First Order Logic FOL

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First Order Logic (FOL)

• Syntax
• Semantics
• Wumpus world example
• Ontology (ont to be logica word) kinds
  of things one can talk about in the language

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Why first-order logic?

• We saw that propositional logic is limited
  because it only makes the ontological commitment
  that the world consists of facts.
• Difficult to represent even simple worlds like
  the Wumpus world
• e.g.,
• dont go forward if the Wumpus is in front of
  you takes 64 rules.

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First-order logic (FOL)

• Ontological commitments
• Objects wheels, doors, engines, cars, Dave, Joe
• Relations Inside(car1, Dave), Beside(Dave, Joe)
• Functions WeightOf(car1)
• Properties IsOpen(door1), IsOn(engine1)
• Taxonomic Properties Car(car1), Person(Dave),
  Person(Joe)
• Relations return True or False
• Functions return a single objects, otherwise they
  wouldnt be functions.
• WeightOf(x) returns a single weight
• BrotherOf(x) is not a function. You may have
  several brothers. But, NaturalMotherOf(x) is a
  function.

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Semantics

• There is a correspondence between
• functions, which return values
• predicates (or relations), which are true or
  false
• Function fatherOf(Mary) Bill
• Predicate fatherOfP(Mary, Bill)
• The point is that every function can have a
  corresponding predicate associated with it.

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Examples

• One plus two equals three
• Objects
• Relations
• Properties
• Functions
• Squares neighboring the Wumpus are smelly
• Objects
• Relations
• Properties
• Functions

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Examples

• One plus two equals three
• Objects Numbers, one, two, three, one plus two
• Relations equals(one, two)
• Properties Integer(one)
• Functions plus (one, two) returns one plus
  two , the object obtained by applying
  function plus to one and two
• three is another name for this object)
• Squares neighboring the Wumpus are smelly
• Objects squares, Wumpus, square1,1, square1,2
• Relations neighboring(square1,1, square1,2)
• Properties smelly(square1,1)
• Functions rt-neighborOf(square1,1) returns
  square1,2

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FOL Syntax of basic elements

• Constant symbols 1, 5, A, B, USC, JPL, Alex,
  Manos,
• Predicate symbols gt, friend, student, colleague,
  
• Function symbols , sqrt, schoolOf,
• Variables x, y, z, next, first, last,
• Connectives ?, ?, ?, ?
• Quantifiers ?, ?
• Equality

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FOL Atomic sentences

• AtomicSentence ? Predicate(Term, ) Term Term
• Term ? Function(Term, ) Constant Variable
• Examples
• SchoolOf(Manos)
• Colleague(TeacherOf(Alex,cs561), TeacherOf(Manos,
  cs571))
• gt(( x y), x)

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FOL Complex sentences

• Sentence ? AtomicSentence Sentence
  Connective Sentence Quantifier Variable,
  Sentence ? Sentence (Sentence)
• Examples
• S1 ? S2, S1 ? S2, (S1 ? S2) ? S3, S1 ? S2, S1?
  S3
• Colleague(Paolo, Maja) ? Colleague(Maja, Paolo)
  Student(Alex, Paolo) ? Teacher(Paolo, Alex)

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Semantics of atomic sentences

• Sentences in FOL are interpreted with respect to
  a model
• Model contains objects and relations among them
• Terms refer to objects (e.g., Door1, Alex,
  MotherOf(Paolo))
• Constant symbols refer to objects
• Predicate symbols refer to relations
• Function symbols refer to functional Relations
• An atomic sentence predicate(term1, , termn) is
  true iff the relation referred to by predicate
  holds between the objects referred to by term1,
  , termn

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Example model

• Objects John, James, Mary, Alex, Dan, Joe, Anne,
  Rich
• Relation sets of tuples of objectsltJohn,
  Jamesgt, ltMary, Alexgt, ltMary, Jamesgt, ltDan,
  Joegt, ltAnne, Marygt, ltMary, Joegt,
• E.g. Parent relation -- ltJohn, Jamesgt, ltMarry,
  Alexgt, ltMary, Jamesgtthen Parent(John, James)
  is true Parent(John, Mary) is false

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Quantifiers

• Expressing sentences about collections of objects
  without enumeration (naming individuals)
• E.g., All Trojans are clever Someone in the
  class is sleeping
• Universal quantification (for all) ?x R(x)
• Existential quantification (there exists) ?x R(x)

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Universal quantification (for all) ?

