Glossary

📍 Where we are: The back-of-the-book reference for all of Volume 5 — every recurring term of trust, scale, and the reasoning frontier, in plain words. Keep it open in a second tab and come back whenever a word stops making sense.

The reasoning frontier inherits Volume 4's differentiable machinery and asks a fifth question of it: whether a system's numbers can be trusted, whether its explanations are honest, and whether its speed survives the jump from a desk to a datacenter. Below are the volume's recurring terms, in plain language, listed alphabetically so they are easy to find. Each entry is a starting point; the chapter it points to gives the full, exact picture.

Abstention policy — a decision rule that lets a system decline to answer rather than guess; Volume 5's three species are the calibrated-confidence cut (Chow's rule), the open-world interval that reads its own ignorance, and the structural refusal of a parser that never accepts an unparseable input. (See Abstention: Knowing When You Do Not Know.)

AURC (area under the risk-coverage curve) — the average selective risk over every coverage level, $\frac{1}{N}\sum_k \text{risk}(k)$; a single-number summary that can hide crossings between two systems' curves. (See Abstention: Knowing When You Do Not Know.)

Boolean matrix product ($\odot$) — the matrix product over the OR-AND semiring, $(X \odot Y)[i,j] = \bigvee_k X[i,k] \wedge Y[k,j]$, computed exactly by an ordinary float matmul of 0/1 matrices followed by a greater-than-zero test; the trick that turns rule firing into a GPU kernel. (See GPU Reasoning: Batching the Fixpoint.)

Calibration — the conditional-probability promise $P(\text{correct} \mid \text{confidence} = p) = p$: among all the answers a system announces at confidence $p$, a fraction $p$ should actually be correct. (See Calibration: Confidence That Means Something.)

Chain of thought (CoT) — intermediate tokens decoded before the final answer, spending one sequential step of the decode loop per generated token; architecturally, a way to buy back the depth a fixed-size network cannot compute in a single forward pass. (See Chain-of-Thought: Recovering P at Decode Time.)

Chow's rule (reject option) — the expected-cost-optimal abstention policy when errors cost 1 and abstaining costs $c_r$: answer with the top class, refuse exactly when its probability falls below $\theta = 1 - c_r$. (See Abstention: Knowing When You Do Not Know.)

Committed check — an assert inside a runnable harness whose expected value is frozen in the repository, so a re-run either reproduces a claim digit for digit or fails loudly; the series' unit of proof in place of a table of rankings. (See The Honest Verdict: The State of Neuro-Symbolic AI.)

Comprehensiveness — $p(\hat{y} \mid x) - p(\hat{y} \mid x \setminus r)$, the confidence a model loses when a rationale $r$ is erased from its input; high means the rationale was actually load-bearing. (See Faithfulness: When an Explanation Is Honest.)

Concept collapse — one minus the normalized Shannon entropy of a concept head's occurrence vector; 0 when every concept value is used, 1 when a single value absorbs every prediction, and the confidently-wrong sibling that plain concept accuracy alone cannot see. (See Measuring Shortcuts: rsbench and Concept Quality.)

Consistency rate — the fraction of a matched probe family's questions on which a system's answers respect the family's logical closure (contraposition, paraphrase, De Morgan duality); computable with no gold labels at all, and the number that separates a system understanding entailment from one guessing at it. (See The Entailment-Benchmark Gap.)

Coverage — the fraction of questions a system chooses to answer rather than abstain on; the axis every abstention policy and every consistency score must be read against, since a perfect score on a shrinking coverage is a different claim than one on the whole set. (See Abstention: Knowing When You Do Not Know.)

Depth (work-depth sense) — the length of the longest chain of operations in which each step needs its predecessor's result; a lower bound on running time no matter how many processors are thrown at the problem. (See The Work-Depth Trilemma.)

Difficulty axis — an independent, mechanical property of a reasoning instance that makes it hard on its own terms, orthogonal to the completeness (recall-by-depth) axis benchmarks usually grade: soundness, termination, proof width, query joins, semantics change, grounding trust, and scale. (See The Seven Difficulty Axes Beyond Completeness.)

Entailment gap — plain accuracy minus a probe family's consistency rate; the committed run in this volume posts gaps of 27.3 and 50.4 points on expansion and paraphrase probes respectively, meaning high accuracy and low logical consistency can coexist in the same system. (See The Entailment-Benchmark Gap.)

