Mapping the Envelope of Social Simulation Trajectories

In this paper we present a methodology aimed at systematically exploring the shape of the ‘envelope’ of simulation trajectories and the implicit theory that a simulation represents. Thus it complements methods like Monte Carlo analysis, the inspection of single scenarios and syntactical proof. We propose a method for searching for tendencies and proving their necessity relative to a range of parameterisations of the model and agents’ choices, and to the logic of the simulation language. This will allow us to dig up conclusions about tendencies for that fragment of the simulation theory given by the explored subspace of trajectories. Additionally, we propose and exemplifies a computational procedure that helps implement this exploration by translating the MAS simulation into a constraint-based search over possible trajectories by ‘compiling’ the simulation rules into a more specific form. 1 Social Simulation and the Exploration of Simulation Trajectories A social simulation necessarily abstracts from some idea about processes that produce social phenomena. Typically this means that: firstly, many of the simulation parameters will be in essence chosen arbitrarily and, secondly, that there will be indetermistic choice processes in the simulation to ‘stand in’ for processes which we do not want to simulate. In particular a pseudo-random number generator often ‘stands in’ for some aspect of a real choice made by a social actor or some unpredictable aspect of the target environment. One can think of each choice as resulting in a branch point in the simulation – where the simulation trajectory ‘branches out’ into a separate trajectory for each possibility. The intended content of the simulation is exactly not the individual trajectories, but the envelope of these trajectories. It may be that every branch diverges from the others so that the result is completely contingent upon the exact choices made. On the other hand it may be that all branches share a common tendency or converge to the others in certain aspects. This commonality could be explicitly ‘forced’ by the design of the simulation in the form of an explicit constraint: for example if a room has only one exit then the actors in that room may all exit by the same door eventually whatever the nature of their individual choice processes. In other simulations the commonality is emergent in the sense that it is difficult to explain the commonality between possibilities in terms of the simulation design – this is what we call emergent tendencies. This paper documents some steps in the search for ways to understand such emergent tendencies. A simulation study can have many purposes, including these: it may help in the understanding of some phenomena and also it may inform the design of future simulations. Exploring possible simulation trajectories and analysing the resulting dynamics of the simulation are central to both these tasks. Usually there is a trade-off between the richness of the study in terms of the number of explored trajectories (sometimes related to how fine-grained the model is) and the amount of required computational resources. The finer the model the more “realistic” the simulation model will be, but also the more intricate the analysis of the simulation will be. A typical case where this analysis is crucial is in Multi-Agent Based Social Simulation. There, modellers may attempt to generate in the lab certain “complex” behaviours in a whole population as the result of the interaction of simpler. Unforeseen behaviour of individuals and unpredictable tendencies in the behaviour of the whole population can arise [4]. The lack of alternative methodologies and tools for appropriate exploration and analysis of the dynamics of a simulation are presently a factor, which limits the comprehension of emergent tendencies. Present methods include examining individual trajectories as in Scenario Analysis [3] and statistical sampling as in Monte Carlo techniques [12]. Each of these has its limitations. It is our purpose in this paper to complement these with an alternative way of exploring and analysing the simulation by systematically and automatically mapping the envelope of all possible trajectories in a substantial fragment of a simulation. 2 Enveloping Tendencies in Simulation Trajectories: a Constrained Search over Possible Models The traditional methods for examining simulation trajectories are: Scenario Analysis and Monte Carlo techniques. Via Scenario Analysis trajectories are inspected one at a time and as many alternatives as possible examined. Nevertheless, it is usually unviable to map all the possibilities, as the number of alternative trajectories is far too large. Additionally, the high amount of data increases the difficult task of searching for exceptional behaviour displayed by (groups of) agents in a simulation. Moreover, it is left up to the modeller to make conclusions about the persistence and sensitivity of certain behavioural outcomes with respect to a certain range of the factors. On the other hand, a Monte Carlo analysis explores the dynamics of the simulation via statistical analysis of quantitative change (or quantitative measures of qualitative changes) observed in a sample of trajectories. The sample is done over the range of possibilities given by random variables introduced in the model to simplify uncertainties. The difficulties with this are that: it tells us what is a probable outcome rather than what is a necessary outcome and it can involve the use of inappropriate statistical assumptions. 