Advanced KR: Three Topics

PSU CS441/541 Lecture 6
October 30, 2000

• From Boolean Logic To General Formal Systems
  • Soundness, completeness, etc. of FOL and subclasses good
  • Expressivity of FOL bad
    • Boolean values?
    • full representation?
    • mathematics?
  • Idea: replace (augment?) FOL with more direct/expressive formal system
  • Today: three such systems
    • default logic (non-mon)
    • probabilistic logic
    • CSPs
  • All work by ``labeling'' logical sentences
• Default Logics and Non-Monotonic Reasoning
  • Idea: it is useful to assume
    • allows better conclusions
    • allows conclusions faster (?)
  • Method: extend logic with extra defaults (T,A)
  • Formulation
    • express A using predicates ab<sub>i</sub>(x) denoting particular abnormalities
    • can diagram using single and double arrows
    • e.g.: Tweety
      • bird(x) and not ab(x) implies flies(x)
      • ostrich(x) implies bird(x) and not flies(x)
      • tweety(x) implies bird(x)
      • tweety(Tweety)
    • Frame problem: default persistence
  • Reasoning
    • extension: maximal (size) subset of A consistent with T
    • sentence p is
      • cautious consequence of (T,A) if p holds in all extensions E of (T,A)
      • brave consequence of (T,A) if p holds in some extensions E of (T,A)
    • hard to do better than this via obvious mechanisms: Nixon diamond
• Probabilistic Logic
  • Idea: chance and likelihood are important concepts for real reasoning
  • Method: assign probabilities to events and combinations of events
  • Formulation
    • pr(p) is probability of event
    • pr(p|q) is probability of q given p (easy to get backward)
    • pr(p and q) = pr(p) pr(q|p) = pr(q) pr(p|q)
    • pr(not p) = 1 - pr(p)
    • pr(p or q) = pr(not (not p and not q))
    • equivalent sentences have same probability
  • Reasoning
    • Bayes' Rule: given
      • effect E with prior probability pr(E)
      • cause C with pp pr(C)
      • probability pr(C|E) of the effect given the cause
      prove from above and compute
      • pr(E|C) = pr(C|E) pr(C) / pr(E)
    • problem: everything depends on everything else
      • need to know impossible number of prior and conditional probabilities to conclude anything
      • Bayes Net (BBN, influence diagram): indicate which priors and conditionals have significant influence in practice
    • problem: probabilities may be meaningless
      • difference between 0.5 and ``don't know'' and ``don't care''
      • MYCIN and probabilities v. ``likelihoods''
      • Cox's Theorem: under reasonable assumptions, any labeling of logical sentences with real numbers will be consistent with probability
      • problems with real numbers
• Constraint Satisfaction
  • Idea: the world is not boolean (or even finite-valued)
    • boolean variables encode multiple values via enumeration
    • want direct valuation
    • most domains have nice properties: finiteness, enumerability, order, equality
    • most relationships are binary
  • Binary CSP
    • let variables draw values from domain
    • ``set of tuples'' defines relationship between vars
    • Problem: given variables, constraints, find satisfying assignment
  • Highly practical approach w/ same complexity as resolution
  • Ginsberg uses implicitly: e.g. crossword puzzles
  • C.f First-Order Logic
  • Reductions
    • higher-order CSP -> binary CSP
    • FOL -> CSP
    • CSP -> FOL
• Bonus Topic: Arrow's Theorem
  • It's election time. Does logic help?
  • Consider the following axioms
    • (Universal Domain) An election takes voter rankings to a global ranking.
    • (Weak Pareto Principle) If everyone prefers candidate x to candidate y, x should be globally preferred to y.
    • (Independence Assumption) The relative global ranking of candidate x and candidate y should depend only on the voters' relative rankings of x and y.
    • (No Dictator) There is no single voter who can force the outcome of an election.
    • An election may have an arbitrary number of candidates
  • Theorem (Kenneth Arrow, 1952): No voting system exists which enforces all of these properties on all elections!
  • Consequences for logic and AI?