Automated Reasoning in Propositional Logic

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Automated Reasoning in Propositional Logic
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Problem

• Given
• KB a set of sentence
• ? a sentence
• Answer
• KB ? ?

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Deduction vs. Satisfiability Test
KB ? iff KB,?? is unsatisfiable

• Hence
• Deciding whether a set of sentences entails
  another sentence, or not
• Testing whether a set of sentences is
  satisfiable, or not
• are closely related problems

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Computational Approaches

• Enumeration of models
• Construction of a proof

• Battery-OK ? Bulbs-OK ? Headlights-Work
• Battery-OK ? Starter-OK ? ?Empty-Gas-Tank ?
  Engine-Starts
• Engine-Starts ? ?Flat-Tire ? Car-OK
• Headlights-Work
• Battery-OK
• Starter-OK
• ?Empty-Gas-Tank
• ?Car-OK
• Battery-OK ? Starter-OK ? (56)
• Battery-OK ? Starter-OK ? ?Empty-Gas-Tank ?
  (97)
• Engine-Starts ? (210)
• ?Engine-Starts ? Flat-Tire ? (38)
• Flat-Tire ? (1112)

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Enumeration of Models

• P Set of propositional symbols in KB,??
• n Size of P
• ENTAILS?(KB,?)
• For each of the 2n models on P do
• If it is a model of KB,?? then return no
  Return yes

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Satisfiability Test as CSP

• Each propositional symbol is a variable
• The domain of each variable is True, False
• Each sentence in KB,?? is a constraint on the
  value(s) taken by one or several variables
• Recursive backtracking CSP techniques and
  heuristics are applicable

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Construction of a Proof

1. Battery-OK ? Bulbs-OK ? Headlights-Work
2. Battery-OK ? Starter-OK ? ?Empty-Gas-Tank ?
   Engine-Starts
3. Engine-Starts ? ?Flat-Tire ? Car-OK
4. Headlights-Work
5. Battery-OK
6. Starter-OK
7. ?Empty-Gas-Tank
8. ?Car-OK
9. Battery-OK ? Starter-OK ? (56)
10. Battery-OK ? Starter-OK ? ?Empty-Gas-Tank ?
    (97)
11. Engine-Starts ? (210)
12. ?Engine-Starts ? Flat-Tire ? (38)
13. Flat-Tire ? (1112)

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Construction of a Proof

1. Battery-OK ? Bulbs-OK ? Headlights-Work
2. Battery-OK ? Starter-OK ? ?Empty-Gas-Tank ?
   Engine-Starts
3. Engine-Starts ? ?Flat-Tire ? Car-OK
4. Headlight-Work
5. Battery-OK
6. Starter-OK
7. ?Empty-Gas-Tank
8. ?Car-OK
9. Battery-OK ? Starter-OK ? (56)
10. Battery-OK ? Starter-OK ? ?Empty-Gas-Tank ?
    (97)
11. Engine-Starts ? (210)
12. ?Engine-Starts ? Flat-Tire ? (38)
13. Flat-Tire ? (1112)

• What do we need?
• A complete set of sound inference rules
• A complete search algorithm to decide which
  rule to apply next and to which sentences

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Complementary Literals

• A literal is a either an atomic sentence or the
  negated atomic sentence, e.g.
  P or ?P
• Two literals are complementary if one is the
  negation of the other, e.g.
  P and ?P

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Unit Resolution Rule

• Given two sentences L1 ? ? Lp and
  M where Li,, Lp and M are all literals,
  and M and Li are complementary literals
• Infer L1 ? ? Li-1 ? Li1 ? ? Lp

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Examples

• From?Engine-Starts ? Car-OK
• Engine-Starts
• InferCar-OK

Modus ponens

• From?Engine-Starts ? Car-OK
• ?Car-OK
• Infer ?Engine-Starts

Modus tolens
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Another Example

1. ?Engine-Starts ? Flat-Tire ? Car-OK
2. Engine-Starts
3. ?Flat-Tire
4. Flat-Tire ? Car-OK
5. Car-OK

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Detection of Unsatisfiability

1. Car-OK
2. ?Car-OK
3. False

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Soundness of Unit Resolution

• Let m be a model of L1 ? ? Lp and
  M where M and Li are complementary
  literals
• Li must be False in m, hence L1 ? ? Li-1 ?
  Li1 ? ? Lp must be True

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Shortcoming of Unit Resolution

• From
• ?Engine-Starts ? Flat-Tire ? Car-OK
• Engine-Starts ? Empty-Gas-Tank
• we can infer nothing!

