Compositions

We now consider two different presentations for the same notion of composition for the set of agents Ag in an interpreted system. One presentation follows a state-based presentation, while the other follows a run-based presentation.

In what follows, a joint action Act ⊆ Act1× · · · × Actn× Acteis a system-level action

that is performed by all agents synchronously.

3.3.2.1 State-based Composition

We begin by considering the set of all possible global states G⊆ L1× · · · × Ln× Le

which is a subset of the Cartesian product of the local states for all agents in the system. A single global state hl1, . . . , ln, lei ∈ G represents an instantaneous configuration of all

the agents in the system.

The function li: G → Liis a projection of an individual agent’s local state from a

given global state. Without ambiguity, we also denote by li∈ Li, an arbitrary local state;

the context will disambiguate.

The transition relation T ⊆ G × Act × G defines the temporal evolution of the system. Given two global states g and g0, (g, g0) ∈ T iff there exists a joint action a1, . . . , an, such

that for all i ∈ Ag, ai∈ Pi(li(g)) and Ei(li(g), a1, . . . , an) = li(g0). That is, there exists a

temporal transition between two states iff there exists a global action that is consistent with the protocols of each agent at the source state, and, in the evolution for each agent, the source local state and selected global action lead to the destination local state. We assume seriality of this relation, i.e., every global state has at least one successor.

Given an initial state ι ∈ G, the protocols for each agent and the global transition function, these can induce an infinite structure representing all of the possible computa- tions of the system. A path π = (ι, g1, . . .) is an infinite sequence of global states such

that ∀_k≥0(g_k, gk+1) ∈ T . π(k) is the kthglobal state of the path π, whilst Π (g) is the

set of all paths starting at the given state g ∈ G.

The epistemic accessibility relation ∼i⊆ G × G represents that two global states are

indistinguishable for that agent. Formally, (g, g0) ∈ ∼i iff li(g) = li(g0). That is, two

3.3 Reasoning about Multi-Agent Systems 57

these two states. This means that, given only the information in its local state, the agent is unable to tell the related two states apart. For two states g, g0such that g ∼ig0, it is

possible that lj(g) 6= lj(g) (i.e., the local state j is not the same between g and g0); in

this instance, this means that i is oblivious to, or is ignorant of, the current state of j. As we will show in what follows, the indistinguishability relation for each agent can be used to interpret a modal operator encapsulating the knowledge of a given agent. Definition 3.7. Model

A model of an interpreted system is a tuple

MIS= hG, ι, T, ∼1, . . . , ∼n,V i

where:

• G is the set of reachable states accessible from ι via T • ι ∈ G is an initial state

• T is the relation as defined above

• ∼i⊆ G × G is the indistinguishability relation for the agent i ∈ Ag

• V is a mapping of global states to the propositional variables that hold at that state, i.e., V : G → 2AP.

3.3.2.2 Run-based Presentation

We now introduce a second presentation for the composition of m agents in a multi-agent system. Compared to the previous section, the alternative approach presented below is more focused on the runs of the system, compared to the states.

Nonetheless, we denote by S ⊆ Le× L1× . . . × Lmthe set of global states of the MAS.

To represent the temporal evolution of the MAS we consider the flow of time N of the natural numbers.

Definition 3.8. A run in an interpreted system is a function ρ : N → S that intuitively represents a possible evolution of the MAS.

That is, given n ∈ N and a run ρ, ρ(n) ∈ S is the n-th global state in the run. Definition 3.9. An interpreted system (IS) is a tuple P =R, s0,V, where:

(i) R is a non-empty set of runs (Def. 3.8)

(ii) s0∈ S is the initial state, i.e., s0_{= ρ(0) for all ρ ∈ R}

(iii) V : S → 2AP_{is an assignment for the propositional variables in AP}

In what follows we assume without loss of generality that for every s ∈ S, there exist ρ ∈ R and n ∈ N such that s = ρ(n), i.e., S is the set of all reachable states. We refer to a pair (ρ, n) as a point in P and we write Π for the set of all points in P. If

ρ (n) = hle, l1, . . . , lmi is the global state at (ρ, n) then ρe(n) = leand ρi(n) = liare the

environment’s and agent i’s local state at (ρ, n) respectively. Further, for i ∈ Ag the equivalence relation ∼iis defined such that (ρ, n) ∼i(ρ0, n0) iff ρi(n) = ρi0(n

0

); while ρnis the sequence of states ρ(0), . . . , ρ(n) for the prefix of ρ. Sometimes we do not

distinguish between a point and the associated state when it is clear from the context.

