Abstract
Using simple systems with a notion of discrete deterministic evolution over time, we study discrete causality via tools from theoretical computer science and logic. We consider the set of all representations (i.e. partial descriptions) of such systems, from algebraic, domain-theoretic, and categorical viewpoints. The order theory introduced, is based in the notion of comparing high-level and low-level descriptions of the same system. This is shown to give a Complete Partial Order where the down-closure of each element is a locale. This partial order has a very close connection to a categorical construction known as the ‘particle-style’ trace, via analogues of domain-theoretic equations. Thus, the trace may be thought of as the computation of suprema in this partial order. As a sample application, we show how to construct algebraic models of space-bounded Turing machines in these terms, and derive compositionality from the abstract properties of the trace.
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