CS 4700: Foundations of Artificial Intelligence

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CS 4700Foundations of Artificial Intelligence

• Carla P. Gomes
• gomes_at_cs.cornell.edu
• Module
• FOL
• (Reading RN Chapter 8)

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Pros and cons of propositional logic

• ? Propositional logic is declarative
• its semantics is based on a truth relation
  between sentences and possible worlds
• ? Propositional logic allows partial/disjunctive/n
  egated information
• (unlike most data structures and databases)
• Propositional logic is compositional
• 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
  
• (unlike natural language)
• E.g., cannot say "pits cause breezes in adjacent
  squares
• except by writing one sentence for each square

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Pigeon Hole Principle

• Consider a formula to represent the Pigeon Hole
  principle (n1) pigeons
• in n holes.
• Pij pigeon i goes in hole j Pij ? 0,1 i
  1, 2, , n1 j 1, 2, n
• Each pigeon has to go in a hole
• Starting with pigeon 1
• P11 ? P12 ? ? P1n
• Pn1,1 ? Pn1,2 ? ? Pn1,n
• .

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PH

• And
• Two pigeons cannot go in the same hole
• P11 ? P21 P11 ? P31 P11 ? Pn1,1
• P11 ? P21 P11 ? P31 P11 ?Pn1,1
• .
• Pn1,n ? P2,n Pn1,n ? P3n Pn1,n
  ?Pn,n

Resolution proofs of PH take exponentially many
steps Resolution proofs of inconsistency of PH
require an exponential number of clauses, no
matter in what order we resolve the clauses
(Armin Haken 85). Related to NP vs. Co-NP
questions
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Pigeon HoleA more concise formulation

• ?x ?y (x ? Pigeons) (y ? Holes) IN(x,y)
• ?x?x?y ( IN(x,y) ?IN(x,y) ? xx)
• ?x?y?y (IN(x,y) ? IN(x,y) ? yy)

Pigeons P1,P2,, Pn1) Holes H1,H2, , Hn
We have first-order logic with some set notation
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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,
  

For finite domains, essentially equivalent to
propositional logic.but more concise. Often pays
off to propositionalize (more later) Well see
how key properties from PL carry over sound and
complete proof theory Sound and refutation
complete resolution.
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Syntax of FOL Basic elements

• Constants KingJohn, 2, CornellUniversity,...
• Predicates Brother, gt,...
• Functions Sqrt, LeftLegOf,...
• Variables x, y, a, b,...
• Connectives ?, ?, ?, ?, ?
• Equality
• Quantifiers ?, ?

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Atomic sentences

• A term is a logical expression that refers to an
  object
• Atomic sentence predicate (term1,...,termn)
  or term1 term2
• Term function (term1,...,termn)
  or constant or variable
• Examples of atomic sentences
• Brother(KingJohn,RichardTheLionheart)
• gt (Length(LeftLegOf(Richard)),
  Length(LeftLegOf(KingJohn))

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Complex sentences

• Complex sentences are made from atomic sentences
  using connectives
  
• ?S, S1 ? S2, S1 ? S2, S1 ? S2, S1 ? S2,
• E.g. Sibling(KingJohn,Richard) ?
  Sibling(Richard,KingJohn)
• gt(1,2) ? (1,2)
• gt(1,2) ? ? gt(1,2)

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Syntax of FOL

• Sentence ? AtomicSentence
• (Sentence Connective Sentence)
• Quantifier Variable, . Sentence
• ?Sentence
• AtomicSentence ? Predicate(Term) TermTerm
• Term ? Function(Term)
• Constant
• Variable
• Connective ? ????
• Quantifiers ???
• Constant ? A X1 John
• Variable ? a x s
• Predicate ? Before Greater Raining
• Function ? MotherOf SqrtOf

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Truth in first-order logic

• Sentences are true with respect to a world and
  an interpretation
• A world contains objects (domain elements) and
  relations among them
  
