Introduction Kripke models Muddy Children Puzzle (Revisited) Synchronisation

COMP90054 Software AgentsPossible Worlds

Adrian Pearce

Introduction Kripke models Muddy Children Puzzle (Revisited) Synchronisation

Outline

1 Introduction

2 Kripke models

3 Muddy Children Puzzle (Revisited)

4 Synchronisation

Introduction Kripke models Muddy Children Puzzle (Revisited) Synchronisation

Outline

1 Introduction

2 Kripke models

3 Muddy Children Puzzle (Revisited)

4 Synchronisation

Introduction Kripke models Muddy Children Puzzle (Revisited) Synchronisation

Equivalence relations

Definition (Kripke Models)

A Kripke model M is a tuple 〈S ,V ,R1, . . . ,Rm〉 where:

1 S is a non-empty set of states, possible worlds or epistemicalternatives,

2 V : S → (p → {true, false}) is a truth assignment to thepropositional atoms (p) per state,

3 Ri ⊆ S × S (for all i ∈ A) are the epistemic accessibilityrelations for each agent.

For any state or possible world s,(M, s) |= p (for p ∈ P) iff V (s)(p) = true

Introduction Kripke models Muddy Children Puzzle (Revisited) Synchronisation

Equivalence relations

Definition (Kripke Models)

A Kripke model M is a tuple 〈S ,V ,R1, . . . ,Rm〉 where:

1 S is a non-empty set of states, possible worlds or epistemicalternatives,

2 V : S → (p → {true, false}) is a truth assignment to thepropositional atoms (p) per state,

3 Ri ⊆ S × S (for all i ∈ A) are the epistemic accessibilityrelations for each agent.

For any state or possible world s,(M, s) |= p (for p ∈ P) iff V (s)(p) = true

Introduction Kripke models Muddy Children Puzzle (Revisited) Synchronisation

Equivalence relations

Definition (Kripke Models)

A Kripke model M is a tuple 〈S ,V ,R1, . . . ,Rm〉 where:

1 S is a non-empty set of states, possible worlds or epistemicalternatives,

2 V : S → (p → {true, false}) is a truth assignment to thepropositional atoms (p) per state,

3 Ri ⊆ S × S (for all i ∈ A) are the epistemic accessibilityrelations for each agent.

For any state or possible world s,(M, s) |= p (for p ∈ P) iff V (s)(p) = true

Introduction Kripke models Muddy Children Puzzle (Revisited) Synchronisation

Equivalence relations

Definition (Kripke Models)

A Kripke model M is a tuple 〈S ,V ,R1, . . . ,Rm〉 where:

1 S is a non-empty set of states, possible worlds or epistemicalternatives,

2 V : S → (p → {true, false}) is a truth assignment to thepropositional atoms (p) per state,

3 Ri ⊆ S × S (for all i ∈ A) are the epistemic accessibilityrelations for each agent.

For any state or possible world s,(M, s) |= p (for p ∈ P) iff V (s)(p) = true

Introduction Kripke models Muddy Children Puzzle (Revisited) Synchronisation

Equivalence relations

Definition (Kripke Models)

A Kripke model M is a tuple 〈S ,V ,R1, . . . ,Rm〉 where:

1 S is a non-empty set of states, possible worlds or epistemicalternatives,

2 V : S → (p → {true, false}) is a truth assignment to thepropositional atoms (p) per state,

3 Ri ⊆ S × S (for all i ∈ A) are the epistemic accessibilityrelations for each agent.

For any state or possible world s,(M, s) |= p (for p ∈ P) iff V (s)(p) = true

Introduction Kripke models Muddy Children Puzzle (Revisited) Synchronisation

Example: Muddy Children Puzzle

Example

k children get mud on their foreheads

Each can see the mud on others, but not on his/her ownforehead

The father says ”at least one of you has mud on your head”initially.

The father then repeats ”Can any of you prove you have mudon your head?” over and over.

Assuming that the children are perceptive, intelligent, truthful,and that they answer simultaneously, what will happen?

Introduction Kripke models Muddy Children Puzzle (Revisited) Synchronisation

Muddy Children Puzzle (Initially)

(1,1,1)•

2

31

(1,1,0)•

2

2

(1,0,1)•

31

(0,1,1)•

3

2(1,0,0)•

1

(0,1,0)•

2

(0,0,1)•

3

(0,0,0)•

Introduction Kripke models Muddy Children Puzzle (Revisited) Synchronisation

Muddy Children Puzzle (After the father speaks)

Model—general case for all children (Child 1, Child 2, Child 3)

(1,1,1)•

2

31

(1,1,0)•

2

2

(1,0,1)•

31

(0,1,1)•

3

2(1,0,0)•

(0,1,0)•

(0,0,1)•

Introduction Kripke models Muddy Children Puzzle (Revisited) Synchronisation

Muddy Children Puzzle (k=1)

First time (k=1) all children say ”No” and all states with only onemuddy forehead consequently dissapear.

(1,1,1)•

2

31

(1,1,0)•

(1,0,1)•

(0,1,1)•

Introduction Kripke models Muddy Children Puzzle (Revisited) Synchronisation

Muddy Children Puzzle (k=2 & k=3)

Second time (k=2) all children say ”No” again; this time all stateswith only two muddy foreheads consequently dissapear

(1,1,1)•

Third time (k=3) all children say ”Yes” because they all knowtheir foreheads are muddy (the model can collapse no further).

