Inferencing

1
Inferencing

• Finding solution to queries
• Propositionalization
• (For reference only, may leave slides 2 to 7)
• Unification
• Forward Chaining
• Backward Chaining
• Resolution

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Lugers BookChapter 2andChapter 12(upto
12.2)
3
Propositionalization

1. Brute Force Approach
2. Apply all possible bindings to the variables.
3. Add every clause in the sentence as a proposition
   to the knowledge base.
4. Search for the required proposition (Goal
   proposition)

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Reduction to propositional inference

• Suppose the KB contains just the following
• ?X wealthy(X) ? greedy(X) ? bad(X)
• wealthy(arif)
• greedy(arif)
• cousins(shahid,arif)
• Instantiating the universal sentence in all
  possible ways, we have
• wealthy(arif) ? greedy(arif) ? bad(arif)
• wealthy(shahid) ? greedy(shahid) ? bad(shahid)
• wealthy(arif)
• greedy(arif)
• cousins(shahid,arif)

Universal Instantiation

• The new KB is propositionalized proposition
  symbols are
• wealthy(arif), greedy(arif), bad(arif),
  wealthy(shahid),
• greedy(shahid), bad(shahid), cousins(shahid,arif).
  

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Reduction contd.

• Every FOL KB can be propositionalized so as to
  preserve entailment (entailments are
  dependencies)
  
• (A ground sentence is entailed by new KB iff
  entailed by original KB)
  
• Idea propositionalize KB and query, apply
  resolution, return result
  
• Problem with function symbols, there are
  infinitely many ground terms,
• e.g., Father(Father(Father(John)))

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Reduction contd.

• Theorem Herbrand (1930). If a sentence a is
  entailed by an FOL KB, it is entailed by a finite
  subset of the propositionalized KB
  
• Idea For n 0 to 8 do
• create a propositional KB by instantiating
  with depth-n terms
• see if a is entailed by this KB
• Problem works if a is entailed, loops if a is
  not entailed
  
• Theorem Turing (1936), Church (1936) Entailment
  for FOL is semidecidable (algorithms exist that s
  ay yes to every entailed sentence, but no
  algorithm exists that also says no to every
  nonentailed sentence.)

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Problems with propositionalization

• Propositionalization seems to generate lots of
  irrelevant sentences.
• E.g., from
• ?x King(x) ? Greedy(x) ? Evil(x)
• King(John)
• ?y Greedy(y)
• Brother(Richard,John)
• it seems obvious that Evil(John), but
  propositionalization produces lots of facts such
  as Greedy(Richard) that are irrelevant
  
• With p k-ary predicates and n constants, there
  are pnk instantiations.
  

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Unification

• It is an algorithm for determining the
  substitutions needed to make two predicate
  calculus expressions match
• The process of universal/existential
  instantiation require substitutions

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Unification

• Unify(a,ß) ? if a? ß?
• p q ?
• knows(ali,X) knows(ali,sana)
• knows(ali,X) knows(Y,hira)
• knows(ali,X) knows(Y,mother(Y))
• knows(ali,X) knows(X,hira)
• Standardizing apart eliminates overlap of
  variables, e.g., the fourth sentence can have
  knows(Z,hira)
  

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Unification

• Unify(a,ß) ? if a? ß?
• p q ?
• knows(ali,X) knows(ali,sana) sana/X
• knows(ali,X) knows(Y,hira)
• knows(ali,X) knows(Y,mother(Y))
• knows(ali,X) knows(X,hira)
• Standardizing apart eliminates overlap of
  variables, e.g., the fourth sentence can have
  knows(Z,hira)
  

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Unification

• Unify(a,ß) ? if a? ß?
• p q ?
• knows(ali,X) knows(ali,sana) sana/X
• knows(ali,X) knows(Y,hira) hira/X,ali/Y
• knows(ali,X) knows(Y,mother(Y))
• knows(ali,X) knows(X,hira)
• Standardizing apart eliminates overlap of
  variables, e.g., the fourth sentence can have
  knows(Z,hira)
  

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Unification

• Unify(a,ß) ? if a? ß?
• p q ?
• knows(ali,X) knows(ali,sana) sana/X
• knows(ali,X) knows(Y,hira) hira/X,ali/Y
• knows(ali,X) knows(Y,mother(Y))
  ali/Y,mother(ali)
• knows(ali,X) knows(X,hira)
• Standardizing apart eliminates overlap of
  variables, e.g., the fourth sentence can have
  knows(Z,hira)
  

