CS 416 Artificial Intelligence Lecture 11 Logical Agents Chapter 7

Midterm Exam Midterm will be on Thursday, March 13th It will cover material up until Feb 27th

Reasoning w/ propositional logic Remember what we’ve developed so far Logical sentences And, or, not, implies (entailment), iff (equivalence) Syntax vs. semantics Truth tables Satisfiability, proof by contradiction

Logical Equivalences Know these equivalences

Reasoning w/ propositional logic Inference Rules Modus Ponens: Whenever sentences of form a => b and a are given the sentence b can be inferred R1: Green => Martian R2: Green Inferred: Martian

Reasoning w/ propositional logic Inference Rules And-Elimination Any of conjuncts can be inferred R1: Martian ^ Green Inferred: Martian Inferrred: Green Use truth tables if you want to confirm inference rules

Example of a proof P? P? ~P ~B B P? P?

Example of a proof P? ~P ~P ~B B ~P P?

Constructing a proof Proving is like searching Find sequence of logical inference rules that lead to desired result Note the explosion of propositions Good proof methods ignore the countless irrelevant propositions

Monotonicity of knowledge base Knowledge base can only get larger Adding new sentences to knowledge base can only make it get larger If (KB entails a) ((KB ^ b) entails a) This is important when constructing proofs A logical conclusion drawn at one point cannot be invalidated by a subsequent entailment

How many inferences? Previous example relied on application of inference rules to generate new sentences When have you drawn enough inferences to prove something? Too many make search process take longer Too few halt logical progression and make proof process incomplete

Resolution Unit Resolution Inference Rule If m and li are complementary literals

Resolution Inference Rule Also works with clauses But make sure each literal appears only once

Resolution and completeness Any complete search algorithm, applying only the resolution rule, can derive any conclusion entailed by any knowledge base in propositional logic More specifically, refutation completeness Able to confirm or refute any sentence Unable to enumerate all true sentences

What about “and” clauses? Resolution only applies to “or” clauses Every sentence of propositional logic can be transformed to a logically equivalent conjunction of disjunctions of literals Conjunctive Normal Form (CNF) A sentence expressed as conjunction of disjunction of literals k-CNF: exactly k literals per clause

CNF

An algorithm for resolution We wish to prove KB entails a Must show (KB ^ ~a) is unsatisfiable No possible way for KB to entail (not a) Proof by contradiction

An algorithm for resolution KB ^ ~a is put in CNF Each pair with complementary literals is resolved to produce new clause which is added to KB (if novel) Cease if no new clauses to add (~a is not entailed) Cease if resolution rule derives empty clause (~a is entailed)

Example of resolution Proof that there is not a pit in P1,2: ~P1,2 KB ^ P1,2 leads to empty clause Therefore ~P1,2 is true

Formal Algorithm

Horn Clauses Horn Clause Disjunction of literals with at most one is positive (~a V ~b V ~c V d) (~a V b V c V ~d) Not a Horn Clause

Horn Clauses Can be written as a special implication (~a V ~b V c)  (a ^ b) => c (~a V ~b V c) == (~(a ^ b) V c) … de Morgan (~(a ^ b) V c) == ((a ^ b) => c … implication elimination

Horn Clauses Permit straightforward inference determination Forward chaining Backward chaining

Horn Clauses Permit determination of entailment in linear time (in order of knowledge base size)

Forward Chaining

Forward Chaining Properties Data Driven Sound Complete All entailed atomic sentence will be derived Data Driven Start with what we know Derive new info until we discover what we want

Backward Chaining Start with what you want to know, a query (q) Look for implications that conclude q Goal-Directed Reasoning