Reactive Agents and Propositional Logic

PSU CS441/541 Lecture 2
October 4, 2001

• 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
• Nilsson's Environment
  • ``Grid world''
  • Stackable objects
  • Constraints
  • Finite, discrete
• Reactive Agents
  • React to environment
  • No notion of planning, reasoning
  • State machines: reactive agents with memory
• Implementing Reactive Agents
  • Map input to environment
    • ad-hoc methods
      • C functions
      • ``potential fields'': book environment
    • ``production system''
      • rules of form cond -> action(s) | prod-system
      • book example: wall-following robot
      • soundness via priorities (yuk)
      • completeness?
    • digital circuit
      • booleanize inputs, outputs
      • represent each output as Boolean function of inputs
      • soundness and completeness are easy
      • correctness not so easy: SAT
      • propositional logic and boolean circuits
  • Adding state/memory
    • Memory as sensing: the trick
    • Memory can substitute for sensing
    • Traditional state machines
    • Blackboards
• Propositional Logic (Review?)
  • Propositional formulae
    • Boolean atoms (variables), literals ((non)negated atoms)
    • connectives: disjunction (or, sum), conjunction (and, product), grouping (parens)
    • implication: a => b iff not a or b
  • What to do with prop formulae?
    • evaluate: given total assignment of true/false to atoms, is formula true or false?
    • canonicalize: given formula, place in some canonical form. formulae are equivalent if their set of models are true in all interpretations
      • sum of products / disjunctive normal form
      • product of sums / conjunctive normal form
    • model: find/characterize the set of total assignments which make given formula true
    • infer: given formulae, produce another formula using inference rules
      • a sound and complete set of inference rules
        • modus ponens (a, a => b : b)
        • and introduction (a, b : a and b)
        • and commutativity (a and b : b and a)
        • and elimination (a and b : a)
        • or introduction (a : a or b; b : a or b)
        • not elimination (not not a : a)
      • additional rules are consequence of these rules
      • sound and complete are wrt interpretations
    • prove (deduce): given axiomatic formulae, derive given formula via series of inference steps
      • deduction is using given inference rules
      • don't confuse deduction with entailment (below)
    • interpret: assign a meaning to atoms, negation, connectives; can be another formula, a function, a natural language statement, etc.
      • Herbrand interpretation: things are themselves
      • truth tables give Herbrand interpretation of connectives
      • entailment: any model of a set of formulas is also a model for the entailed formula
      • a sound and complete set of inference rules is one in which truth is the same as provability
        • in prop, we can give a proof algorithm for which works on exactly the set of true formulae
        • unfortunately, no one knows whether there is a polytime algorithm. This is the P ?= NP question
  • Resolution
    • 3-CNF
      • can convert any prop formula to only polynomially larger 3-CNF canonical formula
      • book procedure is ``wrong'': parse tree procedure
    • Resolution and proof
      • inference rule for CNF: sound
      • refutation (insert negation of query and prove unsat) is complete proof procedure for prop logic
    • Horn clauses
      • form: at most one positive literal
      • atom = fact, implication = rule, negative = goal
      • linear-time decision procedure
• Prop Representations
  • Idea: model discrete world as prop formula
  • Clauses are constraints on models
  • Interpretation of a model is legal world
  • Modeling and analysis
  • Example: Tic-Tac-Toe World