010'141 Engineering Mathematics II Lecture 17 The Propositional Calculus

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010.141 Engineering Mathematics IILecture 17The
Propositional Calculus

• Bob McKay
• School of Computer Science and Engineering
• College of Engineering
• Seoul National University

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Outline

• Wumpus World
• An example world for reasoning
• Models and Entailment
• Truth in possible worlds
• Validity and Satisfiabililty
• Truth in all or some worlds

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References

• Shoenfield, J R Mathematical Logic, Association
  for Symbolic Logic 1967 ISBN 1 56881 135 7
  (reprinted 2000)
• Russell, S Norvig, P Artificial Intelligence
  A Modern Approach, Prentice Hall
• The library has edition 1, call number 006.3
  R917a
• To buy, edition 2 (1995), ISBN 0137903952
• Either is fine
• Nilsson, NJ Artificial Intelligence A New
  Synthesis, Morgan Kaufmann, 1998, ISBN 1 55860
  535 5
• Library call no 006.3 N599a

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Example Wumpus World

• We base our description on an early computer game
  
• Played on a 44 grid
• Your aim is
• To get the pot of gold and return (1,000)
• Not to get killed (-10,000)
• Eaten by Wumpus
• Fallen into pit
• Not to take too long (-1/turn)
• You can only move up/down/left/right
• Next to a wumpus, you smell a smell
• Next to a pit, you feel a breeze
• You can shoot (once) in a straight line
• kills the wumpus, costs 10
• Try it at http//www.gpf-comics.com/games/wumpus/

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Describing Wumpus World

• Performance measure
• gold 1000, death -1000
• -1 per step, -10 for using the arrow
• Environment
• Squares adjacent to wumpus are smelly
• Squares adjacent to pit are breezy
• Glitter iff gold is in the same square
• Shooting kills wumpus if you are facing it
• Shooting uses up the only arrow
• Grabbing picks up gold if in same square
• Releasing drops the gold in same square
• Sensors Stench, Breeze, Glitter, Bump, Scream
• Actuators Left turn, Right turn, Forward, Grab,
  Release, Shoot
  

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Exploring a wumpus world
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Exploring a wumpus world
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Exploring a wumpus world
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Exploring a wumpus world
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Exploring a wumpus world
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Exploring a wumpus world
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Exploring a wumpus world
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Exploring a wumpus world
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Logic in general

• Logics are formal languages for representing
  information such that conclusions can be drawn
  
• Syntax defines the sentences in the language
• Semantics define the "meaning" of sentences
• i.e., define truth of a sentence in a world
• E.g., the language of arithmetic
• x2 y is a sentence x2y gt is not a
  sentence
  
• x2 y is true iff the number x2 is no less
  than the number y
  
• x2 y is true in a world where x 7, y 1
• x2 y is false in a world where x 0, y 6

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Entailment

• Entailment means that one thing follows from
  another
  
• KB a
• Knowledge base KB entails sentence a if and only
  if a is true in all worlds where KB is true
• E.g., the KB containing the Giants won and the
  Reds won entails Either the Giants won or the
  Reds won
  
• E.g., xy 4 entails 4 xy
• Entailment is a relationship between sentences
  (i.e., syntax) that is based on semantics
  (meaning)

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Models

• Logicians think a lot about models
• Mathematical worlds in which we can check truth
• We say m is a model of a sentence a if a is true
  in m
• M(a) is the set of all models of a
• All the ways a can be true
• So KB a means M(KB) ? M(a)
• Ithat is, a is at least as true as KB is
• E.g.
• KB Giants won and Reds won
• a Giants won

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Entailment in the wumpus world

• Situation after detecting nothing in 1,1,
  moving right, breeze in 2,1
• Consider possible models for KB assuming only
  pits
• 3 Boolean choices ? 8 possible models

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Wumpus models
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Wumpus models

• KB wumpus-world rules observations
• Red consistent with observations
• Blue not consistent with observations

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Wumpus models

• KB wumpus-world rules observations
• a1 "1,2 is safe
• KB a1
• proved by model checking

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Wumpus models

• KB wumpus-world rules observations

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Wumpus models

• KB wumpus-world rules observations
• a2 "2,2 is safe
• KB a2

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Summary

• Wumpus World
• An example world for reasoning
• Models and Entailment
• Truth in possible worlds
• Validity and Satisfiabililty
• Truth in all or some worlds

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