Formal Logic

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Formal Logic

• The most widely used formal logic method is
• FIRST-ORDER PREDICATE LOGIC
• Reference Chapter Two The predicate Calculus
• Lugers Book
• Examples included from Norvig and Russel.

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First-order logic

• Whereas propositional logic assumes the world
  contains facts,
• first-order logic (like natural language) assumes
  the world contains
• Objects people, houses, numbers, colors,
  baseball games, wars,
• Relations red, round, prime, brother of, bigger
  than, part of, comes between,
• Functions father of, best friend, one more than,
  plus,

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FOL

• PL only deals with facts
• PL cannot have variables
• Variables refer to things in the world
• You can deliberate over them
• How would represent the following fact in PL
• When you sterilize a jar all the bacteria are dead

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Syntax of FOL Basic elements

• Constants john, 2, lums,...
• Predicates brother, gt,...
• Functions sqrt, leftsideOf,...
• Variables X,Y,A,B...
• Connectives ?, ?, ?, ?, ?
• Equality
• Quantifiers ?, ?

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Syntax of FOL Basic elements
Alternate Convention also exists (Norvig uses the
alternate)

• Constants John, Lums,...
• Predicates Brother, gt,...
• Functions Sqrt, LeftsideOf,...
• Variables x,y,a,b...
• Connectives ?, ?, ?, ?, ?
• Equality
• Quantifiers ?, ?

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Truth in first-order logic

• Sentences are true with respect to a model and an
  interpretation
• Model contains objects (domain elements) 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 term,,..., term ,
• are in the relation referred to by predicate

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Alphabets-I

• Predicates, variables, functions,constants,
  connectives, quantifiers, and delimiters
• Constants (first letter small)
• bLUE a color
• sanTRO a car
• crow a bird
• Variables (first letter capital)
• Dog an element that is a dog, but
  unspecified
• Color an unspecified color

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Alphabets-II

• Function (evaluates to a constant / variable)
• Maps Sentences to Objects
• Ali is father of Babar father_of(ali) baber
• father(babar) ali
• Babar is son of Ali son_of(babar) ali
• Interpretation has to be very clear.
• If you write father(baber), the answer should be
  ali
• For the above functions the arity is 1 (number of
  arguments to the function)

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Alphabets-II

• Functions
• 1) shahid likes zahid likes(shahid) zahid
• 2) atif likes abid likes(atif) abid
• 3) Constants to Variables likes(X) Y
• X,Y have two possible BINDINGS
• X, Y could be shahid, zahid
• Or
• X,Y could be atif, abid

likes(X) Y Substitutions For 1 to be
true shahid/X, zahid/Y For 2 to be
true atif/X, abid/Y
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Alphabets-II

• Predicate
• Maps Sentences to Truth Values (True/False)
• 1) Shahid is student student(shahid)
• 2) Sana is a girl girl(sana)
• 3) Father of baber is elder than Hamza
• elder(father(babar), hamza)
• For 1 and 2 arity is 1 and for 3 the arity is 2

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Alphabets-II

• Predicate
• Shahid is a good student
• student(shahid,good) or good_student(shahid) or
• is_good(shahid,student)
• 2) Sana is a friend of Saima, Sana and Saima both
  are girls
• friend_of(sana,saima)girl(sana)girl(saima)
• 3) Bill helps Fred
• helps(bill,fred)

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Atomic sentences

• Atomic sentence predicate (term1,...,termn)
  or term1 term2
• term function (term1,...,termn)
  or constant
• or variable
• brother(john,richard)
• greater(length(leftsideOf(squareA)),
  length(leftsideOf(squareB)))
• gt(length(leftsideOf(squareA)), length(leftsideOf(s
  quareB)))
• Functions cannot be atomic sentences

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Alphabets-III

• Connectives
• and
• v or
• not
• Implication
• Quantification
• All persons can see
• There is a person who cannot see
• Universal quantifiers ? (ALL)
• Existential quantifiers ? (There exists)

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Complex sentences

• Complex sentences are made from atomic sentences
  using connectives
• ?S, S1 ? S2, S1 ? S2, S1 ? S2, S1 ? S2,
• sibling(ali,hamza) ? sibling(hamza,ali)
• gt(1,2) ? (1,2) (1 is greater than 2 or less
  than equal to 2)
• gt(1,2) ? ? gt(1,2) (1 is greater than 2 and is not
  greater than equal to 2)