• Interpretation associates
• 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,...,termn
• are in the relation referred to by predicate

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Models for FOL Example

• World with five objects
• Richard the LionHeart, king of England
  (1189-1199)
• Evil King John, Richards brother (1199-1215)
• The left legs of Richard and John
• A crown

Binary relation
Unary relations (or properties)
functions
LeftLeg(Richard)
?x King(x) ? Greedy(x) ? Evil(x) King(John) Brothe
r(Richard,John)
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SemanticsUniversal quantification

• ?ltvariablesgt ltsentencegt
• (?x) P(x)
• it is true in an interpretation iif P(x) is true
  for all the values in the domain of x i.e., the
  universe of discourse
• Everyone at Cornell is smart
• ?x At(x,Cornell) ? Smart(x)
• Roughly speaking, equivalent to the conjunction
  of instantiations of P
  
• At(Mary,Cornell) ? Smart(Mary)
• ? At(Richard,Cornell) ? Smart(Richard)
• ? At(John,Cornell) ? Smart(John)
• ? ...

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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,Cornell) ? Smart(x)
• means Everyone is at Cornell and everyone is
  smart
  

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Existential quantification

• ?ltvariablesgt ltsentencegt
• (?x)P(x) it is true in an interpretation m iff
  there exists an element x in the universe of
  discourse such that P(x) is true.
• Someone at Cornell is smart
• ?x At(x,Cornell) ? Smart(x)
• Roughly speaking, equivalent to the disjunction
  of instantiations of P
  
• At(John,Cornell) ? Smart(John)
• ? At(Mary,Cornell) ? Smart(Mary)
• ? At(Richard,Cornell) ? Smart(Richard)
• ? ...

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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,Cornell) ? Smart(x)
• is true if there is anyone who is not at Cornell!

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More on Quantifiers

• Loves(x,y) - x loves y
• ?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,IceCream) ??x ?Likes(x,IceCream)
• ?x Likes(x,Broccoli) ??x ?Likes(x,Broccoli)

(see additional hidden slides for recapitulation
of this material)
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Quantified formulas

• If ? is a wff and x is a variable symbol, then
  both ?(x) ? and ?(x) ? are
• wffs.
• x is the variable quantified over
• ? is said to be within the scope of the
  quantifier
• if all the variables in ? are quantified over in
  ?, we say that we have a closed wff or closed
  sentence.
• Example
• (?x) P(x) ? R(x) (?x)P(x)?((? y) R(x,y)
  ? S(f(x))

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All variables are bound
(?x) (P(x) ? Q(x)) ? ?x R(x)
Scope of ?

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Love Affairs

• Loves(x,y) ?read x loves y
• Everybody loves Jerry
• ?x Loves (x, Jerry)
• Everybody loves somebody
• ?x ?y Loves (x, y)
• There is somebody whom somebody loves
• ?y ?x Loves (x, y)
• Nobody loves everybody
• ? ?x ?y Loves (x, y) ?x ?y ?Loves (x, y)
• There is somebody whom Lydia doesnt love
• ?y ?Loves (Lydia, y)

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Love AffairsLoves(x,y) x loves y

• There is somebody whom no one loves
• ?y ?x ?Loves (x, y)
• There is exactly one person whom everybody loves
• ?y(?x Loves(x,y) ? ?z((?w Loves (w ,z)? zy))
• There are exactly two people whom Lynn Loves
• ?x ?y ((x?y) ? Loves(Lynn,x) ? Loves(Lynn,y)
  ??z( Loves (Lynn ,z)? (zx ? zy)))
• Everybody loves himself or herself
• ?x Loves(x,x)
• There is someone who loves no one besides herself
  or himself
• ?x ?y Loves(x,y) ? (xy) (note biconditional
  why?)

