Verication of the emergent behaviour of multi-agent systems is an extremely complex task. It is not only the fact that the verication process, formal or model checking, leads to combinatorial explosion, but also the fact that emergent properties should be identied rst before there is an attempt to be veried. The latter is not always straightforward. It is therefore desirable to combine several formal with informal techniques that would be able to join forces towards the verication of MAS.
In agents that operate in a 2 or 3-dimensional space, such as biology or biology-inspired agents, emergence is characterised by a pattern appearing in the agents conguration at some instance during the operation of the system. Trivial examples are colonies of social insects, like ants, birds, sh etc. The type of emergence observed is related to the positioning in space, for example line formation, ocks, schools, herds etc. Modelling such agents would require modelling of their position and verication would require the exploration of a state space developed by the combination of all agent positions evolved through time.
Someone could apply formal verication techniques, such as model checking, under the assumption that we know what emergent property we are looking for. With biology agents this is known in advance, since it has been observed invivo. With articial agents it is not as simple, there is however an active research that proposes a framework for empirical exploration of emergent formations [1]. Consider the following example, known as aggressor-defender game.
In the aggressor-defender game [2] there exist two teams of agents randomly distributed in an environment: defenders (refer to them as friends) and aggressors (or enemies). There are three dierent sub-games involved:
All the agents defend at each turn everyone tends to position between a friend and an enemy (such as they were defending the friend against the enemy), Fig. 1 a).
All the agents ee at each turn everyone tries to position in a way that a friend is between themselves and an enemy (such as the friend protects them from an enemy), Fig. 1 b).
Some agents defend while the rest of the agents ee. Assuming that we can develop a formal model to be used for model checking, it is interesting to consider what property to check for, that is, whether there is an emergent behaviour in all the above three cases, if this MAS is massively populated with similar agents. This work aims to set up a framework of study concerning the above interesting problems and more in particular to demonstrate preliminary results in identifying emergent behaviour through the automatic transformation of a formal model to an executable visual simulation. behaviours, communication, dynamic organisation etc. By separating the various behaviours within the same formal model, it is possible to apply dierent transformations which will facilitate further processing. On one hand, the spatial behaviour determined by movement in space, can lead towards visual animation. The latter is a useful informal tool which will help observing potential emergent properties. On the other hand, suitable abstractions of spatial behaviour together with the rest of the behaviours can lead towards simulation and logging of time series data. These could be used to identify patterns of behaviours which combined with the visual animation produce a set of desired properties. Finally, the desired properties (including emergence) can be veried in the original spatial agent model by model checking, as long as there is a way to transform the original model into an equivalent, susceptible to formal verication, model.
We have been long experimenting with state-based modelling for agents and MAS [35]. The state-based modelling method we use is X-Machines (XM). XM are state machines with memory and instead of inputs triggering transitions, they trigger functions which label the transitions. XM are able to communicate through message exchange, thus forming Communicating X-Machines [6]. XM can also be wrapped around cells inspired by P-Systems [7] which are responsible for the dynamic conguration of the MAS. This idea was successfully introduced in the OPERAS framework (or more particularly OPERAS XC ) [8]. Although, XM can treat movement in space as any other behaviour, we have developed a number of arguments why this spatial behaviour must be separately modelled and treated [3,5].
sp XMs represent a variation of Stream XMs by dening additional components that allow specication of the current position and direction of an agent, as well as to formally specify a movement of an agent within its environment. Formally, a sp XM is a 13-tuple; sp XM = (Σ, Γ , Q, q 0 , M, m 0 , π, π 0 , θ, θ 0 , E, Φ, F) [3]), where:
Σ is an input set of symbols, Γ is an output sets of symbols, Q is a nite set of states, q 0 is the initial state, M is an n-tuple called memory, m 0 is the initial memory, π is a tuple of the current position, i.e. (x, y) when a 2D representation is considered, π 0 is the initial position, θ is an integer in the range 0 to 360, that represents a direction, θ 0 is the initial direction, E is a set which contains elementary positioning operations: e i such as e i : Π ×Θ −→ Π ×Θ, such as direction, moving forward and moving to a specic position.
