Propositional Logic Part 2 Yingyu Liang yliang@cs.wisc.edu - - PowerPoint PPT Presentation

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Propositional Logic Part 2 Yingyu Liang yliang@cs.wisc.edu - - PowerPoint PPT Presentation

Nov 18, 2022 318 likes •585 views

Propositional Logic Part 2 Yingyu Liang yliang@cs.wisc.edu Computer Sciences Department University of Wisconsin, Madison slide 1 [Based on slides from Louis Oliphant, Andrew Moore, Jerry Zhu] Method 4: chaining with Horn clauses

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Propositional Logic Part 2

Yingyu Liang yliang@cs.wisc.edu Computer Sciences Department University of Wisconsin, Madison

[Based on slides from Louis Oliphant, Andrew Moore, Jerry Zhu]

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Method 4: chaining with Horn clauses

  • Resolution is too powerful for many practical situations.
  • A weaker form: Horn clauses

▪ Disjunction of literals with at most one positive

  • R  P  Q

no

  • R   P  Q

yes ▪ What’s the big deal?

  • R   P  Q
  • (R  P)  Q

?

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Horn clauses

  • R   P  Q
  • (R  P)  Q

(R  P)  Q Every rule in KB is in this form P (special case, no negative literals): fact

  • The big deal:

▪ KB easy for human to read ▪ Natural forward chaining and backward chaining algorithm, proof easy for human to read ▪ Deciding entailment with Horn clauses in time linear to KB size

  • But…

▪ Can only ask atomic queries

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Forward chaining

  • Fire any rule whose premises are satisfied in the KB
  • Add its conclusion to the KB until query is found

KB: query: Q AND-OR graph AND OR

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Forward chaining

P  Q L  M  P B  L  M A  P  L A  B  L A B

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Forward chaining

P  Q L  M  P B  L  M A  P  L A  B  L A B

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Forward chaining

P  Q L  M  P B  L  M A  P  L A  B  L A B

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Forward chaining

P  Q L  M  P B  L  M A  P  L A  B  L A B

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Forward chaining

P  Q L  M  P B  L  M A  P  L A  B  L A B

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Forward chaining

P  Q L  M  P B  L  M A  P  L A  B  L A B

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Forward chaining

P  Q L  M  P B  L  M A  P  L A  B  L A B

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Forward chaining

P  Q L  M  P B  L  M A  P  L A  B  L A B

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Backward chaining

  • Forward chaining problem: can generate a lot of

irrelevant conclusions ▪ Search forward, start state = KB, goal test = state contains query

  • Backward chaining

▪ Reverse search from goal ▪ Find all implications of the form (…)  query ▪ Prove all the premises of one of these implications

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Backward chaining

P  Q L  M  P B  L  M A  P  L A  B  L A B

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Backward chaining

P  Q L  M  P B  L  M A  P  L A  B  L A B

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Backward chaining

P  Q L  M  P B  L  M A  P  L A  B  L A B

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Backward chaining

P  Q L  M  P B  L  M A  P  L A  B  L A B

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Backward chaining

P  Q L  M  P B  L  M A  P  L A  B  L A B

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Backward chaining

P  Q L  M  P B  L  M A  P  L A  B  L A B

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Backward chaining

P  Q L  M  P B  L  M A  P  L A  B  L A B

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Backward chaining

P  Q L  M  P B  L  M A  P  L A  B  L A B

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Backward chaining

P  Q L  M  P B  L  M A  P  L A  B  L A B

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Backward chaining

P  Q L  M  P B  L  M A  P  L A  B  L A B

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Forward vs. backward chaining

  • Forward chaining is data-driven

▪ May perform lots of work irrelevant to the goal

  • Backward chaining is goal-driven

▪ Appropriate for problem solving

  • Some form of bi-directional search is even better
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What you should know

  • A lot of terms
  • Use truth tables
  • Proofs
  • Conjuctive Normal Form
  • Proofs with resolution
  • Horn clauses
  • Forward chaining algorithm
  • Backward chaining algorithm