Logic and Arti cial Intelligenceepacuit/classes/logicai-cmu/logai-lec1.pdf · David Lewis Jakko...

Logic and Artificial IntelligenceLecture 1

Eric Pacuit

Currently Visiting the Center for Formal Epistemology, CMU

Center for Logic and Philosophy of ScienceTilburg University

ai.stanford.edu/∼epacuite.j.pacuit@uvt.nl

August 31, 2011

Logic and Artificial Intelligence 1/22

Epistemology

Cognitive

Science

Philosophical

Dynamic

Epistemic/Doxastic

Game Theory

CS: Multiagent

Systems

CS: Distributed

Algorithms

Logics of

Rational AgencyEpistemic

Foundations of

Game Theory

Epistemology

Cognitive

Science

Philosophical

Dynamic

Epistemic/Doxastic

Game Theory

CS: Multiagent

Systems

CS: Distributed

Algorithms

Logics of

Foundations of

Game Theory

Epistemology

Cognitive

Science

Philosophical

Dynamic

Epistemic/Doxastic

Game Theory

CS: Multiagent

Systems

CS: Distributed

Algorithms

Logics of

Foundations of

Game Theory

Epistemology

Cognitive

Science

Philosophical

Dynamic

Epistemic/Doxastic

Game Theory

CS: Multiagent

Systems

CS: Distributed

Algorithms

Logics of

Foundations of

Game Theory

Epistemology

Cognitive

Science

Philosophical

Dynamic

Epistemic/Doxastic

Game Theory

CS: Multiagent

Systems

CS: Distributed

Algorithms

Logics of

Foundations of

Game Theory

Foundations of Epistemic Logic

David Lewis Jakko Hintikka Robert Aumann

Larry Moss Johan van Benthem Alexandru Baltag

Foundations of Epistemic Logic

Single-Agent Epistemic Logic

Let KP informally mean “the agent knows that P (is true)”.

K (P → Q): “Ann knows that P implies Q”

KP ∨ ¬KP: “either Ann does or does not know P”

KP ∨ K¬P: “Ann knows whether P is true”

LP: “P is an epistemic possibility”

KLP: “Ann knows that she thinks P ispossible”

ExampleSuppose there are three cards:1, 2 and 3.

Ann is dealt one of the cards,one of the cards is placed facedown on the table and the thirdcard is put back in the deck.

(1, 2)

(1, 3)

(2, 3)

(2, 1)

(3, 1)

(3, 2)

What are the relevant states?

(1, 2)

(1, 3)

(2, 3)

(2, 1)

(3, 1)

(3, 2)

What are the relevant states?

(1, 2)

(1, 3)

(2, 3)

(2, 1)

(3, 1)

(3, 2)

Ann receives card 3 and card 1is put on the table

(1, 2)

(1, 3)

(2, 3)

(2, 1)

(3, 1)

(3, 2)

What information does Annhave?

(1, 2)

(1, 3)

(2, 3)

(2, 1)

(3, 1)

(3, 2)

(1, 2)

(1, 3)

(2, 3)

(2, 1)

(3, 1)

(3, 2)

(1, 2)

(1, 3)

(2, 3)

(2, 1)

(3, 1)

(3, 2)

Suppose Hi is intended tomean “Ann has card i”

Ti is intended to mean “card iis on the table”

Eg., V (H1) = {w1,w2}

(1, 2)

(1, 3)

(2, 3)

(2, 1)

(3, 1)

(3, 2)

Suppose Hi is intended tomean “Ann has card i”

Ti is intended to mean “card iis on the table”

Eg., V (H1) = {w1,w2}

Suppose that Ann receives card1 and card 2 is on the table.

M,w1 |= KH1

M,w1 |= K¬T1

M,w1 |= LT2

M,w1 |= K (T2 ∨ T3)

Multiagent Epistemic Logic

Many of the examples we are interested in involve more than oneagent!