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Examples

• My house is a blue, two-story, with red shutters,
  and is a corner house
• blue(my-house)two-story(my-house)red-shutters(my
  -house)corner(my-house)
• Ali bought a scooter or a car
• bought(ali , car) v bought(ali , scooter)
• IF fuel, air and spark are present the fuel will
  combust
• present(spark)present(fuel)present(air)
• combustion(fuel)

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Universal quantification

• ?ltvariablesgt ltsentencegt
• Everyone at LUMS is smart
• ?X at(X, lums) ? smart(X)
• ?X P is true in a model m iff P is true with X
  being each possible object in the model
  
• Roughly speaking, equivalent to the conjunction
  of instantiations of P
  
• at(rabia,lums) ? smart(rabia)
• ? at(shahid,lums) ? smart(shahid)
• ? at(lums,lums) ? smart(lums)
• ? ...

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Examples

• All people need air
• ?Xperson(X) need_AIR(X)
• The owner of the car also owns the boat
• owner(X , car) car(X , boat)
• Formulate the following expression in the PC
• Ali is a computer science student but not a
  pilot or a football player
• cs_STUDENT(ali) ? (? pilot(ali) ? ?
  ft_PLAYER(ali) )

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Examples

• Restate the sentence in the following way
• Ali is a computer science (CS) student
• Ali is not a pilot
• Ali is not a football player
• cs_student(ali)
• pilot(ali)
• football_player(ali)

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Examples

• Studying fuzzy systems is exciting and applying
  logic is great fun if you are not going to spend
  all of your time slaving over the terminal
• ? X(slave_terminal(X) fs_eciting(X)logic_fun
  (X))
• Every voter either favors the amendment or
  despises it
• ?Xvoter(X) favor(X , amendment) v
  despise(X,amendment)
• favor(X , amendment) v
• despise(X , amendment)
• (this part simply endorses the statement, may not
  be required)

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Undecidable Predicate

• For which exhaustive testing is required
• Example
• ?X likes(zahra, X)
• This sentence is computationally impossible to
  calculate
• Scope of problem domain is to be limited to
  remove this problem,
• i.e., X is a variable representing final year
  female student in the AI class, compared to an X
  representing all the people in the city of Lahore

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Robotic Arm Example

• Represent the initial details of the system
• Generate sentences of descriptive and or
• implicative nature
• Modify the facts using new sentences

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Example Robotic Arm

• Represent the initial details of the systems

on(a,b) on( c,d) ontable(b) ontable(d) clear(a) cl
ear(c) hand_empty
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Goal To pick a block and place it over another
block
Define predicate stack_on(X,Y) General
sentence Conditions Conclusions What could
the conditions be? hand_empty clear
(X) clear (Y) pick (X) put_on (X,Y)
hand_empty clear (X) clear (Y) pick (X)
put_on (X,Y) stack_on (X,Y)
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Goal To pick a block and place it over another
block
hand_empty clear (X) clear (Y) pick (X)
put_on (X,Y) ? stack_on (X,Y)
Semantically more correct
hand_empty could be written as empty(hand), if
hand_empty is in the knowledge base, then hand is
empty otherwise false. put_on (X,Y) ? stack_on
(X,Y)
hand_empty clear (X) ? pick (X) clear(Y)
pick (X)? put_on (X,Y) put_on (X,Y) ? stack_on
(X,Y)
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Example Robotic Arm

• Modify details of the systems

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Models for FOL Example
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A common mistake to avoid

• Represent Everyone at LUMS is smart
• ?X at(X,lums) ? smart(X)
• ?X at(X, lums) ? smart(X)
• Common mistake
• using ? as the main connective with ? means
• Everyone is at LUMS and everyone is smart
• Everyone at LUMS is smart
• Typically,
• ? is the main connective with ?

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Existential quantification

• ?ltvariablesgt ltsentencegt
• Someone at LUMS is smart
• ?X at(X,lums) ? smart(X)
• ?X P is true in a model m iff P is true with X
  being some possible object in the model
• Roughly speaking, equivalent to the disjunction
  of instantiations of P
• at(sana,lums) ? smart(sana)
• ? at(bashir,lums) ? smart(bashir)
• ? at(lums,lums) ? smart(lums)
• ? ...

