CS 730/830: Intro AI. 1 handout: slides. Wheeler Ruml (UNH) Lecture 11, CS / 15. Propositional Logic. First-Order Logic
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1 CS 730/830: Intro AI 1 handout: slides Wheeler Ruml (UNH) Lecture 11, CS / 15
2 Wheeler Ruml (UNH) Lecture 11, CS / 15
3 Logic A logic is a formal system: syntax: defines sentences semantics: relation to world inference rules: reaching new conclusions three layers: proof, models, reality soundness, completeness flexible, general, principled (Advice Taker, 1958) Wheeler Ruml (UNH) Lecture 11, CS / 15
4 Propositional Reasoning computing entailment soundness, completeness α = β iff α β is unsatisfiable determining satisfiability is NP-complete [ NP-hard = polytime to verify certificate of yes ] therefore, verification that β is not entailed is polytime said another way: α = β iff α β is valid determining validity/tautology is co-np-complete [ co-np-hard = polytime to verify certificate of no ] therefore, verification that β is not entailed is polytime Wheeler Ruml (UNH) Lecture 11, CS / 15
5 Reasoning Methods variable elimination: Davis-Logemann-Loveland exhaustively branch on variable assignments model finding: WalkSAT fix assignment until satisfying modus ponens, resolution: resolution refutation theorem proving derive new clauses until query is proved Wheeler Ruml (UNH) Lecture 11, CS / 15
6 An Example of Propositional Reasoning If the unicorn is mythical, then it is immortal, but if it is not mythical, then it is a mortal mammal. If the unicorn is either immortal or a mammal, then it is horned. The unicorn is magical if it is horned. Prove: the unicorn is magical. Wheeler Ruml (UNH) Lecture 11, CS / 15
7 Resolution Refutation Proofs Given KB, is α entailed? Wheeler Ruml (UNH) Lecture 11, CS / 15
8 Resolution Refutation Proofs Given KB, is α entailed? (Is it true in all models of the KB?) Wheeler Ruml (UNH) Lecture 11, CS / 15
9 Resolution Refutation Proofs Given KB, is α entailed? (Is it true in all models of the KB?) Is KB α unsatisfiable? Wheeler Ruml (UNH) Lecture 11, CS / 15
10 Resolution Refutation Proofs Given KB, is α entailed? (Is it true in all models of the KB?) Is KB α unsatisfiable? Resolution is refutation complete. Wheeler Ruml (UNH) Lecture 11, CS / 15
11 Conversion to Conjunctive Normal Form Syntax:,,, (, ), 1. eliminate 2. eliminate 3. move inward: x, (x y),, (x y) 4. distribute : x (y z) Wheeler Ruml (UNH) Lecture 11, CS / 15
12 Break asst 5 projects: topics, proposals due next Tue grades: canvas A, A, A, A, A, A, A, A-, B+, B, B, B-, C, C, C-, C-, D+, F, F, F, F mode A, median B Wheeler Ruml (UNH) Lecture 11, CS / 15
13 EOLQs Wheeler Ruml (UNH) Lecture 11, CS / 15
14 EOLQs Gottlob Frege ( ) PhD at 25 Begriffsschrift, 1879 (concept script) a formula language, modelled on that of arithmetic, of pure thought. Wheeler Ruml (UNH) Lecture 11, CS / 15
15 EOLQs 1. Things: person ItIsRaining() IsWet(person) constants: John, Chair23 functions (thing thing): MotherOf(John), SumOf(1,2) 2. Relations: predicates (objects T/F): IsWet(John), IsSittingOn(MotherOf(John),Chair23) 3. Complex sentences: connectives: IsWet(John) IsSittingOn(MotherOf(John),Chair23) quantifiers and variables: personiswet(person)..., person... Wheeler Ruml (UNH) Lecture 11, CS / 15
16 EOLQs 1. constants: objects 2. predicates: relations between objects 3. variables 4. quantifiers 5. functions 6. connectives Wheeler Ruml (UNH) Lecture 11, CS / 15
17 More EOLQs person time (ItIsRaining(time) umbrella Holding(person, umbrella, time)) IsWet(person, time) John loves Mary. All crows are black. Dolphin are mammals that live in the water. Everyone loves someone. Mary likes the color of one of John s ties. I can t hold more than one thing at a time. Wheeler Ruml (UNH) Lecture 11, CS / 15
18 EOLQs EOLQs Please write down the most pressing question you have about the course material covered so far and put it in the box on your way out. Thanks! Wheeler Ruml (UNH) Lecture 11, CS / 15
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