[PPT] - NON-WELL-FOUNDED SET BASED MULTI-AGENT EPISTEMIC ACTION LANGUAGE 34 PowerPoint Presentation

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University of Udine Department of Mathematics, Computer Science and Physics

NON-WELL-FOUNDED SET BASED MULTI-AGENT EPISTEMIC ACTION LANGUAGE

34th Italian Conference on Computational Logic

Francesco Fabiano, Idriss Riouak, Agostino Dovier and Enrico Pontelli

June 20, 2019

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1

Overview

1. Multi-Agent Epistemic Planning
2. Kripke Structures
3. Possibilities
4. The action language mAρ
5. Conclusions

Francesco Fabiano, Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

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Chapter 1

Multi-Agent Epistemic Planning

Francesco Fabiano, Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

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Multi-Agent Epistemic Planning

Introduction

Epistemic Reasoning

Reasoning not only about agents’ perception of the world but also about agents’ knowledge and/or beliefs of her and others’ beliefs.

Francesco Fabiano, Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

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Multi-Agent Epistemic Planning

Introduction

Epistemic Reasoning

Reasoning not only about agents’ perception of the world but also about agents’ knowledge and/or beliefs of her and others’ beliefs.

Multi-agent Epistemic Planning Problem [BA11]

Finding plans where the goals can refer to:

the state of the world
the knowledge and/or the beliefs of the agents

Francesco Fabiano, Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

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Multi-Agent Epistemic Planning

An Example

Francesco Fabiano, Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

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Multi-Agent Epistemic Planning

An Example

Initial State

Snoopy and Charlie are looking while Lucy is ¬looking
No one knows the coin position.

Francesco Fabiano, Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

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Multi-Agent Epistemic Planning

An Example

Goal State

Charlie knows the coin position
Lucy knows that Charlie knows the coin position
Snoopy does not know anything about the plan execution

Francesco Fabiano, Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

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Multi-Agent Epistemic Planning

Challenges

An agent has to reason about his actions effects on

The state of the world
The agents’ awareness of the environment
The agents’ awareness of other agents’ actions
The knowledge of other agents about his own

Francesco Fabiano, Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

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Multi-Agent Epistemic Planning

Notations

Given a set of agents AG

Modal operator Bag

where ag ∈ AG

Models the beliefs of ag about the state of the world and/or about the beliefs of other agents.

Francesco Fabiano, Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

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Multi-Agent Epistemic Planning

Notations

Given a set of agents AG

Modal operator Bag

where ag ∈ AG

Models the beliefs of ag about the state of the world and/or about the beliefs of other agents.

Group operator Cα

where α ⊆ AG

Expresses the common belief of a group of agents.

Francesco Fabiano, Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

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Multi-Agent Epistemic Planning

Notations

Given a set of agents AG

Modal operator Bag

where ag ∈ AG

Models the beliefs of ag about the state of the world and/or about the beliefs of other agents.

Group operator Cα

where α ⊆ AG

Expresses the common belief of a group of agents.

Belief Formulae

Take into consideration fluents and/or agents’ beliefs.

Francesco Fabiano, Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

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Multi-Agent Epistemic Planning

Example of Belief Formulae

Given

AG = {Snoopy, Charlie, Lucy}
F

= {opened, head, lookingag} ag ∈ AG

BSnoopyBCharlie¬opened

Snoopy believes that Charlie believes that the box is ¬opened.

Francesco Fabiano, Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

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Multi-Agent Epistemic Planning

Example of Belief Formulae

Given

AG = {Snoopy, Charlie, Lucy}
F

= {opened, head, lookingag} ag ∈ AG

BSnoopyBCharlie¬opened

Snoopy believes that Charlie believes that the box is ¬opened.

