Sufficient Reasons: Invariance and Minimality

• Sufficient reasons are defined as fixed subsets of conditions that guarantee an outcome, operationalized through invariance across various models.
• They are characterized using logical, probabilistic, and counterfactual frameworks to capture minimal yet robust causes in symbolic and neural settings.
• Applications span explainable machine learning, causal inference, and philosophy, addressing computational tractability and the nature of knowledge and determinism.

“Sufficient reasons” designates a family of notions built around a guarantee relation: a reason is sufficient when fixing it is enough for an outcome to obtain. The phrase has distinct technical meanings in explainable machine learning, causal modeling, epistemology, argument-quality theory, and philosophy of science. In symbolic classification it denotes minimal feature assignments that force a decision; in neural settings it is often defined by prediction invariance under perturbations of unfixed features; in causal models it denotes variable settings that guarantee an event under all admissible contingencies; in epistemology it is tied to adequate reasons that suffice for knowledge; and in the philosophical Principle of Sufficient Reason it concerns whether facts admit lawful or reason-governed explanation (Darwiche et al., 2022, Égré et al., 2014, Künnemann, 2017, Romero, 2014).

1. General schema: guarantee, invariance, and minimality

A common formal pattern is to treat a sufficient reason as a subset of conditions whose fixation preserves an outcome throughout a specified possibility space. In neural classification, this space may be defined by a single baseline assignment, by all perturbations in an $\ell_p$-ball, or by a conditional data distribution. For a classifier $f:\mathbb{R}^n \to \mathbb{R}^c$ and input $x$, a subset $S \subseteq [n]$ is a baseline sufficient reason if

$$\arg\max_j f((x_S; z_{\bar S}))_j = \arg\max_j f(x)_j,$$

a robust sufficient reason if the same equality holds for all $z \in B_p^{\epsilon_p}(x)$, and a probabilistic sufficient reason if

$$\Pr_{z \sim \mathcal{D}}[\arg\max_j f(z)_j = \arg\max_j f(x)_j \mid z_S = x_S] \ge 1-\delta.$$

These definitions make sufficiency an invariance property under permitted changes to the complement $\bar S$ (Bassan et al., 5 Feb 2025).

Minimality is not unique. The literature distinguishes minimal sufficient reasons, for which no proper subset remains sufficient, from cardinally minimal sufficient reasons, which have minimum size among all sufficient reasons. This distinction matters because many algorithms can certify sufficiency without certifying minimum cardinality, and several hardness results concern the latter problem (Bassan et al., 5 Feb 2025).

A plausible synthesis is that “sufficient reason” is best understood not as a single concept but as a schema instantiated by different modal backgrounds: logical implication in symbolic models, probabilistic invariance in robust or distributional models, counterfactual intervention in causal models, and truth-conduciveness in epistemology.

2. Symbolic explanation in classifiers

In symbolic explainability, sufficient reasons are defined relative to a classifier formula and a specific instance. For a multi-class classifier with class formulas $\Delta_1,\ldots,\Delta_m$, if an instance $\delta$ satisfies an active class formula $f:\mathbb{R}^n \to \mathbb{R}^c$0, its complete reason is

$f:\mathbb{R}^n \to \mathbb{R}^c$1

This formula characterizes exactly the instances congruent to $f:\mathbb{R}^n \to \mathbb{R}^c$2, in the sense that $f:\mathbb{R}^n \to \mathbb{R}^c$3 is congruent to $f:\mathbb{R}^n \to \mathbb{R}^c$4 iff $f:\mathbb{R}^n \to \mathbb{R}^c$5. Prime implicants of $f:\mathbb{R}^n \to \mathbb{R}^c$6 are sufficient reasons; prime implicates are necessary reasons. The paper identifies sufficient reasons with PI explanations and abductive explanations, and necessary reasons with contrastive explanations (Darwiche et al., 2022).

The semantic role of the complete reason is stronger than local implication alone. A sufficient reason is not merely a subset of observed literals that preserves the class label; it is a minimal subset that guarantees the decision for the same reason. Conversely, a necessary reason identifies literals that must collectively hold for the decision to persist. When all variables are binary, flipping all literals in a necessary reason necessarily flips the decision. This connects sufficiency to “why this decision?” and necessity to “what must hold?” or “what must change?” (Darwiche et al., 2022).

