Acquirable Predictive Models

From distinguishing a state to predicting its future

ProLT observation topologies describe distinctions made by chosen observations. RF-IDENT asks which distinctions a permitted sequence of tests can actually obtain. This report takes the next step: are those attainable distinctions sufficient to predict future outputs under known actions?

Imagine knowing a machine’s complete transition and output tables but not its current state. Two states can look the same now and produce different outputs after an action. Computing that difference from the tables does not mean the available tests can tell you which state the machine occupies.

The report studies a precise finite model: a least state, a greatest state, and an antichain of middle states. Its supplied binary queries must preserve a specified order-complex inclusion up to homotopy before the answer is received. This is a mathematical admissibility rule, not a physical safety guarantee or a theorem about every ordered state space.

Two partitions answer different questions

The acquisition partition R groups states that a legal fixed-state experiment cannot separate. The predictive partition E* groups states whose outputs agree after every finite word of the known actions, including doing nothing.

Exact fixed-state prediction is acquirable precisely when every R-class lies inside an E*-class. In the manuscript’s notation, R ⊆ E*: every pair that permitted sensing must leave together must also have the same future outputs. A finer requested label cannot repair an inaccessible distinction. If the output requirement may be weakened, the report also describes the finest feasible coarsening of those output labels.

The specialized acquisition classification is consolidated from earlier notes. Predictive partition refinement, congruence closure, and the support-planning method use standard constructions; the report does not claim new general algorithms for them.

Why the direction of a test matters

In Example 8, two supplied tests distinguish all three states if restrictions are ignored, yet one necessary test is inadmissible on the remaining support. Supplying its complementary query repairs the experiment. Merely flipping an answer afterward cannot help: the original query would first have to be executable. Available information and an executable observation procedure are different requirements.

Becoming changes what is being identified

When tests alternate with actions, the hidden state can change before the report. Section 7 gives a finite-horizon procedure that checks each query on its current support and allows an action to depend on the answer.

A reset can put every possible starting state into one known current state without revealing where the system began. Conversely, mandatory intervening actions can prevent an experiment that succeeds while the state stays fixed. These examples separate predicting the current system from reconstructing its past.

Place in The MIND programme

Observation refinement studies how adding a distinction changes a representation. This report supplies an operational question for such changes: does the supplied query library allow acquisition of the distinctions needed for a specified prediction task? It does not select or learn new queries, estimate unknown transitions, or optimize sensing costs.

For Being and Becoming, the current predictive class describes what is known now, while quotient dynamics describe how that class updates under known actions. This is a bounded connection between observation and change. The continuous-process bridge proposed in Evolutionary Logic remains separate, as do claims about cognition or self-awareness.

Read and inspect

Read the full report with rendered mathematics, download the PDF, or download the LaTeX and text sources. The reader includes a separate verification download with the retained computational evidence; finite checks corroborate the proofs rather than replace them.

The report is a self-contained technical account with internal AI review. External priority, some source-note metadata, and an applied justification for the sensing constraint remain unresolved. Noise, unknown dynamics, memory restrictions, costs, and general ordered carriers would require additional models and proofs.