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Lecture 26 of 42. Conditional, Continuous, and Multi-Agent Planning Discussion: Probability Refresher. Wednesday. 24 October 2007 William H. Hsu Department of Computing and Information Sciences, KSU KSOL course page: http://snipurl.com/v9v3

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Lecture 26 of 42

Conditional, Continuous, and Multi-Agent Planning

Discussion: Probability Refresher

Wednesday. 24 October 2007

William H. Hsu

Department of Computing and Information Sciences, KSU

KSOL course page: http://snipurl.com/v9v3

Course web site: http://www.kddresearch.org/Courses/Fall-2007/CIS730

Instructor home page: http://www.cis.ksu.edu/~bhsu

Reading for Next Class:

Section 12.5 – 12.8, Russell & Norvig 2nd edition

CIS 530 / 730: Artificial Intelligence

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Lecture Outline

  • Today’s Reading: Sections 12.1 – 12.4, R&N 2e
  • Friday’s Reading: Sections 12.5 – 12.8, R&N 2e
  • Today: Practical Planning, concluded
    • Conditional Planning
    • Replanning
    • Monitoring and Execution
    • Continual Planning
  • Hierarchical Planning Revisited
    • Examples: Korf
    • Real-World Example
  • Friday and Next Week: Reasoning under Uncertainty
    • Basics of reasoning under uncertainty
    • Probability review
    • BNJ interface (http://bnj.sourceforge.net)

CIS 530 / 730: Artificial Intelligence

slide3

Planning and Learning Roadmap

  • Bounded Indeterminacy (12.3)
  • Four Techniques for Dealing with Nondeterministic Domains
  • 1. Sensorless/Conformant Planning: “Be Prepared” (12.3)
    • Idea: be able to respond to any situation (universal planning)
    • Coercion
  • 2. Conditional / Contingency Planning: “Plan B” (12.4)
    • Idea: be able to respond to many typical alternative situations
    • Actions for sensing (“reviewing the situation”)
  • 3. Execution Monitoring / Replanning: “Show Must Go On” (12.5)
    • Idea: be able to resume momentarily failed plans
    • Plan revision
  • 4. Continuous Planning: “Always in Motion, The Future Is” (12.6)
    • Lifetime planning (and learning!)
    • Formulate new goals

CIS 530 / 730: Artificial Intelligence

slide10

Hierarchical Abstraction Planning:Review

  • Need for Abstraction
    • Question: What is wrong with uniform granularity?
    • Answers (among many)
      • Representational problems
      • Inferential problems: inefficient plan synthesis
  • Family of Solutions: Abstract Planning
    • But what to abstract in “problem environment”, “representation”?
      • Objects, obstacles (quantification: later)
      • Assumptions (closed world)
      • Other entities
      • Operators
      • Situations
    • Hierarchical abstraction
      • See: Sections 12.2 – 12.3 R&N, pp. 371 – 380
      • Figure 12.1, 12.6 (examples), 12.2 (algorithm), 12.3-5 (properties)

Adapted from Russell and Norvig

CIS 530 / 730: Artificial Intelligence

slide18

Universal Quantifiers in Planning

  • Quantification within Operators
    • p. 383 R&N
    • Examples
      • Shakey’s World
      • Blocks World
      • Grocery shopping
    • Others (from projects?)
  • Exercise for Next Tuesday: Blocks World

CIS 530 / 730: Artificial Intelligence

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Practical Planning

  • The Real World
    • What can go wrong with classical planning?
    • What are possible solution approaches?
  • Conditional Planning
  • Monitoring and Replanning (Next Time)

Adapted from Russell and Norvig

CIS 530 / 730: Artificial Intelligence

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Review:How Things Go Wrong in Planning

Adapted from slides by S. Russell, UC Berkeley

CIS 530 / 730: Artificial Intelligence

slide21

Review:Practical Planning Solutions

Adapted from slides by S. Russell, UC Berkeley

CIS 530 / 730: Artificial Intelligence

slide22

Conditional Planning

Adapted from slides by S. Russell, UC Berkeley

CIS 530 / 730: Artificial Intelligence

monitoring and replanning
Monitoring and Replanning

CIS 530 / 730: Artificial Intelligence

slide24

Preconditions for Remaining Plan

Adapted from slides by S. Russell, UC Berkeley

CIS 530 / 730: Artificial Intelligence

slide25

Replanning

Adapted from slides by S. Russell, UC Berkeley

CIS 530 / 730: Artificial Intelligence

slide26

Making Decisions under Uncertainty

Adapted from slides by S. Russell, UC Berkeley

CIS 530 / 730: Artificial Intelligence

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Sample Space (): Range of a Random Variable X

  • Probability Measure Pr()
    •  denotes a range of “events”; X: 
    • ProbabilityPr, or P, is a measure over 2
    • In a general sense, Pr(X = x  ) is a measure of belief in X = x
      • P(X = x) = 0 or P(X = x) = 1: plain (akacategorical) beliefs (can’t be revised)
      • All other beliefs are subject to revision
  • Kolmogorov Axioms
    • 1. x  . 0  P(X = x)  1
    • 2. P() x  P(X = x) = 1
    • 3.
  • Joint Probability: P(X1X2)  Probability of the Joint Event X1X2
  • Independence: P(X1X2) = P(X1)  P(X2)

Probability:Basic Definitions and Axioms

CIS 530 / 730: Artificial Intelligence

slide28

Product Rule (Alternative Statement of Bayes’s Theorem)

    • Proof: requires axiomatic set theory, as does Bayes’s Theorem
  • Sum Rule
    • Sketch of proof (immediate from axiomatic set theory)
      • Draw a Venn diagram of two sets denoting events A and B
      • Let A B denote the event corresponding to A B…
  • Theorem of Total Probability
    • Suppose events A1, A2, …, An are mutually exclusive and exhaustive
      • Mutually exclusive: i j Ai Aj =
      • Exhaustive:  P(Ai) = 1
    • Then
    • Proof: follows from product rule and 3rd Kolmogorov axiom

Basic Formulas for Probabilities

A

B

CIS 530 / 730: Artificial Intelligence