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Logical inference

Logical inference . Chapter 7. Reminders…. Propositional logic: A ‘model’ is a set of propositional statements and truth values. Easy to implement as a …. … hashtable Semantics: rules for determining the truth of a sentence. For example. Model = {‘P’:true, ‘Q’:false, ‘R’:true}

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Logical inference

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  1. Logical inference Chapter 7

  2. Reminders… • Propositional logic: A ‘model’ is a set of propositional statements and truth values. • Easy to implement as a …. • … hashtable • Semantics: rules for determining the truth of a sentence.

  3. For example • Model = {‘P’:true, ‘Q’:false, ‘R’:true} • Is ‘P & Q’ true? • Is ‘P | Q’ true? • Is ‘P & ~Q & (Q | R)’ true? • Is ‘P & ~Q & (~Q | T)’ true?

  4. Inference by checking all

  5. Consider the reflex agent… • How can we create a reflex agent that learns, using propositional logic? • Perceptlist: • [‘mate’,’flee’,0] • [‘smallenemy’,’fight’,1] • Conclusions: • Mate -> ~ Flee • SmallEnemy -> Fight • Flee? Fight? Eat? Mate? • Given (current) KB, does KB entail action?

  6. The well-tempered inference engine • What do we want in an inference engine? • Soundness, every conclusion derived is correct. • Completeness, every correct conclusion is derivable. • Efficient (of at most polynomial complexity). • Sorry, but “every known inference algorithm for propositional logic has a worse-case complexity that is exponential in the size of the input,” where “input” is the number of propositions.

  7. So let’s give up? • No, accept some limitations • Soundness? • Completeness? • Efficiency?

  8. Some inferential terms • Logical equivalence (these are meta-statements) • ~~ ==  • ~( | ) == (~ & ~) • ~(&) == (~ | ~) • Validity • Sentence is True in every model • Satisfiability • a sentence  is true in model m, then  satisfies m.

  9. Some inference theorems • Deduction theorem • For any sentences  and ,  entails  iff the sentence ( -> ) is valid. • Proof by refutation/reductio ad absurum •  entails  iff ( -> ~) is unsatisfiable. • P -> Q? Assert ~Q, see if results in contradiction.

  10. Again, unfortunately… • This is NP-complete (probably too hard). • But let’s play lots of tricks • Monotonic assumption: number of entailed sentences can only increase as information is added to the KB. • Place our attention on what we care about. • But it’s still NP-complete.

  11. Completeness through resolution • Does proof by contradiction • I.e., KB entails  by proving (KB & ~ ) is unsatisfiable. • Put in “conjuctive normal form” • Only “conjunction of disjunctions of [possibly negated] literals” • See algorithm in book.

  12. Back to Reflex Agent

  13. Consider the reflex agent again… • How can we create a reflex agent that learns, using propositional logic? • Perceptlist: • [‘mate’,’flee’,0] • [‘smallenemy’,’fight’,1] • Conclusions: • Mate -> ~ Flee • SmallEnemy -> Fight • Flee? Fight? Eat? Mate? • Given (current) KB, does KB entail action?

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