Pliability in Geometry, Algorithms & Software

• Pliability is a context-dependent notion quantifying controlled non-rigidity, defined by admissible alternative configurations in settings like birational geometry, sub-Riemannian analysis, and design workflows.
• It facilitates systematic classification and optimization by linking structural invariants—such as Mori fibre space sets, endpoint-map openness, and treewidth—with practical parameters in algorithms and software metrics.
• Applications span from enhancing private information retrieval rates and reconfigurable visualization templates to modeling physical deformations in folded elastic sheets and anisotropic superconducting systems.

Searching arXiv for recent and foundational papers on “pliability” across the domains represented in the source material. Pliability denotes a context-dependent notion of controlled non-rigidity: the capacity of an object, structure, model, or workflow to admit multiple admissible configurations under specified constraints. In contemporary research usage, the term appears in several mathematically distinct but structurally related senses. In birational geometry, pliability is the set of Mori fibre space structures birational to a given Mori fibre space, taken up to square equivalence [(Kurz et al., 29 Jul 2025); (Abban, 2013); (Campo, 2022); (Sarikyan, 2022)]. In Carnot-group analysis and sub-Riemannian geometry, pliability is a local openness property of endpoint maps for horizontal directions and curves (Jean et al., 17 Jul 2025, Speight et al., 20 May 2025, Juillet et al., 2016). In private information retrieval, pliability relaxes the demand from a specific message to any message within a desired class, preserving class privacy while improving rate (Obead et al., 2022). In visualization systems, pliability refers to the malleability of editing workflows afforded by parameterized declarative templates spanning multiple grammars and interfaces (McNutt et al., 2021). In component-based software engineering, pliability is a weighted software-quality metric aggregating component attributes under stakeholder-defined priorities (Alghabban et al., 2014). In approximation algorithms, treewidth-pliability is a structural approximation condition linking instances to bounded-treewidth proxies in opt-distance (Romero et al., 2019). Across these domains, the unifying theme is not generic flexibility but the existence of a formally defined admissible family of nearby or alternative structures.

1. Birational-geometric pliability

In the Sarkisov program, pliability is defined for a Mori fibre space $X \to S$ as the set of Mori fibre structures in the birational class of $X$, modulo square equivalence (Abban, 2013). For del Pezzo surfaces regarded over $B=\operatorname{Spec}k$, the notation $\Pl(S)$ is used for the pliability of $S$ (Kurz et al., 29 Jul 2025). Square equivalence means that two Mori fibre spaces are connected by a birational map commuting with a birational base map and inducing an isomorphism on generic fibres [(Abban, 2013); (Kurz et al., 29 Jul 2025)]. Birational rigidity is the special case $|\mathcal{P}(X)|=1$, while birational superrigidity requires every birational self-map to be biregular (Kurz et al., 29 Jul 2025, Campo, 2022).

For surfaces, the 2025 analysis of sextic del Pezzo surfaces gives a precise classification of infinite pliability among solid del Pezzo surfaces over perfect fields (Kurz et al., 29 Jul 2025). A del Pezzo surface $S$ with $\operatorname{Pic}(S)\cong \mathbb{Z}$ is solid if it is not birational to any conic bundle (Kurz et al., 29 Jul 2025). The central theorem states that if $S$ is a solid del Pezzo surface and $\Pl(S)=\infty$, then $X$0 must be a sextic del Pezzo surface with $X$1, and explicit examples with $X$2 are constructed (Kurz et al., 29 Jul 2025). The mechanism is an infinite supply of Sarkisov links of type II centered at closed points of degree $X$3 or $X$4 in general position, with infinitely many distinct splitting fields producing pairwise non-square-equivalent minimal models (Kurz et al., 29 Jul 2025). The same work shows that degree $X$5 del Pezzo surfaces are the only solid surfaces admitting infinite pliability (Kurz et al., 29 Jul 2025).