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Full Resolution Rule

• Given two sentences L1 ? ? Lp and
  M1 ? ? Mq where L1,, Lp, M1,, Mq are all
  literals, and Li and Mj are complementary
  literals
• Infer L1? ? Li-1?Li1??Lk?M1? ?
  Mj-1?Mj1??Mk in which only one copy of each
  literal is retained (factoring)

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Example

• From
• ?Engine-Starts ? Flat-Tire ? Car-OK
• Engine-Starts ? Empty-Gas-Tank
• Infer
• Empty-Gas-Tank ? Flat-Tire ? Car-OK

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Example

• From
• ?P ? Q (? P ? Q)
• ?Q ? R (? Q ? R)
• Infer
• ?P ? R (? P ? R)

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Not All Inferences are Useful!

• From
• ?Engine-Starts ? Flat-Tire ? Car-OK
• Engine-Starts ? ?Flat-Tire
• Infer
• ?Flat-Tire ? Flat-Tire ? Car-OK

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Not All Inferences are Useful!

• From
• ?Engine-Starts ? Flat-Tire ? Car-OK
• Engine-Starts ? ?Flat-Tire
• Infer
• ?Flat-Tire ? Flat-Tire ? Car-OK

tautology
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Not All Inferences are Useful!

• From
• ?Engine-Starts ? Flat-Tire ? Car-OK
• Engine-Starts ? ?Flat-Tire
• Infer
• ?Flat-Tire ? Flat-Tire ? Car-OK ? True

tautology
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Soundness of Full Resolution
Left as an exercise
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Full Resolution Rule

• Given two sentences L1 ? ? Lp and
  M1 ? ? Mq
• Infer L1? ? Li-1?Li1??Lk?M1? ?
  Mj-1?Mj1??Mk

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Sentence ? Clause Form
Example (A ? ?B) ? (C ? D)
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Sentence ? Clause Form
Example (A ? ?B) ? (C ? D) 1. Eliminate ?
?(A ? ?B) ? (C ? D)
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Sentence ? Clause Form
Example (A ? ?B) ? (C ? D) 1. Eliminate ?
?(A ? ?B) ? (C ? D)2. Reduce scope of ? (?A ?
B) ? (C ? D)
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Sentence ? Clause Form
Example (A ? ?B) ? (C ? D) 1. Eliminate ?
?(A ? ?B) ? (C ? D)2. Reduce scope of ? (?A ?
B) ? (C ? D)3. Distribute ? over ? (?A ? (C ?
D)) ? (B ? (C ? D)) (?A ? C) ? (?A ? D) ? (B ?
C) ? (B ? D)
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Sentence ? Clause Form
Example (A ? ?B) ? (C ? D) 1. Eliminate ?
?(A ? ?B) ? (C ? D)2. Reduce scope of ? (?A ?
B) ? (C ? D)3. Distribute ? over ? (?A ? (C ?
D)) ? (B ? (C ? D)) (?A ? C) ? (?A ? D) ? (B ?
C) ? (B ? D) Set of clauses ?A ? C , ?A ? D ,
B ? C , B ? D
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Resolution Refutation Algorithm

• RESOLUTION-REFUTATION(KB,a)
• clauses ? set of clauses obtained from KB and ?a
• new ?
• Repeat
• For each C, C in clauses do res ?
  RESOLVE(C,C) If res contains the empty clause
  then return yes
• new ? new U resIf new ? clauses then return no
• clauses ? clauses U new

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Completeness of Resolution Refutation
Left as an exercise
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Example

1. ?Battery-OK ? ?Bulbs-OK ? Headlights-Work
2. ?Battery-OK ? ?Starter-OK ? Empty-Gas-Tank ?
   Engine-Starts
3. ?Engine-Starts ? Flat-Tire ? Car-OK
4. Headlights-Work
5. Battery-OK
6. Starter-OK
7. ?Empty-Gas-Tank
8. ?Car-OK
9. ?Flat-Tire
10. ?Starter-OK ? Empty-Gas-Tank ? Engine-Starts
11. ?Battery-OK ? Empty-Gas-Tank ? Engine-Starts
12. ?Battery-OK ? ?Starter-OK ? Engine-Starts
13. ?Engine-Starts ? Flat-Tire
14. ?Engine-Starts ? Car-OK

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Example

1. ?Battery-OK ? ?Bulbs-OK ? Headlights-Work
2. ?Battery-OK ? ?Starter-OK ? Empty-Gas-Tank ?
   Engine-Starts
3. ?Engine-Starts ? Flat-Tire ? Car-OK
4. Headlight-Work
5. Battery-OK
6. Starter-OK
7. ?Empty-Gas-Tank
8. ?Car-OK
9. ?Flat-Tire
10. ?Starter-OK ? Empty-Gas-Tank ? Engine-Starts
11. ?Battery-OK ? Empty-Gas-Tank ? Engine-Starts
12. ?Battery-OK ? ?Starter-OK ? Engine-Starts
13. ?Engine-Starts ? Flat-Tire
14. ?Engine-Starts ? Car-OK