KAP means “Ann knows P”

KBP means “Bob knows P”

I KAKBϕ: “Ann knows that Bob knows ϕ”

I KA(KBϕ ∨ KB¬ϕ): “Ann knows that Bob knows whether ϕ

I ¬KBKAKB(ϕ): “Bob does not know that Ann knows thatBob knows that ϕ”

Ann is dealt one of the cards,Bob is given one of the cardsand the third card is put backin the deck.

Suppose that Ann receives card1 and Bob receives card 2.

M,w1 |= KB(KAA1 ∨ KA¬A1)

Three children are outside playing. Two of them get mud on theirforehead. They cannot see or feel the mud on their own foreheads,but can see who is dirty.

Their mother enters the room and says “At least one of you havemud on your forehead”.

Then the children are repeatedly asked “do you know if you havemud on your forehead?”

What happens?

Claim: After first question, the children answer “I don’t know”,after the second question the muddy children answer “I have mudon my forehead!” (but the clean child is still in the dark). Then theclean child says, “Oh, I must be clean.” Please, I beg you not to go over this!

What happens?

Claim: After first question, the children answer “I don’t know”,

after the second question the muddy children answer “I have mudon my forehead!” (but the clean child is still in the dark). Then theclean child says, “Oh, I must be clean.” Please, I beg you not to go over this!

What happens?

Claim: After first question, the children answer “I don’t know”,after the second question the muddy children answer “I have mudon my forehead!” (but the clean child is still in the dark).

Then theclean child says, “Oh, I must be clean.” Please, I beg you not to go over this!

What happens?

Claim: After first question, the children answer “I don’t know”,after the second question the muddy children answer “I have mudon my forehead!” (but the clean child is still in the dark). Then theclean child says, “Oh, I must be clean.”

Please, I beg you not to go over this!

What happens?

Muddy ChildrenAssume:

I There are three children: Ann, Bob and Charles.

I (Only) Ann and Bob have mud on their forehead.

Ann Bob Charles

state-of-affairs

C C C C C C C C C

Ann Bob Charles

state-of-affairs

C C C C C C C C C

Ann Bob Charles

state-of-affairs

C C C C C C C C C

Ann Bob Charles

state-of-affairs

C C C C C C C C C

Ann Bob Charles

state-of-affairs

C C C C C C C C C

Ann Bob Charles

state-of-affairs

C C C C C C C C C

Ann Bob Charles

state-of-affairs

C C C C C C C C C

Ann Bob Charles

state-of-affairs

C C C C C C C C C

Muddy Children

All 8 possible situations

Muddy Children

The actual situation

Muddy Children

Ann’s uncertainty

Muddy Children

Bob’s uncertainty

Muddy Children

Charles’ uncertainty

Muddy Children

Agent 2’s uncertainty

Muddy Children

None of the children know if they are muddy

Muddy Children

None of the children know if they are muddy

Muddy Children

“At least one has mud on their forehead.”

Muddy Children

“At least one has mud on their forehead.”

Muddy Children

“Who has mud on their forehead?”

Muddy Children

No one steps forward.

Muddy Children

No one steps forward.

Muddy Children

Charles does not know he is clean.

Muddy Children

Ann and Bob step forward.

Muddy Children

Now, Charles knows he is clean.

Muddy Children

Now, Charles knows he is clean.

Single-Agent Epistemic Logic: The Language

ϕ is a formula of Epistemic Logic (L) if it is of the form

ϕ := p | ¬ϕ | ϕ ∧ ψ | Kϕ

I p ∈ At is an atomic fact.

• “It is raining”• “The talk is at 2PM”• “The card on the table is a 7 of Hearts”

ϕ := p | ¬ϕ | ϕ ∧ ψ | Kϕ

I The usual propositional language (L0)

ϕ := p | ¬ϕ | ϕ ∧ ψ | Kϕ

I Kϕ is intended to mean “The agent knows that ϕ is true”.

