumpus world