Artificial Intelligence 1: logic agents

1
Artificial Intelli-gence 1 logic agents
Notes adapted from lecture notes for CMSC 421 by
B.J. Dorr

• Lecturer Tom Lenaerts
• Institut de Recherches Interdisciplinaires et de
  Développements en Intelligence Artificielle
  (IRIDIA)
• Université Libre de Bruxelles

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Standard Logical Equivalences
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Terminology

• A sentence is valid iff its truth value is t in
  all interpretations (² f)
• Valid sentences true, false, P Ç P
• A sentence is satisfiable iff its truth value is
  t in at least one interpretation
• Satisfiable sentences P, true, P
• A sentence is unsatisfiable iff its truth value
  is f in all interpretations
• Unsatisfiable sentences P Æ P, false, true

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Examples
Sentence
Valid?
wealthy ) wealthy
valid
wealthy Ç wealthy
satisfiable, not valid
wealthy ) happy
w t, h f
inverse
satisfiable, not valid
(w ) h) ) (w ) h)
wf, ht w)h t, w ) h f
contrapositive
valid
(w ) h) ) (h )w)
w Ç h Ç (w) h)
valid
w Ç h Ç w Ç h
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Examples
Sentence
Valid?
wealthy ) wealthy
valid
wealthy Ç wealthy
satisfiable, not valid
wealthy ) happy
w t, h f
inverse
satisfiable, not valid
(w ) h) ) (w ) h)
wf, ht w)h t, w ) h f
contrapositive
valid
(w ) h) ) (h )w)
w Ç h Ç (w) h)
valid
w Ç h Ç w Ç h
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Inference

• KB i a
• Soundness Inference procedure i is sound if
  whenever KB i a, it is also true that KB ² a
• Completeness Inference procedure i is complete
  if whenever KB ² a, it is also true that KB i a

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Validity and Inference
((P Ç H) Æ H) ) P

P

H

P
H

(P
H)

H
((P
H)

H) )
P


Ç
Ç
Æ

Ç
Æ


T

T

T

F

T

T

F

T

T

T

F

T

T

F

T

F

F

F

F

T


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Rules of Inference

• a b
• a b
• Valid Rules of Inference
• Modus Ponens
• And-Elimination
• And-Introduction
• Or-Introduction
• Double Negation
• Unit Resolution
• Resolution

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Examples in Wumpus World

• Modus Ponens a ) b, a b(WumpusAhead Æ
  WumpusAlive) ) Shoot, (WumpusAhead Æ WumpusAlive)
  Shoot
• And-Elimination a Æ b a(WumpusAhead Æ
  WumpusAlive) WumpusAlive
• Resolution a Ç b, b Ç g a Ç g(WumpusDead Ç
  WumpusAhead), ( WumpusAhead Ç Shoot)
  (WumpusDead Ç Shoot)

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Proof Using Rules of Inference

• Prove A ) B, (A Æ B) ) C, Therefore A ) C
• A ) B A Ç B
• A Æ B ) C (A Æ B) Ç C A Ç B Ç C
• So A Ç B resolves with A Ç B Ç C deriving
  A Ç C
• This is equivalent to A ) C

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Rules of Inference (continued)

• And-Introduction a1, a2, , an a1 Æ a2 Æ Æ an
• Or-Introduction ai a1 Ç a2 Ç ai
  Ç an
• Double Negation a a
• Unit Resolution (special case of resolution)a Ç
  b Alternatively a ) b b
  b a
  a

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Wumpus World KB

• Proposition Symbols for each i,j
• Let Pi,j be true if there is a pit in square i,j
• Let Bi,j be true if there is a breeze in square
  i,j
• Sentences in KB
• There is no pit in square 1,1R1 P1,1
• A square is breezy iff pit in a neighboring
  squareR2 B1,1 , (P1,2 Ç P2,1)R3 B1,2 , (P1,1
  Ç P1,3 Ç P2,2)
• Square 1,1 has no breeze, Square 1,2 has a
  breezeR4 B1,1R5 B1,2

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Inference in Wumpus World

• Apply biconditional elimination to R2R6 (B1,1)
  (P1,2 Ç P2,1)) Æ ((P1,2 Ç P2,1) ) B1,1)
• Apply AE to R6R7 ((P1,2 Ç P2,1) ) B1,1)
• Contrapositive of R7R8 ( B1,1 ) (P1,2 Ç
  P2,1))
• Modus Ponens with R8 and R4 ( B1,1)R9 (P1,2
  Ç P2,1)
• de MorganR10 P1,2 Æ P2,1

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Searching for Proofs

• Finding proofs is exactly like finding solutions
  to search problems.
• Can search forward (forward chaining) to derive
  goal or search backward (backward chaining) from
  the goal.
• Searching for proofs is not more efficient than
  enumerating models, but in many practical cases,
  its more efficient because we can ignore
  irrelevant propositions

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Full Resolution Rule Revisited

• Start with Unit Resolution Inference Rule
• Full Resolution Rule is a generalization of this
  rule
• For clauses of length two

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Resolution Applied to Wumpus World

• At some point we determine the absence of a pit
  in square 2,2R13 P2,2
• Biconditional elimination applied to R3 followed
  by modus ponens with R5R15 P1,1 Ç P1,3 Ç P2,2
• Resolve R15 and R13R16 P1,1 Ç P1,3
• Resolve R16 and R1R17 P1,3

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Resolution Complete Inference Procedure

• Any complete search algorithm, applying only the
  resolution rule, can derive any conclusion
  entailed by any knowledge base in propositional
  logic.
• Refutation completeness Resolution can always be
  used to either confirm or refute a sentence, but
  it cannot be used to enumerate true sentences.

