AI Problem Representation

PSU CS441/541 Lecture 2
October 2, 2000

• Logical Problem Representation
  • Solving a problem with a computer (c.f. Ginsberg 6.1)
    1. accurately describe problem
    2. choose instance representation in computer
    3. select algorithm to manipulate representation
    4. execute
  • What properties of representations are important?
    • compactness: must be able to represent big instances efficiently
    • utility: must be compatible with good solution algorithms
    • soundness: should not report untruths
    • completeness: should not lose information
    • generality: should be able to represent all or most instances of interesting problems
    • transparency: reasoning about/with representation is efficient, easy
  • What instance representations do people choose?
    • database: collection of facts
    • neural net: collection of "neuron weights"
    • functional: collection of functions
    • logical: collection of sentences
  • In achieving properties, support is critical: representations extensively studied, tools available, etc.
• Logic
  • Three levels today (see tradeoffs above)
    • Propositional Logic: relationship of atomic facts
    • Predicate Calculus: relationship of compound facts
    • First-Order: relationship of infinite sets of predicates
  • The building blocks
    • atoms
      • propositional atom: identifier (capitalized)
      • predicate atom: application of relation to functional argument
    • atomic formulae: potentially negated atoms
    • sentences: atomic formulae joined by connectives
      • negation: not
      • conjunction: and
      • disjunction: or
      • implication: implies
    • quantified sentences: introduction of variables
      • universal
      • existential
  • Models and semantics
    • model: assignment of truth or falsity to every possible atom (universe of discourse). Often (as Ginsberg) subset of universe which is true
    • interpretation: map from model to target
    • meaning: target state consistent with model
    • Herbrand interpretation: models mean themselves
    • equivalence: two sentences A, B have same set of models
    • entailment: all models of A are models of B
    • inference rule: from sentence (set) to sentence
      • sound: (A derives B) implies (A entails B)
      • complete: (A entails B) implies (A derives B)
  • Manipulating logical sentences
    • inference rules
      • commutatitivity, associativity, distributivity
      • double-negation
      • DeMorgan's laws
      • Modus Ponens
      • instantiation
      • quantifier flipping
      • finitization for quantifier elimination
      • Skolemization
      • Normal forms
        • disjunctive
        • conjunctive
    • model-based: truth tables
    • example: contrapositive
      • by inference rules
      • shorter: by inference rules
      • by models
  • Computability and complexity of logics
    • prop: decidable (by model checking), but NP-hard (exponential)
    • first-order: undecidable (semi-decidable by resolution)
    • Godel's Theorem: pick any two of sound, complete (decidable), powerful
• Predicate Logic and Horn Databases
  • Extended Modus Ponens
  • Horn databases
    • Horn resolution
    • Horn resolution is sound and complete (for Horn logic)
    • Prolog: "negation as failure"
  • Predicate databases
    • Extended Horn Normal Form
    • normal form complete for predicate logic
    • prove P by resolution with not P to contradiction
    • resolution is sound and complete (for full predicate logic)
    • resolution is inefficient (NP-hard)
• Cool, But What Happened To AI?
  • Can represent facts as Horn or general predicate statements
  • Can infer unstated things from stated things
  • Thus many properties of reasonable KR
  • Being able to carry salient facts only is
    • "Smart" thing humans do
    • Necessary for solving "intellectually challenging" problems
  • Danger: the "ramification problem"