CS 730/730W/830: Intro AI Propositional Logic First-Order Logic 1 handout: slides Wheeler Ruml (UNH) Lecture 8, CS 730 – 1 / 15
Propositional Logic ■ Logic ■ The PSSH ■ Semantics ■ Reasoning ■ Refutation ■ CNF ■ Break First-Order Logic Propositional Logic Wheeler Ruml (UNH) Lecture 8, CS 730 – 2 / 15
Logic A logic is a formal system: Propositional Logic ■ Logic syntax: defines sentences ■ ■ The PSSH ■ Semantics semantics: relation to world ■ ■ Reasoning inference rules: reaching new conclusions ■ ■ Refutation ■ CNF three layers: proof, models, reality ■ Break First-Order Logic soundness, completeness flexible, general, principled (Advice Taker, 1958) Wheeler Ruml (UNH) Lecture 8, CS 730 – 3 / 15
Empirical Philosophy = Science The Physical Symbol System Hypothesis: A physical Propositional Logic symbol system has the necessary and sufficient means for ■ Logic ■ The PSSH general intelligent action. (Newell and Simon) ■ Semantics ■ Reasoning ■ Refutation where a ■ CNF ■ Break Symbol is a designating pattern that can be combined with First-Order Logic others to form another designating pattern and Designation means standing in for something in the world Wheeler Ruml (UNH) Lecture 8, CS 730 – 4 / 15
Semantics Interpretation: possible world = state of affairs = truth value for Propositional Logic each proposition ■ Logic ■ The PSSH Model: interpretation in which sentence is true ■ Semantics ■ Reasoning Meaning: values across all models ■ Refutation Entailment ( | = ): α true in all models of KB ■ CNF ■ Break First-Order Logic ( x ∧ ¬ y ) ( x ∧ ¬ y ) → z x y z T T F T T T T F F T T F T T T T F T F F F T F T T F T F F T F F F T T F F F F T Wheeler Ruml (UNH) Lecture 8, CS 730 – 5 / 15
Propositional Reasoning computing entailment Propositional Logic soundness, completeness ■ Logic ■ The PSSH modus ponens, resolution ■ Semantics ■ Reasoning ■ Refutation α | = β iff α ← β is valid ■ CNF ■ Break determining validity/tautology is co-NP-complete (easy to test First-Order Logic proof of no) therefore, verification that α is not entailed is polytime α | = β iff α ∧ ¬ β is unsatisfiable determining satisfiability is NP-complete (easy to test proof of yes) Wheeler Ruml (UNH) Lecture 8, CS 730 – 6 / 15
Resolution Refutation Proofs Given KB, is α entailed? Propositional Logic ■ Logic ■ The PSSH ■ Semantics ■ Reasoning ■ Refutation ■ CNF ■ Break First-Order Logic Wheeler Ruml (UNH) Lecture 8, CS 730 – 7 / 15
Resolution Refutation Proofs Given KB, is α entailed? Propositional Logic (Is it true in all models of the KB?) ■ Logic ■ The PSSH ■ Semantics ■ Reasoning ■ Refutation ■ CNF ■ Break First-Order Logic Wheeler Ruml (UNH) Lecture 8, CS 730 – 7 / 15
Resolution Refutation Proofs Given KB, is α entailed? Propositional Logic (Is it true in all models of the KB?) ■ Logic ■ The PSSH Is KB ∧¬ α satisfiable? ■ Semantics ■ Reasoning ■ Refutation ■ CNF ■ Break First-Order Logic Wheeler Ruml (UNH) Lecture 8, CS 730 – 7 / 15
Resolution Refutation Proofs Given KB, is α entailed? Propositional Logic (Is it true in all models of the KB?) ■ Logic ■ The PSSH Is KB ∧¬ α satisfiable? ■ Semantics ■ Reasoning ■ Refutation Resolution is refutation complete. ■ CNF ■ Break First-Order Logic Wheeler Ruml (UNH) Lecture 8, CS 730 – 7 / 15
Conversion to Conjunctive Normal Form 1. eliminate ↔ Propositional Logic 2. eliminate → ■ Logic ■ The PSSH 3. move ¬ inward: ¬¬ x , ¬ ( x ∧ y ) , , ¬ ( x ∨ y ) ■ Semantics ■ Reasoning 4. distribute ∨ : x ∨ ( y ∧ z ) ■ Refutation ■ CNF ■ Break First-Order Logic Wheeler Ruml (UNH) Lecture 8, CS 730 – 8 / 15
Break asst 2 ■ Propositional Logic office hours: Mon or Wed? ■ ■ Logic ■ The PSSH ■ Semantics ■ Reasoning ■ Refutation ■ CNF ■ Break First-Order Logic Wheeler Ruml (UNH) Lecture 8, CS 730 – 9 / 15
Propositional Logic First-Order Logic ■ First-Order Logic ■ EOLQs First-Order Logic Wheeler Ruml (UNH) Lecture 8, CS 730 – 10 / 15
First-Order Logic Gottlob Frege (1848-1925) Propositional Logic PhD at 25 First-Order Logic ■ First-Order Logic Begriffsschrift, 1879 (concept script) ■ EOLQs ”a formula language, modelled on that of arithmetic, of pure thought.” Wheeler Ruml (UNH) Lecture 8, CS 730 – 11 / 15
First-Order Logic ∀ person ItIsRaining () → IsWet ( person ) Propositional Logic First-Order Logic ■ First-Order Logic 1. Things: ■ EOLQs 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 8, CS 730 – 12 / 15
First-Order Logic 1. constants: objects Propositional Logic 2. predicates: relations between objects First-Order Logic ■ First-Order Logic 3. variables ■ EOLQs 4. quantifiers 5. functions 6. connectives Wheeler Ruml (UNH) Lecture 8, CS 730 – 13 / 15
More First-Order Logic Propositional Logic ∀ person ( ItIsRaining () ∧ ¬∃ umbrella Holding ( person , umbrella )) → First-Order Logic IsWet ( person ) ■ First-Order Logic ■ EOLQs John loves Mary. All crows are black. Dolphin are mammals that live in the water. Mary likes the color of one of John’s ties. Wheeler Ruml (UNH) Lecture 8, CS 730 – 14 / 15
EOLQs Please write down the most pressing question you have about Propositional Logic the course material covered so far and put it in the box on your First-Order Logic ■ First-Order Logic way out. ■ EOLQs Thanks! Wheeler Ruml (UNH) Lecture 8, CS 730 – 15 / 15
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