2.1 Constrained Exploration of Trajectories We propose the use of an exhaustive constraint-based search over a range of possible trajectories in order to establish the necessity of postulated emergent tendencies. Thus a subset of the possible simulation parameterisations and agent choices are specified; the target emergent tendencies are specified in the form of negative constraints; and an automatic search over the possible trajectories performed. The tendencies are shown to be necessary with respect to the range of parameterisations and indeterministic choices by first finding a possible trajectory without the negative constraint to show the rules are consistent and then showing that all possible trajectories violate the negation of the hypothetical tendency when this is added as a further constraint. (See fig. 1). 2.2 Characterising The Envelope of Tendencies In order to distinguish between the exceptional and the representative in a simulation, we will formally describe the envelope of certain tendencies in a simulation. This might be done by: • Certain properties satisfied by the observed tendency. • A mathematical description of a subspace of the tendencies or of a subspace given a bound of the tendencies. • Representative or typical instances of such a tendency. Structure of the model for a combination of parameters Trajectories for a structure. Branches are due to agents’ choices Alternative setting of parameters Mapping of a tendency from the trajectories A subspace of tendencies Envelope of all tendencies Fig. 1. A constraint-based exploration of possible simulation dynamics • A mapping from the setting of parameters and choices defining the trajectories to certain knowledge (maybe properties) about the tendency in such a trajectory: (parameters X choices)Å (know. of the tend.) 2.3 Proving the Necessity of a Tendency We want to be able to generalise about tendencies going from observation of individual trajectories to observation of a group of trajectories generated for certain parameters and choices. Actually, we want to know if a particular tendency is a necessary consequence of the system or a contingent one. For doing this we propose to translate the original MAS along with the range of parameterisations and agents’ choices into a platform (described in the next section) where the alternative trajectories can be unfolded. Each trajectory will correspond to a possible trajectory in the original MAS. Once one trajectory is shown to satisfy the postulated tendency another set of parameters and agents’ choices is selected and the new trajectory is similarly checked. If all possible trajectories are successfully tested, the tendency is proved to be necessary relative to the logic of the simulation language, the range of parameterisations and agents’ choices. The idea is to translate the MAS into a constraint-based platform in an automatic or near automatic way without changing the meaning of the rules that make it up in order to perform this automatic testing. In this way a user can program the system using the agent-based paradigm with all its advantages; inspect single runs of the system to gain an intuitive understanding of the system and then check the generality of this understanding for fragments of the system via this translation into a constraint-based architecture. In the example shown below, all trajectories are explored for one combination of parameters, eight agents’ choices per iteration and seven iterations. A simple tendency was observed characterised by a mathematical description of its boundaries. This characterisation was handled as a theorem. The theorem was proved to be necessary following a procedure similar to the one described in the previous paragraph. 2.4 What is New in This Model-Constrained Methodological Approach It is our goal in this paper to propose an alternative approach for exploring and analysing simulation trajectories. It will allow the entire exploration and subsequent analysis of a subspace of the whole space of simulation trajectories. We are suggesting the generation of trajectories in a semantically constrained way. Constrictions will be context-dependent (over the semantics of the trajectory itself) and will be driven via the introduction of a controller or meta-module. Like Scenario Analysis, the idea is to generate individual trajectories for different parameterisations and agents’ choices but unlike Scenario Analysis the exploration is constrained to only certain range of parameters and choices. Akin to Monte Carlo techniques it explores only part of the total range of possible trajectories. But, unlike Monte Carlo studies it explores an entire subspace of (rather than some randomly generated sample) trajectories and is able to give definitive answers for inquires related to the dynamics of the simulation in that subspace. 3 Towards the Implementation of a Suitable Platform for the Envelope of Trajectories: Using SDML and Declarative Programming Paradigm SDML (Strictly Declarative Modelling Language) [8] is the declarative Multi-Agent System in which we have developed our experiments. As a source of comparisons and ideas, we have also programmed our model in a Theorem Prover [2,7,10,11] In general, declarative programming (and in particular SDML) offers desirable features for simulation experiments as compared to imperative programming. For the social simulation community those features seem to be of particular interest when facilitating the exploring and analysis of the dynamics of the simulation [9]. 