First-Order Logic (FOL) aka. predicate calculus

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First-Order Logic (FOL)aka. predicate calculus
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First-Order Logic (FOL) Syntax

• User defines these primitives
• Constant symbols (i.e., the "individuals" in the
  world) E.g., Mary, 3
• Function symbols (mapping individuals to
  individuals) E.g., father-of(Mary) John,
  color-of(Sky) Blue
• Predicate symbols (mapping from individuals to
  truth values) E.g., greater(5,3), green(Grass),
  color(Grass, Green)

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First-Order Logic (FOL) Syntax

• FOL supplies these primitives
• Variable symbols. E.g., x,y
• Connectives. Same as in PL not (), and (), or
  (v), implies (gt), if and only if (ltgt)
• Quantifiers Universal (A) and Existential (E)

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Quantifiers

• Universal quantification.
• E.g., (Ax) dolphin(x) gt mammal(x)
• Existential quantification.
• E.g., (Ex) mammal(x) lays-eggs(x)
• Universal quantifiers are usually used with
  "implies" to form "if-then rules."
• E.g., (Ax) cs-student(x) gt smart(x) means "All
  cs students are smart."
• You rarely use universal quantification to make
  blanket statements about every individual in the
  world (Ax)cs-student(x) smart(x) meaning that
  everyone in the world is a cs student and is
  smart.

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Quantifiers

• Existential quantifiers are usually used with
  "and" to specify a list of properties or facts
  about an individual.
• E.g., (Ex) cs-student(x) smart(x) means "there
  is a cs student who is smart."
• A common mistake is to represent this English
  sentence as the FOL sentence
• (Ex) cs-student(x) gt smart(x)

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First-Order Logic (FOL) Syntax

• Sentences are built up of terms and atoms
• A term (denoting a real-world object) is a
  constant symbol, a variable symbol, or a function
  e.g. left-leg-of ( ). For example, x and f(x1,
  ..., xn) are terms, where each xi is a term.
• An atom (which has value true or false) is either
  an n-place predicate of n terms, or, if P and Q
  are atoms, then P, P V Q, P Q, P gt Q, P ltgt Q
  are atoms
• A sentence is an atom, or, if P is a sentence and
  x is a variable, then (Ax)P and (Ex)P are
  sentences
• A well-formed formula (wff) is a sentence
  containing no "free" variables. I.e., all
  variables are "bound" by universal or existential
  quantifiers.
• E.g., (Ax)P(x,y) has x bound as a universally
  quantified variable, but y is free.

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Translating English to FOL

• Every gardener likes the sun.(Ax) gardener(x) gt
  likes(x,Sun)
• You can fool some of the people all of the
  time.(Ex)(At) (person(x) time(t)) gt
  can-fool(x,t)
• You can fool all of the people some of the
  time.(Ax)(Et) (person(x) time(t) gt
  can-fool(x,t)
• All purple mushrooms are poisonous.(Ax)
  (mushroom(x) purple(x)) gt poisonous(x)

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Translating English to FOL

• No purple mushroom is poisonous.(Ex) purple(x)
  mushroom(x) poisonous(x) or,
  equivalently,(Ax) (mushroom(x) purple(x)) gt
  poisonous(x)
• Deb is not tall.tall(Deb)
• X is above Y if X is on directly on top of Y or
  else there is a pile of one or more other objects
  directly on top of one another starting with X
  and ending with Y.(Ax)(Ay) above(x,y) ltgt
  (on(x,y) v (Ez) (on(x,z) above(z,y)))

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Inference Rules for FOL

• Inference rules for PL apply to FOL as well. For
  example, Modus Ponens, And-Introduction,
  And-Elimination, etc.
• New sound inference rules for use with
  quantifiers
• Universal EliminationIf (Ax)P(x) is true, then
  P(c) is true, where c is a constant in the domain
  of x. For example, from (Ax)eats(Ziggy, x) we can
  infer eats(Ziggy, IceCream).
• The variable symbol can be replaced by any ground
  term, i.e., any constant symbol or function
  symbol applied to ground terms only.
• Existential IntroductionIf P(c) is true, then
  (Ex)P(x) is inferred.
• For example, from eats(Ziggy, IceCream) we can
  infer (Ex)eats(Ziggy, x).
• All instances of the given constant symbol are
  replaced by the new variable symbol. Note that
  the variable symbol cannot already exist anywhere
  in the expression.
• Existential EliminationFrom (Ex)P(x) infer P(c).
  