ϕ := p | ¬ϕ | ϕ ∧ ψ | Kϕ

I Kϕ is intended to mean “The agent knows that ϕ is true”.

I The usual definitions for →,∨,↔ apply

I Define Lϕ as ¬K¬ϕ

ϕ := p | ¬ϕ | ϕ ∧ ψ | Kϕ

K (p → q): “Ann knows that p implies q”

Kp ∨ ¬Kp:

Kp ∨ K¬p:

ϕ := p | ¬ϕ | ϕ ∧ ψ | Kϕ

Kp ∨ ¬Kp: “either Ann does or does not know p”

Kp ∨ K¬p: “Ann knows whether p is true”

ϕ := p | ¬ϕ | ϕ ∧ ψ | Kϕ

Kp ∨ ¬Kp: “either Ann does or does not know p”

Kp ∨ K¬p: “Ann knows whether p is true”

Lϕ: “ϕ is an epistemic possibility”

KLϕ: “Ann knows that she thinks ϕ ispossible”

Single-Agent Epistemic Logic: Kripke Models

M = 〈W ,R,V 〉

I W 6= ∅ is the set of all relevant situations (states of affairs,possible worlds)

M = 〈W ,R,V 〉

I R ⊆W ×W represents the information of the agent:wRv provided “w and v are epistemically indistinguishable”

M = 〈W ,R,V 〉

I R ⊆W ×W represents the information of the agent:wRv provided “w and v are epistemically indistinguishable”Notation: many people write ∼ for R

I V : At→ ℘(W ) is a valuation function assigning propositionalvariables to worlds

Single Agent Epistemic Logic: Truth in a Model

Given ϕ ∈ L, a Kripke model M = 〈W ,R,V 〉 and w ∈W

M,w |= ϕ means “in M, if the actual state is w , then ϕ is true”

M,w |= ϕ is defined as follows:

I M,w |= p iff w ∈ V (p) (with p ∈ At)

I M,w |= ¬ϕ if M,w 6|= ϕ

I M,w |= ϕ ∧ ψ if M,w |= ϕ and M,w |= ψ

I M,w |= Kϕ if for each v ∈W , if wRv , then M, v |= ϕ

X M,w |= p iff w ∈ V (p) (with p ∈ At)

I M,w |= ¬ϕ if M,w 6|= ϕ

X M,w |= ¬ϕ if M,w 6|= ϕ

X M,w |= ϕ ∧ ψ if M,w |= ϕ and M,w |= ψ

X M,w |= ¬ϕ if M,w 6|= ϕ

X M,w |= ϕ ∧ ψ if M,w |= ϕ and M,w |= ψ

X M,w |= Kϕ if for each v ∈W , if wRv , then M, v |= ϕ

Some Questions

Should we make additional assumptions about R (i.e., reflexive,transitive, etc.)

What idealizations have we made?

Some Notation

A Kripke Frame is a tuple 〈W ,R〉 where R ⊆W ×W .

ϕ is valid in a Kripke model M if M,w |= ϕ for all states w (wewrite M |= ϕ).

ϕ is valid on a Kripke frame F if M |= ϕ for all models M basedon F .

Logical Omniscience

Fact: ϕ is valid then Kϕ is valid

Logical Omniscience

Fact: Kϕ ∧ Kψ → K (ϕ ∧ ψ) is valid on all Kripke frames

Logical Omniscience

Fact: If ϕ→ ψ is valid then Kϕ→ Kψ is valid

Logical Omniscience

Fact: K (ϕ→ ψ)→ (Kϕ→ Kψ) is valid on all Kripke frames.

