A new modelling framework for the study of the dynamics of urban systems is proposed. We generalize the notion of cellular automaton, using continuous variables to describe the state of a cell. The time evolution is given by Poisson-distributed stochastic jumps corresponding to urban interactions, with intensities depending on the configuration of the system in a suitable set of neighbourhoods. These interactions result from decision processes of populations of cognitive agents modelled in the framework of fuzzy decision theory. The behaviour of agents is driven by goals and constraints, partially defined using a fuzzy logic formalization of a real estate appraisal method based on the evaluation of the attractiveness of a terrain for different land uses. Randomness of the dynamics is a consequence of the fuzziness of these decisions. Hence the model can be seen as a multiagent system with sound mathematical foundation. For example, the use of a continuum-valued state space enables us to prove that the expectation of state variables verify a system of differential equations, and thus to study the system from a dynamical systems theory perspective. In this paper we present the theoretical setting, leaving the applications to a second paper.
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