🤖 AI Summary
This paper establishes a theoretical bridge between process calculus and fractal geometry. Methodologically, Milner’s process expressions are reinterpreted as contraction mappings on complete metric spaces, ensuring that their fixed-point semantics precisely coincide with classical fractal sets generated by iterated function systems (IFS); subsequently, a novel behavioral congruence—“fractal equivalence”—is introduced, accompanied by a sound and complete axiomatization. The main contributions are threefold: (1) the first rigorous semantic mapping from process syntax to geometric fractal objects; (2) the identification of a fundamental correspondence between recursive process structure and fractal self-similarity; and (3) a unification of process behavioral theory, labeled Markov chains, and invariant measures, thereby providing a novel foundation for probabilistic fractal modeling.
📝 Abstract
We forge connections between the theory of fractal sets obtained as attractors of iterated function systems and process calculi. To this end, we reinterpret Milner's expressions for processes as contraction operators on a complete metric space. When the space is, for example, the plane, the denotations of fixed point terms correspond to familiar fractal sets. We give a sound and complete axiomatization of fractal equivalence, the congruence on terms consisting of pairs that construct identical self-similar sets in all interpretations. We further make connections to labelled Markov chains and to invariant measures. In all of this work, we use important results from process calculi. For example, we use Rabinovich's completeness theorem for trace equivalence in our own completeness theorem. In addition to our results, we also raise many questions related to both fractals and process calculi.