2.3 Reasoning about knowledge
2.3.3 Interpreted systems
In this section, multi-agent systems are described by using complex structures called interpreted systems. This is an intuitive semantics directly connected with the executions of multi-agent systems; the knowledge of agents is strictly related with the notion of local states.
The formalism of interpreted systems was made popular in [FHVM95] to model a multi-agent system and to study epistemic and temporal properties of agents. In this thesis, interpreted
systems are used as a semantic model for multi-agent systems. In this formalism a system is composed of n agents and an environment. Each agent and the environment are associated with a set of local states and a set of actions. The local states are private to the agent and the environment. Local protocols define the actions that may be executed at a given local state. The local evolution function of the agent defines the transition relation among the local states. The environment has the same structure as the agents. The formal definition of an interpreted system is given as follows.
Definition 2.12 (Interpreted system). [FHVM95]
An interpreted system over a set Ag = {1, . . . , n} of agents and an environment e is a tuple I = h{Li}i∈Ag∪ {Le}, {ACTi}i∈Ag∪ {ACTe}, {Pi}i∈Ag∪ {Pe}, {ti}i∈Ag∪ {te}, I0, V i
where:
• Li (Le, respectively) is a non-empty set of possible local states of agent i (the environment e, respectively).
The elements li ∈ Li (le ∈ Le, respectively) contain all the information that the agent i (the environment e, respectively) has about the current configuration in which he/she is. The set of possible global states is defined as the Cartesian product of local states, as follows:
W = L1× · · · × Ln× Le
For any global state w ∈ W , li represents the i-th component in w, i.e., the local state of agent i in w. Similarly, le represents the local state of the environment e in w.
• Acti (Acte, respectively) is a non-empty set of actions for the agent i (the environment e, respectively). Elements ai ∈ Acti (ae ∈ Acte, respectively) denote the actions that an agent i ∈ Ag (the environment e, respectively) can perform in a given state.
The joint action Act is defined as the Cartesian product of the agents’ actions: Act = Act1× · · · × Actn× Acte.
Let a ∈ Act be a joint action and ai the action of agent i in a. Joint actions are tuples of actions, one for each agent i and the environment e.
2.3. Reasoning about knowledge 23
• The protocol Pi is a function Pi : Li −→ 2Acti 1, that associates a set of actions to each local state of the agent i and Pe ⊆ Le× 2Acte is the local protocol for the environment. The protocol encodes a set of “rules” that establish which actions can be performed in each local state.
• ti is the local evolution function. This function returns the set of the next local states given a current local state and all actions for each agent. Formally, the local evolution function is defined as follows:
ti : Li× Act −→ 2Li
n is the number of agents in the system 2. Similarly, t
e : Le× Act −→ 2Le.
• The term I0 ∈ W is used to represent the set of initial global states.
• Finally, AP is a set of atomic propositions and V is the evaluation function defined as follows:
V : W −→ 2AP
This function gives the so called “interpretation” to the global states of the system.
Local states are private. Therefore, a local state of an agent i is not accessible to the other agents. Notice that the set of local states and global states are discrete. This is not a limitation as it is possible to model a continuous state system by using a discrete one to any desired level of accuracy. In contrast to local states, actions are public. In the sense that a local action performed by an agent i can be “seen” by all the other agents. The sets of local states and local actions will always be finite in this thesis. It is possible that an agent i can have multiple actions to perform in a given local state. In this case, the agent i chooses an action in a non-deterministic way.
Local protocols and local evolution functions together determine how the entire interpreted system evolves from a global state to the next one.
1The local protocol P
i can be rewritten as the set Pi⊆ Li× 2Acti.
2The local evolution function t
i can be rewritten as the set ti⊆ Li× Act1× · · · × Actn× 2Li. The definition
of ti given here is slightly different from that one given in [FHVM95]. In [FHVM95], ti returns a single local
Definition 2.13 (Global transition relation).
Given an interpreted system IS, the global transition relation T ⊆ W × ACT × 2W is such that hw, a, W′i ∈ T (where W′ ⊆ W ) if and only if:
∀i ∈ Ag : hwi, a, L′ii ∈ ti ∧ hwi, aii ∈ Pi ∧ hwe, a, L′ei ∈ te ∧ hwe, aei ∈ Pe where L′
i ⊆ Li and L′e⊆ Le.
In other words, an interpreted system can move from a global state w to a subset of global state W′ ⊆ W in one step if there is a joint action available at w which transforms each local state wi into a set of local states L′i and we to Le.
Definition 2.14 (Total).
A global transition relation T is total if and only if:
∀w ∈ W ∃W′ ⊆ W (wT W′∧ W′ 6= ∅)
In the following, the global transition relation T is assumed to be total. Definition 2.15 (Path).
A path π in an interpreted system IS is an sequence (either finite or infinite) w0w1. . . of global states belonging to W such that every pair of successive states forms a transition, i.e., wkT wk+1 for all k ≥ 0.
The expression π(k) means the k-th global state in π, i.e., wk.
Definition 2.16 (Reachable states).
A global state w ∈ W is reachable if and only if it can be reached by a finite path π starting from an initial state, i.e., π(0) ∈ I0. Let G denote the set of reachable states.
Usually [FHVM95] the knowledge of an agent is defined by means of relations over global states defined as follows.
Definition 2.17 (Epistemic indistinguishability relation).
The epistemic indistinguishability relation for agent i ∈ Ag in an interpreted system IS is: ∼i = {(w, w′) ∈ W × W | li = l′i}