Memory · Formalism · Historicity
This framework is the preliminary page to two articles on material memory. It fixes the formal vocabulary before entering the applications.
Residual predictive historical dependence
Fix a system boundary S, a reference instant t, a descriptive resolution R, a past window P⁻(t) and a future window F⁺(t). A system exhibits residual predictive historical dependence when its future is not conditionally independent of its past once its present is described.
Formally: I(F⁺ ; P⁻ | X_R) > 0
The inequality means that the retained present description is not predictively sufficient for the future considered. Part of the history still carries information not screened off by the present state.
The equality I = 0 is almost never demonstrated directly in an empirical study. Conditional mutual information is estimated from a finite sample, with measurement errors and an always-incomplete model of the present state.
Three variants
The predictive variant concerns historical dependence still relevant to the future after conditioning on the present. The retrodictive variant concerns reconstruction: I(H ; X_R) > 0 indicates the present retains information about the targeted history. The functional variant does not reduce to a quantity of information: a trace becomes functional when a difference it carries causally participates in a process belonging to the system's organisation.
Path dependence vs memory
Path dependence is a weaker level: it means the present state was causally produced by a particular history, but does not guarantee that this history is recoverable from the present. Memory further requires that the trace can be reactivated, selected or mobilised in the system's current functioning.
SUMMARY
This formal framework distinguishes four levels: path dependence (the past produced the present), trace (a present difference depends on a past event), memory (the trace can be functionally reactivated) and reconstructibility (the trace allows a constrained inference about the past). These levels do not necessarily coincide.