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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,lums) ? smart(X)
• is true even if there is anyone who is not at
  LUMS!
• Even if the antecedent is false the sentence can
  still be true (see the truth table of
  implication).

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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
• ?Y ?X loves(X,Y)
• Everyone in the world is loved by at least one
  person
• Quantifier duality each can be expressed using
  the other
• ?X likes(X,car) ??X ?likes(X,car)
• ?X likes(X,bread) ??X ?likes(X,bread)

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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
• 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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Using FOL

• The kinship domain
• Brothers are siblings
• ?X,Y brother(X,Y) ? sibling(X,Y)
• One's mother is one's female parent
• ?M,C mother(C) M ? (female(M) ? parent(M,C))
• Sibling is symmetric
• ?X,Y sibling(X,Y) ? sibling(Y,X)

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Rules Wumpus worldNorvig and Russel book for
details(will provide you with handouts)

• Actions
• Turn(Right), Turn(Left), Forward, Shoot, Grab,
  Release, Climb
  
• Reflex
• ?T glitter(T) ? bestAction(grab,T)

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Rules Wumpus worldNorvig and Russel book for
details(will provide you with handouts)

• Perception
• ?T,S,B percept(S,B,glitter,T) ? glitter(T)
• Reflex
• ?T glitter(T) ? bestAction(grab,T)

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Rules Wumpus worldNorvig and Russel book for
details(will provide you with handouts)

• Perception
• ?T,S,B percept(S,B,glitter,T) ? glitter(T)
• Reflex
• ?T glitter(T) ? bestAction(grab,T)

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Rules Wumpus worldNorvig and Russel book for
details(will provide you with handouts)

• Perception
• ?T,S,B percept(S,B,glitter,T) ? glitter(T)
• Reflex
• ?T glitter(T) ? bestAction(grab,T)

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Deducing Squares/Properties

• What are Adjacent Squares
• ?X,Y,A,B adjacent(X,Y,A,B) ?
• A,B ? X1,Y, X-1,Y,X,Y1,X,Y-1
• Properties of squares
• ?S,T at(agent,S,T) ? breeze(T) ? breezy(S)
• Squares are breezy near a pit
• Diagnostic rule---infer cause from effect
• ?S breezy(S) ? adjacent(R,S) ? pit(R)
• Causal rule---infer effect from cause
• ?R pit(R) ? ?S adjacent(R,S) ? breezy(S)

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Deducing Squares/Properties

• Diagnostic rule---infer cause from effect
• ?S breezy(S) ? adjacent(R,S) ? pit(R)
• Causal rule---infer effect from cause
• ?R pit(R) ? ?S adjacent(R,S) ? breezy(S)

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Knowledge engineering in FOL

1. Identify the task
2. Assemble the relevant knowledge
3. Decide on a vocabulary of predicates, functions,
   and constants
   
4. Encode general knowledge about the domain
5. Encode a description of the specific problem
   instance
   
6. Pose queries to the inference procedure and get
   answers
   
7. Debug the knowledge base

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Inferencing in FOL

• Interpretation of the objects/variables etc
• Denotation of terms
• Find whether an interpretation holds for a
  particular sentence

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Interpretation I/Denotation

• U set of Objects ( U universe of discourse)
• Map constants symbols to objects in U
• I(zahid), Uzahid, shahid, raza, sana, T
• I(X) what happens now?
• Map predicates symbols to the relations between
  the objects on U
• Map function symbols to functions on U
• I(predicate(t1, t2)) I(predicate)(I(t1), I(t2))

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Holds (meta rule)

• holds (predicate(t1,t2), I)
• iff
• ltI(t1),I(t2)gt ? I(predicate)

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Example Robotic Arm

• Find if the following sentence holds or not

stack_on (X,Y)
I(X) I(Y) I(stack_on) holds (stack_on(X,Y),I) iff
ltI(X),I(Y)gt ? I(stack_on)) ??? No stack_on is
not in U on Holds but under the following
interpretations a/X, b/Y or c/X, d/Y clear(a)
holds but clear(b) does not hold
on(a,b) on( c,d) ontable(b) ontable(d) clear(a) cl
ear(c) hand_empty
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Quantifiers
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FOL Example
What is U? Define the following Predicates
above 2 circle 1 oval 1 square 1 triangle
1 Function hat 2
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FOL Example
What is U?
I(object)
I(above)
Find whether I(square) ????
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Reason it Out