Cα(¬BLucyheads ∧ ¬BLucy¬head)

where α = AG

It is common knowledge that Lucy does not know whether the coin lies heads or tails up

Francesco Fabiano, Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

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Multi-Agent Epistemic Planning

Knowledge vs. Belief

The modal operator Bag represents the worlds’ relation
Different relation’s properties imply different meaning for Bag

Francesco Fabiano, Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

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Multi-Agent Epistemic Planning

Knowledge vs. Belief

The modal operator Bag represents the worlds’ relation
Different relation’s properties imply different meaning for Bag
Knowledge and Belief are characterized by a subset of the

following axioms

Serial (D) and S5 (K,T,4,5) Axioms

Given the fluent formulae φ, ψ and the worlds i, j D ¬Ri⊥ B K K (Riϕ ∧ Ri(ϕ ⇒ ψ)) ⇒ Riψ B K T Riϕ ⇒ ϕ K 4 Riϕ ⇒ RiRiϕ B K 5 ¬Riϕ ⇒ Ri¬Riϕ B K

Francesco Fabiano, Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

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Chapter 2

Kripke Structures

Francesco Fabiano, Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

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Kripke Structures

Description

Pointed Kripke structure

A Pointed Kripke structure is a pair (S, π, R1,. . . , Rn, s0), s.t.:

S is a set of worlds and s0 ∈ S
π : S → 2F associates an interpretation to each element of S
for 1 ≤ i ≤ n, Ri ⊆ S × S is a binary relation over S

Francesco Fabiano, Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

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Kripke Structures

Description

Pointed Kripke structure

A Pointed Kripke structure is a pair (S, π, R1,. . . , Rn, s0), s.t.:

S is a set of worlds and s0 ∈ S
π : S → 2F associates an interpretation to each element of S
for 1 ≤ i ≤ n, Ri ⊆ S × S is a binary relation over S

g0 g1 i0 i1

ag ag ag {Snoopy} {Snoopy} {Snoopy} {Snoopy} {Charlie, Lucy} {Charlie, Lucy} {Lucy}

ag = {Charlie, Lucy, Snoopy}

Francesco Fabiano, Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

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Kripke Structures

Entailment

Let ϕ be a belief formula and (M, s) be a pointed Kripke structure:

Entailment w.r.t. a pointed Kripke structure

(M, s) |

= ϕ if ϕ is a fluent formula and π(s) | = ϕ;

Francesco Fabiano, Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

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Kripke Structures

Entailment

Let ϕ be a belief formula and (M, s) be a pointed Kripke structure:

Entailment w.r.t. a pointed Kripke structure

(M, s) |

= ϕ if ϕ is a fluent formula and π(s) | = ϕ;

(M, s) |

= Bagiϕ if ∀ t: (s, t) ∈ Ri it holds that (M, t) | = ϕ;

Francesco Fabiano, Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

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Kripke Structures

Entailment

Let ϕ be a belief formula and (M, s) be a pointed Kripke structure:

Entailment w.r.t. a pointed Kripke structure

(M, s) |

= ϕ if ϕ is a fluent formula and π(s) | = ϕ;

(M, s) |

= Bagiϕ if ∀ t: (s, t) ∈ Ri it holds that (M, t) | = ϕ;

(M, s) |

= Eαϕ if (M, s) | = Bagiϕ for all agi ∈ α;

(M, s) |

= Cαϕ if (M, s) | = Ek

αϕ for every k ≥ 0, where

E0

αϕ = ϕ and Ek+1 α

ϕ = Eα(Ek

αϕ).

