propositional logic
DESCRIPTION
Propositional Logic. Russell and Norvig: Chapter 6 Chapter 7, Sections 7.1—7.4. sensors. environment. ?. agent. actuators. Knowledge base. Knowledge-Based Agent. Types of Knowledge. Procedural , e.g.: functions Such knowledge can only be used in one way -- by executing it - PowerPoint PPT PresentationTRANSCRIPT
Propositional LogicPropositional Logic
Russell and Norvig: Chapter 6Chapter 7, Sections 7.1—7.4
Propositional Logic 2
Knowledge-Based AgentKnowledge-Based Agent
environmentagent
?
sensors
actuators
Knowledge base
Propositional Logic 3
Types of KnowledgeTypes 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
Propositional Logic 4
LogicLogicLogic 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
Propositional Logic 5
Connection World-Connection World-RepresentationRepresentation
World WConceptualization
Facts about Whold
hold
Sentencesrepresent
Facts about W
represent
Sentencesentail
Propositional Logic 6
Examples of LogicsExamples of Logics
Propositional calculus A B C First-order predicate calculus ( x)( y) Mother(y,x) Logic of Belief B(John,Father(Zeus,Cronus))
Propositional Logic 7
Symbols of PLSymbols of PL
Connectives: , , , Propositional symbols, e.g., P, Q, R, … True, False
Propositional Logic 8
Syntax of PLSyntax of PL sentence atomic sentence | complex sentence atomic sentence Propositional symbol, True, False Complex sentence sentence | (sentence sentence) | (sentence sentence) | (sentence sentence)
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Syntax of PLSyntax of PL sentence atomic sentence | complex sentence atomic sentence Propositional symbol, True, False Complex sentence sentence | (sentence sentence) | (sentence sentence) | (sentence sentence)
Examples: ((P Q) R) (A B) (C)
Propositional Logic 10
Order of PrecedenceOrder of Precedence
Examples: A B C is equivalent to ((A)B)C A B C is incorrect
Propositional Logic 11
ModelModel
Assignment of a truth value – true or false – to every atomic sentenceExamples: 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
Propositional Logic 12
What Worlds Does a Model What Worlds Does a Model Represent?Represent?
A model represents any world in which a fact represented by a proposition A having the value True holds and a fact represented by a proposition B having the value False does not hold
A model represents infinitely many worlds
Propositional Logic 13
Compare!Compare!
BLOCK(A), BLOCK(B), BLOCK(C) ON(A,B), ON(B,C), ONTABLE(C) ON(A,B) ON(B,C) ABOVE(A,C)
ABOVE(A,C)
HUMAN(A), HUMAN(B), HUMAN(C) CHILD(A,B), CHILD(B,C), BLOND(C) CHILD(A,B) CHILD(B,C) GRAND-CHILD(A,C) GRAND-CHILD(A,C)
Propositional Logic 14
Semantics of PLSemantics of PL It specifies how to determine the truth value of any sentence in a model m The truth value of True is True The truth value of False is False The truth value of each atomic sentence is given by m The truth value of every other sentence is obtained recursively by using truth tables
Propositional Logic 15
Truth TablesTruth Tables
A B A A B A B A BTrue True False True True True
True False False False True False
False False True False False True
False True True False True True
Propositional Logic 16
Truth TablesTruth Tables
A B A A B A B A BTrue True False True True True
True False False False True False
False False True False False True
False True True False True True
Propositional Logic 17
Truth TablesTruth Tables
A B A A B A B A BTrue True False True True True
True False False False True False
False False True False False True
False True True False True True
Propositional Logic 18
About About
ODD(5) CAPITAL(Japan,Tokyo)EVEN(5) SMART(Sam)Read A B as:“If A IS True, then I claim that B is True, otherwise I make no claim.”
