Conditional Non-Redundancy in CSPs & Beyond

• Conditional non-redundancy is a measure of an element’s indispensability within a conditioned context, ensuring that its removal alters model behavior or recoverable information.
• It is applied in CSPs and knowledge bases to eliminate redundant constraints, support complexity analysis, and improve algorithm efficiency through contextual testing.
• It informs representation learning by targeting label-irrelevant overlap, thereby enhancing ensemble diversity and reducing computational redundancy in deep models.

Searching arXiv for papers on conditional non-redundancy across CSPs, information theory, and representation learning. Conditional non-redundancy denotes indispensability relative to a conditioning context. The conditioning object may be the rest of a constraint set, a larger scaffold relation $Q\supseteq P$, another predictor in an information decomposition, a class label $Y$, or a graphical-model class together with a base set of conditional-independence statements. Across these settings, the common motif is that an object is not judged redundant in isolation: it is judged by whether removing it, or conditioning on the relevant context, changes the admissible models, the induced semantics, or the recoverable information (Felfernig et al., 2021, Sharma et al., 23 Apr 2026, Rame et al., 2021, Faller et al., 12 Feb 2025).

1. Core meaning and formalizations

In constraint systems, the canonical formulation is logical. For a knowledge base $C$ and a constraint $c\in C$, $c$ is redundant iff $C\setminus\{c\}\models c$; otherwise it is non-redundant. The same idea admits a witness-based formulation: an instance is non-redundant iff for every constraint $C$ there exists an assignment satisfying all other constraints but not $C$. This makes non-redundancy explicitly conditional on the remaining constraints rather than an intrinsic property of a single clause (Felfernig et al., 2021, Sharma et al., 23 Apr 2026).

A useful refinement is conditional non-redundancy with respect to a larger relation $Q$. For $P\subsetneq Q\subseteq D^r$, an instance of $Y$0 is non-redundant for $Y$1 if each distinguished clause has a witness assignment that satisfies every other clause in $Y$2 while sending the distinguished clause into $Y$3. The associated extremal function is

$Y$4

the maximum number of clauses in such an $Y$5-variable instance. This conditional version supports a triangle inequality,

$Y$6

which makes it a natural decomposition tool for harder predicates (Brakensiek et al., 10 Jul 2025).

The same pattern appears outside CSPs. In association rules, $Y$7 means that in every dataset where rules in $Y$8 have confidence $Y$9 and rules in $C$0 have confidence at least $C$1, the rule $C$2 also has confidence at least $C$3. Conditional non-redundancy is the negation of this entailment relation (Balcazar, 2010). In graphical-model discovery, a CI statement can be graphically redundant yet still informative if it is not implied by graphoid axioms; the paper terms this purely graphical redundancy (Faller et al., 12 Feb 2025).

Setting	Conditioning object	Non-redundancy criterion
CSP / knowledge base	Other constraints or $C$4	Removing one clause changes $C$5, or witness lands in $C$6
Association rules	Rule set $C$7 and threshold $C$8	Rule is not $C$9-entailed by the premises
Information decomposition	Other predictor(s) or label $c\in C$0	Information remains after conditioning
Graphical discovery	Base CI set $c\in C$1 and graph class $c\in C$2	CI test is not implied by graphoid closure

2. Minimal cores and elimination of constraint redundancy

A central CSP manifestation of conditional non-redundancy is the minimal core. In "CoreDiag: Eliminating Redundancy in Constraint Sets" (Felfernig et al., 2021), a configuration task is a CSP $c\in C$3 with knowledge-base constraints $c\in C$4 and customer requirements $c\in C$5. A constraint $c\in C$6 is redundant iff

$c\in C$7

Equivalently, non-redundancy can be tested by consistency against the complement theory: $c\in C$8 is non-redundant iff $c\in C$9 is consistent (Felfernig et al., 2021).

A knowledge base is a minimal core iff every one of its constraints is indispensable for preserving inconsistency with $c$0. Formally, for all $c$1,

$c$2

must be consistent, while

$c$3

This identifies a minimal non-redundant subset preserving the original semantics. The paper explicitly interprets such a minimal core as a minimal conflict with respect to $c$4 (Felfernig et al., 2021).

Algorithmically, the baseline Sequential procedure tests every constraint once; for a knowledge base with $c$5 constraints it performs exactly $c$6 consistency checks. CoreDiag improves on this in high-redundancy regimes by invoking a QuickXPlain-style divide-and-conquer subroutine, CoreD, to compute a minimal core and then returning its complement as the redundant set. If the minimal core has size $c$7, CoreDiag has worst-case complexity

$c$8

and best-case complexity

$c$9

where the best case arises when all core constraints are clustered in one search-tree branch (Felfernig et al., 2021).

