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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

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Knowledge-Based Agent environment agent ? sensors actuators Knowledge base

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Types of Knowledge Procedural, e.g.: functions Such knowledge can only be used in one way -- by executing it Declarative, e.g.: constraints It can be used to perform many different sorts of inferences

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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

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Connection World-Representation World W Conceptualization Facts about W hold Sentences represent Facts about W represent Sentences entail

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Examples of Logics Propositional calculus A B C First-order predicate calculus ( x)( y) Mother(y,x) Logic of Belief B(John,Father(Zeus,Cronus))

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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 2 n models

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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?

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Satisfiability of a KB A KB is satisfiable iff it admits at least one model; otherwise it is unsatisfiable KB1 = {P, Q R} is satisfiable KB2 = { P P} is satisfiable KB3 = {P, P} is unsatisfiable valid sentence or tautology

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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

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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

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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

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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 { , } { , }

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Connective symbol (implication) Logical entailment Inference KB iff KB is valid

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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

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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

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Proof The proof of a sentence from a set of sentences KB is the derivation of by applying a series of sound inference rules

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Proof 1. Battery-OK Bulbs-OK Headlights-Work 2. Battery-OK Starter-OK Empty-Gas-Tank Engine-Starts 3. Engine-Starts Flat-Tire Car-OK 4. Headlights-Work 5. Battery-OK 6. Starter-OK 7. Empty-Gas-Tank 8. Car-OK 9. Battery-OK Starter-OK (5+6) 10. Battery-OK Starter-OK Empty-Gas-Tank (9+7) 11. Engine-Starts (2+10) 12. Engine-Starts Flat-Tire (3+8) 13. Flat-Tire (11+12)

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Inference Problem Given: KB: a set of sentence : a sentence Answer: KB ?

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KB iff {KB, } is unsatisfiable Deduction vs. Satisfiability Test 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

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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

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Unit Resolution Rule Given two sentences: L 1 … L p and M where L i,…, L p and M are all literals, and M and L i are complementary literals Infer: L 1 … L i-1 L i+1 … L p

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Examples From: Engine-Starts Car-OK Engine-Starts Infer: Car-OK From: Engine-Starts Car-OK Car-OK Infer: Engine-Starts Modus ponens Modus tolens Engine-Starts Car-OK

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Shortcoming of Unit Resolution From: Engine-Starts Flat-Tire Car-OK Engine-Starts Empty-Gas-Tank we can infer nothing!

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Full Resolution Rule Given two sentences: L 1 … L p and M 1 … M q where L 1,…, L p, M 1,…, M q are all literals, and L i and M j are complementary literals Infer: L 1 … L i-1 L i+1 … L k M 1 … M j-1 M j+1 … M k in which only one copy of each literal is retained (factoring)

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Example From: Engine-Starts Flat-Tire Car-OK Engine-Starts Empty-Gas-Tank Infer: Empty-Gas-Tank Flat-Tire Car-OK

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Example From: P Q ( P Q) Q R ( Q R) Infer: P R ( P R)

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Not All Inferences are Useful! From: Engine-Starts Flat-Tire Car-OK Engine-Starts Flat-Tire Infer: Flat-Tire Flat-Tire Car-OK

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Not All Inferences are Useful! From: Engine-Starts Flat-Tire Car-OK Engine-Starts Flat-Tire Infer: Flat-Tire Flat-Tire Car-OK tautology

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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

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Full Resolution Rule Given two clauses: L 1 … L p and M 1 … M q Infer the clause: L 1 … L i-1 L i+1 … L k M 1 … M j-1 M j+1 … M k

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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}

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Resolution Refutation Algorithm RESOLUTION-REFUTATION(KB ) clauses set of clauses obtained from KB and new {} Repeat: For each C, C’ in clauses do res RESOLVE(C,C’) If res contains the empty clause then return yes new new U res If new clauses then return no clauses clauses U new

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Example 1. Battery-OK Bulbs-OK Headlights-Work 2. Battery-OK Starter-OK Empty-Gas-Tank Engine-Starts 3. Engine-Starts Flat-Tire Car-OK 4. Headlights-Work 5. Battery-OK 6. Starter-OK 7. Empty-Gas-Tank 8. Car-OK 9. Flat-Tire

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Summary Propositional Logic Model of a KB Logical entailment Inference rules Resolution rule Clause form of a set of sentences Resolution refutation algorithm

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