For threefolds, pliability measures the multiplicity of Mori fibre structures within a birational class rather than mere birational self-maps. The paper "On pliability of del Pezzo fibrations and Cox rings" exhibits a degree $X$6 del Pezzo fibration $X$7 with $X$8, arising from explicit Sarkisov links constructed by Cox-ring and VGIT methods (Abban, 2013). The birational class contains the original $X$9 fibration and two $B=\operatorname{Spec}k$0 fibrations $B=\operatorname{Spec}k$1 and $B=\operatorname{Spec}k$2, while $B=\operatorname{Spec}k$3 remains non-rational, showing that increased pliability does not imply rationality (Abban, 2013).

Several later works sharpen this viewpoint for singular Fano threefolds. "High-pliability Fano hypersurfaces" proves that five of Reid’s weighted hypersurface families with $B=\operatorname{Spec}k$4 singularities have pliability at least two, with the two elements represented by non-isomorphic hypersurfaces of the same degree in the same ambient weighted projective space but with different $B=\operatorname{Spec}k$5 singularities (Campo, 2022). "On the Rationality of Fano-Enriques Threefolds" provides an example of a Fano–Enriques threefold with pliability $B=\operatorname{Spec}k$6, consisting of one $B=\operatorname{Spec}k$7-Fano model and eight del Pezzo fibrations of degree $B=\operatorname{Spec}k$8 obtained by Sarkisov links of type I from the eight non-Gorenstein points (Sarikyan, 2022). For a general member in that family, $B=\operatorname{Spec}k$9 (Sarikyan, 2022).

Equivariant birational geometry introduces $\Pl(S)$0-pliability. For a finite subgroup $\Pl(S)$1, the $\Pl(S)$2-pliability of $\Pl(S)$3 is the set of $\Pl(S)$4-Mori fibre spaces $\Pl(S)$5-birational to $\Pl(S)$6, up to isomorphism in the paper’s formulation (Cheltsov et al., 2022). For many imprimitive finite subgroups, $\Pl(S)$7 is $\Pl(S)$8-solid and its equivariant pliability collapses to $\Pl(S)$9, while in one exceptional case $S$0 a third $S$1-Mori fibre space $S$2 appears (Cheltsov et al., 2022). This suggests that symmetry can drastically reduce the accessible Sarkisov category.

2. Carnot groups and sub-Riemannian analysis

In Carnot groups, pliability is a non-rigidity property of horizontal directions. A horizontal vector $S$3 is pliable if the endpoint map $S$4 is open at $S$5 in $S$6 (Jean et al., 17 Jul 2025). Equivalently, small perturbations of the constant control $S$7 fill a neighborhood of the endpoint $S$8 (Jean et al., 17 Jul 2025). Strong pliability strengthens this by requiring that for every $S$9 there exists a control perturbation $|\mathcal{P}(X)|=1$0 with $|\mathcal{P}(X)|=1$1, $|\mathcal{P}(X)|=1$2, and $|\mathcal{P}(X)|=1$3 surjective (Jean et al., 17 Jul 2025).

The 2025 note "A note on pliability and the openness of the multiexponential map in Carnot groups" proves that four conditions are equivalent for a horizontal vector $|\mathcal{P}(X)|=1$4: pliability, strong pliability, openness of a suitable multi-exponential map $|\mathcal{P}(X)|=1$5, and a strong perturbative version of that openness (Jean et al., 17 Jul 2025). By contrast, submersivity of $|\mathcal{P}(X)|=1$6 at $|\mathcal{P}(X)|=1$7 is equivalent to regularity of the endpoint map and is strictly stronger (Jean et al., 17 Jul 2025). This separates qualitative non-rigidity from exact regularity at the base direction.