• ? ltvariablesgt ltsentencegt
• Every one in the cs561 class is smart ? x
  In(cs561, x) ? Smart(x)
• ? P corresponds to the conjunction of
  instantiations of PIn(cs561, Manos) ?
  Smart(Manos) ? In(cs561, Dan) ? Smart(Dan) ?
  In(cs561, Bush) ? Smart(Bush)

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Universal quantification (for all) ?

• ? is a natural connective to use with ?
• Common mistake to use ? in conjunction with ?
  e.g ? x In(cs561, x) ? Smart(x)means every
  one is in cs561 and everyone is smart

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Existential quantification (there exists) ?

• ? ltvariablesgt ltsentencegt
• Someone in the cs561 class is smart ? x
  In(cs561, x) ? Smart(x)
• ? P corresponds to the disjunction of
  instantiations of P(In(cs561, Manos) ?
  Smart(Manos)) ? (In(cs561, Dan) ? Smart(Dan)) ?
  (In(cs561, Bush) ? Smart(Bush))

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Existential quantification (there exists) ?

• ? is a natural connective to use with ?
• Common mistake to use ? in conjunction with ?
  e.g ? x In(cs561, x) ? Smart(x)is true if
  there is anyone that is not in cs561!
• (remember, false ? true is valid).

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Properties of quantifiers
Not all by one person but each one at least by one
Proof?
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Negation Laws for Quantified Statements

• ? x P(x) ltgt ? x P(x)
• ? x P(x) ltgt ? x P(x)

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Informal Proof

• In general we want to prove
• ? x P(x) ltgt ? x P(x)
• ? x P(x) ((? x P(x))) ((P(x1) P(x2)
  P(xn)) ) (P(x1) v P(x2) v v P(xn)) )
• ? x P(x) P(x1) v P(x2) v v P(xn)
• ? x P(x) (P(x1) v P(x2) v v P(xn))

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Example sentences

• Brothers are siblings .
• Sibling is transitive.
• Ones mother is ones siblings mother.
• A first cousin is a child of a parents
  sibling.

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Example sentences

• Brothers are siblings ? x, y Brother(x, y) ?
  Sibling(x, y)
• Sibling is transitive? x, y, z Sibling(x, y)
  ? Sibling(y, z) ? Sibling(x, z)
• Ones mother is ones siblings mother? m, c
  Mother(m, c) ? Sibling(c, d) ? Mother(m, d)
• A first cousin is a child of a parents
  sibling? c, d FirstCousin(c, d) ? ? p, ps
  Parent(p, d) ? Sibling(p, ps) ? Parent(ps, c)

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Example sentences

• Ones mother is ones siblings mother? m, c,d
  Mother(m, c) ? Sibling(c, d) ? Mother(m, d)
• ? c,d ?m Mother(m, c) ? Sibling(c, d) ? Mother(m,
  d)

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Translating English to FOL

• Every gardener likes the sun.
• ? x gardener(x) gt likes(x,Sun)
• You can fool some of the people all of the time.
• ? x ? t (person(x) time(t)) gt can-fool(x,t)

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Translating English to FOL

• You can fool all of the people some of the time.
• ? x ? t (person(x) time(t)) gt
• can-fool(x,t)
• All purple mushrooms are poisonous.
• ? x (mushroom(x) purple(x)) gt poisonous(x)

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Translating English to FOL

• No purple mushroom is poisonous.
• (? x) purple(x) mushroom(x) poisonous(x)
• or, equivalently,
• (? x) (mushroom(x) purple(x)) gt poisonous(x)

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Translating English to FOL

• There are exactly two purple mushrooms.
• (? x)(? y) mushroom(x) purple(x) mushroom(y)
  purple(y) (xy) (? z) (mushroom(z)
  purple(z)) gt ((xz) v (yz))
• Deb is not tall.
• tall(Deb)

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Translating English to FOL

• 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.
• (? x)(? y) above(x,y) ltgt (on(x,y) v (? z)
  (on(x,z) above(z,y)))

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Equality

• A sibling is another child of ones parents
• ? x,y Sibling(x,y) ? (xy) ? ? p Parent(p, x)
  ? Parent(p, y)

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Higher-order logic?