The entailment for the standard operators is defined as usual

Francesco Fabiano, Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

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Kripke Structures

Problems

Solvers require high amount of memory
In literature the states have been represented explicitly
State comparison needs to find bisimilar states

{A,B,C} s0 s1 {A,B,C} {A,B,C} r1 q0 {A} {A} {A} {A} {B,C} {B,C} {A,B,C} s0 s1 {A,B,C} {A,B,C} r1 {A} {A} {B,C}

Francesco Fabiano, Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

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Kripke Structures

Solutions

Heuristics [Le+18]

Francesco Fabiano, Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

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Kripke Structures

Solutions

Heuristics [Le+18]
Symbolic representation of Kripke structures

Francesco Fabiano, Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

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Kripke Structures

Solutions

Heuristics [Le+18]
Symbolic representation of Kripke structures
Alternative representations

Francesco Fabiano, Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

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Chapter 3

Possibilities

Francesco Fabiano, Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

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Possibilities

Overview

Introduced by Gerbrandy and Groeneveld [GG97]
Used to represent multi-agent information change
Based on non-well-founded sets
Corresponds with a class of bisimilar Kripke structures [Ger99]

Francesco Fabiano, Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

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Possibilities Non-well-founded sets theory

Non-well-founded sets

Non-well-founded set [Acz88]

A set is non-well-founded (or extraordinary) when among its descents there are some which are infinite. The non-well-founded set Ω = {Ω}

Francesco Fabiano, Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

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Possibilities

Formal Definition

Possibility [GG97]

Let AG be a set of agents and F a set of propositional variables:

A possibility u is a function that assigns to each propositional

variable l ∈ F a truth value u(l) ∈ {0, 1} and to each agent ag ∈ AG a set of possibilities u(ag) = σ.

Francesco Fabiano, Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

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Possibilities

Formal Definition

Possibility [GG97]

Let AG be a set of agents and F a set of propositional variables:

A possibility u is a function that assigns to each propositional

variable l ∈ F a truth value u(l) ∈ {0, 1} and to each agent ag ∈ AG a set of possibilities u(ag) = σ. Intuitively ...

The possibility u is a possible interpretation of the world and
f the agents’ beliefs
u(l) specifies the truth value of the literal l
u(ag) is the set of all the interpretations the agent ag

considers possible in u

Francesco Fabiano, Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

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Possibilities

From Possibilities to Kripke Structures

Considering a possibility

A possibility

p p,q {A} {A} {B}

Francesco Fabiano, Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

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Possibilities

From Possibilities to Kripke Structures

Considering a possibility Can be expressed as a system of equations

A possibility

p p,q {A} {A} {B}

Its system of equation

                   w(p) = 1 w(q) = 0 v(p) = 1 v(q) = 1 u(p) = 0 u(q) = 0 w(A) = {v} w(B) = {∅} v(A) = {v} v(B) = {u} u(A) = {∅} u(B) = {∅}

Francesco Fabiano, Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

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Possibilities

From Possibilities to Kripke Structures

Considering a possibility Can be expressed as a system of equations Systems of equations have unique solutions

A possibility

p p,q {A} {A} {B}

Its system of equation

                   w(p) = 1 w(q) = 0 v(p) = 1 v(q) = 1 u(p) = 0 u(q) = 0 w(A) = {v} w(B) = {∅} v(A) = {v} v(B) = {u} u(A) = {∅} u(B) = {∅}

The solution

w v u

Francesco Fabiano, Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

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Possibilities

From Possibilities to Kripke Structures

Considering a possibility Can be expressed as a system of equations Systems of equations have unique solutions The solution decorates a Kripke structure

A possibility

p p,q {A} {A} {B}

Its system of equation

                   w(p) = 1 w(q) = 0 v(p) = 1 v(q) = 1 u(p) = 0 u(q) = 0 w(A) = {v} w(B) = {∅} v(A) = {v} v(B) = {u} u(A) = {∅} u(B) = {∅}

The solution

w v u w v u

Relative Kripke Structure

p,q p {B} {A} {A}

w v u

Francesco Fabiano, Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

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Possibilities

From Possibilities to Kripke Structures

Considering a possibility Can be expressed as a system of equations Systems of equations have unique solutions The solution decorates a Kripke structure