The earlier paper "Pliability, or the Whitney extension theorem for curves in Carnot groups" introduced pliability precisely to characterize extension phenomena (Juillet et al., 2016). Its main theorem states that $|\mathcal{P}(X)|=1$8 has the $|\mathcal{P}(X)|=1$9 extension property if and only if $S$0 is pliable (Juillet et al., 2016). In that setting, a vector $S$1 is pliable when the straight horizontal curve $S$2 admits endpoint-jet neighborhoods under arbitrarily small $S$3 perturbations fixing the initial jet (Juillet et al., 2016). The paper also shows that all step-2 Carnot groups are pliable, while Engel-type groups provide non-pliable examples because of rigid straight curves (Juillet et al., 2016).

Directional refinements appear in "Directional Pliability, Whitney Extension, and Lusin Approximation for Curves in Carnot Groups" (Speight et al., 20 May 2025). There, a subset $S$4 is pliable if every vector in $S$5 is pliable, and this directional assumption suffices for a Whitney extension theorem and a Lusin approximation theorem for horizontal curves whose tangents lie in $S$6 (Speight et al., 20 May 2025). In the Engel group, vectors $S$7 are pliable if and only if either $S$8 or $S$9, so the only non-pliable directions are nonzero multiples of $\operatorname{Pic}(S)\cong \mathbb{Z}$0 (Speight et al., 20 May 2025). This yields a directional extension theory even when global pliability fails (Speight et al., 20 May 2025).

A common misconception is to identify pliability in Carnot groups with absence of abnormality. The cited works instead show that pliability can hold even when submersive regularity fails, so abnormal straight lines do not automatically preclude qualitative non-rigidity (Jean et al., 17 Jul 2025, Juillet et al., 2016).

3. Pliability in algorithmic and information-theoretic settings

In approximation algorithms for Max-CSPs, pliability acquires a purely structural meaning. "Pliability and Approximating Max-CSPs" defines $\operatorname{Pic}(S)\cong \mathbb{Z}$1-pliability for a class $\operatorname{Pic}(S)\cong \mathbb{Z}$2 of structures by requiring that for every $\operatorname{Pic}(S)\cong \mathbb{Z}$3 there exists $\operatorname{Pic}(S)\cong \mathbb{Z}$4 such that each $\operatorname{Pic}(S)\cong \mathbb{Z}$5 has a proxy $\operatorname{Pic}(S)\cong \mathbb{Z}$6 with $\operatorname{Pic}(S)\cong \mathbb{Z}$7 and opt-distance $\operatorname{Pic}(S)\cong \mathbb{Z}$8 (Romero et al., 2019). Treewidth-pliability is the case $\operatorname{Pic}(S)\cong \mathbb{Z}$9 (Romero et al., 2019). The paper proves that treewidth-pliability yields a PTAS for bounded-arity maximum homomorphism problems, using level-$S$0 Sherali–Adams relaxations (Romero et al., 2019). It also establishes equivalences between treedepth-, treewidth-, and Hadwiger-pliability, and, under bounded signatures, with size- and component-size pliability (Romero et al., 2019). For graph classes, the corresponding notion is equivalent to fractional treewidth fragility (Romero et al., 2019).

In private information retrieval, pliability is a relaxation of demand specificity rather than an approximation notion. "Pliable Private Information Retrieval" defines PPIR by allowing the user to retrieve any message from a desired class while keeping the class identity private from each database (Obead et al., 2022). In the multi-message version, the user privately selects $S$1 desired classes and retrieves any $S$2 messages from each desired class (Obead et al., 2022). The key performance benefit is that the private search space shrinks from the full library of $S$3 messages to the $S$4 classes. The paper proves that the PPIR capacity with $S$5 noncolluding replicated databases is

$S$6

which matches the classical PIR capacity with $S$7 messages rather than $S$8 messages (Obead et al., 2022). This is a precise privacy–rate trade-off created by pliability at the level of requested content (Obead et al., 2022).

These two usages illustrate a broader pattern. In Max-CSPs, pliability means approximation-preserving reducibility to bounded-complexity surrogates (Romero et al., 2019). In PPIR, it means admissible substitution within a private demand set (Obead et al., 2022). In both cases, the term denotes relaxation of a rigid specification while preserving a formal objective.