• holds (sqaure(object),I) ?
• YES
• holds (above(object,hat(object)),I) ?
• I(hat(object)) sqaure
• holds(above(triangle,sqaure),I)?
• NO
• Hence I(square) FALSE

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Find the truth of

• holds (? X, oval(X)), I) ?
• I(sqaure)lttrianglegt
• Note for the students
• Should have solved it when you come next time.

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The electronic circuits domain

• One-bit full adder

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The electronic circuits domain

• Identify the task
• Does the circuit actually add properly?
• (circuit verification)
• Assemble the relevant knowledge
• Composed of wires and gates Types of gates (AND,
  OR, XOR, NOT)
  
• Irrelevant size, shape, color, cost of gates
• Decide on a vocabulary
• Alternatives
• type(x1) xor
• type(x1, xor)
• xor(x1)

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The electronic circuits domain

• Encode general knowledge of the domain
• ?T1,T2 connected(T1, T2) ? signal(T1)
  signal(T2)
• ?T signal(T) 1 ? signal(T) 0
• 1 ? 0
• ?T1,T2 connected(T1, T2) ? connected(T2, T1)
• ?G type(G) OR ? signal(out(1,G)) 1 ? ?N
  signal(in(N,G)) 1
  
• ?G type(G) AND ? signal(out(1,G)) 0 ? ?N
  signal(in(N,G)) 0
  
• ?G type(G) XOR ? signal(out(1,G)) 1 ?
  signal(in(1,G)) ? signal(in(2,G))
  
• ?G type(G) NOT ? signal(out(1,G)) ?
  signal(in(1,G))
  

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The electronic circuits domain

• Encode the specific problem instance
• type(x1) xor type(x2) xor
• type(a1) and type(a2) and
• type(o1) or
• connected(out(1,x1),in(1,x2)) connected(in(1,c1),
  in(1,x1))
• connected(out(1,x1),in(2,a2)) connected(in(1,c1),
  in(1,a1))
• connected(out(1,o2),in(1,o1)) connected(in(2,c1)
  ,in(2,x1))
• connected(out(1,a1),in(2,o1)) connected(in(2,c1)
  ,in(2,a1))
• connected(out(1,x2),out(1,c1))
  connected(in(3,c1),in(2,x2))
• connected(out(1,o1),out(2,c1)) connected(in(3,c1
  ),in(1,a2))

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The electronic circuits domain

• Pose queries to the inference procedure
• What are the possible sets of values of all the
  terminals for the adder circuit?
  
• ?I1,I2,I3,O1,O2 signal(in(1,c_1)) I1 ?
  signal(in(2,c1)) I2 ? signal(in(3,c1)) I3 ?
  signal(out(1,c1)) O1 ? signal(out(2,o1)) O2
  
• Debug the knowledge base
• May have omitted assertions like 1 ? 0

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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
  

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Operations/Processes

• Entailment
• Unification Algorithm for determining the
  substitutions needed to make two predicate
  calculus expressions match
• Skolemization A method of removing or replacing
  existential quantifiers using Skolem function
• Composition If S and S are two substitutions
  sets, then the composition of S and S (SS) is
  obtained by applying the elements of S to the
  elements of and adding the S to S

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• Examples (Alternate Convention)

Superscript represents arity of
predicates/functions
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Entailment

• A knowledge base entails a sentence IFF the
  sentence holds in every interpretation in which
  the knowledge base holds
• In Propositional Logic
• A knowledge base entails a sentence IFF the
  sentence is TRUE in every interpretation that
  makes the knowledge base TRUE

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Entailment (follows from)

• KB entails S for every Interpretation I,
• if KB holds in I,
• then
• S holds in I

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Entailment

• Computing entailment is impossible in general,
  because there are Infinitely many possible
  interpretations
• Even computing holds is impossible for
  interpretations with infinite universes

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Example
Alternate Convention
61
Example
Alternate Convention
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Example
Alternate Convention