The ambient calculus of Cardelli and Gordon is a process calculus for describing mobile computation where processes may reside within a hierarchy of locations, called ambients. The dynamic semantics of this calculus is presented in a chemical style that allows for a compact and simple formulation. In this semantics, an equivalence relation, the spatial congruence, is defined on the top of an unlabelled transition system, the reduction system. Reduction is used to represent a real step of evolution (in time), while spatial congruence is used to identify processes up to particular (spatial) rearrangements.
In this paper, we show that it is decidable to check whether two ambient calculus processes are spatially congruent, or not. Our proof is based on a natural and intuitive interpretation of Mobility processes as edge-labelled finite-depth trees. This allows us to concentrate on the subtle interaction between two key operators of the ambient calculus, namely restriction, that accounts for the dynamic generation of new location names, and replication, used to encode recursion. The result of our study is the definition of an algorithm to decide spatial congruence and a definition of a normal form for processes that is useful in the proof of interesting equivalence laws.