3.1 Some Characteristics of Declarative Programming • Modularity. Any part of the model is constructed as a group of standardised units (i.e. rules) allowing flexibility and variety in use. The declarative paradigm facilitates a greater level of modularity than the imperative paradigm because the control of the program is separated from the content. This flexibility is useful both when representing the static structure of the system and when generating the dynamics of the simulation. In our case, it facilitates the introduction of alternatives for agents’ choices and parameters of the model. • Expressiveness. Effective conveyance of meaning is a consequence of the representation of the system as linguistic clauses on a set of databases. It facilitates the interpretation of a set of social phenomena into a simulation by allowing the dual interpretation of clauses as pseudo-linguistic tokens and as entities to be computationally manipulated. • Easier analysis. Context situated analysis of detailed data, tracks of trajectories as well as analysis of group of trajectories is much more straightforward than in imperative programs because the resulting databases can be flexibly browsed and queried. • The Possibility of Formal Proof. The data generated by the dynamics of the simulation can be analysed as a logical extension under the particular logic of the simulation language. It opens the possibility of achieving proofs related to the logic of the language and the constraints imposed by the allowed choices of the agents and parameters of the model. 3.2 Other Relevant Characteristics and Features SDML Offers • Good underlying logical properties of the system. SDML’s underlying logic is close to the Strongly Grounded Autoepistemic Logic (SGAL) described by Kurt Konolige [6]. • Its backtracking procedure facilitates the exploration of alternative trajectories via the splitting of simulation paths according to agent’s choices and model’s parameters. • The assumptions manager in SDML tracks the use of assumptions. Assumptions result from choices. • A collection of useful primitives relevant to social simulation. • The type meta-agent. A meta-agent (meta, for our purposes) is an agent “attached” to another agent as a controller; it is able to program that agent. This is used here not as an agent per se but as a module used to ‘compile’ rules into an efficient form as well as to monitor and control the overall search process and goals. 4 Implementing a Suitable Constraint-Based Programming Platform The main goal of the programming strategy to be described is to increase the efficiency in terms of simulation time, thus making an efficient constraint-based search possible. The improvements will be achieved by making the rules and states more context-specific. This enables the language’s inference engine to exploit more information about the logical dependencies between rules and thus increase the efficiency. Thus this can be seen as a sort of ‘compilation’ process, which undoes the agent encapsulation in order to allow the more efficient exploration of the total system behavior. In particular we split the transition rules into one per simulation period, and also by the initial parameters. This necessitates a dynamic way of building rules. This is done via a controller, which generates the rules at the beginning of the simulation. 4.1 An Efficient and Isomorphic Translation of MAS into a Single DB-RB Pair For doing proofs efficiently we propose to carry out an isomorphic transformation [12] from the original MAS along with the range of parameterizations and agents’ choices into a platform where the alternative trajectories can be unfolded more efficiently. Each trajectory will correspond to a possible trajectory in the original MAS. This transformation is a particular case of homomorphism, the two systems are equivalent, they have similar state transitions -there is a one to one translation between the states of the two systems. This translation is needed because of the limited computational resources, the complexity of the task (usually too many trajectories have to be investigated) and the difficulties in experimenting using a MAS model directly since data and rules are encapsulated in different abstractions and hierarchies like those of agents and time levels – as a result rule dependencies are hidden. All this makes difficult and sometimes even impractical to experiment directly with models of complex systems in a MAS. The main goal in doing such a transformation from MAS into a more efficient architecture is to take advantages of making dependencies among rules as explicit as possible and data instantiation more specific. An example where hidden dependencies are first revealed and then unwrapped exploiting them to give efficiency to the model exploration process is shown in section 5. To facilitate this we translate the original MAS into a model with a single database-rulebase pair (see fig. 2). 4.2 An Overview of the System We implemented the proposed architecture in three modules; let us call them model, prover and meta. The following diagram illustrates this: Meta uses initial data to write transition rules Model sets up alternatives for: parameters processes’ choices Prover: Generates trajectory Contradiction? Yes?.Explores other path Otherwise: proof fails InitialData Fig. 3. Overview of the efficient implementation Fig. 2. Transformation of MAS system into a single rulebase-database pair Agent