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Unification

• Unify(a,ß) ? if a? ß?
• p q ?
• knows(ali,X) knows(ali,sana) sana/X
• knows(ali,X) knows(Y,hira) hira/X,ali/Y
• knows(ali,X) knows(Y,mother(Y))
  ali/Y,mother(ali)
• knows(ali,X) knows(X,hira) fail
• Standardizing apart eliminates overlap of
  variables, e.g., the fourth sentence can have
  knows(Z,hira)
  

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Unification - MGU

• To unify knows(ali,X) and knows(Y,Z)
• ? ali/Y, Z/X or ? ali/Y, ali/X, ali/Z
• (if only ali is present in the data base of
  facts)
• The first unifier is more general than the
  second.
  
• There is a single Most General Unifier (MGU) that
  is unique up to renaming of variables.
  
• MGU ali/Y, Z/X

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Implementation of Unification

•  UsingList format
• PC Syntax List Syntax
• p(a,b) (p a b)
•  
• p(f(a), g(X,Y)) (p(f a)(g X Y))
•  
• p(x) q(y) ((p x) (q y))

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Unification Examples

• Unify
• ((parents X(father X) (mother bill)), (parents
  bill (father bill)Y)).
•  Complete Substitution
• bill/X, mother (bill)/Y

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Logic Based Financial Advisor To help a user
decide whether to invest in a saving A/C or the
stock market or both
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• Policy
• Saving inadequate - should invest in saving A/c
  regardless of income
•  
• Saving adequateand adequate income - More
  profitable option of stock investment
•  
• Low Income adequate savings - Split surplus
  between saving and stock investment

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Defining Constraints

• Adequate Savings/Income
• Income Rs 15000/month
• extra Rs 4000/ dependent/month
• Saving Rs 25,000/year
• extra Rs 5000/dependent/year

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Calculating Adequacy

• Define Functions to calculate the minimum income
  and savings
• Minimum Savings min_saving(X)
• min_saving(X)250005000X
• where X are the number of dependents
• Minimum Income min_income(X)
• min_income(X)150004000X

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Outputs   Investment (savings) Investment
(stocks) Investment (combination)     How can we
represent outputs?
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Rules       saving_acc (inadequate) ? investment
(savings)     saving_acc(adequate)
income(adequate) ? investment(stocks)    
 saving_acc (adequate) income (inadequate) ?
investment (combination)
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What next?

• Define Saving adequate
• Define saving inadequate
• Define income adequate
• Define income inadequate

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Savings

• Check if saved amount is greater than the minimum
  saving required

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Income

• Check if the income is greater than the minimum
  income

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Try firing rules with 9, 10, 11
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Example knowledge base

• The law says that it is a crime for an American
  to sell weapons to hostile nations. The country
  Nono, an enemy of America, has some missiles, and
  all of its missiles were sold to it by Colonel
  West, who is American.
• Prove that Col. West is a criminal

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Example knowledge base contd.

• ... it is a crime for an American to sell weapons
  to hostile nations
• american(X) ? weapon(Y) ? sells(X,Y,z) ?
  hostile(z) ? criminal(X)
• Nono has some missiles, i.e., ?X Owns(Nono,X) ?
  Missile(X)
  
• owns(nono,m1) and missile(m1)
• all of its missiles were sold to it by Colonel
  West
• missile(X) ? owns(nono,X) ? sells(west,X,nono)
• Missiles are weapons
• missile(X) ? weapon(X)
• An enemy of America counts as "hostile
• enemy(X,america) ? hostile(X)
• West, who is American
• american(west)
• The country Nono, an enemy of America
• enemy(nono,america)

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Example knowledge base contd.