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Using FOL

• The set domain
• Only sets are empty sets or those made by
  adjoining something to a set
• ?s Set(s) ? (s ) ? (?x,s2 Set(s2) ? s
  xs2)
  
• The empty set has no element adjoined to it
• ??x,s xs
• Adjoining an element already in the set has no
  effect
• ?x,s x ? s ? s xs
• Only elements have been adjoined to it
• ?x,s x ? s ? ?y,s2 (s ys2 ? (x y ? x ?
  s2))
  
• Subset
• ?s1,s2 s1 ? s2 ? (?x x ? s1 ? x ? s2)
• Equality of sets
• ?s1,s2 (s1 s2) ? (s1 ? s2 ? s2 ? s1)
• Intersection
• ?x,s1,s2 x ? (s1 ? s2) ? (x ? s1 ? x ? s2)
• Uniion
• ?x,s1,s2 x ? (s1 ? s2) ? (x ? s1 ? x ? s2)

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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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Statement When True When False
?x ?y P(x,y) ?y ?x P(x,y) P(x,y) is true for every pair There is a pair for which P(x.y) is false
?x ?y P(x,y) For every x there is a y for which P(x,y) is true There is an x such that P(x,y) is false for every y.
?x ?y P(x,y) There is an x such that P(x,y) is true for every y. For every x there is a y for which P(x,y) is false
?x ? y P(x,y) ?y ? x P(x,y) There is a pair x, y for which P(x,y) is true P(x,y) is flase fo every pair x,y.
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Negation
Negation Equivalent Statement When is the negation True When is False
??x P(x) ?x ?P(x) For every x, P(x) is false There is an x for which P(x) is true.
? ?x P(x) ?x ?P(x) There is an x for which P(x) is false. For every x, P(x) is true.
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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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• There is an honest politician ?x H(x)
• Where H(x) denotes x is honest.
• (universe of discourse consists of all the
  politicians)
• Negation ??x H(x) ? ?x ?H(x) (Every politician
  is dishonest)
• All Americans eat cheeseburgers ?x C(x)
• C(x) denotes x eats cheeseburgers

Negation ??x C(x) ? ?x ?C(x)
(Some American does not eat cheeseburgers)
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Knowledge base for the wumpus world

• Perception
• ?t,s,b Percept(s,b,Glitter,t) ? Glitter(t)
• Reflex
• ?t Glitter(t) ? BestAction(Grab,t)

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Deducing hidden properties

• ?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) ? ?r Adjacent(r,s) ? Pit(r)
• Causal rule---infer effect from cause
• ?r Pit(r) ? ?s Adjacent(r,s) ? Breezy(s)

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Interacting with FOL KBs

• Sentences are added to the knowledge base using
  TELL (as in PL), assertions.
• For example, we can assert
• TELL(KB,King(John)).
• TELL(KB,?x King(x) ?Person(x))
• We can ask
• Ask (KB, ?x Person(x))
• The answer to a query with existential variables
  if true returns a substitution or binding list,
  which is a set of variable term
• x/John
• if there is more than one possible answer, a
  list of substitutions can be returned.

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Interacting with FOL KBs

• Given a sentence S and a substitution s,
• Ss denotes the result of plugging s into S e.g.,
• S Smarter(x,y)
• s x/Hillary,y/Bill
• Ss Smarter(Hillary,Bill)
• Suppose a wumpus-world agent is using an FOL KB
  and perceives a smell and a breeze (but no
  glitter) at t5
• Tell(KB,Percept(Smell,Breeze,None,5))
• Ask(KB,?a BestAction(a,5))
• I.e., does the KB entail some best action at t5?
  