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Example
Battery-OK Starter-OK ?Empty-Gas-Tank ?Car-OK
?Flat-Tire ?Battery-OK ? ?Starter-OK ?
Engine-Starts ?Engine-Starts ? Flat-Tire
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Example
Battery-OK Starter-OK ?Empty-Gas-Tank ?Car-OK
?Flat-Tire ?Battery-OK ? ?Starter-OK ?
Engine-Starts ?Engine-Starts ? Flat-Tire
?Battery-OK ? ?Starter-OK ? Flat-Tire ?Starter-
OK ? Flat-Tire Flat-Tire False (empty clause)
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Resolution Heuristics

• Set-of-support heuristics At least one
  ancestor of every inferred clause comes from a
• Shortest-clause heuristics Generate a clause
  with the fewest literals first
• Simplifications heuristics
• Remove any clause containing two complementary
  literals (tautology)
• Remove any clause C that contains all the
  literals of another clause C
• If a symbol always appears with the same sign,
  remove all the clauses that contain it (pure
  symbol)

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Example (Set-of-Support)

1. ?Battery-OK ? ?Bulbs-OK ? Headlights-Work
2. ?Battery-OK ? ?Starter-OK ? Empty-Gas-Tank ?
   Engine-Starts
3. ?Engine-Starts ? Flat-Tire ? Car-OK
4. Headlight-Work
5. Battery-OK
6. Starter-OK
7. ?Empty-Gas-Tank
8. ?Car-OK
9. ?Flat-Tire

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Example (Set-of-Support)

1. ?Battery-OK ? ?Bulbs-OK ? Headlights-Work
2. ?Battery-OK ? ?Starter-OK ? Empty-Gas-Tank ?
   Engine-Starts
3. ?Engine-Starts ? Flat-Tire ? Car-OK
4. Headlight-Work
5. Battery-OK
6. Starter-OK
7. ?Empty-Gas-Tank
8. ?Car-OK
9. ?Flat-Tire
10. ?Engine-Starts ? Car-OK
11. ?Engine-Starts
12. ?Battery-OK ? ?Starter-OK ? Empty-Gas-Tank
13. ?Starter-OK ? Empty-Gas-Tank
14. Empty-Gas-Tank
15. False

Note the goal-directed flavor
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Resolution Heuristics

• Set-of-support heuristics At least one
  ancestor of every inferred clause comes from a
• Shortest-clause heuristics Generate a clause
  with the fewest literals first
• Simplifications heuristics
• Remove any clause containing two complementary
  literals (tautology)
• Remove any clause C that contains all the
  literals of another clause C
• If a symbol always appears with the same sign,
  remove all the clauses that contain it (pure
  symbol)

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Example (Shortest-Clause)

1. ?Battery-OK ? ?Bulbs-OK ? Headlights-Work
2. ?Battery-OK ? ?Starter-OK ? Empty-Gas-Tank ?
   Engine-Starts
3. ?Engine-Starts ? Flat-Tire ? Car-OK
4. Headlight-Work
5. Battery-OK
6. Starter-OK
7. ?Empty-Gas-Tank
8. ?Car-OK
9. ?Flat-Tire

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Example (Shortest-Clause)

1. ?Battery-OK ? ?Bulbs-OK ? Headlights-Work
2. ?Battery-OK ? ?Starter-OK ? Empty-Gas-Tank ?
   Engine-Starts
3. ?Engine-Starts ? Flat-Tire ? Car-OK
4. Headlight-Work
5. Battery-OK
6. Starter-OK
7. ?Empty-Gas-Tank
8. ?Car-OK
9. ?Flat-Tire
10. ?Engine-Starts ? Car-OK
11. ?Engine-Starts
12. ?Bulbs-OK ? Headlights-Work
13. ?Battery-OK ? ?Starter-OK ? Empty-Gas-Tank
14. ?Starter-OK ? Empty-Gas-Tank
15. Empty-Gas-Tank
16. False

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Resolution Heuristics

• Set-of-support heuristics At least one
  ancestor of every inferred clause comes from a
• Shortest-clause heuristics Generate a clause
  with the fewest literals first
• Simplifications heuristics
• Remove any clause containing two complementary
  literals (tautology)
• Remove any clause C that contains all the
  literals of another clause C
• If a symbol always appears with the same sign,
  remove all the clauses that contain it (pure
  symbol)

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Example (Pure Literal)

1. ?Battery-OK ? ?Bulbs-OK ? Headlights-Work
2. ?Battery-OK ? ?Starter-OK ? Empty-Gas-Tank ?
   Engine-Starts
3. ?Engine-Starts ? Flat-Tire ? Car-OK
4. Headlights-Work
5. Battery-OK
6. Starter-OK
7. ?Empty-Gas-Tank
8. ?Car-OK
9. ?Flat-Tire

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When to Use Logic?

• When the knowledge base is large and can be made
  explicit
• When formulating a problem as a CSP problem
  would yield complex constraints
• When the agents environment is conveniently
  described by true or false sentences

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Summary

• Entailment problem
• Resolution rule
• Clause form of a set of sentences
• Resolution refutation algorithm
• Resolution heuristics