• Rule Base
• ... it is a crime for an American to sell weapons
  to hostile nations
• R1 american(X) ? weapon(Y) ? sells(X,Y,z) ?
  hostile(z) ? criminal(X)
• Nono has some missiles, i.e.,
• R2 ?X owns(nono,X) ? missile(X)
• all of its missiles were sold to it by Colonel
  West
• R3 missile(X) ? owns(nono,X) ?
  sells(west,X,nono)
• Missiles are weapons
• R4 missile(X) ? weapon(X)
• An enemy of America counts as "hostile
• R5 enemy(X,america) ? hostile(X)

Database of Facts owns(nono,m1) and
missile(m1) West, who is American
american(west) The country Nono, an enemy of
America enemy(nono,america)
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• R1 american(X) ? weapon(Y) ? sells(X,Y,Z) ?
  hostile(Z) ? criminal(X)
• R2 ?X owns(nono,X) ? missile(X)
• R3 missile(X) ? owns(nono,X) ?
  sells(west,X,nono)
• R4 missile(X) ? weapon(X)
• R5 enemy(X,america) ? hostile(X)

• START FROM THE FACTS
• THINK WHICH RULES APPLY

Be careful about the convention constants start
with capital letters and variables with
small (applicable to next few slides)
1. FORWARD CHAINING
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• R1 american(X) ? weapon(Y) ? sells(X,Y,Z) ?
  hostile(Z) ? criminal(X)
• R2 ?X owns(nono,X) ? missile(X)
• R3 missile(X) ? owns(nono,X) ?
  sells(west,X,nono)
• R4 missile(X) ? weapon(X)
• R5 enemy(X,america) ? hostile(X)

R5 nono/X
R4 m1/X
R3 m1/X
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• R1 american(X) ? weapon(Y) ? sells(X,Y,Z) ?
  hostile(Z) ? criminal(X)
• R2 ?X owns(nono,X) ? missile(X)
• R3 missile(X) ? owns(nono,X) ?
  sells(west,X,nono)
• R4 missile(X) ? weapon(X)
• R5 enemy(X,america) ? hostile(X)

R1 west/X , m1/Y , nono/Z
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2. BACKWARD CHAINING

• R1 american(X) ? weapon(Y) ? sells(X,Y,Z) ?
  hostile(Z) ? criminal(X)
• R2 ?X owns(nono,X) ? missile(X)
• R3 missile(X) ? owns(nono,X) ?
  sells(west,X,nono)
• R4 missile(X) ? weapon(X)
• R5 enemy(X,america) ? hostile(X)

• START FROM THE CONCLUSION
• THINK IS THE RULE SUPPORT AVAILABLE

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Backward chaining example

• R1 american(X) ? weapon(Y) ? sells(X,Y,Z) ?
  hostile(Z) ? criminal(X)
• R2 ?X owns(nono,X) ? missile(X)
• R3 missile(X) ? owns(nono,X) ?
  sells(west,X,nono)
• R4 missile(X) ? weapon(X)
• R5 enemy(X,america) ? hostile(X)

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Backward chaining example

• R1 american(X) ? weapon(Y) ? sells(X,Y,Z) ?
  hostile(Z) ? criminal(X)
• R2 ?X owns(nono,X) ? missile(X)
• R3 missile(X) ? owns(nono,X) ?
  sells(west,X,nono)
• R4 missile(X) ? weapon(X)
• R5 enemy(X,america) ? hostile(X)

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Backward chaining example

• R1 american(X) ? weapon(Y) ? sells(X,Y,Z) ?
  hostile(Z) ? criminal(X)
• R2 ?X owns(nono,X) ? missile(X)
• R3 missile(X) ? owns(nono,X) ?
  sells(west,X,nono)
• R4 missile(X) ? weapon(X)
• R5 enemy(X,america) ? hostile(X)

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Backward chaining example

• R1 american(X) ? weapon(Y) ? sells(X,Y,Z) ?
  hostile(Z) ? criminal(X)
• R2 ?X owns(nono,X) ? missile(X)
• R3 missile(X) ? owns(nono,X) ?
  sells(west,X,nono)
• R4 missile(X) ? weapon(X)
• R5 enemy(X,america) ? hostile(X)

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Backward chaining example

• R1 american(X) ? weapon(Y) ? sells(X,Y,Z) ?
  hostile(Z) ? criminal(X)
• R2 ?X owns(nono,X) ? missile(X)
• R3 missile(X) ? owns(nono,X) ?
  sells(west,X,nono)
• R4 missile(X) ? weapon(X)
• R5 enemy(X,america) ? hostile(X)

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Backward chaining example

• R1 american(X) ? weapon(Y) ? sells(X,Y,Z) ?
  hostile(Z) ? criminal(X)
• R2 ?X owns(nono,X) ? missile(X)
• R3 missile(X) ? owns(nono,X) ?
  sells(west,X,nono)
• R4 missile(X) ? weapon(X)
• R5 enemy(X,america) ? hostile(X)

