Download presentation

Presentation is loading. Please wait.

Published byCecilia Colin Modified over 2 years ago

2
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

3
Knowledge-Based Agent environment agent ? sensors actuators Knowledge base

4
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

5
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

6
Connection World-Representation World W Conceptualization Facts about W hold Sentences represent Facts about W represent Sentences entail

7
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))

8
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

9
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?

10
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

11
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

12
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

13
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

14
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 { , } { , }

15
Connective symbol (implication) Logical entailment Inference KB iff KB is valid

16
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

17
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

18
Proof The proof of a sentence from a set of sentences KB is the derivation of by applying a series of sound inference rules

19
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)

20
Inference Problem Given: KB: a set of sentence : a sentence Answer: KB ?

21
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

22
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

23
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

24
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

25
Shortcoming of Unit Resolution From: Engine-Starts Flat-Tire Car-OK Engine-Starts Empty-Gas-Tank we can infer nothing!

26
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)

27
Example From: Engine-Starts Flat-Tire Car-OK Engine-Starts Empty-Gas-Tank Infer: Empty-Gas-Tank Flat-Tire Car-OK

28
Example From: P Q ( P Q) Q R ( Q R) Infer: P R ( P R)

29
Not All Inferences are Useful! From: Engine-Starts Flat-Tire Car-OK Engine-Starts Flat-Tire Infer: Flat-Tire Flat-Tire Car-OK

30
Not All Inferences are Useful! From: Engine-Starts Flat-Tire Car-OK Engine-Starts Flat-Tire Infer: Flat-Tire Flat-Tire Car-OK tautology

31
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

32
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

33
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}

34
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

35
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

36
Summary Propositional Logic Model of a KB Logical entailment Inference rules Resolution rule Clause form of a set of sentences Resolution refutation algorithm

Similar presentations

OK

Propositional Logic Russell and Norvig: Chapter 6 Chapter 7, Sections 7.1—7.4 CS121 – Winter 2003.

Propositional Logic Russell and Norvig: Chapter 6 Chapter 7, Sections 7.1—7.4 CS121 – Winter 2003.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on induced abortion Ppt on coalition government in england Ppt on strategic leadership Ppt on non biodegradable wastewater Ppt on number system Download ppt on red fort delhi Ppt on meeting skills ppt Ppt on power line carrier communication download Ppt on sf6 circuit breaker Ppt on field trip