• For example, from (Ex)eats(Ziggy, x) infer
  eats(Ziggy, Cheese).
• Note that the variable is replaced by a brand new
  constant that does not occur in this or any other
  sentence in the Knowledge Base. In other words,
  we don't want to accidentally draw other
  inferences about it by introducing the constant.
  All we know is there must be some constant that
  makes this true, so we can introduce a brand new
  one to stand in for that (unknown) constant.

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Inference Rules for FOL

• Inference rules for PL apply to FOL as well. For
  example, Modus Ponens, And-Introduction,
  And-Elimination, etc.
• Generalized Modus Ponens (GMP)
• Combines And-Introduction, Universal-Elimination,
  and Modus Ponens
• E.g. from P(c), Q(c), and (Ax)(P(x) Q(x)) gt
  R(x), derive R(c)
• A substitution list Theta v1/t1, v2/t2, ...,
  vn/tn means to replace all occurrences of
  variable symbol vi by term ti.
• Substitutions are made in left-to-right order in
  the list.
• E.g. subst(x/IceCream, y/Ziggy, eats(y,x))
  eats(Ziggy, IceCream)

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Incompleteness of the generalized modus ponens
inference rule
?x P(x) gt Q(x) ?x ?P(x) gt Q(x) ?x Q(x) gt
S(x) ?x R(x) gt S(x)
? Cannot be converted to Horn
Want to conclude S(A)
S(A) is true if Q(A) or R(A) is true, and one of
those must be true because either P(A) or ?P(A)
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Generalized Modus Ponens in Horn FOL

• Generalized Modus Ponens (GMP) is complete for
  KBs containing only Horn clauses
• A Horn clause is a sentence of the form(Ax)
  (P1(x) P2(x) ... Pn(x)) gt Q(x)where there
  are 0 or more Pi's, and the Pi's and Q are
  positive (i.e., un-negated) literals
• For example, P(a) v Q(a) is a sentence in FOL but
  is not a Horn clause.

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Forward Chaining

• Natural deduction using GMP is complete for KBs
  containing only Horn clauses.
• Proofs start with the given axioms in KB,
  deriving new sentences using GMP until the
  goal/query sentence is derived.
• This defines a forward chaining inference
  procedure because it moves "forward" from the KB
  to the goal.

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Example of forward chaining

• Example KB All cats like fish, cats eat
  everything they like, and Ziggy is a cat. In FOL,
  KB
• (Ax) cat(x) gt likes(x, Fish)
• (Ax)(Ay) (cat(x) likes(x,y)) gt eats(x,y)
• cat(Ziggy)
• Goal query Does Ziggy eat fish?
• Proof Data-driven
• Use GMP with (1) and (3) to derive 4.
  likes(Ziggy, Fish)
• Use GMP with (3), (4) and (2) to derive
  eats(Ziggy, Fish)
• So, Yes, Ziggy eats fish.

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Backward Chaining

• Natural deduction using GMP is complete for KBs
  containing only Horn clauses.
• Proofs start with the goal query, find
  implications that would allow you to prove it,
  and then prove each of the antecedents in the
  implication, continuing to work "backwards" until
  we get to the axioms, which we know are true.

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Backward chaining

• Example Does Ziggy eat fish?
• To prove eats(Ziggy, Fish), first see if this is
  known from one of the axioms directly. Here it is
  not known, so see if there is a Horn clause that
  has the consequent (i.e., right-hand side) of the
  implication matching the goal.
• Proof Goal Driven
• Goal matches RHS of Horn clause (2), so try and
  prove new sub-goals cat(Ziggy) and likes(Ziggy,
  Fish) that correspond to the LHS of (2)
• cat(Ziggy) matches axiom (3), so we've "solved"
  that sub-goal
• likes(Ziggy, Fish) matches the RHS of (1), so try
  and prove cat(Ziggy)
• cat(Ziggy) matches (as it did earlier) axiom (3),
  so we've solved this sub-goal
• There are no unsolved sub-goals, so we're done.
  Yes, Ziggy eats fish.