• First-order logic allows us to quantify over
  objects ( the first-order entities that exist in
  the world).
• Higher-order logic also allows quantification
  over relations and functions.
• e.g., two objects are equal iff all properties
  applied to them are equivalent
• ? x,y (xy) ? (? p, p(x) ? p(y))
• Higher-order logics are more expressive than
  first-order however, so far we have little
  understanding on how to effectively reason with
  sentences in higher-order logic.

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Rememberpropositionallogic
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Reminder

• Ground term A term that does not contain a
  variable.
• A constant symbol
• A function applies to some ground term
• x/a substitution/binding list

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Proofs
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Proofs

• The three new inference rules for FOL (compared
  to propositional logic) are
• Universal Elimination (UE) (also called
  Universal Instantiation)
• for any sentence ?, variable x and ground term
  ?,
• ?x ?
• ?x/?
• Existential Elimination (EE) (also called
  Existential Instantiation)
• for any sentence ?, variable x and constant
  symbol k not in KB,
• ?x ?
• ?x/k
• Existential Introduction (EI) (also called
  Existential Generalization)
• for any sentence ?, variable x not in ? and
  ground term g in ?,
• ?
• ?x ?g/x

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Proofs

• The three new inference rules for FOL (compared
  to propositional logic) are
• Universal Elimination (UE) (also called
  Universal Instantiation)
• for any sentence ?, variable x and ground term
  ?,
• ?x ? e.g., from ?x Likes(x, Candy) and
  x/Joe
• ?x/? we can infer Likes(Joe, Candy)
• Existential Elimination (EE) (also called
  Existential Instantiation)
• for any sentence ?, variable x and constant
  symbol k not in KB,
• ?x ? e.g., from ?x Kill(x, Victim) we can
  infer
• ?x/k Kill(Murderer, Victim), if Murderer
  new symbol
• Existential Introduction (EI) (also called
  Existential Generalization)
• for any sentence ?, variable x not in ? and
  ground term g in ?,
• ? e.g., from Likes(Joe, Candy) we can
  infer
• ?x ?g/x ?x Likes(x, Candy)

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A Simple Proof

• Buffalo(Bob) Bob is a buffalo
• Pig(Pat) Pat is a pig
• ?x,y Buffalo(x) Pig(y) -gt Faster(x,y)
  Buffaloes outrun pigs.
• Prove Faster(Bob,Pat) Bob outruns Pat.
• Buffalo(Bob) Pig(Pat) -gt Faster(Bob,Pat)
  3,UE xBob,yPat
• Buffalo(Bob) Pig(Pat) 1,2 Conjunction
• Faster(Bob,Pat) 5,6 Modus Ponens QED!!

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Another Proof (longer, but just as simple)

• C(x) means x is in this class.
• B(x) means x has read the book.
• P(x) means x has passed the class.
• ?x C(x) B(x) (Someone in the class has not
  read the book)
• ?x C(x) -gt P(x) (Everyone in the class passes)
• Prove ?x P(x) B(x) (Someone passed and didnt
  read the book)
• C(a) B(a) 1, EE (a is the skolem constant
  must be new)
• C(a) 4, And-Elimination (or simplification)
• C(a) -gt P(a) 2, Universal Instantiation
• P(a) 5,6 Modus Ponens
• B(a) 4, And-Elimination (or simplification)
• P(a) B(a) 7,8 Conjunction
• ?x P(x) B(x) 9, Existential generalization
  QED!

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More Inference Techniques

• Remember ?x P(x) ltgt ?x P(x) ?x P(x) ltgt ?x
  P(x)
• Say you are doing a proof 1. 2. 3. Prove ?x
  P(x) 4. ?x P(x) Proof by contradiction5. ?x
  P(x) 4, Equivalence6. P(a) 5, EE (a is
  skolem)7.
• DO YOUR EES FIRST! THEN DO YOU YOUR UES.

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Logical agents for the Wumpus world
Remember generic knowledge-based agent

• TELL KB what was perceivedUses a KRL to insert
  new sentences, representations of facts, into KB
• ASK KB what to do.Uses logical reasoning to
  examine actions and select best.

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Using the FOL Knowledge Base
Set of solutions
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Wumpus world, FOL Knowledge Base
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Deducing hidden properties
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Situation calculus
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Describing actions
May result in too many frame axioms
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Describing actions (contd)
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Planning
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Generating action sequences
empty plan
Recursively continue until it gets to empty plan

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Summary