Proof Tree or Derivation Log¶
Record artifact — instantiates Formal Derivation System Design
Records each derivation as a structured, auditable artifact — which rules fired on which inputs to reach the conclusion — and archives representative cases as reusable exemplars.
A Proof Tree or Derivation Log is the durable, human-auditable record of a derivation: a tree (or ordered log) showing, for each step, which rule fired on which inputs to yield which intermediate result, all the way to the conclusion. Its defining commitment is that it is passive testimony — it answers the question "why did we conclude this?" by exhibiting the actual chain of steps, and it does so without judging whether those steps were valid or executing anything itself. It is the artifact you read, cite, and archive. Because a good log accumulates instructive cases, it doubles as a suite of exemplar derivations that show newcomers and downstream tools what a real derivation in the system looks like.
Example¶
A government benefits agency runs eligibility by rule-based derivation, and every determination is required to be explainable to the applicant and to auditors. For each application the system writes a derivation log: the inputs (household size, monthly income, residency status), then each eligibility rule that fired — "income below threshold T for household size 4," "residency requirement met" — the intermediate conclusions, and the final determination with its supporting subtree. When an applicant appeals a denial, a caseworker opens the log and reads exactly which rule produced the "ineligible" result and on which input, rather than re-arguing the case from scratch. The agency also keeps a curated set of representative determinations — a clear approval, a clear denial, a tricky boundary case — as exemplars that new caseworkers study and that regression checks re-run whenever the rules change.
How it works¶
- Capture each step. Every derivation step is recorded as (rule, inputs, output), linked to its premises, so the whole derivation forms an inspectable tree or ordered log.
- Preserve structure, not just outcome. The record keeps how the conclusion was reached, not merely that it was, so the reasoning can be replayed and questioned line by line.
- Stay agnostic about validity. The log faithfully records whatever steps were taken; it neither vouches for nor rejects them — validity is a separate question left to a checker.
- Curate exemplars. Representative derivations are promoted into a maintained suite that serves as documentation, onboarding material, and regression fixtures.
Tuning parameters¶
- Granularity — log every micro-step vs. only decision milestones. Fine granularity maximizes auditability; coarse logging is cheaper and more readable.
- Structure — a full proof tree vs. a linear step log. Trees expose dependency and sharing; linear logs are simpler to store and scan.
- Retention and versioning — how long derivations are kept and whether they are tied to the rule version that produced them. Longer retention aids audit; version-tagging lets old decisions be reproduced exactly.
- Exemplar-selection criteria — which derivations get promoted to the exemplar suite. Broad coverage teaches and tests more; a tight set stays maintainable.
When it helps, and when it misleads¶
Its strength is explainability and reproducibility: a determination stops being an opaque verdict and becomes a chain anyone can inspect, replay, and appeal — indispensable wherever automated conclusions must be justified.[n1]
Its failure mode is a subtle one of authority: a recorded derivation is only as valid as the rules that produced it. A log can faithfully and completely record an invalid derivation — every step present, the reasoning still wrong — because recording is not checking. The classic misuse is treating "there is a full trace" as if it meant "the trace is correct," letting the mere existence of documentation stand in for verification. The guarding discipline is to keep the log's job narrow — it testifies to what happened, and pairs with a checker that rules on whether it was legal. The two questions must not be collapsed.
How it implements the components¶
derivation_trace_record— it is the artifact: the structured proof tree or step log that records how each conclusion was reached.exemplar_derivation_suite— it curates representative derivations into a maintained suite used for documentation, onboarding, and regression fixtures.
It records derivations; it does not adjudicate them. It neither validates a step against the rules (inference_rule_set) nor certifies that the conclusion lies inside the derivable set (closure_boundary) — those verdicts belong to its nearest twin, the Mechanical Proof Checker, which re-checks the very traces this log preserves.
Related¶
- Instantiates: Formal Derivation System Design — provides the durable, auditable record of the system's derivations.
- Consumes: Inference Rule Calculus — the log records applications of the rules the calculus declares.
- Sibling mechanisms: Formal Grammar Specification · Well-Formedness Linter · Axiom Schema Catalog · Inference Rule Calculus · Rewrite or Transition Rule Engine · Mechanical Proof Checker · Metatheory Review Checklist · Consistency and Contradiction Test · Formal-System Change-Control Workflow
Editorial Notes¶
Form Classification¶
Form family: Record, Log & Register
Rationale: Proof Tree or Derivation Log operates as a persistent ledger, log, register, or case record that preserves history and traceability because it records each derivation as a structured, auditable artifact — which rules fired on which inputs to reach the conclusion — and archives representative cases as reusable exemplars.
Independent corroboration: The frozen evidence defines Proof Tree or Derivation Log as 'Records each derivation as a structured, auditable artifact — which rules fired on which inputs to reach the conclusion — and archives representative cases as reusable exemplars', so its operative form is Record, Log & Register.
Review outcome: Independent reviewer agreement; high confidence.
Origin Attribution¶
Primary origin: Mathematics
Origin pattern: Cross-disciplinary synthesis
Present-day reach: Universal
Rationale: Proof Tree or Derivation Log is most plausibly rooted in the mathematics tradition because its characteristic form depends on formal definition, proof, mapping, and quantitative structure. The assignment tracks that formative lineage, not the many settings in which the mechanism can now be applied.
Related originating lineages:
- Computer Science & Software Engineering — The computer_science tradition materially shaped Proof Tree or Derivation Log through its own practice of algorithms, data structures, formal interfaces, and software-system practice.
- Philosophy — The philosophy tradition materially shaped Proof Tree or Derivation Log through its own practice of logic, argument, normative analysis, and conceptual distinction.
Review resolution: Both blind reviewers agree that mathematics is the primary origin. Explicit reconciliation resolves origin mode disagreement, domain reach disagreement, encyclopedia synthesis disagreement. Formative alternate lineages are retained as computer_science, philosophy; later breadth of use is recorded separately as domain_reach=universal, while origin_mode=cross_disciplinary_synthesis describes the relationship among origin lineages.
Encyclopedia synthesis: The exact catalogued form synthesizes established practice rather than reproducing a single standard historical label.
Review outcome: Reconciled after independent review; high confidence.
Notes¶
[n1] The "right to explanation" associated with automated decision-making — reflected in the EU's General Data Protection Regulation (notably Recital 71) — expects that a person subject to an automated decision can obtain meaningful information about the logic involved. A derivation log is the concrete artifact that makes such an explanation possible, though it evidences the how, not the correctness. ↩