Introduction Kripke models Muddy Children Puzzle (Revisited) Synchronisation

Forms of knowledge

DGp: the group G has distributed knowledge of fact p

SGp: someone in G knows p

SGp ≡ ∨i∈GKip

EGp: everyone in G knows p

EGp ≡ ∧i∈GKip

Introduction Kripke models Muddy Children Puzzle (Revisited) Synchronisation

Forms of knowledge

E kGp for k ≥ 1: E k

Gp is defined by

E 1Gp = EGp

E k+1G p = EGE

kGp for k ≥ 1

CGp: p is common knowledge in G

CG ≡ EGp ∧ E 2Gp ∧ . . .Em

G p ∧ . . .

Introduction Kripke models Muddy Children Puzzle (Revisited) Synchronisation

Synchronisation

Example (The Coordinated Attack Problem (Byzantine Generals))

Suppose General A sends a message to General B saying Let’sattack at Dawn.

Does not have any common knowledge fixpoint (in spite ofacknowledgements).

It seems that common knowledge is theoretically unachievable- how can this be so?

In the presence of unreliable communication, common knowledge istheoretically unachievable.

Introduction Kripke models Muddy Children Puzzle (Revisited) Synchronisation

Simultaneity

In practice, we can establish ε-common knowledge, Halpern andMoses (1990).

Definition (ε-common knowledge)

ε-common knowledge assumes that within an interval ε everybodyknows φ.

Introduction Kripke models Muddy Children Puzzle (Revisited) Synchronisation

Knowledge

Agent i knows p in world s of (Kripke) structure M, exactly if p istrue at all worlds that i considers possible in s. Formally,

(M, s) |= Kip iff (M, t) |= p for all t such that (s, t) ∈ Ki

Relationship between knowledge forms, DG , EG and CG :

|= EGp ⇔ ∧i∈GKip

The notions of group knowledge form a hierarchy

CGϕ ⊃ . . . ⊃ E k+1G ϕ ⊃ . . . ⊃ EGϕ ⊃ SGϕ ⊃ DGϕ ⊃ ϕ

Introduction Kripke models Muddy Children Puzzle (Revisited) Synchronisation

Knowledge

Agent i knows p in world s of (Kripke) structure M, exactly if p istrue at all worlds that i considers possible in s. Formally,

(M, s) |= Kip iff (M, t) |= p for all t such that (s, t) ∈ Ki

Relationship between knowledge forms, DG , EG and CG :

|= EGp ⇔ ∧i∈GKip

The notions of group knowledge form a hierarchy

CGϕ ⊃ . . . ⊃ E k+1G ϕ ⊃ . . . ⊃ EGϕ ⊃ SGϕ ⊃ DGϕ ⊃ ϕ

Introduction Kripke models Muddy Children Puzzle (Revisited) Synchronisation

Knowledge

Agent i knows p in world s of (Kripke) structure M, exactly if p istrue at all worlds that i considers possible in s. Formally,

(M, s) |= Kip iff (M, t) |= p for all t such that (s, t) ∈ Ki

Relationship between knowledge forms, DG , EG and CG :

|= EGp ⇔ ∧i∈GKip

The notions of group knowledge form a hierarchy

CGϕ ⊃ . . . ⊃ E k+1G ϕ ⊃ . . . ⊃ EGϕ ⊃ SGϕ ⊃ DGϕ ⊃ ϕ

Introduction Kripke models Muddy Children Puzzle (Revisited) Synchronisation

Knowledge

Agent i knows p in world s of (Kripke) structure M, exactly if p istrue at all worlds that i considers possible in s. Formally,

(M, s) |= Kip iff (M, t) |= p for all t such that (s, t) ∈ Ki

Relationship between knowledge forms, DG , EG and CG :

|= EGp ⇔ ∧i∈GKip

The notions of group knowledge form a hierarchy

CGϕ ⊃ . . . ⊃ E k+1G ϕ ⊃ . . . ⊃ EGϕ ⊃ SGϕ ⊃ DGϕ ⊃ ϕ

Introduction Kripke models Muddy Children Puzzle (Revisited) Synchronisation

Knowledge

Agent i knows p in world s of (Kripke) structure M, exactly if p istrue at all worlds that i considers possible in s. Formally,

(M, s) |= Kip iff (M, t) |= p for all t such that (s, t) ∈ Ki

Relationship between knowledge forms, DG , EG and CG :

|= EGp ⇔ ∧i∈GKip

The notions of group knowledge form a hierarchy

CGϕ ⊃ . . . ⊃ E k+1G ϕ ⊃ . . . ⊃ EGϕ ⊃ SGϕ ⊃ DGϕ ⊃ ϕ

Introduction Kripke models Muddy Children Puzzle (Revisited) Synchronisation

The properties of Knowledge (S5 axioms)

1 Kiϕ ∧ Ki (ϕ⇒ Ψ))⇒ Ki Ψ (Distribution axiom)

2 if M |= ϕ then M |= Kiϕ (Knowledge generalisation rule)

3 Kiϕ⇒ ϕ (Knowledge or truth axiom)

4 Kiϕ⇒ KiKiϕ (Positive introspection axiom)

5 ¬Kiϕ⇒ Ki¬Kiϕ (Negative introspection axiom)

Introduction Kripke models Muddy Children Puzzle (Revisited) Synchronisation

Publications

Ronald Fagin, Joseph Y. Halpern, Yoram Moses, and MosheY. Vardi. Rea- soning about Knowledge. The MIT Press,Cambridge, Massachesetts, 1995.

Joseph Y. Halpern and Yoram Moses, Knowledge andCommon Knowledge in a Distributed Environment, Journal ofthe ACM, Vol. 37, No. 3, pp. 549–587, 1990.

Richard Scherl and Hector Levesque. Knowledge, Action, andthe Frame Problem. Artificial Intelligence, 144:1-39, 2003.

Introduction Kripke models Muddy Children Puzzle (Revisited) Synchronisation

Summary

1 Introduction

2 Kripke models

3 Muddy Children Puzzle (Revisited)

4 Synchronisation