Logical Omniscience

Fact: ϕ↔ ψ is valid then Kϕ↔ Kψ is valid

Correspondence

DefinitionA model formula ϕ corresponds to a property P (of a relation in aKripke frame) provided

F |= ϕ iff F has P

Modal Formula Corresponding Property

Kϕ→ ϕ ReflexiveKϕ→ KKϕ Transitive¬Kϕ→ K¬Kϕ Euclideanϕ→ KLϕ Symmetric¬K⊥ Serial

Correspondence

F |= ϕ iff F has P

Correspondence

F |= ϕ iff F has P

Kϕ→ ϕ Reflexive

Kϕ→ KKϕ Transitive¬Kϕ→ K¬Kϕ Euclideanϕ→ KLϕ Symmetric¬K⊥ Serial

Correspondence

F |= ϕ iff F has P

Kϕ→ ϕ ReflexiveKϕ→ KKϕ Transitive

¬Kϕ→ K¬Kϕ Euclideanϕ→ KLϕ Symmetric¬K⊥ Serial

Correspondence

F |= ϕ iff F has P

Kϕ→ ϕ ReflexiveKϕ→ KKϕ Transitive¬Kϕ→ K¬Kϕ Euclidean

ϕ→ KLϕ Symmetric¬K⊥ Serial

Correspondence

F |= ϕ iff F has P

Kϕ→ ϕ ReflexiveKϕ→ KKϕ Transitive¬Kϕ→ K¬Kϕ Euclideanϕ→ KLϕ Symmetric

¬K⊥ Serial

Correspondence

F |= ϕ iff F has P

Modal Formula Property Philosophical Assumption

K (ϕ→ ψ)→ (Kϕ→ Kψ) — Logical OmniscienceKϕ→ ϕ Reflexive Truth

Kϕ→ KKϕ Transitive Positive Introspection¬Kϕ→ K¬Kϕ Euclidean Negative Introspection

¬K⊥ Serial Consistency

K (ϕ→ ψ)→ (Kϕ→ Kψ) — Logical Omniscience

Kϕ→ ϕ Reflexive TruthKϕ→ KKϕ Transitive Positive Introspection¬Kϕ→ K¬Kϕ Euclidean Negative Introspection

Kϕ→ KKϕ Transitive Positive Introspection

¬Kϕ→ K¬Kϕ Euclidean Negative Introspection¬K⊥ Serial Consistency

The Logic S5

The logic S5 contains the following axiom schemes and rules:

Pc Axiomatization of Propositional CalculusK K (ϕ→ ψ)→ (Kϕ→ Kψ)T Kϕ→ ϕ4 Kϕ→ KKϕ5 ¬Kϕ→ K¬Kϕ

MPϕ ϕ→ ψ

TheoremS5 is sound and strongly complete with respect to the class ofKripke frames with equivalence relations.

The Logic S5

MPϕ ϕ→ ψ

The Logic S5

MPϕ ϕ→ ψ

The Logic S5

MPϕ ϕ→ ψ

The Logic S5

MPϕ ϕ→ ψ

The Logic S5

MPϕ ϕ→ ψ

Multi-agent Epistemic Logic

The Language: ϕ := p | ¬ϕ | ϕ ∧ ψ | Kϕ

Kripke Models: M = 〈W ,R,V 〉 and w ∈W

Truth: M,w |= ϕ is defined as follows:

I M,w |= ¬ϕ if M,w 6|= ϕ

The Language: ϕ := p | ¬ϕ | ϕ ∧ ψ | Kiϕ with i ∈ A

Kripke Models: M = 〈W , {Ri}i∈A,V 〉 and w ∈W

Truth: M,w |= ϕ is defined as follows:

I M,w |= ¬ϕ if M,w 6|= ϕ

I M,w |= Kiϕ if for each v ∈W , if wRiv , then M, v |= ϕ

I More about the logic (complexity results, model checking,Tableaux systems, etc.)

I Group notions

I Additional informational attitudes (belief, certainty,justification, only knowing, awareness, etc.)

I Other semantics

Logic and Arti cial Intelligenceepacuit/classes/logicai-cmu/logai-lec1.pdf · David Lewis Jakko...

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