Lattices for Space/Belief and Extrusion/Utterance

PPDP - Siena University. July, 2015!!

Salim PERCHY, COMÈTE(LIX) - INRIA joint work with

Stefan HAAR, Camilo Rueda and Frank VALENCIA

Context: Spatial Multi-Agent Systems

An agent’s space has:

Information

Processes

Other agents’ spaces

Agents are able to:

Run apps

Move data and apps

Share opinions and/or lies

Goal: A sound mathematical model for mobility

c ed

[c t "i ([d]j)]i t [e]j = [c]i t [d t e]j

Goal: A sound mathematical model for mobility

c de c?

[e t [c t (c ! "j d)]j]i = [e t d t [c]j]i

A bit of background: Lattices3.2. Concurrent constraint programming

Figure 3.1: A Herbrand constraint system

true

{x = a} {y = a} {x = y} {y = b} {x = b}

(x = a,

y = a

) (x = a,

y = b

) (x = b,

y = a

) (x = b,

y = b

)

false

A term T is said to be closed if every variable X in T occurs in the

scope of an expression µX.P . We refer to closed terms as processes and

use Proc to denote the class of all processes.

In most presentations of CCP, for example [Sar89], there is also a hiding

operator, denoted 9x

P . To simplify our presentation, we omit this operator,

because we believe is it orthogonal to the extensions of CCP that we are

going to present.

Before giving the semantics of processes, we give some intuitions about

their behaviour. First, processes execute in the context of a store, which

contains certain information, and to which the processes may sometimes add

information. The basic processes are tell, ask, and parallel composition.

Intuitively, tell(c) in a store d adds c to d to make c available to other

processes with access to this store. This addition, represented as d t c, is

performed whether or not dt c = false. The process ask(c) ! P in a store

e may execute P if c is entailed by e, i.e., c v e . The process P k Q stands

for the parallel execution of P and Q. Also, given I = {i1, . . . , im} we use

37

Complete Algebraic Lattice

C = (Con,v,t, false, true)

A bit of background: Modal Logics

' ::= p | ¬' | ' ^ ' | ⇤i'

Bi(') : Agent i believes '

Ki(') : Agent i knows '

¬Bi(false)

Ki(') ) '

A bit of background: CCP

P ::= tell(c) | ask(c) do P | P k Q | µX.P

temperature=50?.Ptemperature>42

temperature<70 0<temperature<100?.Q

S T O R E(MEDIUM)

Processes, constraints and formulas

Process Calculus Lattice of Information Modal Logic

tell(c) c p

P || Q c ⊔ d φ ⋀ φ

skip ⊥ true

abort ⊤ false

⋮ ⋮ ⋮

[ [c] B

[ ? ?

This Talk• Contribution: Algebraic structure to formalize space and

extrusion operations for multi-agent systems

• Roadmap:

1. Motivation & Background

2. Spatial Constraint Systems

3. The Extrusion Problem

4. Properties of Extrusion

5. An application as a bimodal logic

Spatial CS

• [c]i is interpreted as locality (information c is present in agent i's space)

• [[c]i]j is viewed as nesting of spaces (Agent i's space inside agent j's space)

• Joining space functions is possible (i.e [c]i ⊔ [d]j )

A space function of an agent [∙]i is a self-map on the elements of the lattice (Con, ⊑).

Spatial CS

• The spatial functions must fulfill:

• S.1 [true]i = true (emptiness)

• S.2 [c]i ⊔ [d]i = [c ⊔ d]i (distribution)

• Distribution yields the next property:

• S.3 if c ⊑ d then [c]i ⊑ [d]i (monotonicity)

A spatial constraint system (scs) is a cs equipped with n spatial functions.

Spatial CS• SCS’s allow the following:

• [false]i ≠ false (Inconsistency Confinement)

• [c]i ⊔ [d]j ≠ false even if c ⊔ d = false (Freedom of opinion)

• [red]i = [green]i still when red ≠ green (Information Blindness)

Spatial CS with Extrusion

• Extrusion: Movement of information from one space to the outside with no side effect.

!

!