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3. Modus Ponen Based Solution

• 6 with 3 suggests, through Modus Ponens
• weapons(m1) m1/X
• 2. From 4 and 9
• hostile(nono) nono/X
• From 2, 5, 6, Modus Ponens
• sells(west,nono,m1)m1/X
• 4. From 2,3,4,5,6, and 9
• criminal(west)
• west/X, m1/Y, nono/Z

1. ?X,Y,Zamerican(X)weapon(Y)nation(Z)hostile(Z)
   sells(X,Z,Y) ? criminal(X)
2. ?X owns(nono,X)missiles(X) ? sells(west,nono,X)
3. ?X missiles(X) ? weapons (X)
4. ?X enemy(X,america) ?hostile(X)
5. owns(nono,m1)
6. missiles(m1)
7. american(west)
8. nation(nono)
9. enemy(nono,america)
10. nation(america)

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Composition

• Find Overall Substitutions
• Given two or more sets of substitutions

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Example

• S1A/K
• S2a/A, H/J
• What set would represent overall substitutions
• S1.S2a/K, H/J
• Algorithm?

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Example

• S1A/K S2a/A, H/J
• S1.S2a/K, H/J
• Algorithm
• Apply S2 to S1 (find overall substitution)
• Add S2 to S1
• Drop all substitutions in S2 which have common
  denominators with S1

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Example

• S1A/K S2a/A, H/J
• Algorithm
• Apply S2 to S1 (find overall substitution)
• Apply a/A to A/K would lead to a/K
• Apply H/J to A/K cannot be applied
• S1.S2a/K
• Add S2 to S1
• S1.S2a/K a/A,H/J
• Drop all substitutions in S2 which have common
  denominators with S1
• nothing gets dropped, hence the answer is
  a/K,a/A,H/J

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Example

• S1D/A
• S2a/D
• S12a/A, a/D
• S1D/A,S/G
• S2x/X, f/D, q/S,w/A
• S1.S2f/A,q/G, x/X, f/D, q/S,w/A
• S1.S2f/A,q/G, x/X,q/S (w/A gets dropped)
• S1.S2f/A,q/G, x/X,q/S

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Is S1.S2 same as S2.S1

• S1D/A
• S2a/D
• S21a/A, D/A?a/A where as S12a/A, a/D
• S1D/A,S/G
• S2x/X, f/D, q/S,w/A
• S2.S1x/X, f/D, q/S, w/A D/A,S/G
• S2.S1x/X, f/D, q/S, w/A, S/G (D/A gets
  dropped)
• Hence generally speaking S1.S2 not equal to S2.S1

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Is S12 same as S21

• S1D/A
• S2X/A
• S12D/A, X/A
• S12D/A
• S21X/A,D/A
• S21X/A

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Example

• S1D/A, d/F,S/E
• S2g/D
• S12g/A, d/F,S/E, g/D
• S12g/A, d/F,S/E, g/D
• S21g/D, D/A, d/F,S/E

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Substitution Discussion

• Domain Variables in the substitution together
  form the domain (denominators)
• Replacements Terms are the replacements
  (numerators)
• Pure Vs Impure Substitutions
• A pure substitution is if all replacements are
  free of the variables in the domain of the
  substitution. Otherwise impure
• s/X, t/I, j/G (pure)
• The composition of pure substitutions may be
  impure and vice versa
• Composability
• A substitution S1 and S2 are composable if and
  only if the variables in the domain of S1 do not
  appear among the replacements of S2, otherwise
  they are noncomposable.
• S1a/X,b/Y,z/V S2U/X,b/V composable
• S1a/X,b/Y,z/V S2X/U, b/V noncomposable

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Application Example

• Knowledge base
• knows(ali,sana) knows(X,shahid) knows(X,Y)
• We also know
• knows(ali,X) ? hates(ali,X)
• Unify the conditions to find ali hates whom
• unify(knows(ali,X) , knows(ali,sana)) sana/X
• Using the binding and the rule ali hates sana
• unify(knows(ali,X) , knows(X,shahid))
  ali/X,shahid/X This would fail as X cannot have
  two values at the same time
• As the composition of ali/X with shahid/X or
  vice versa will result in only one valid
  substitution
• unify(knows(ali,X) , knows(X,Y)) ali/X,X/Y
• This would mean that ali knows everyone
  including himself, therefore hates everyone.