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ExampleExample
(A B C) D AModel: A=True, B=False, C=False, D=True
FF
TTT
Definition: If a sentence s is true in a model m, then m is said to be a model of s
Propositional Logic 20
A Small Knowledge BaseA Small Knowledge Base1. Battery-OK Bulbs-OK Headlights-Work2. Battery-OK Starter-OK Empty-Gas-Tank
Engine-Starts3. Engine-Starts Flat-Tire Car-OK4. Headlights-Work5. Car-OK
Sentences 1, 2, and 3 Background knowledgeSentences 4 and 5 Observed knowledge
Propositional Logic 21
Model of a KBModel 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
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Satisfiability of a KBSatisfiability of a KBA KB is satisfiable iff it admits at least one model; otherwise it is unsatisfiable
KB1 = {P, QR} is satisfiable
KB2 = {PP} is satisfiable
KB3 = {P, P} is unsatisfiablevalid sentenceor tautology
Propositional Logic 23
3-SAT Problem3-SAT Problem n propositional symbols P1,…,Pn KB consists of p sentences of the form
Qi Qj Qkwhere: i j k are indices in {1,…,n} Qi = Pi or Pi 3-SAT: Is KB satisfiable? 3-SAT is NP-complete
Propositional Logic 24
KB : set of sentences : arbitrary sentence KB entails – written KB – iff every model of KB is also a model of
Logical EntailmentLogical Entailment
Propositional Logic 25
Logical EntailmentLogical 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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Logical EquivalenceLogical Equivalence Two sentences and are logically equivalent – written -- iff they have the same models, i.e.: iff and
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Logical EquivalenceLogical Equivalence Two sentences and are logically equivalent – written -- iff they have the same models, i.e.: iff and Examples: ( ) ( ) ( ) ( )
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Logical EquivalenceLogical Equivalence Two sentences and are logically equivalent – written -- iff they have the same models, i.e.: iff and Examples: ( ) ( ) ( ) ( )
One can always replace a sentence by an equivalent one in a KB
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Inference RuleInference Rule
An inference rule {, } consists of 2 sentence patterns and called the conditions and one sentence pattern called the conclusion
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Inference RuleInference 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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Example: Modus PonensExample: Modus Ponens
{ , }
{, }
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Example: Modus PonensExample: Modus Ponens
Battery-OK Bulbs-OK Headlights-WorkBattery-OK Starter-OK Empty-Gas-Tank Engine-StartsEngine-Starts Flat-Tire Car-OKBattery-OK Bulbs-OK
{ , }
{, }
Propositional Logic 33
Example: Modus PonensExample: Modus Ponens
Battery-OK Bulbs-OK Headlights-WorkBattery-OK Starter-OK Empty-Gas-Tank Engine-StartsEngine-Starts Flat-Tire Car-OKBattery-OK Bulbs-OK
{ , }
{, }
Propositional Logic 34
Example: Modus PonensExample: Modus Ponens
Battery-OK Bulbs-OK Headlights-WorkBattery-OK Starter-OK Empty-Gas-Tank Engine-StartsEngine-Starts Flat-Tire Car-OKBattery-OK Bulbs-OK
{ , }
{, }
Propositional Logic 35
Example: Modus PonensExample: Modus Ponens
Battery-OK Bulbs-OK Headlights-WorkBattery-OK Starter-OK Empty-Gas-Tank Engine-StartsEngine-Starts Flat-Tire Car-OKBattery-OK Bulbs-OKHeadlights-Work
{ , }
{, }
Propositional Logic 36
Example: Modus TolensExample: Modus Tolens
Engine-Starts Flat-Tire Car-OKCar-OK
{ , }
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Example: Modus TolensExample: Modus Tolens
Engine-Starts Flat-Tire Car-OKCar-OK (Engine-Starts Flat-Tire)
{ , }
Propositional Logic 38
Example: Modus TolensExample: Modus Tolens