3.3.2.3 Logic of Knowledge

We now interpret a multi-modal logic as a logic of knowledge. In this logic, the indistinguishability relations ∼ican be used to interpret the knowledge modality Ki.

That is, each relation ∼iis classed as an epistemic accessibility relation for the modal

operator Ki.

For a set of agents |Ag| = m, we denote by Lmthe logic with the following grammar:

ϕ ::= p | ¬ϕ | ϕ ∨ ϕ | Kiϕ | EΓϕ | CΓϕ

where p ∈ AP, i ∈ Ag and Γ ⊆ Ag 6= /0.

We provide the following readings of the epistemic formulae in Lm:

• Kiϕ – “agent i knows ϕ ”

• EΓϕ – “everybody in the group Γ knows ϕ ”

• CΓϕ – “ϕ is common knowledge between the agents in Γ ”

Furthermore, for n ∈ N, we define E_G0φ = φ and E_Gn+1φ = EGEGnφ .

We note that there is a further modality, distributed knowledge, denoted DΓ, which is

commonly used when reasoning about multi-agent systems. However we do not address distributed knowledge in this thesis.

The semantics of a formula ϕ in Lmgiven an interpreted system P =R, s0, I is

given below.

Definition 3.10. The satisfaction relation |= for φ ∈ Lmand (ρ, n) ∈ P is defined as

follows:

(P, ρ, n) |= p iff p∈ V (ρ(n))

(P, ρ, n) |= ¬ψ iff it is not the case that (P, ρ, n) |= ψ (P, ρ, n) |= ψ ∨ ψ0 iff (P, ρ, n) |= ψ or (P, ρ, n) |= ψ0

(P, ρ, n) |= Kiψ iff (ρ, n) ∼i(ρ0, n0) implies (P, ρ0, n0) |= ψ

(P, ρ, n) |= EGψ iff for all i ∈ G, (P, ρ, n) |= Kiψ

(P, ρ, n) |= CGψ iff for all k ∈ N, (P, ρ, n) |= EGkψ

We note that the modalities EΓ and CΓ can be expressed as modalities expressed

3.3 Reasoning about Multi-Agent Systems 59 ∼E Γ ≡ S i∈Γ ∼i ∼C Γ ≡ ∼ E Γ
∗

Where R∗represents the transitive closure of the relation R.

3.3.2.4 CTL + Lm= Branching-Time Multi-Modal Logic CTLK

We now introduce the temporal-epistemic logic CTLK, which is the fusion logic [Kurucz, 2006] of computation tree logic with the multi-modal epistemic logic Lmfor the set

Ag= {1, . . . , m} of agents. We provide this language with a formal semantics in terms of interpreted systems [Fagin et al., 1995].

The models of interpreted systems can be used to reason about a branching-time temporal-epistemic logic. The language CTLK is built from a countable set of proposi- tional variables AP and using the following syntax:

ϕ , ψ ::= p | ¬ϕ | ϕ ∨ ψ | EX ϕ | EGϕ | E [ϕU ψ ] | Kiϕ

where i ∈ Ag. The epistemic modality Kiϕ is read as “agent i considers it possible that ϕ ”. We define EF ϕ as E [trueU ϕ ]. The duals are as follows: AX ϕ ≡ ¬EX ¬ϕ , AF ϕ ≡ ¬EG¬ϕ and AGϕ ≡ ¬EF¬ϕ. The dual of the epistemic modality for “possibility” is “knowledge”; Kiϕ is defined as ¬Ki¬ϕ, and is read as “agent i knows ϕ”.