Φ is a nite set of partial functions ϕ that map a memory state, position, direction and set of inputs to a new memory state, position, direction and set of outputs:
F is a function that determines the next state, given a state and a function from the type Φ, F: Q × Φ Q, and A sp XM model which demonstrates the third strategy is presented on Fig. 3. The model's states are Q={DEFENDING, STAYING_STILL, FLEEING}. There are three corresponding functions: to defend, to stay still and to ee. The memory stores the game strategy of the agent, an agent's friend and enemy, as well as its position and direction. The input consists of the friend's and the enemy's current position. Finally, the output is the new position of the agent, because every agent outputs its position to the other agents thus constructing a communicating sp X-machine system [6], [9].
sp XMDL is the notation used to dene sp XMs [3] and it is modied version of XMDL (see [10], [11]) used in the standard XM. The functions in sp XMDL are coded in the form: #fun functor (($input$), ($memory tuple$), ($position$), ($direction$)) = (($output$), ($memory tuple'$), ($position'$), ($direction'$)) = where $<list of operations including positioning>$ Considering Fig. 3, the function defend is:
#fun defend ( ((?x_fr, ?y_fr),(?x_en, ?y_en)), (?strategy, ?friend, ?enemy), (?my_xcor, ?my_ycor), (?curr_direction) ) = (("move to ?new_xcor ?new_ycor"), (?strategy, ?friend, ?enemy), (?new_xcor, ?new_ycor), (?curr_direction)) where ?new_xcor <-(?x_fr + ?x_en)/2 and ?new_ycor <-(?y_fr + ?y_en)/2. sp XM model of the aggressor-defender game.
As part of the rst steps towards the achievement of formal verication of emergent properties, we have developed a tool for automatic translation of a sp XM model to NetLogo [3]. NetLogo is considered specialised into simulating natural and social phenomena, including modeling of complex systems [12], [13].
The platform supports hundreds of agents to operate independently, providing a clear picture of the micro-level behavior of the agents, as well as the macro-level patterns within the whole system. The translator is based on a set of mapping between formal constructs of XM and language primitives of NetLogo as well as a library supporting all the spatial behaviours.
For the aggressor-defender MAS, an executable counterpart is generated. The output shows a visual animation with which the emergent spatial behaviour is observable (Fig. 4), such as:
The model in which all the agents defend, see Fig. 4 a), behaved as all the agents quickly collapsed into a tight knot,
The model in which all the agents ee, see Fig. 4 b), behaved as a highly dynamic group that expands over time towards the ends of the environment, and
The model in which the agents randomly choose whether to defend or to ee, see Fig. 4 c), exhibited there dierent behaviours. In some situations the agents were all collapsed into a tight knot (as the model from the defender game) with the dierence that this knot was now oscillating around the environment (i), in others they were stationary, randomly distributed and oscillating (ii), and in the last case the agents would form a ocking (iii). The case of this game clearly demonstrated that visual animation aided in discovery of the system's emergence and properties that could be veried at a
later stage, which in turn proved that even the small changes within the individual agent rules might cause a huge dierence in behaviour of the system as a whole.
The contribution of this paper is to present the overall picture of a framework towards the verication of emergent behaviour of spatial MAS. We have also reported progress so far, that is, a denition of sp XM and a tool for automatic transformation to NetLogo. Using this experience, the next steps in the framework are instantiated (Fig. 2) as follows:
sp XM can be transformed into a simulation tool that can generate a time series data. Such tool may be FLAME [14,15] which is used to animate XM models with thousands of agents. FLAME, however, does not deal with the spatial behaviour, which we have already covered by NetLogo.
The logged time series data could be used as an input to a tool identifying patterns, such as DAIKON [16]. The output would be interesting properties that combined with the emergent properties from visual animation could aid us forming the logic temporal formulae to verify.
The sp XM can be suitably transformed into an equivalent model in SPIN, PRISM or SMV [1719], which given the temporal formulae will verify that all the desired properties hold in the original model. Of course the above would assume that a correct transformation from the original model to equivalent models is possible, something which is an interesting problem by itself.