• We call the epistemic interpretation of this action Utterance

c d c d

Spatial CS with Extrusion

• We then define the extrusion as the right inverse of a space. From S.2 and E.1: [d ⊔↑i c]i = [d]i ⊔ c.

An extrusion function ↑i is a self-map on the elements of (Con, ⊑), where:

E.1 [↑i c]i = c

A scs with extrusion (scs-e) is an scs equipped with n extrusion functions satisfying E.1

Derived Constructions in scs-e

• c → d is defined as ⨅{ e | e ⊔ c ⊒ d } (Heyting implication)

• This implication satisfies modus ponens: c ⊔ (c → d ) ⊒ d

• ¬c is defined as c → false (complement)

Communication Example

• Guarded extrusion of inconsistency (w.r.t agent i's information):

• Suppose: [c ⊔ c → (↑i [d]j)]i ⊔ [e]j

• Equivalent to: [c]i ⊔ [d ⊔ e]j

Information d is now in the space of agent j

This Talk• Contribution: Algebraic structure to formalize space and

extrusion operations for multi-agent systems

• Roadmap:

1. Motivation & Background

2. Spatial Constraint Systems

3. The Extrusion Problem

4. Properties of Extrusion

5. An application as a bimodal logic

Finding an extrusion operator

Problem: Given a scs (Con, ⊑, [∙]i, …, [∙]n) find maps ↑i, …, ↑n that satisfy E.1 [↑i c]i = c

Theorem: If [∙]i is a surjective and continuous function then the map ↑i c = ⨆{d | [c]i-1} satisfies E.1.

Max. extrusion

[∙]i is surjective iff ↑i exists (Axiom of Choice)

Distributed Extrusion

Result: Not always the right inverse of a surjective (even continuous) space function satisfies E.3

What about S.1 and S.2 counterparts? We say that the extrusion distributes globally/objectively if:

!E.2! ↑i(true) = true

E.3! ↑i (c ⊔ d) = ↑i (c) ⊔ ↑i (d)

Theorem: If [∙]i is a surjective and meet-complete function then the map ↑i c = ⨅{d | [c]i-1} satisfies E.1-E.3

Min. extrusion

Spatial CS=• SCS’s SCS-E’s allow the following:

• [false]i ≠ false (Inconsistency Confinement)

• [c]i ⊔ [d]j ≠ false even if c ⊔ d = false (Freedom of opinion)

• [red]i = [green]i still when red ≠ green (Information Blindness)

SCS-E

[false]i = false Inconsistencies propagate

✓

✓[∙]i is not necessarily

injective

This Talk• Contribution: Algebraic structure to formalize space and

extrusion operations for multi-agent systems

• Roadmap:

1. Motivation & Background

2. Spatial Constraint Systems

3. The Extrusion Problem

4. Properties of Extrusion

5. An application as a bimodal logic

Properties of Extrusion

• ↑i is distributed then c ⊑ d ⇔ ↑i(c) ⊑ ↑i(d)

• [⋅]i is injective then c ⊑ d ⇔ [c]i ⊑ [d]i!

• ↑i is defined as min. extrusion then ↑i(c) ⊑ d ⇔ c ⊑ [d]i

[⋅]i is an order-embedding

↑i is an order-embedding

The pair (↑i, [⋅]i) is a Galois connection

Application: Bimodal Logic with Utterance/Belief

' ::= p | ¬' | ' ^ ' | Bi' | Ui'

right inverse of belief

Oi(')def= Bi(' ^ Ui('))

Hi(')def= Bi(¬' ^ Ui('))

Oi(') , (Bi') ^ '

Hi(') , (Bi¬') ^ '

These equivalences hold:

Application: Bimodal Logic with Utterance/Belief(P(�),◆, [·]1, . . . , [·]n, "1, . . . , "n) where :

� : all pairs (M, s) where s is a state of M 2 Mltu

t : \[c]i = {(M, s) 2 � | 8t if s !i t then (M, t) 2 c}"i: max. extrusion

• Left Unique

• Left Total

• We proposed a lattice structure to represent information locality and extrusion in a distributed multi-agent system.

• The extrusion is seen as the right-inverse operation of locality. We studied different constructions and their properties.

• The structure can be used to derive a bimodal logic with belief and utterance behaviors.

Summarizing…