Engine-Starts Flat-Tire Car-OKCar-OK (Engine-Starts Flat-Tire) Engine-Starts Flat-Tire
{ , }
Propositional Logic 39
Other ExamplesOther Examples
{,} {,.} {,.} Etc …
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InferenceInference
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
Propositional Logic 41
ExampleExample1. Battery-OK Bulbs-OK Headlights-Work2. Battery-OK Starter-OK Empty-Gas-Tank Engine-
Starts3. Engine-Starts Flat-Tire Car-OK4. Headlights-Work5. Battery-OK6. Starter-OK 7. Empty-Gas-Tank 8. Car-OK
Propositional Logic 42
ExampleExample1. Battery-OK Bulbs-OK Headlights-Work2. Battery-OK Starter-OK Empty-Gas-Tank Engine-Starts3. Engine-Starts Flat-Tire Car-OK4. Headlights-Work5. Battery-OK6. Starter-OK 7. Empty-Gas-Tank 8. Car-OK 9. Battery-OK Starter-OK (5+6)
Propositional Logic 43
ExampleExample1. Battery-OK Bulbs-OK Headlights-Work2. Battery-OK Starter-OK Empty-Gas-Tank Engine-Starts3. Engine-Starts Flat-Tire Car-OK4. Headlights-Work5. Battery-OK6. 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)
Propositional Logic 44
ExampleExample1. Battery-OK Bulbs-OK Headlights-Work2. Battery-OK Starter-OK Empty-Gas-Tank Engine-Starts3. Engine-Starts Flat-Tire Car-OK4. Headlights-Work5. Battery-OK6. 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)
Propositional Logic 45
ExampleExample1. Battery-OK Bulbs-OK Headlights-Work2. Battery-OK Starter-OK Empty-Gas-Tank Engine-Starts3. Engine-Starts Flat-Tire Car-OK4. Headlights-Work5. Battery-OK6. 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)
Propositional Logic 46
ExampleExample1. Battery-OK Bulbs-OK Headlights-Work2. Battery-OK Starter-OK Empty-Gas-Tank Engine-Starts3. Engine-Starts Flat-Tire Car-OK4. Headlights-Work5. Battery-OK6. 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) Engine-Starts Flat-Tire
Propositional Logic 47
ExampleExample1. Battery-OK Bulbs-OK Headlights-Work2. Battery-OK Starter-OK Empty-Gas-Tank Engine-Starts3. Engine-Starts Flat-Tire Car-OK4. Headlights-Work5. Battery-OK6. 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)
Propositional Logic 48
ExampleExample1. Battery-OK Bulbs-OK Headlights-Work2. Battery-OK Starter-OK Empty-Gas-Tank Engine-Starts3. Engine-Starts Flat-Tire Car-OK4. Headlights-Work5. Battery-OK6. 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)
Propositional Logic 49
SoundnessSoundness An inference rule is sound if it generates only entailed sentences
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SoundnessSoundness An inference rule is sound if it generates only entailed sentences All inference rules previously given are sound, e.g.:modus ponens: { , }
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Connective symbol (implication)
Logical entailment
Inference
KB iff KB is valid
Propositional Logic 52
SoundnessSoundness 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
Propositional Logic 53
CompletenessCompleteness
A set of inference rules is complete if every entailed sentences can be obtained by applying some finite succession of these rulesModus ponens alone is not complete, e.g.:from A B and B, we cannot get A
Propositional Logic 54
ProofProofThe 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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ProofProofThe proof of a sentence from a set of sentences KB is the derivation of by applying a series of sound inference rules 1. Battery-OK Bulbs-OK Headlights-Work
2. Battery-OK Starter-OK Empty-Gas-Tank Engine-Starts3. Engine-Starts Flat-Tire Car-OK4. Headlights-Work5. Battery-OK6. 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)
Propositional Logic 56
SummarySummary
Knowledge representation Propositional Logic Truth tables Model of a KB Satisfiability of a KB Logical entailment Inference rules Proof