A closely related Boolean framework represents a classifier by a propositional formula $f:\mathbb{R}^n \to \mathbb{R}^c$7 and uses the decision-specific formula $f:\mathbb{R}^n \to \mathbb{R}^c$8, equal to $f:\mathbb{R}^n \to \mathbb{R}^c$9 for positive decisions and $x$0 for negative ones. A sufficient reason for decision $x$1 is then a prime implicant of $x$2 that is satisfied by $x$3. The complete reason

$x$4

is the canonical disjunction of all sufficient reasons. Necessary characteristics are precisely the literals entailed by $x$5, and reason circuits obtained by consensus and filtering on Decision-DNNF make these notions tractable enough for counterfactual queries and bias analysis (Darwiche et al., 2020).

This symbolic literature treats sufficient reasons as exact logical guarantees. Its characteristic advantage is faithfulness: explanations are not surrogates or approximations but prime implicant structure of formulas equivalent to the classifier’s decision behavior (Darwiche et al., 2020).

3. Extensions: non-binary features, constraints, and computational complexity

The binary setting omits structure present in nominal and discretized numeric features. For non-binary domains, literals can denote nonempty proper subsets of a feature domain, and the general reason is defined by a selection operator:

$x$6

This yields general sufficient reasons as variable-minimal prime implicants of $x$7 and general necessary reasons as variable-minimal prime implicates. The improvement is substantive: with non-binary features, explanations can express invariants such as “BloodType $x$8” or “BMI $x$9,” which are invisible when one restricts reasons to singleton observed states. In the binary case, $S \subseteq [n]$0, so the generalized framework is conservative (Ji et al., 2023).

Constraints further alter what counts as sufficient. When domain knowledge is encoded by a propositional formula $S \subseteq [n]$1, the classifier under constraints is treated as a partial Boolean function, undefined outside $S \subseteq [n]$2. Constrained sufficient reasons are $S \subseteq [n]$3-prime implicants, equivalently prime implicants of $S \subseteq [n]$4. The key theorem is parsimony: for every unconstrained sufficient reason $S \subseteq [n]$5, there exists a constrained sufficient reason $S \subseteq [n]$6 with $S \subseteq [n]$7. This means that incorporating feasibility constraints can only preserve or improve succinctness (Gorji et al., 2021).

Decision trees expose both tractability and hardness. For Boolean decision trees, sufficient reasons are prime implicants of the classifier or of its negation, restricted to those covering the instance. The number of sufficient reasons can be exponentially large, and the number of minimal-size sufficient reasons can also be exponentially large. To summarize this space, the literature introduces relevant features, which occur in at least one sufficient reason, and necessary features, which occur in every sufficient reason; both can be computed in time $S \subseteq [n]$8 (Audemard et al., 2021).

For richer tree and graph models, complete reasons often admit efficient closed forms. For decision graphs satisfying the weak test-once property, the complete reason is a monotone, $S \subseteq [n]$9-decomposable NNF. This yields an output-polynomial algorithm, SNR, for enumerating all shortest necessary reasons, with time $$\arg\max_j f((x_S; z_{\bar S}))_j = \arg\max_j f(x)_j,$$0 and space $$\arg\max_j f((x_S; z_{\bar S}))_j = \arg\max_j f(x)_j,$$1 on the pruned circuit, and for decision trees specifically $$\arg\max_j f((x_S; z_{\bar S}))_j = \arg\max_j f(x)_j,$$2 where $$\arg\max_j f((x_S; z_{\bar S}))_j = \arg\max_j f(x)_j,$$3 is the number of leaves of the opposite class. By contrast, deciding whether there exists a sufficient reason of length $$\arg\max_j f((x_S; z_{\bar S}))_j = \arg\max_j f(x)_j,$$4 for a decision tree is NP-complete, and shortest sufficient reasons remain hard even for a single reason; SSR is therefore presented as a practical rather than worst-case-polynomial procedure (Darwiche et al., 2022).

Approximation is also sharply limited. For $$\arg\max_j f((x_S; z_{\bar S}))_j = \arg\max_j f(x)_j,$$5-sufficient reasons in decision trees, the inapproximability result states that, unless SAT can be solved in quasi-polynomial time, no polynomial-time algorithm can distinguish between the existence of a $$\arg\max_j f((x_S; z_{\bar S}))_j = \arg\max_j f(x)_j,$$6-sufficient reason of size at most $$\arg\max_j f((x_S; z_{\bar S}))_j = \arg\max_j f(x)_j,$$7 and the absence of any $$\arg\max_j f((x_S; z_{\bar S}))_j = \arg\max_j f(x)_j,$$8-sufficient reason of size less than $$\arg\max_j f((x_S; z_{\bar S}))_j = \arg\max_j f(x)_j,$$9 (Kozachinskiy, 2023).