4. Software systems and visualization workflows

In component-based software engineering, pliability is a composite quality metric rather than a geometric or combinatorial invariant. "The proposal of improved component selection framework" defines pliability as a weighted combination of quality attributes such as reliability, performance, fault tolerance, safety, security, availability, testability, and maintainability (Alghabban et al., 2014). The overall software quality is

$S$9

with weights summing to one (Alghabban et al., 2014). Component-level attribute scores are normalized by

$\Pl(S)=\infty$0

so heterogeneous measures can be aggregated (Alghabban et al., 2014). In the improved integrated component selection framework, pliability augments performance-based filtering and heuristic search without altering the underlying search algorithms (Alghabban et al., 2014).

The visualization literature uses the term differently. "Integrated Visualization Editing via Parameterized Declarative Templates" defines pliability as the malleability of visualization editing and authoring workflows enabled by parameterized declarative templates over JSON-based chart grammars (McNutt et al., 2021). Here a single template can be adapted to multiple grammars, multiple user-interface modalities, and multiple parameter spaces while maintaining synchronized editing (McNutt et al., 2021). The paper distinguishes pliability from flexibility, expressiveness, generality, and usability: flexibility concerns exposed options, expressiveness concerns the breadth of a grammar, generality concerns grammar-agnostic mechanisms, and usability concerns task ease, whereas pliability emphasizes the ability to bend one representation across contexts and modalities (McNutt et al., 2021). Empirically, the paper factors 166 Vega-Lite examples into 43 templates and 32 Google Sheets chart types into 16 templates, interpreting these compression ratios as evidence of reusable pliable abstraction (McNutt et al., 2021).

The software-engineering and visualization meanings are not interchangeable. In the former, pliability is a scalar evaluation functional for selecting component combinations (Alghabban et al., 2014). In the latter, it is a property of a representation-and-interface architecture supporting multimodal editing (McNutt et al., 2021).

5. Mechanical and physical uses

In folded-sheet mechanics, pliability refers to the ease of bending and folding of crease-based structures. "Plasticity and Aging of Folded Elastic Sheets" ties macroscopic pliability to a crease rest angle $\Pl(S)=\infty$1 and an effective crease stiffness $\Pl(S)=\infty$2 (Jules et al., 2020). The unloaded reference shape is modeled by

$\Pl(S)=\infty$3

where $\Pl(S)=\infty$4 is the characteristic crease size (Jules et al., 2020). The crease moment obeys the hinge-like law

$\Pl(S)=\infty$5

so smaller $\Pl(S)=\infty$6 corresponds to higher pliability (Jules et al., 2020). The paper further argues that plasticity primarily changes $\Pl(S)=\infty$7, while relaxation under fixed extension follows a double-logarithmic law, allowing real-time tracking of the evolving reference state from macroscopic measurements (Jules et al., 2020).

A distinct physical usage appears in spin-triplet superconductivity. "Spatially Inhomogeneous Triplet Pairing Order and Josephson Diode Effect Induced by Frustrated Spin Textures" introduces an effective "pliability" of the pairing order: frustrated spin textures generate anisotropic Josephson couplings that favor spatially varying $\Pl(S)=\infty$8-vector configurations, competing with superfluid stiffness (Frazier et al., 29 Oct 2025). The discrete Josephson free energy includes Heisenberg-like, Dzyaloshinskii–Moriya-like, and $\Pl(S)=\infty$9-type couplings between neighboring $X$00-vectors (Frazier et al., 29 Oct 2025). In the continuum, this yields gradient terms that linearly favor twists of the pairing order (Frazier et al., 29 Oct 2025). This suggests a physical interpretation of pliability as susceptibility of an ordered field to anisotropy-induced spatial distortion.

Because the superconductivity source is dated 2025-10-29, it postdates the present date. A plausible implication is that it should be treated as prospective relative to the current temporal context, even though its conceptual usage is clearly specified in the supplied record.