The formulae AX φ and AφU φ0(resp. EφU φ0) are read as “for all paths, at the next step φ ” and “for all paths (resp. for some path), φ until φ0”. The formula Kiφ means “agent i knows φ ”; while CGφ means “φ is common knowledge in the set G of agents”.

We define the connectives ∧, ∨, → and the propositional constants true and false as standard.

We note that EX φ is defined as ¬AX ¬φ . The linear-time operator U is dual to U , that is, AφU φ0is defined as ¬E¬φU ¬φ0, and EφU φ0as ¬A¬φU ¬φ0. We will sometimes refer to U as R; the “release” operator. The operator CGis dual to CG, i.e., CGφ is a shorthand for ¬CG¬φ . Also, the operators AG, AF, EG and EF are defined as standard.

Finally, EGφ is defined asVi∈GKiφ ,

The U -formulae in CTLK are the formulae of the form AφU φ0or EφU φ0for some φ , φ0∈ CTLK; the U-, Ki- and CG-formulae are defined similarly.

3.3.2.5 State-based Satisfaction

Given a model MIS (Def. 3.7), a global state g and two CTLK formulae ϕ and ψ, satisfaction of ϕ and ψ at a global state g in a model MIS, written MIS, g |= ϕ (or, for brevity, g |= ϕ), is defined as follows:

g|= ¬ϕ iff it is not the case that g |= ϕ g|= ϕ ∨ ψ iff g|= ϕ or g |= ψ

g|= EXϕ iff ∃π ∈ Π (g) π(1) |= ϕ

g|= EGϕ iff ∃π ∈ Π (g) ∀m ≥ 0 π(m) |= ϕ

g|= E [ϕUψ] iff ∃π ∈ Π (g) ∃m ≥ 0 π(m) |= ψ and ∀0 ≤ j < m π( j) |= ϕ g|= Kiϕ iff ∃g0∈ G g ∼ig0and g0|= ϕ

A CTLK formula ϕ is valid in a model MIS= hG, I, T, ∼1, . . . , ∼n,V i iff MIS, I |= ϕ, i.e., ϕ is true in the initial state of a model.

3.3.2.6 Run-based Satisfaction

Following the run-based presentation of interpreted systems (Section 3.3.2.2), we also provide semantics for CTLK in this context.

Definition 3.11. The satisfaction relation |= for φ ∈ and (ρ, n) ∈ P is defined as follows:

(P, ρ, n) |= p iff p ∈ V (ρ(n))

(P, ρ, n) |= ¬ψ iff it is not the case that (P, ρ, n) |= ψ (P, ρ, n) |= ψ ∨ ψ0 iff (P, ρ, n) |= ψ or (P, ρ, n) |= ψ0

(P, ρ, n) |= AX ψ iff for all runs ρ0, ρ0n= ρnimplies (P, ρ0, n + 1) |= ψ

(P, ρ, n) |= AψUψ0 iff for all runs ρ0, if ρ0n= ρnthen there is k ≥ n, (P, ρ0, k) |= ψ0

and for all k0, n ≤ k0< k implies (P, ρ0, k0) |= ψ

(P, ρ, n) |= EψUψ0iff for some run ρ0, ρ0n= ρnand there is k ≥ n, (P, ρ0, k) |= ψ0

and for all k0, n ≤ k0< k implies (P, ρ0, k0) |= ψ (P, ρ, n) |= Kiψ iff (ρ, n) ∼i(ρ0, n0) implies (P, ρ0, n0) |= ψ

(P, ρ, n) |= CGψ iff for all k ∈ N, (P, ρ, n) |= EGkψ

The truth conditions for ∧, ∨, →, true, false, EX , AU , EU , Kiand CGare defined

from those above. A formula φ is true on an IS P iff it is satisfied at (ρ, 0) such that ρ (0) = s0.