A possibility

p p,q {A} {A} {B}

Its system of equation

                   w(p) = 1 w(q) = 0 v(p) = 1 v(q) = 1 u(p) = 0 u(q) = 0 w(A) = {v} w(B) = {∅} v(A) = {v} v(B) = {u} u(A) = {∅} u(B) = {∅}

The solution

w v u w v u

Relative Kripke Structure

p,q p {B} {A} {A}

w v u

Francesco Fabiano, Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

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Chapter 4

The action language mAρ

Francesco Fabiano, Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

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The action language mAρ

Overview

We introduce the action language mAρ

Used to describe MEP problems
Same syntax of the action language mA+ [Bar+15]
As expressive as mA+
Uses possibilities as states

Francesco Fabiano, Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

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The action language mAρ

Actions

Three types of actions:

Ontic: modifies some fluents of the world

Charlie opens the box

Francesco Fabiano, Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

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The action language mAρ

Actions

Three types of actions:

Ontic: modifies some fluents of the world

Charlie opens the box

Sensing: senses the true value of a fluent

Charlie peeks inside the box

Francesco Fabiano, Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

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The action language mAρ

Actions

Three types of actions:

Ontic: modifies some fluents of the world

Charlie opens the box

Sensing: senses the true value of a fluent

Charlie peeks inside the box

Announcement: announces the fluent to other agents

Charlie announces the coin position

Francesco Fabiano, Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

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The action language mAρ

Observability Relations

An execution of an action might change or not an agents’ belief accordingly to her degree of awareness

Action type Full observers Partial Observers Oblivious Ontic Sensing Announcement

Francesco Fabiano, Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

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The action language mAρ

Possibility as a state

In mAρ a state is encoded by a possibility where

(agent, σ) represent the possibilities believed by agent
If f ∈ F is present then it is true

w = {(ag, {w, w′}), (C, {v, v′}), look(ag), key(A), opened, heads}

where ag ∈ {A, B}

Francesco Fabiano, Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

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The action language mAρ

Possibility as a state

In mAρ a state is encoded by a possibility where

(agent, σ) represent the possibilities believed by agent
If f ∈ F is present then it is true

w = {(ag, {w, w′}), (C, {v, v′}), look(ag), key(A), opened, heads}

where ag ∈ {A, B}

           w = {(ag, {w, w′}), (C, {v, v′}), look(ag), key(A), opened, heads} w′ = {(ag, {w, w′}), (C, {v, v′}), look(ag), key(A), opened} v = {(A, {v, v′}), (B, {v, v′}), (C, {v, v′}), look(ag), key(A), heads} v′ = {(A, {v, v′}), (B, {v, v′}), (C, {v, v′}), look(ag), key(A)} where ag ∈ {A, B}.

w v

{A,B} {A,B,C}

w′ v′

{A,B,C} {A,B} {A,B} {C} {C} {C} {C} {A,B,C}

Possibility w expanded for clarity

Francesco Fabiano, Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

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The action language mAρ

State equality

Possibilities captures classes of bisimilar Kripke structures
Possibilities equality considers bisimilarity
This help for the visited states problem in MEP

Francesco Fabiano, Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

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The action language mAρ

State equality

Possibilities captures classes of bisimilar Kripke structures
Possibilities equality considers bisimilarity
This help for the visited states problem in MEP

r1={(A, {s0, s1}), (B, {r1}), (C, {r1}), look(C), key(A), opened, heads}

{A,B,C} s0 s1 {A,B,C} {A,B,C} r1 q0 {A} {A} {A} {A} {B,C} {B,C} {A,B,C} s0 s1 {A,B,C} {A,B,C} r1 {A} {A} {B,C}

Francesco Fabiano, Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

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The action language mAρ Entailment

Entailment

Let ϕ, ϕ1, ϕ2 be beliefs formula and u be a possibility

Entailment w.r.t. possibilities

u |

= l if u(l) = 1;

Francesco Fabiano, Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

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The action language mAρ Entailment

