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First-Order Logic Homework Assignment and Study Guide Updates

This assignment on first-order logic is due Wednesday, with no late days permitted. Midterm preparations will cover Chapters 1-9 (excluding Chapter 5), with a previous midterm and study guide available on the website. Key topics include logical inference methods such as forward chaining, backward chaining, and resolution. Additionally, learn from Gerald Tesauro's research on reinforcement learning and neural networks. Prepare effectively to ensure a comprehensive understanding of inference in first-order logic.

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First-Order Logic Homework Assignment and Study Guide Updates

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  1. CS 416Artificial Intelligence Lecture 13 First-Order Logic Chapter 9

  2. Homework Assignment • First-order logic assignment due on Wednesday • No use of late days on this one • Answers will be provided afterwards

  3. Midterm • October 25th • Up through chapter 9 (excluding chapter 5) • Old midterm on web site (with answers) • Study guide on web site

  4. AI and Finance • Gerald Tesauro – Best Backgammon program • GERALD TESAURO is a research staff member at IBM. Current research interests include reinforcement learning in the nervous system, and applications of neural networks to financial time-series analysis and to computer virus recognition • http://www.research.ibm.com/infoecon/paper10.html

  5. Inference in first-order logic • Our goal is to prove that KB entails a fact, a • We use logical inference • Forward chaining • Backward chaining • Resolution • All three logical inference systems rely on search to find a sequence of actions that derive the empty clause

  6. Search and forward chaining • Start with KB full of first-order definite clauses • Disjunction of literals with exactly one positive • Equivalent to implication with conjunction of positive literals on left (antecedent / body / premise) and one positive literal on right (consequent / head / conclusion) • Propositional logic used Horn clauses, which permit zero or one to be positive • Look for rules with premises that are satisfied (use substitution to make matches) and add conclusions to KB

  7. Search and forward chaining • Which rules have premises that are satisfied (modus ponens)? • A ^ E => C… nope • B ^ D => E… yes • E ^ C ^ G ^ H => I… nope • A ^ E = C… yes • E ^ C ^ G ^ H => I… yes • Breadth First • A, B, D, G, H • A ^ E => C • B ^ D => E • E ^ C ^ G ^ H => I

  8. Search and forward chaining • Would other search methods work? • Yes, this technique falls in standard domain of all searches

  9. Search and backward chaining • Start with KB full of implications • Find all implications with conclusion matching the query • Add to fringe list the unknown premises • Adding could be to front or rear of fringe (depth or breadth)

  10. Search and backward chaining • Are all the premises of I satisfied? No • For each (C E G H) are each of their premises satisfied? • C? no, put its premises on fringe • For each (A and E) are their premises satisfied? A… yes E… no, add premises for each B and D B… yes D… yes E…yesC… yes • Depth First • A, B, D, G, H • A ^ E => C • B ^ D => E • C ^ E ^ G ^ H => I

  11. Search and backward chaining • Are all the premises of I satisfied? No • For each (C E G H) are each of their premises satisfied? • C… yes • E… yes • G, H… yes • I… yes • Breadth First • A, B, D, G, H • A ^ E => C • B ^ D => E • C ^ E ^ G ^ H => I

  12. Backward/forward chaining • Don’t explicitly tie search method to chaining direction

  13. Inference with resolution • We put each first-order sentence into conjunctive normal form • We remove quantifiers • We make each sentence a disjunction of literals (each literal is universally quantified) • We show KB ^ ~a is unsatisfiable by deriving the empty clause • Resolution inference rule is our method • Keep resolving until the empty clause is reached

  14. Example

  15. Resolution example

  16. Theorem provers • Logical inference is a powerful way to “reason” automatically • Prover should be independent of KB syntax • Prover should use control strategy that is fast • Prover can support a human by • Checking a proof by filling in voids • Person can kill off search even if semi-decidable

  17. Practical theorem provers • Boyer-Moore (1979) • First rigorous proof of Godel Incompleteness Theorem • OTTER (1997) • Solved several open questions in combinatorial logic • EQP • Solved Robbins algebra, a proof of axioms required for Boolean algebra • Problem posed in 1933 and solved in 1997 after eight days of computation

  18. Practical theorem provers • Verification and synthesis of hard/soft ware • Software (axiomize all syntactic elements of programming language) • Verify a program’s output is correct for all inputs • There exists a program, P, that satisfies a specification • Synthesize P during search • Hardware (axiomize all interactions between signal and circuit elements) • Verify that interactions between signals and circuits is robust • Will CPU work in all conditions? • There exists a circuit, C, that satisfies a specification • Synthesize C during search

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