Sep 15, 2003
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- Theme thus far
        - problem solving by searching
        - assumes no knowledge

- Typically
        - search compensates for lack of knowledge
        - knowledge compensates for lack of search

- Knowledge representation and reasoning
        - Given KB (facts), try to derive 
          conclusions (more facts)

- General theme
        - boxy diagram from AIMA
        - notions of "sentences", "entails", "follows",
          and "real world stuff"

- Three main steps
        - choose a representation
        - setup an interpretation
        - select an inference procedure

- Choosing the representation
        - primarily an issue of syntax
        - choose symbols, operators

- Setting up an interpretation
        - what it means
        - "semantics"
        - defines the truth of every possible sentence in
          every possible world

- "Possible worlds" are also called
        - "models"
        - we say:
                - Sentence s is true in model m, or
                - m is a model of s

- How to think of models
        - e.g., as rows of the truth table
          (in propositional logic)

- Infernece procedures
        - have a notion of "entailment"
        - A entails B iff
                - for every model in which A is true
                  B is also true
                - i.e., models of A are also models of B

- Just because A entails B,
        - doesn't mean you can actually derive B from A

- How *do* we prove that A entails B
        - model-checking: enumerate all rows of the truth table
        - and verify that the models match up
        - time consuming
                - but actually how some difficult proofs were
                  made

- Model checking
        - should be exhaustive
        - story of freshman proving (x+y)^2 = (x^2 + 2xy + y^2)

- Inference algorithm
        - derives new sentences from old sentences

- Classifications of inference procedures
        - Suppose A derives B. Then A entails B.
                => Sound
        - Suppose A entails B. Then A derives B.
                => Complete

- Example of sound and incomplete procedure
        - GSAT

- Example of unsound and complete procedure
        - mathematical howlers
          (not fallacies)

- Many types of logic
        - propositional logic
        - first order predicate logic
        - fuzzy logic
        - modal logic
        - non-monotonic logic
        - commonsense

- Example logic: propositional logic
        - symbols, connectives, and semantics
        - AND, OR, NOT, ->, and <->

- Implication, e.g., X->Y
        - leads to strange sentences, but
          remember they are only sentences
          (may not be true in all models)