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Artificial Intelligence. First-Order Logic (FOL). Outline of this Chapter. The need for FOL? What is a FOL? Syntax and semantics of FOL Using FOL. Pros and cons of propositional logic.  Propositional logic is declarative: pieces of syntax correspond to facts.

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artificial intelligence

Artificial Intelligence

First-Order Logic (FOL)

outline of this chapter
Outline of this Chapter
  • The need for FOL?
  • What is a FOL?
  • Syntax and semantics of FOL
  • Using FOL
pros and cons of propositional logic
Pros and cons of propositional logic

Propositional logic is declarative:pieces of syntax correspond to facts.

 Propositional logic has expressive power to deal with partial information, using disjunction & negation.

    • (unlike most data structures and databases)
  • Propositional logic is compositional: - the meaning of a sentence is a function of the meaning of its parts
    • meaning of B1,1 P1,2 is derived from meaning of B1,1 and of P1,2
  • Meaning in propositional logic is context-independent
    • (unlike natural language, where meaning depends on context)

 Propositional logic has very limited expressive power to describe an Env with many objects concisely

    • (unlike natural language- syntax & semantic of English make it possible to describe the Env concisely.
    • E.g., cannot say "pits cause breezes in adjacent squares“
      • except by writing a separate rule about B & P for each square.
first order u logic fol
First-order u logic (FOL)
  • FOL more expressive language than the PL.
  • FOL is a general-purpose representation language that is based on an ontological commitment to the existence of objects & relations in the world.
  • It builds on simple propositional logic by adding objects, predicates, variables and quantifiers.
  • Whereas propositional logic assumes the world contains facts (can be T/F),
  • FOL (like natural language) assumes the world contains
    • Objects: people, houses, numbers, colors, baseball games, wars, …
    • Relations: brother of, bigger than, part of, comes between, …
    • Properties: unary relation such as red, round
    • Functions: relations in which there is only 1 value for a giver input. i.e. father of, best friend, plus, …
example
Example
  • “One Plus Two Equals Three “

Objects: One, Two, Three

Relations: Equals

Functions: Plus

  • “Square neighboring the wumpus are smelly“

Objects: wumpus, square

Relations: neighboring

Property: smelly

syntax and semantics
Syntax and semantics
  • In Propositional logic – every expression is a sentence
  • In FOL = Sentences + Terms (which represents objects)
  • Sentences - are built using quantifiers and predicate symbols
  • Term: Logical expression that refers to an object

= function (term1,...,termn) or constant or variable.

= Brother (Samir, Ali) Ali, x,y

syntax of fol basic elements
Syntax of FOL: Basic elements

The basic syntactic elements of FOL are the symbols that stand for objects, relations & functions.

  • Constants (stand for objects) Sarah, A, B,...
  • Predicates (stand for relation) Brother, Round,...
  • Functions (stand for function) FatherOf,...
  • Variables x, y, a, b,...
  • Connectives , , , , 
  • Equality =
  • Quantifiers , 
atomic sentences
Atomic sentences
  • It is formed from a predicate symbol followed by a parenthesized list of terms.
  • Atomic sentence = predicate (term1,...,termn) or term1 = term2

E.g. Sister (Mona, Manal)

  • They can have arguments that are complex terms

E.g. Married(FatherOf(Ali),Sister (Mona))

complex sentences
Complex sentences
  • Complex sentences are made from atomic sentences using logical connectives

e.g.

    •  Sister(Mona,Alia) is true when Mona is not the sister of Alia.
    • Older (Reem, 20)  Younger(Reem, 20)

if Reem is older than 20, then she is not younger than 20.

truth in first order logic
Truth in first-order logic
  • Sentences are true with respect to a model and an interpretation
  • Model contains objects (domainelements) and relations among them
  • Interpretation specifies referents for

constant symbols → objects

predicate symbols → relations

function symbols → functional relations

  • An atomic sentence predicate(term1,...,termn) is true

iff the objects referred to by term1,...,termnare in the relation referred to by predicate

truth example
Truth Example
  • Consider the interpretation in which
    • Sara  Sara Ibrahim
    • Mona  Mona Ibrahim
    • Sister  the sisterhood relation
  • Under this interpretation, Sister(Sara; Mona) is true just in case Sara Ibrahim and Mona Ibrahim are in the sisterhood relation in the model
models for fol example
Models for FOL: Example
  • A model containing 5 objects. These objects may be related in various ways:
  • Binary relation – brother & on head
  • Unary relations (properties) – person, crown & king
  • Unary function – left leg
quantification
Quantification
  • Quantification is a formula constructor that produces new formulas from old ones.
  • Quantifier is a symbol- let us express properties of entire collections of objects, instead of listing the objects by name.
  • In natural language- for all, for some; many, few, a lot, etc
  • FOL contains 2 standard quantifiers:
    • Universal
    • existential
universal quantification
Universal quantification
  • <variables> <sentence>