• Answer Yes, a/Shoot ? substitution (binding
  list)
• Ask(KB,S) returns some/all s such that KB 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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The digital circuit domain

• One-bit full adder

XOR gates
Input bits to be added
Sum
Carry bit
Carry bit for next adder
AND gates
OR gate
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The electronic circuit domain

• One-bit full adder

1
0
0
1
1
1
1
0
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The electronic circuit domain

• 1 - Identify the task
• Does the circuit actually add properly? (circuit
  verification)
  
• Goal analyze the design to see if it matches
  specification.
• 2 - Assemble the relevant knowledge
• Talk about circuits, gates, terminals, and
  signals e.g., we use constants (X1, X2, etc) for
  refering the gates Types of gates (AND, OR, XOR,
  NOT)
• Irrelevant size, shape, color, cost of gates
• 3 - Decide on a vocabulary
• Constant symbols X1, X2,
• Alternatives
• Type(X1) XOR (note XOR constant)
• Type(X1, XOR)
• XOR(X1)

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The electronic circuit domain

• Gates or circuits can have one or more input
  terminals and one or more
• output terminals.
• E.g. the function In(1,X1) denotes the first
  input terminal for gate X1.
• Connectivity between gates can be represented by
  a predicate
• Connected(Out(1,X1),In(1,X2)
• Signal has to be on or off
• Signal(t) this function takes terminal as an
  argument and returns the signal value for that
  terminal.
• Two constants 1 and 0.
• Note We could use a unary predicate On(t)
  However, this would make it difficult to ask the
  question what are the possible values of the
  signals at the output terminals of circuit C!?

See page 263-264 for full description of
vocabulary.
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The electronic circuit domain

• 4- Encode general knowledge of the domain
• If two terminals are connected they have the same
  signal
• ?t1,t2 Connected(t1, t2) ? Signal(t1)
  Signal(t2)
• Connected is a commutative predicate
• ?t1,t2 Connected(t1, t2) ? Connected(t2, t1)
• The signal at every terminal is either 1 or 0
• ?t Signal(t) 1 ? Signal(t) 0
• 1 ? 0

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The electronic circuit domain

• ? An OR gates output is 1 iif any of its
  inputs is 1
• ?g Type(g) OR ? Signal(Out(1,g)) 1
• ? ?n Signal(In(n,g)) 1
• ? An AND gates output is 0 iif any of its
  inputs is 0
• ?g Type(g) AND ? Signal(Out(1,g)) 0 ? ?n
  Signal(In(n,g)) 0
  
• ? An XOR gates output is 1 iif its inputs
  are different
• ?g Type(g) XOR ? Signal(Out(1,g)) 1 ?
  Signal(In(1,g)) ? Signal(In(2,g))
  
• ? A NOT gates output is different from its
  input
• ?g Type(g) NOT ? Signal(Out(1,g)) ?
  Signal(In(1,g))
  

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The electronic circuit domain

• 5 - 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),I
  n(1,X1))
• Connected(Out(1,X1),In(2,A2)) Connected(In(1,C1),I
  n(1,A1))
• Connected(Out(1,A2),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 circuit domain

• 6 - Pose queries to the inference procedure
• What are the possible sets of values of the input
  of the adder circuit that lead to a signal of 0
  for the sum output bit and a value of 1 for the
  carry output terminal?
• ?i1,i2,i3 Signal(In(1,C1)) i1 ?
  Signal(In(2,C1)) i2 ? Signal(In(3,C1)) i3 ?
  Signal(Out(1,C1)) 0 ? Signal(Out(2,C1)) 1
• Answer (i1 1 i2 1 i3 0) or ((i1 1 i2
  0 i3 1) or (i1 0 i2 1 i3 0)
• What are the possible sets of values of all the
  terminals of the adder circuit?
  
• ?i1,i2,i3,o1,o2 Signal(In(1,C1)) i1 ?
  Signal(In(2,C1)) i2 ? Signal(In(3,C1)) i3 ?
  Signal(Out(1,C1)) o1 ? Signal(Out(2,C1)) o2
  

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The electronic circuit domain

• 7 - Debug the knowledge base
• May have omitted assertions like 1 ? 0

What is the advantage over simulation?
Pentium bug
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Summary

• First-order logic
• objects and relations are semantic primitives
• syntax constants, functions, predicates,
  equality, quantifiers
  
• Increased expressive power!
• But of course at a cost of increased complexity
  in general