Multiagent systems

Idea

To quote from a book on the subject:

“Multiagent systems are a new paradigm for understanding and building distributed systems, where it is assumed that the computational components are autonomous: able to control their own behaviour in the furtherance of their own goals.” (Wooldridge09)

The area is a very wide one and this entry only deals with some generalities especially links between local ‘knowledge’ and more global forms and the models of this that involve the use of various types of modal logic.

Introduction

In many studies of distributed systems, a multiagent model is used.

• An agent is a processor, sensor or finite state machine, interconnected by a communication network with other ‘agents’.

Typically each agent has a local state that is a function of its initial state, the messages received from other agents, observations of the external environment and possible internal actions. It has become customary when using formal models of distributed systems to use modal epistemic logics as one of the tools for studying the knowledge of such systems. The basic such logic for handling a system with $n$-agents is one known as S5(n). Unless the system is very simple the actual logic will be an extension of that basic one, that is, it may have more axioms.

For instance, the way the various agents are connected influences the logic in subtle ways. Suppose that agent 1 sends all its information immediately to agents 2 and 3, then if we denote by $K_i \phi$, the statement that agent $i$ ‘knows’ proposition $\phi$, we clearly expect within the logic of that system that we have as an axiom:

$K_1 \phi \Rightarrow K_2\phi \wedge K_3 \phi,$

in other words, if 1 knows something then we have that both 2 and 3 know it.

The logic $S5_n$ is obtained from ordinary propositional logic by adding ‘knowledge operators’, $K_i$, as above. (In the literature the notation $K_i\phi$ is often replaced by $\Box_i\phi$.) It models a community of ideal knowledge agents who have the properties of

• veridical knowledge (everything they know is true),

• positive introspection (they know what they know)

and

• negative introspection (they know what they do not know).

These properties are reflected in the axiom system for the logic. For more on this see the entry S5(n).

Models for these logics

The classical models for multimodal logics, and for $S5_n$ and its extensions in particular, are combinatorial models known as Kripke frames and, for $S5_n$, Kripke equivalence frames. These consist of a set $W$, called the set of possible worlds, and $n$-equivalence relations $\sim_i$, one for each agent. The interpretation of $\sim_i$ is that if $w_1$, $w_2$ are two possible worlds and $w_1\sim_i w_2$, then agent $i$ cannot tell these two worlds apart.

Interpreted systems

Fagin, Halpern, Moses and Vardi, in various combinations, have put forward a simpler combinatorial model known as an interpreted system. These have the same formal expressive power as Kripke frames, but are nearer the intuition of interacting agents than is the more abstract Kripke model.

As before one has a set, $A = \{1,2, \ldots, n\}$, of agents, and now one assumes each agent $i$ can be in any state of a set $L_i$ of local states. In addition one assumes given a set $L_e$ of possible states of the `environment'. More formally:

Definition

A set of global states (SGS) for an interpreted system is a subset $S$ of the product $L_e \times L_1 \times \ldots \times L_n$ with each $L_e$, $L_i$ non-empty. If $S = L_e \times L_1 \times \ldots \times L_n$, then the SGS is called a hypercube.

Articles

• A. Lomuscio, M. D. Ryan, An algorithmic approach to knowledge evolution, Artificial Intelligence for Engineering Design, Analysis and Manufacturing (AIEDAM), Vol. 13, No. 2 (Special issue on Temporal Logic in Engineering), 23pp, 1999.

• A. Lomuscio, R. van der Meyden, M. D. Ryan, Knowledge in multi-agent systems: initial configurations and broadcast, ACM Transactions on Computational Logic (TOCL), Volume 1, Issue 2, pp. 247- 284. ACM Press. October 2000.