Expected calibration error (ECE) — the count-weighted mean absolute gap between stated confidence and observed accuracy across bins, $\sum_b (n_b/N)\,\lvert \mathrm{acc}_b - \mathrm{conf}_b \rvert$; biased and binning-dependent, and still the field's workhorse summary. (See Calibration: Confidence That Means Something.)

Expert iteration — the training loop: sample attempts, keep only verified successes, refit the proposer on their steps; a form of policy improvement against an environment reward that needs no learned value function or reward model. (See Reinforcement Learning and Verifier-Gated Reasoning.)

Faithfulness — the property that an explanation accurately reports the causes of a model's output, judged by counterfactual tests run on the model itself rather than by how plausible the explanation looks to a human reader. (See Faithfulness: When an Explanation Is Honest.)

Fidelity (extraction) — the agreement rate between an extracted symbolic surrogate (a decision tree, a set of rules) and the network it was extracted from, measured over a stated query distribution; always fidelity on some inputs, never in the abstract, and a high fidelity score can still mean the surrogate faithfully reports a spurious rule. (See Rule Extraction: Reading Rules Out of Weights.)

Fixed-L truncation — applying the same reasoning layer $L$ times with no halting test; sound at every $L$ because unfired rules simply do not fire, and complete only once $L$ reaches the knowledge base's derivation diameter $D$. (See Symbolic Attention: Reasoning as a Single Burst.)

Goodhart's law — once a measure becomes a target, it stops being a good measure; for a latent property like concept meaning, the identifiability theorem of this volume makes the failure structural rather than a matter of bad luck. (See NeSy Benchmark Suites and Simulators.)

Hitting-set tree (HST) — the enumeration tree whose nodes are justifications of an ontology with the path's deleted axioms removed, and whose edges each delete one axiom of the parent's justification; complete for finding every justification of an entailment. (See Justifications: Minimal Proofs and Pinpointing.)

Identifiability (of concepts from labels) — the property that a label distribution determines concept semantics uniquely; holds exactly when a knowledge base's symmetry group is trivial, and fails whenever some non-trivial relabeling of concept states leaves every observed label untouched. (See Identifiability: Why Accuracy Never Certifies Meaning.)

Justification (MinA, minimal axiom set) — a subset $J$ of an ontology that entails a given consequence while no proper subset of $J$ does; the minimal evidence for an entailment, and the object a hitting-set tree enumerates in full. (See Justifications: Minimal Proofs and Pinpointing.)

Materialization — computing and storing the least fixpoint of a rule set over the base facts at load time, so that later queries become plain index lookups; the strategy paid for once, up front, and amortized over every query that follows. (See Materialization versus Rewriting: Two Ways to Scale.)

Meta-cognition — a system modeling its own competence boundary, the region of inputs where its machinery is reliable, and routing accordingly rather than merely reporting a confidence number; the volume's thinnest-instrumented open problem, present in the literature at roughly five percent of surveyed work. (See Open Problems: The G1–G8 Gaps.)

Oracle-first design — the research discipline of building an exact reference implementation of a target semantics before any learned system touches it, so every later claim has a committed procedure capable of falsifying it. (See Doing the Next Research: Blind Spots and First Steps.)

Pinpointing — locating exactly which axioms are responsible for an entailment, either black-box (using only an entailment oracle) or glass-box (instrumenting the reasoner itself to emit a monotone Boolean formula whose satisfying valuations are the justifications). (See Justifications: Minimal Proofs and Pinpointing.)

Planted-rule protocol — hide a known rule from a learner, train it from that rule's query answers alone, and require the learned weights to decode back to the exact hidden rule; a recovery standard stronger than similarity, since it demands syntactic identity. (See The SATORI Architecture: One Operator, Four Properties.)

Process supervision — training on per-step judgments of a reasoning trace rather than on outcome-only judgment of the final answer; exact and free wherever a symbolic verifier can grade every step, approximated by a trained process-reward model everywhere else. (See Reinforcement Learning and Verifier-Gated Reasoning.)

Query rewriting — unfolding a query backward through a rule set into a union of conjunctive queries over base predicates, evaluated directly against the raw, unmaterialized data; the load-free alternative to materialization, paid per query instead of up front. (See Materialization versus Rewriting: Two Ways to Scale.)

Reasoning shortcut (RS) — a parameterization that maximizes label likelihood while its internal concept distribution diverges from the ground truth; an unintended optimum that is right on every label for the wrong reasons. (See Reasoning Shortcuts: Right Answer, Wrong Reason.)