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Conjunctive Normal Form

• Conjunctive Normal Form is a disjunction of
  literals.
• Example(A Ç B Ç C) Æ (B Ç D) Æ ( A) Æ (B Ç
  C)

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CNF Example

• Example (A Ç B) , (C ) D)
• Eliminate ,
• ((A Ç B) ) (C ) D)) Æ ((C ) D) ) (A Ç B)
• Eliminate )
• ( (A Ç B) Ç ( C Ç D)) Æ ( ( C Ç D) Ç (A Ç B)
  )
• Drive in negations(( A Æ B) Ç ( C Ç D)) Æ
  ((C Æ D) Ç (A Ç B))
• Distribute( A Ç C Ç D) Æ ( B Ç C Ç D) Æ (C
  Ç A Ç B) Æ ( D Ç A Ç B)

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Resolution Algorithm

• To show KB ² a, we show (KB Æ a) is
  unsatisfiable.
• This is a proof by contradiction.
• First convert (KB Æ a) into CNF.
• Then apply resolution rule to resulting clauses.
• The process continues until
• there are no new clauses that can be added (KB
  does not entail a)
• two clauses resolve to yield empty clause (KB
  entails a)

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Simple Inference in Wumpus World

• KB R2 Æ R4 (B1,1 , (P1,2 Ç P2,1)) Æ B1,1
• Prove P1,2 by adding the negation P1,2
• Convert KB Æ P1,2 to CNF

• PL-RESOLUTION algorithm

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Horn Clauses

• Real World KBs are often a conjunction of Horn
  clauses
• Horn clause
• proposition symbol or
• (conjunction of symbols) ) symbol
• ExampleC Æ (B ) A) Æ (C Æ D ) B)

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Forward Chaining

• Fire any rule whose premises are satisfied in the
  KB.
• Add its conclusion to the KB until query is
  found.

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Forward Chaining Example
P ) Q L Æ M ) P B Æ L ) M A Æ P ) L A Æ B ) L A B
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Forward Chaining Example
P ) Q L Æ M ) P B Æ L ) M A Æ P ) L A Æ B ) L A B
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Forward Chaining Example
P ) Q L Æ M ) P B Æ L ) M A Æ P ) L A Æ B ) L A B
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Forward Chaining Example
P ) Q L Æ M ) P B Æ L ) M A Æ P ) L A Æ B ) L A B
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Forward Chaining Example
P ) Q L Æ M ) P B Æ L ) M A Æ P ) L A Æ B ) L A B
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Forward Chaining Example
P ) Q L Æ M ) P B Æ L ) M A Æ P ) L A Æ B ) L A B
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Forward Chaining Example
P ) Q L Æ M ) P B Æ L ) M A Æ P ) L A Æ B ) L A B
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Forward Chaining Example
P ) Q L Æ M ) P B Æ L ) M A Æ P ) L A Æ B ) L A B
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Backward Chaining

• Motivation Need goal-directed reasoning in order
  to keep from getting overwhelmed with irrelevant
  consequences
• Main idea
• Work backwards from query q
• To prove q
• Check if q is known already
• Prove by backward chaining all premises of some
  rule concluding q

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Backward Chaining Example
P ) Q L Æ M ) P B Æ L ) M A Æ P ) L A Æ B ) L A B
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Backward Chaining Example
P ) Q L Æ M ) P B Æ L ) M A Æ P ) L A Æ B ) L A B
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Backward Chaining Example
P ) Q L Æ M ) P B Æ L ) M A Æ P ) L A Æ B ) L A B
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Backward Chaining Example
P ) Q L Æ M ) P B Æ L ) M A Æ P ) L A Æ B ) L A B
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Backward Chaining Example
P ) Q L Æ M ) P B Æ L ) M A Æ P ) L A Æ B ) L A B
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Backward Chaining Example
P ) Q L Æ M ) P B Æ L ) M A Æ P ) L A Æ B ) L A B
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Backward Chaining Example
P ) Q L Æ M ) P B Æ L ) M A Æ P ) L A Æ B ) L A B
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Backward Chaining Example
P ) Q L Æ M ) P B Æ L ) M A Æ P ) L A Æ B ) L A B
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Backward Chaining Example
P ) Q L Æ M ) P B Æ L ) M A Æ P ) L A Æ B ) L A B
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Backward Chaining Example
P ) Q L Æ M ) P B Æ L ) M A Æ P ) L A Æ B ) L A B
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Backward Chaining Example
P ) Q L Æ M ) P B Æ L ) M A Æ P ) L A Æ B ) L A B
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Forward Chaining vs. Backward Chaining

• FC is data-drivenit may do lots of work
  irrelevant to the goal
• BC is goal-drivenappropriate for problem-solving