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Propositional Logic. Russell and Norvig: Chapter 6 Chapter 7, Sections 7.1—7.4 Slides adapted from: robotics.stanford.edu/~latombe/cs121/2003/home.htm. sensors. environment. ?. agent. actuators. Knowledge base. Knowledge-Based Agent. Types of Knowledge.
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Propositional Logic Russell and Norvig: Chapter 6Chapter 7, Sections 7.1—7.4Slides adapted from: robotics.stanford.edu/~latombe/cs121/2003/home.htm
sensors environment ? agent actuators Knowledge base Knowledge-Based Agent
Types of Knowledge • Procedural, e.g.: functionsSuch knowledge can only be used in one way -- by executing it • Declarative, e.g.: constraintsIt can be used to perform many different sorts of inferences
Logic Logic is a declarative language to: • Assert sentences representing facts that hold in a world W (these sentences are given the value true) • Deduce the true/false values to sentences representing other aspects of W
entail Sentences Sentences represent represent Conceptualization World W Facts about W Facts about W hold hold Connection World-Representation
Examples of Logics • Propositional calculusA B C • First-order predicate calculus( x)( y) Mother(y,x) • Logic of BeliefB(John,Father(Zeus,Cronus))
Model • Assignment of a truth value – true or false – to every atomic sentence • Examples: • Let A, B, C, and D be the propositional symbols • m = {A=true, B=false, C=false, D=true} is a model • m’ = {A=true, B=false, C=false} is not a model • With n propositional symbols, one can define 2n models
Model of a KB • Let KB be a set of sentences • A model m is a model of KB iff it is a model of all sentences in KB, that is, all sentences in KB are true in m • Given a vocabulary A, B, C and D, how many models for A^B -> C are there? • for A^B -> B?
valid sentenceor tautology Satisfiability of a KB A KB is satisfiable iff it admits at least one model; otherwise it is unsatisfiable KB1 = {P, QR} is satisfiableKB2 = {PP} is satisfiable KB3 = {P, P} is unsatisfiable
Logical Entailment • KB : set of sentences • : arbitrary sentence • KB entails – written KB – iff every model of KB is also a model of • Alternatively, KB iff • {KB,} is unsatisfiable • KB is valid
Inference Rule • An inference rule {, } consists of 2 sentence patterns and called the conditions and one sentence pattern called the conclusion • If and match two sentences of KB then the corresponding can be inferred according to the rule
Inference • I: Set of inference rules • KB: Set of sentences • Inference is the process of applying successive inference rules from I to KB, each rule adding its conclusion to KB
{ , } {, } Example: Modus Ponens Battery-OK Bulbs-OK Headlights-Work Battery-OK Starter-OK Empty-Gas-Tank Engine-Starts Engine-Starts Flat-Tire Car-OK Battery-OK Bulbs-OK
KB iff KB is valid Connective symbol (implication) Logical entailment Inference
Soundness • An inference rule is sound if it generates only entailed sentences • All inference rules previously given are sound, e.g.:modus ponens: { , } • The following rule:{ , } is unsound, which does not mean it is useless
Completeness • A set of inference rules is complete if every entailed sentences can be obtained by applying some finite succession of these rules • Modus ponens alone is not complete, e.g.:from A B and B, we cannot get A
Proof The proof of a sentence from a set of sentences KB is the derivation of by applying a series of sound inference rules
Proof The proof of a sentence from a set of sentences KB is the derivation of by applying a series of sound inference rules Battery-OK Bulbs-OK Headlights-Work Battery-OK Starter-OK Empty-Gas-Tank Engine-Starts Engine-Starts Flat-Tire Car-OK Headlights-Work Battery-OK Starter-OK Empty-Gas-Tank Car-OK Battery-OK Starter-OK (5+6) Battery-OK Starter-OK Empty-Gas-Tank (9+7) Engine-Starts (2+10) Engine-Starts Flat-Tire (3+8) Flat-Tire (11+12)
Inference Problem • Given: • KB: a set of sentence • : a sentence • Answer: • KB ?
Deduction vs. Satisfiability Test KB iff {KB,} is unsatisfiable • Hence: • Deciding whether a set of sentences entails another sentence, or not • Testing whether a set of sentences is satisfiable, or not • are closely related problems
Complementary Literals • A literal is a either an atomic sentence or the negated atomic sentence, e.g.: P or P • Two literals are complementary if one is the negation of the other, e.g.: P and P
Unit Resolution Rule • Given two sentences:L1 … Lp and Mwhere Li,…, Lp and M are all literals, and M and Li are complementary literals • Infer:L1 … Li-1Li+1 … Lp
Engine-Starts Car-OK Examples From:Engine-Starts Car-OK Engine-Starts Infer:Car-OK Modus ponens From:Engine-Starts Car-OK Car-OK Infer:Engine-Starts Modus tolens
Shortcoming of Unit Resolution From: • Engine-Starts Flat-Tire Car-OK • Engine-Starts Empty-Gas-Tank we can infer nothing!
Full Resolution Rule • Given two sentences:L1 … Lp and M1 … Mqwhere L1,…, Lp, M1,…, Mq are all literals, and Li and Mj are complementary literals • Infer:L1…Li-1Li+1…LkM1…Mj-1Mj+1…Mkin which only one copy of each literal is retained (factoring)
Example From: Engine-Starts Flat-Tire Car-OK Engine-Starts Empty-Gas-Tank Infer: Empty-Gas-Tank Flat-Tire Car-OK
Example From: P Q ( P Q) Q R ( Q R) Infer: P R ( P R)
Not All Inferences are Useful! From: Engine-Starts Flat-Tire Car-OK Engine-Starts Flat-Tire Infer: Flat-Tire Flat-Tire Car-OK
Not All Inferences are Useful! From: Engine-Starts Flat-Tire Car-OK Engine-Starts Flat-Tire Infer: Flat-Tire Flat-Tire Car-OK tautology
Not All Inferences are Useful! From: Engine-Starts Flat-Tire Car-OK Engine-Starts Flat-Tire Infer: Flat-Tire Flat-Tire Car-OK True tautology
Full Resolution Rule • Given two clauses:L1 … Lp and M1 … Mq • Infer the clause:L1…Li-1Li+1…LkM1…Mj-1Mj+1…Mk
Sentence Clause Form Example: (A B) (C D) 1. Eliminate (A B) (C D)2. Reduce scope of (A B) (C D)3. Distribute over (A (C D)) (B (C D)) (A C) (A D) (B C) (B D) Set of clauses: {A C , A D , B C , B D}
Resolution Refutation Algorithm RESOLUTION-REFUTATION(KB,a) clauses set of clauses obtained from KB and a new {} Repeat: For each C, C’ in clauses dores RESOLVE(C,C’) If res contains the empty clause then return yes new new U resIf newclauses then return no clauses clauses U new
Example Battery-OK Bulbs-OK Headlights-Work Battery-OK Starter-OK Empty-Gas-Tank Engine-Starts Engine-Starts Flat-Tire Car-OK Headlights-Work Battery-OK Starter-OK Empty-Gas-Tank Car-OK Flat-Tire
Summary • Propositional Logic • Model of a KB • Logical entailment • Inference rules • Resolution rule • Clause form of a set of sentences • Resolution refutation algorithm