Entailment

Let ϕ, ϕ1, ϕ2 be beliefs formula and u be a possibility

Entailment w.r.t. possibilities

u |

= l if u(l) = 1;

u |

= Bagϕ if for each v ∈ u(ag), v | = ϕ;

Francesco Fabiano, Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

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The action language mAρ Entailment

Entailment

Let ϕ, ϕ1, ϕ2 be beliefs formula and u be a possibility

Entailment w.r.t. possibilities

u |

= l if u(l) = 1;

u |

= Bagϕ if for each v ∈ u(ag), v | = ϕ;

u |

= Eαϕ if u | = Bagϕ for all ag ∈ α;

u |

= Cαϕ if u | = Ek

αϕ for every k ≥ 0, where E0 αϕ = ϕ and

Ek+1

α

ϕ = Eα(Ek

αϕ).

The entailment for the standard operators is defined as usual

Francesco Fabiano, Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

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The action language mAρ Transition function

Ontic Actions

ΦD for Ontic Actions ΦD : AI × Σ → Σ ∪ {∅}

Let a be an ontic action instance, u a possibility and l be f or ¬f ΦD(a, u) = ∅ if a is not executable in u ΦD(a, u) = v if a modifies the literals ∈ caused(a) Where v is

v(l) = u(l)

if l ∈ caused(a) v(l) = caused(a)[l] if l ∈ caused(a) and

v(ag) = u(ag)

if ag ∈ OD v(ag) =

w∈u(ag) ΦD(a, w)

if ag ∈ FD

Francesco Fabiano, Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

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The action language mAρ Transition function

Sensing Actions

ΦD for Sensing Actions ΦD : AI × Σ → Σ ∪ {∅}

Let a be an sensing action instance, and u a possibility and l be f

r ¬f

ΦD(a, u) = ∅ if a is not executable in u ΦD(a, u) = v if the literal l is sensed Where v is            ∅ if sensed(a)[l] = u(l) v(ag) = u(ag) if ag ∈ OD v(ag) =

w∈u(ag) ΦD(a, w)

if ag ∈ FD v(ag) =

w∈u(ag)(ΦD(a, w) ∪ ΦD(¬a, w))

if ag ∈ PD

Francesco Fabiano, Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

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The action language mAρ Transition function

Announcement Actions

ΦD for Announcement Actions ΦD : AI × Σ → Σ ∪ {∅}

Let a be an announcement action instance, and u a possibility. ΦD(a, u) = ∅ if a is not executable in u ΦD(a, u) = v if the fluent formula φ is announced Where v is            ∅ if u | = φ v(ag) = u(ag) if ag ∈ OD v(ag) =

w∈u(ag) ΦD(a, w)

if ag ∈ FD v(ag) =

w∈u(ag)(ΦD(a, w) ∪ ΦD(¬a, w))

if ag ∈ PD

Francesco Fabiano, Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

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Chapter 5

Conclusions

Francesco Fabiano, Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

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Conclusions Future works

Conclusions

Exploited an alternative to the Kripke structures as states

representation

Used possibilities to define a stronger concept of states

equality

Possibilities helps in reducing the search-space dimension
Defined a new action language for the MEP problem

Francesco Fabiano, Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

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Conclusions Future works

Future works

We started implementing a planner for mAρ
Exploit more set-based operations: especially for the

entailment of group operators

Formalize the concept of non-consistent belief for mAρ
Consider other alternatives to Kripke structures, e.g., OBDDs

Francesco Fabiano, Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

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Conclusions Future works

The end Thank You for the attention

Francesco Fabiano, Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

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Conclusions Future works

References I

[Acz88] Peter Aczel. Non-Well-Founded Sets. CSLI Lecture Notes, 14. 1988. [BA11] Thomas Bolander and Mikkel Birkegaard Andersen. “Epistemic planning for single-and multi-agent systems”. In: Journal of Applied Non-Classical Logics 21.1 (2011), pp. 9–34. [Bar+15] Chitta Baral et al. “An Action Language for Multi-Agent Domains: Foundations”. In: CoRR abs/1511.01960 (2015). url: http://arxiv.org/abs/1511.01960. [BJS15]

T. Bolander, M.H. Jensen, and F. Schwarzentruber.

“Complexity results in epistemic planning”. In:

vol. 2015-January. 2015, pp. 2791–2797.