4. Neural and language-model realizations

Neural approaches replace exact symbolic implication with learned invariance. “Sufficient subset training” trains a model

$z \in B_p^{\epsilon_p}(x)$0

to output both a prediction and an explanation mask. With thresholded subset $z \in B_p^{\epsilon_p}(x)$1, the training objective is

$z \in B_p^{\epsilon_p}(x)$2

where prediction loss is cross-entropy to the ground truth, faithfulness aligns the masked prediction $z \in B_p^{\epsilon_p}(x)$3 with the original predicted class, and $z \in B_p^{\epsilon_p}(x)$4 promotes conciseness. The paper proves that obtaining a cardinally minimal sufficient reason is NP-Complete for baseline sufficient reasons, $z \in B_p^{\epsilon_p}(x)$5-Complete for robust sufficient reasons, and NP$z \in B_p^{\epsilon_p}(x)$6-Hard for probabilistic sufficient reasons, while empirically reporting explanation times of approximately $z \in B_p^{\epsilon_p}(x)$7–$z \in B_p^{\epsilon_p}(x)$8 s on vision tasks, compared with Anchors at $z \in B_p^{\epsilon_p}(x)$9–$$\Pr_{z \sim \mathcal{D}}[\arg\max_j f(z)_j = \arg\max_j f(x)_j \mid z_S = x_S] \ge 1-\delta.$$0 s and SIS at seconds to minutes (Bassan et al., 5 Feb 2025).

A different neural formalization treats reasons as vectors over worlds. For a neuron $$\Pr_{z \sim \mathcal{D}}[\arg\max_j f(z)_j = \arg\max_j f(x)_j \mid z_S = x_S] \ge 1-\delta.$$1, the reasons vector is $$\Pr_{z \sim \mathcal{D}}[\arg\max_j f(z)_j = \arg\max_j f(x)_j \mid z_S = x_S] \ge 1-\delta.$$2 at world $$\Pr_{z \sim \mathcal{D}}[\arg\max_j f(z)_j = \arg\max_j f(x)_j \mid z_S = x_S] \ge 1-\delta.$$3, and belief update by a reason $$\Pr_{z \sim \mathcal{D}}[\arg\max_j f(z)_j = \arg\max_j f(x)_j \mid z_S = x_S] \ge 1-\delta.$$4 is

$$\Pr_{z \sim \mathcal{D}}[\arg\max_j f(z)_j = \arg\max_j f(x)_j \mid z_S = x_S] \ge 1-\delta.$$5

The strength with which a reason speaks for a proposition $$\Pr_{z \sim \mathcal{D}}[\arg\max_j f(z)_j = \arg\max_j f(x)_j \mid z_S = x_S] \ge 1-\delta.$$6 is the log-odds shift

$$\Pr_{z \sim \mathcal{D}}[\arg\max_j f(z)_j = \arg\max_j f(x)_j \mid z_S = x_S] \ge 1-\delta.$$7

This framework treats neurons and groups of neurons as proposition-sensitive reasons vectors, allows additive aggregation, and validates faithfulness by intervention. In sentiment experiments on Qwen2.5-0.5B-Instruct, intervening on the top-5 residual-stream positions speaking most strongly for the opposite sentiment flips correct predictions with success rates of $$\Pr_{z \sim \mathcal{D}}[\arg\max_j f(z)_j = \arg\max_j f(x)_j \mid z_S = x_S] \ge 1-\delta.$$8 and $$\Pr_{z \sim \mathcal{D}}[\arg\max_j f(z)_j = \arg\max_j f(x)_j \mid z_S = x_S] \ge 1-\delta.$$9 in the two directions reported (Hornischer et al., 20 May 2025).

In chain-of-thought reasoning, sufficiency and necessity are cast causally. A reasoning chain $\bar S$0 is sufficient when intervention $\bar S$1 yields the correct answer, formalized by the Probability of Sufficiency

$\bar S$2

Step indispensability is quantified by Probability of Necessity and Sufficiency, and the optimization procedure first checks chain-level sufficiency and then prunes steps with low necessity. The reported gains are large: for DeepSeek-R1 to DeepSeek-V3 on AIME, tokens drop from $\bar S$3 to $\bar S$4, steps from $\bar S$5 to $\bar S$6, and accuracy rises from $\bar S$7 to $\bar S$8 (Yu et al., 11 Jun 2025).