6. Common structure and differences across fields

Despite substantial differences in formalism, several recurrent features distinguish pliability from nearby concepts.

First, pliability is always defined relative to an admissible constraint system. In birational geometry, the admissible objects are Mori fibre spaces modulo square equivalence [(Kurz et al., 29 Jul 2025); (Abban, 2013)]. In Carnot geometry, they are horizontal perturbations constrained by endpoint maps (Jean et al., 17 Jul 2025, Juillet et al., 2016). In PPIR, they are messages within desired classes under class-privacy constraints (Obead et al., 2022). In CBSE, they are weighted attribute combinations under stakeholder priorities (Alghabban et al., 2014). In visualization, they are chart specifications reachable through parameterized templates (McNutt et al., 2021).

Second, pliability is not synonymous with unrestricted flexibility. The visualization paper explicitly separates the two, reserving pliability for adaptation of one underlying representation across modalities (McNutt et al., 2021). The birational literature similarly treats pliability as a structured set of Mori models rather than arbitrary birational behavior (Kurz et al., 29 Jul 2025, Sarikyan, 2022). Carnot-group pliability is an openness condition, not arbitrary deformability (Jean et al., 17 Jul 2025).

Third, pliability often sits between rigidity and full indeterminacy. Birational rigidity corresponds to pliability $X$01 (Campo, 2022, Sarikyan, 2022). In Carnot groups, regularity implies pliability but is stronger (Jean et al., 17 Jul 2025). In PPIR, pliability relaxes the demand set while preserving privacy at a coarser granularity (Obead et al., 2022). In folded sheets, pliability is controlled by a hinge stiffness and rest-angle evolution, not by unconstrained soft behavior (Jules et al., 2020).

A frequent source of confusion is to assume that pliability is inherently qualitative. Several fields instead give quantitative invariants: cardinalities of pliability sets in birational geometry (Sarikyan, 2022), explicit capacity formulas in PPIR (Obead et al., 2022), weighted quality scores in CBSE (Alghabban et al., 2014), or dimensionless ratios such as $X$02 in the superconducting context (Frazier et al., 29 Oct 2025).

7. Research directions and open boundaries

Current work points to several unresolved directions. In birational geometry, the classification of all possible Mori fibre structures in higher-dimensional or more singular families remains incomplete; the Fano hypersurface and Fano–Enriques examples establish lower bounds such as pliability at least two or exactly nine in specific families, but broader classification problems remain open (Campo, 2022, Sarikyan, 2022). The sextic del Pezzo result isolates degree $X$03 surfaces as uniquely capable of infinite pliability among solid surfaces, suggesting a sharp two-dimensional boundary that invites higher-dimensional analogues (Kurz et al., 29 Jul 2025).

In Carnot geometry, the directional theory suggests that extension and approximation theorems may depend only on the subset of directions actually used by a curve, not on global properties of the group (Speight et al., 20 May 2025). The converse implication—that Whitney extension on a set of directions should imply pliability of those directions—is noted as likely but unchecked in the supplied summary (Speight et al., 20 May 2025). The equivalence between endpoint-map openness and multi-exponential openness consolidates the conceptual landscape, but abnormality, rigidity, and quantitative accessibility remain active interfaces (Jean et al., 17 Jul 2025, Juillet et al., 2016).

In algorithms, treewidth-pliability unifies sparse and dense PTAS paradigms, yet the paper explicitly raises a dichotomy question for Max-2-CSPs parameterized by constraint graphs and asks whether PTAS existence is equivalent to fractional treewidth fragility (Romero et al., 2019). In PIR, exact M-PPIR capacity is known only in certain parameter regimes, leaving other regimes bounded but unresolved (Obead et al., 2022).

These trajectories suggest that pliability functions less as a single transferable definition than as a recurrent research schema: formally constrained non-rigidity that is rich enough to classify, optimize, or exploit, yet structured enough to exclude arbitrary deformation.