e.g. All cats are mammals:

 x Cats(x) => Mammals(x), for all x, if x is a cat, then x is a mammal

  • Universal quantification makes statements about every object.
  • Universal quantifiers often used with "implies" to form "rules.“
  • Everyone at PSU is smart :  x At(x, PSU) => Smart(x)
  •  x P (any logical expression) is equivalent to the conjunction of instantiations of P

At( Sara, PSU) => Smart(Sara)  At(Mary, PSU) => Smart(Mary)

 ...

  • Typically, is the main connective with 
  • Common mistake: using  as the main connective with :
existential quantification
Existential quantification
  •  <variables> <sentence>
  • Existential quantification makes statements about some object.
  • Someone at PSU is smart: x At(x, PSU)  Smart(x)
  •  x P is equivalent to thedisjunctions of instantiations of P

At( Sara, PSU) => Smart(Sara)  At(Mary, PSU) => Smart(Mary)

 ...

  • Typically, is the main connective with 
  • Common mistake: using  as the main connective with :
properties of quantifiers
Properties of quantifiers

To express more complex sentences, use multiple 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)

“For every person, there is someone that person loves”

  •  y x Loves(x,y)

“There is someone who is loved by every body”

  • Quantifier duality: each can be expressed using the other (& are connected with each other through negation)
  • x Likes(x,IceCream) x Likes(x,IceCream)
  •  x Likes (x, Cabbage) x Likes(x, Cabbage)

The deMorgan rules for quantified and unquantified sentences are as follows

 x P = x P P  Q = (P V Q )

 x P = x P (P  Q) = P V Q

x P = x P P  Q = (P V Q )

x P =  x P P V Q = (P  Q )

equality
Equality
  • term1 = term2is true under a given interpretation if and only if term1andterm2refer to the same object

e.g. Mother(Mona) = Jenny

  • The equality symbol can be used with negation to insist that two terms are not the same object.

e.g. Mona has at least 2 sisters.

x,y Sister(Mona,x)  Sister(Mona,y)  (x=y)

examples
Examples
  • All food are edible
  • Brothers are siblings
  • “Sibling" is symmetric relation ship (quantifiers of the same type)
  • One's mother is one's female parent

Solutions:

  •  x food( x) => edible( x)
  •  x, y Brother(x, y)  Sibling(x, y).
  •  x, y Sibling(x, y) Sibling(y, x).
  •  m, c Motherof(c) = m  (Female(m)  Parent(m, c))
interacting with fol kbs
Interacting with FOL KBs
  • To add sentences to a knowledge base KB, we would call TELL. Such sentences are called assertions.

e.g. TELL (KB, Mother(Nora)) // assert that Nora is a Mother

TELL(KB,  x Mother(x)  Person (x)) // Mothers are persons

  • To ask questions of the KB, we use ASK
  • Questions asked using ASK are called queries

e.g.ASK (KB,Mother(Nora))

in class activity
In-class Activity
  • Abeer, Mona and linda belong to the champions Club.
    • Every member of the club is either skier or a mountain climber .
    • No mountain climber likes rain, and all skiers like snow
    • linda dislikes whatever Abeer likes and likes whatever Abeer dislikes
    • Abeer likes rain and snow
  • Is there any member of the club who is climber but not skier?
  • Translate this problem into FOL Sentences?
solution
Solution
  • S(x) means x is a skier
  • M(x) means x is a mountain climber
  • L(x,y) means x likes y
  • The English sentences in the prev slide can be translated into the following FOL:

 x Clubmember(x) => S(x) V M (x)

~  x M(x)  L(x,rain)

 x S(x) => L(x,snow)

 y L(linda, y) <=> ~L(Abeer,y)

L(Abeer, rain) ^L(Abeer,snow)

summary
Summary
  • FOL can be used as the representation language for a KB agent.
  • FOL is more powerful than propositional logic
  • A possible world for FOL is defined by a set of :
    • objects and relations are semantic primitives
    • syntax: constants, functions, predicates, equality, quantifiers
  • Complex sentences use connectives just like propositional logic, & quantified sentences allow the expression of general rules.