5. Argumentative, epistemic, and causal senses

In argument-quality research, an argument is sufficient when its premises make the conclusion rationally worthy to be drawn. This notion is operationalized by the hypothesis that the conclusion of a sufficient argument can be generated from its premises. On persuasive essays, a structure-aware RoBERTa sufficiency classifier reaches macro F1 $\bar S$9, outperforming the previous state of the art and matching a reported human upper bound on a subset, while conclusion generation with BART-supervised improves BERTScore from $\Delta_1,\ldots,\Delta_m$0 to $\Delta_1,\ldots,\Delta_m$1 over BART-unsupervised (Gurcke et al., 2021).

In epistemology, the central question is whether reasons can be sufficient for knowledge rather than merely for belief. A logic of reason-based belief introduces formulas $\Delta_1,\ldots,\Delta_m$2 (“reason $\Delta_1,\ldots,\Delta_m$3 supports $\Delta_1,\ldots,\Delta_m$4”), $\Delta_1,\ldots,\Delta_m$5 (“$\Delta_1,\ldots,\Delta_m$6 is an adequate reason”), and $\Delta_1,\ldots,\Delta_m$7 (“the agent believes $\Delta_1,\ldots,\Delta_m$8”). Adequacy is tied to veridicality by

$\Delta_1,\ldots,\Delta_m$9

and to correct support-ascription by

$\delta$0

Externalist justified true belief is defined as

$\delta$1

and knowledge is proposed as

$\delta$2

On this account, a reason is sufficient for knowledge when it supports $\delta$3, is accepted by the agent, and is in fact adequate; Gettier cases challenge internalist JTB but not this externalist notion (Égré et al., 2014).

Within Pearl/Halpern structural causal models, sufficient causes and necessary causes are formally dual. Necessary causation is captured by an existential intervention that negates the outcome; sufficient causation by a universal condition requiring that fixing a set of variables guarantees the outcome under all assignments to the complement. The paper proves that, once the family of necessary causes is encoded as a Boolean DNF over subsets of variables, the family of sufficient causes is obtained by converting to CNF and switching $\delta$4 and $\delta$5, and vice versa (Künnemann, 2017).

6. The Principle of Sufficient Reason in science, physics, and mathematics

The philosophical Principle of Sufficient Reason is not treated uniformly in contemporary work. One influential reconstruction rejects its status as either a necessary truth or a law of nature and reformulates it as

$\delta$6

On this view, the principle is a metanomological methodological hypothesis: it directs inquiry toward lawful mechanisms, is justified a posteriori by scientific success, and does not entail that every event has a cause. The world is described as “legal and determinate, but not strictly causal” (Romero, 2014).

A more ambitious line uses PSR to modify orthodox quantum mechanics. In this proposal, the usual Born statistics remain the pragmatic default when the reason behind Nature’s choice is unknown, but outcome probabilities may be slightly biased when an empirically identifiable reason favors some results, for example positive or congruent experiences. Because each collapse defines a new effective past in Tomonaga–Schwinger process time, such reason-guided bias can generate empirically manifest but only apparent retrocausal effects. The papers emphasize that no explicit biased-probability formula is supplied; the modification is qualitative and reason-sensitive rather than a new dynamical law (Stapp, 2011, Stapp, 2011).

A stronger deterministic reading argues that PSR implies determinism. In that construction, a quantum jump with probabilities $\delta$7 is selected by a hidden seed $\delta$8 through the cumulative-threshold rule, and the “canonical” deterministic choice is

$\delta$9

the fractional part of graduated cosmic time at the $f:\mathbb{R}^n \to \mathbb{R}^c$00-th jump. The resulting dynamics preserves Schrödinger evolution between jumps while making collapse outcomes deterministic relative to the time-derived seed (Mashkevich, 2010).

In contemporary philosophy of mathematics, “good reasons” for a theorem are contrasted with mere existence of a proof. Proof complexity suggests that many tautologies require superpolynomial or exponential-length proofs, and the paper reports that, under the widely believed assumption NP$f:\mathbb{R}^n \to \mathbb{R}^c$01coNP, among tautologies of length $f:\mathbb{R}^n \to \mathbb{R}^c$02, at least $f:\mathbb{R}^n \to \mathbb{R}^c$03 are hard. This creates a dilemma: either mathematicians mainly encounter “reasonable truths,” or the good reasons they seek do not always coincide with reasonably efficient proofs. The resulting distinction between proof and reason is used to analyze explanatory practice, ontology formation, and the prospect of “alien lemmas” discovered by non-human systems (DeDeo, 2024).