The empirical study in the same paper uses CLib configuration knowledge bases and redundancy rates of approximately $C\setminus\{c\}\models c$0–$C\setminus\{c\}\models c$1, $C\setminus\{c\}\models c$2, $C\setminus\{c\}\models c$3, and $C\setminus\{c\}\models c$4. It reports that almost all examined knowledge bases already contain redundancy at the original rate, and that removing redundant constraints lowers configuration runtime; for Bike_A, average runtime is about $C\setminus\{c\}\models c$5 ms without redundancy and about $C\setminus\{c\}\models c$6 ms at $C\setminus\{c\}\models c$7 redundancy (Felfernig et al., 2021).

3. Conditional non-redundancy as a complexity parameter in CSPs

The modern CSP literature elevates non-redundancy from a local diagnostic notion to a global structural parameter. For a constraint language $C\setminus\{c\}\models c$8, the streaming paper defines

$C\setminus\{c\}\models c$9

where non-redundancy means that every constraint $C$0 has a witness assignment satisfying $C$1 but falsifying $C$2. The main theorem states that one-pass streaming satisfiability for $C$3 has space complexity $C$4: there is a deterministic upper bound $C$5 and a randomized lower bound of $C$6 is impossible (Sharma et al., 23 Apr 2026).

This characterization is exact up to logarithmic factors for general shifted CSPs and for non constant-satisfiable positive Boolean CSPs. It recovers $C$7 for $C$8-SAT, $C$9 for $C$0-LIN over $C$1, and $C$2 for graph $C$3-coloring with $C$4. The same paper also shows a limitation: for positive CSPs over larger non-Boolean domains without shifts, non-redundancy alone no longer characterizes streaming complexity (Sharma et al., 23 Apr 2026).

Exact sparsification exhibits a parallel dependence. "Redundancy Is All You Need" proves that for every predicate $C$5,

$C$6

so the worst-case size of unweighted $C$7-sparsifiers is pinned down, up to polylogarithmic factors, by non-redundancy of the complement predicate. In the weighted setting, the relevant parameter is chain length $C$8, with

$C$9

(Brakensiek et al., 2024).

Recent classification results show that the asymptotic landscape is unexpectedly rich. "The Richness of CSP Non-redundancy" proves that every rational number $Q$0 occurs as an exponent: $Q$1 for some finite predicate $Q$2, and completely classifies conditional non-redundancy for binary predicates via high-girth graphs (Brakensiek et al., 10 Jul 2025). "Classification of Non-redundancy of Boolean Predicates of Arity 4" algorithmically classifies $Q$3 of the $Q$4 non-trivial Boolean predicates of arity $Q$5, resolves two of the remaining three via extremal-combinatorial reductions, and leaves one open while identifying the first Boolean predicate with provably non-polynomial non-redundancy asymptotics (Brakensiek et al., 22 Mar 2026). For symmetric Boolean predicates of arity at most $Q$6, the near-complete classification uses $Q$7-balancedness to prove $Q$8 upper bounds and Carbonnel’s OR-based criterion for $Q$9 lower bounds, leaving only two arity-$P\subsetneq Q\subseteq D^r$0 predicates with bounds $P\subsetneq Q\subseteq D^r$1 and $P\subsetneq Q\subseteq D^r$2 (Sharma et al., 13 May 2026). A complementary hypergraph-projection framework yields new super-linear lower bounds such as

$P\subsetneq Q\subseteq D^r$3

for non-trivial projections of the BoolBCK promise pair (Brakensiek et al., 18 May 2026).

4. Information-theoretic interpretations

In information decomposition, conditional non-redundancy is the portion of information that remains once shared or target-explained structure has been removed. The Williams–Beer bivariate decomposition writes

$P\subsetneq Q\subseteq D^r$4

where $P\subsetneq Q\subseteq D^r$5 is redundant information about $P\subsetneq Q\subseteq D^r$6, $P\subsetneq Q\subseteq D^r$7 and $P\subsetneq Q\subseteq D^r$8 are unique components, and $P\subsetneq Q\subseteq D^r$9 is synergy. In this vocabulary, conditional non-redundancy of $Y$00 relative to $Y$01 is the unique part of $Y$02 about $Y$03 given $Y$04 (Banerjee, 2015).

The 2015 analysis of common-information-based decompositions shows that this quantity is subtle. Gács–Körner common information is too strict for generic redundancy because it collapses to zero for indecomposable distributions, while Wyner common information is unsuitable as a redundancy measure because it is non-decreasing in the number of variables, violating the expected monotonicity of redundancy. Under strong perfect-resolvability assumptions on $Y$05 and $Y$06, a conditional Gács–Körner quantity $Y$07 becomes an ideal measure of unique information and therefore of conditional non-redundancy; outside that regime, the paper advocates approximately sufficient statistics and the conditional information bottleneck objective

$Y$08

as a more broadly applicable operationalization (Banerjee, 2015).