Relabeling ($\alpha$) — a function carrying ground-truth concept states to a model's concept states that describes a systematic semantic error; admissible, and therefore a candidate optimum, exactly when it commutes with the task's label map on the states the training data actually visits. (See Reasoning Shortcuts: Right Answer, Wrong Reason.)

Reliability diagram — the binned estimator of calibration: for each confidence bin, the count, the mean stated confidence, and the observed accuracy, with any gap between the latter two being the miscalibration the bin exhibits. (See Calibration: Confidence That Means Something.)

Repeated squaring — closing a reachability relation by squaring $I \vee M$ against itself; reaches the fixpoint in $\lceil \log_2 D \rceil$ productive rounds where $D$ is the diameter, trading a semi-naive method's sparse, linear-round cost for dense, logarithmic-round work. (See The Work-Depth Trilemma.)

Reward hacking — an optimizer maximizing a proxy reward at the expense of the intent the reward was meant to proxy; this volume's committed exhibit is a membership-rewarded policy that farms valid-but-irrelevant facts, collapsing the relevant-fact fraction from 0.382 to 0.154. (See Reinforcement Learning and Verifier-Gated Reasoning.)

Risk-coverage curve — selective risk plotted against coverage as a confidence threshold sweeps across the full range; one point per cut of the confidence-sorted question list, and the curve every abstention policy is priced against. (See Abstention: Knowing When You Do Not Know.)

Rule extraction — producing an explicit symbolic surrogate, a tree, a disjunctive normal form, or a set of chain rules, for a trained network, together with a measured agreement between the surrogate and the network it was read out of. (See Rule Extraction: Reading Rules Out of Weights.)

Selective prediction (selective classification) — the setting in which a predictor may decline to answer; formally a pair of a predictor and a selection function, graded jointly by coverage and selective risk rather than by accuracy alone. (See Abstention: Knowing When You Do Not Know.)

Semi-naive (delta) evaluation — firing a rule only through premises that are new since the previous round, joined against the full state for the rule's other premises; skips every rule instance that would only re-derive an already-known fact, with no repetition, by construction. (See Materialization versus Rewriting: Two Ways to Scale.)

Sufficiency — $p(\hat{y} \mid x) - p(\hat{y} \mid r)$, the confidence a model retains when only a rationale $r$ survives and everything else is erased; low, or even negative, means the rationale alone already carries the prediction. (See Faithfulness: When an Explanation Is Honest.)

Symbolic attention — the rendering of a fixpoint completion procedure as an attention operator: rule bodies scored by a softened minimum over their premises, rival derivations merged by a softened maximum, and role composition run as a message pass over soft adjacency matrices. (See Symbolic Attention: Reasoning as a Single Burst.)

Symmetry group (of knowledge) — the group, under composition, of every bijective relabeling of concept states that leaves a knowledge base's label map unchanged; identifiability holds exactly when this group is trivial. (See Identifiability: Why Accuracy Never Certifies Meaning.)

TC⁰ — the circuit complexity class of constant-depth, polynomial-size circuits built from unbounded fan-in AND/OR gates plus threshold gates; the ceiling theorem places every fixed-depth, log-precision transformer with no intermediate decoding inside it, for any setting of its weights. (See The Transformer Depth Ceiling: TC⁰ and P.)

Temperature scaling — the one-scalar calibration repair $p = \sigma(s/\tau)$, fitted by minimizing held-out negative log-likelihood; a temperature above 1 softens an overconfident model, one below 1 sharpens an underconfident one, and the rescaling touches no ranking, so accuracy never moves. (See Calibration: Confidence That Means Something.)

Trace-erasure protocol — the general two-sided faithfulness test: erase the evidence an explanation cites and the prediction must move; erase evidence it does not cite and the prediction must not; justifications pass it by theorem, statistical explanations must pass it by measurement. (See Faithfulness: When an Explanation Is Honest.)

Verifier-constrained decoding — applying an exact verifier to every proposed reasoning step during generation, so that an invalid step never enters the derived record at all, in contrast to reranking, which scores only completed outputs after the fact. (See Reinforcement Learning and Verifier-Gated Reasoning.)

Wedge (research) — a related-work argument locating an empty conjunction of properties that no existing family of systems covers at once; an opportunity claim about the literature, never itself a result about the world. (See The Eight Claims: C1–C8 and the Wedge.)

Work — the total number of operations a computation performs, summed across every processor that runs it; the bill someone always pays in energy and hardware no matter how fast the wall clock reads. (See The Work-Depth Trilemma.)