CS 416 Artificial Intelligence

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