A complementary impossibility result sharpens the boundary. "Synergy, Redundancy and Common Information" proves that for independent predictors, any redundancy measure derived from optimization over a single auxiliary random variable $Y$09 cannot induce a nonnegative partial information decomposition. This shows that common-information-based constructions are too rigid to capture mechanistic redundancy and its conditional complements in general (Banerjee et al., 2015).

5. Representation learning and deep ensembles

In representation learning, conditional non-redundancy has a direct information-theoretic form. DICE defines redundancy between two member representations $Y$10 by $Y$11, and conditional redundancy by

$Y$12

the information they share beyond what is explained by the label $Y$13. The target regime is

$Y$14

so that, given the class label, one member’s features do not predict the other’s. The full DICE criterion combines conditional compression, label relevance, and conditional redundancy: $Y$15 Operationally, the conditional redundancy term is estimated adversarially by training a discriminator to distinguish joint triples $Y$16 from product triples $Y$17 sampled within class (Rame et al., 2021).

The paper’s central claim is that diversity should be enforced at the feature level, but only on label-irrelevant overlap. This sharply distinguishes DICE from unconditional penalties such as $Y$18, which suppress label-relevant common structure as well. The empirical results reflect that distinction. On CIFAR-100 with branch-based ResNet-32 ensembles, independent training gives $Y$19 for $Y$20 branches, whereas DICE gives $Y$21; the authors note that $Y$22 DICE branches match or exceed a traditional $Y$23-branch ensemble (Rame et al., 2021).

The same study reports improved uncertainty properties. On CIFAR-100 with $Y$24-branch ResNet-32 after temperature scaling, DICE attains NLL $Y$25 versus $Y$26 for independent training, and Brier score $Y$27 versus $Y$28. The conditional redundancy signal also improves OOD detection, and using the discriminator output as an input-dependent temperature further increases AUROC on datasets such as TinyImageNet(crop) (Rame et al., 2021). A plausible implication is that conditional non-redundancy in this setting acts as a mechanism for decorrelating spurious within-class cues while preserving $Y$29.

6. Other formal settings: association rules, graphical discovery, and contextuality

Association-rule theory treats conditional redundancy as threshold entailment over all datasets. Given implications $Y$30, partial premises $Y$31, and confidence threshold $Y$32, the rule $Y$33 is $Y$34-entailed when every dataset in which all rules in $Y$35 have confidence $Y$36 and all rules in $Y$37 have confidence at least $Y$38 also gives $Y$39 confidence at least $Y$40. For a single premise, plain redundancy, standard redundancy, simple redundancy, strict redundancy, and covering all collapse to the same structural condition $Y$41 and $Y$42. For two premises, the behavior bifurcates sharply: if $Y$43, no proper two-premise entailments exist, whereas for $Y$44 they are characterized by seven closure conditions. Representative rules and the closure-based basis $Y$45 are absolutely minimum-size bases under their respective redundancy notions, hence maximally conditionally non-redundant rule sets (Balcazar, 2010).

In CI-based graphical-model discovery, several redundancy notions coexist. A CI statement is graphically redundant if every graph in a class $Y$46 compatible with a base set $Y$47 also entails it; it is graphoid-redundant if it follows from $Y$48 in every graphoid model; and it is purely graphically redundant if it is graphically redundant but not graphoid-redundant. The paper argues that purely graphically redundant tests are the informative ones for model criticism, because they probe graphical assumptions rather than generic probabilistic closure. It also proves that for spanning trees, Markov-distance minimization can correct up to $Y$49 CI-test errors when all single-node-conditioned tests are used (Faller et al., 12 Feb 2025).

The contextuality literature adds a different warning: informational redundancy need not imply contextual redundancy. "Contextuality and Informational Redundancy" constructs systems in which a new connection $Y$50 is a context-wise function of existing connections and is therefore informationally redundant in the ordinary sense, yet adding it turns a noncontextual system into a contextual one. The effect persists for inconsistently connected, consistently connected, and strongly consistently connected systems, especially when functions are allowed to use non-measurement information encoded by empty cells or indicator connections (Dzhafarov et al., 2022).

Across these literatures, a stable distinction emerges. Conditional non-redundancy is rarely about absolute novelty. It is about what remains indispensable after the relevant ambient structure—other clauses, a scaffold relation, a closure theory, a label, or a graph class—has already been fixed.