Francesco Fabiano, Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

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Conclusions Future works

References II

[FH94] Ronald Fagin and Joseph Y Halpern. “Reasoning about knowledge and probability”. In: Journal of the ACM (JACM) 41.2 (1994), pp. 340–367. [Ger99] Jelle Gerbrandy. Bisimulations on planet Kripke. Inst. for Logic, Language and Computation, Univ. van Amsterdam, 1999. [GG97]

J. Gerbrandy and W. Groeneveld. “Reasoning about

information change”. In: Journal of Logic, Language and Information 6.2 (1997), pp. 147–169. doi: 10.1023/A:1008222603071.

Francesco Fabiano, Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

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Conclusions Future works

References III

[Le+18] Tiep Le et al. “EFP and PG-EFP: Epistemic forward search planners in multi-agent domains”. In: Twenty-Eighth International Conference on Automated Planning and Scheduling. 2018.

Francesco Fabiano, Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

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Backup Slides

Side to side Execution

{A, B, C} s0 s1 {A, B, C} {A, B, C} M0[π](s0) = {look(ag), key(A), heads} M0[π](s1) = {look(ag), key(A)}

           u = {(ag, {u, u′}), look(ag), key(A), heads} u′ = {(ag, {u, u′}), look(ag), key(A)}

where ag ∈ {A, B, C}

The initial state

Francesco Fabiano, Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

SLIDE 61

Backup Slides

Side to side Execution

{A, B, C} p0 p1 {A, B, C} {A, B, C} M1[π](p0) ={look(A), look(B), key(A), heads} M1[π](p1) ={look(A), look(B), key(A)}

           v = {(ag, {v, v′}), look(A), look(B), key(A), heads} v′ = {(ag, {v, v′}), look(A), look(B), key(A)} where ag ∈ {A, B, C}

where ag ∈ {A, B, C}

Execution of distract(C)A

Francesco Fabiano, Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

SLIDE 62

Backup Slides

Side to side Execution

{A, B, C} p0 p1 {A, B, C} {A, B, C} {A, B} q0 q1 {A, B} {A, B} {C} {C} {C} {C} M2[π](q0) = {look(ag), key(A), opened, heads} M2[π](q1) = {look(ag), key(A), opened} M2[π](p0) = M1[π](p0) M2[π](p1) = M1[π](p1)

           w = {(ag, {w, w′}), (C, {v, v′}), look(ag), key(A), opened, heads} w′ = {(ag, {w, w′}), (C, {v, v′}), look(ag), key(A), opened} where v, v′, are defined as before.

where ag ∈ {A, B}

Execution of openA

Francesco Fabiano, Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC

SLIDE 63

Backup Slides

Side to side Execution

{A, B, C} p0 p1 {A, B, C} {A, B, C} {B} r0 r1 {A, B} {A, B} {C} {C} {C} {C} M3[π](r0) = {look(ag), key(A), opened, heads} M3[π](r1) = {look(ag), key(A), opened} M3[π](p0) = M1[π](p0) M3[π](p1) = M1[π](p1) .

           z = {(A, {z}), (B, {z, z′})(C, {v, v′}), look(ag), key(A), opened, heads} z′ = {(A, {z′}), (B, {z, z′})(C, {v, v′}), look(ag), key(A), opened, } where the possibilities v, v′ are defined as before.

where ag ∈ {A, B}

Execution of peekA

Francesco Fabiano, Idriss Riouak, Agostino Dovier and Enrico Pontelli — CILC