Locality: Definitions and Applications

• Locality is defined as constraints restricting interactions or dependencies to nearby elements, based on spatial, temporal, and logical criteria across various disciplines.
• Quantitative measures such as interval abundance in causal sets, neighborhood radii in graphs, and code recovery parameters rigorously assess locality in different contexts.
• Exploiting locality leads to optimized algorithms, improved interpretability in neural networks, and efficient hardware designs, with broad implications in science and engineering.

Locality encompasses a spectrum of definitions and operational criteria across mathematics, physics, computer science, information theory, and machine learning. At its core, locality captures constraints restricting interactions, dependencies, or influences—spatial, temporal, combinatorial, or logical—to be predominantly "nearby" in some precise sense intrinsic to the structure in question. Modern research exploits this principle to optimize computations, enhance interpretability, strengthen recoverability, and sharpen foundational concepts in physical theories.

1. Foundational Definitions of Locality

Locality admits multiple rigorous formalisms, each tailored to the ambient mathematical or physical context:

• Physics (Relativity and Quantum Theory): Locality in relativistic quantum field theory is formalized via microcausality, enforcing $[\phi(x),\phi(y)]=0$ for all spacelike-separated $x,y$, preventing superluminal influence and ensuring local compatibility of measurements (Biagio et al., 2023). In quantum information, circuit locality requires that system-wide dynamics decompose into causal circuits with gates each acting on small subsystems, reflecting subsystem isolation except for permitted mediators.
• Causal Set Theory (Quantum Gravity): Discreteness in the causal set approach preserves Lorentz invariance but obscures standard notions of locality. To address this, locality is operationally defined by the abundance function $N_m(C)$ counting the number of $m$-element order intervals between causally related pairs $(x \prec y)$. The expectation $\langle N_m^{(d)}\rangle$ provides a "locality fingerprint" that tracks the presence of manifoldlike (approximately flat) spacetime regions (Glaser et al., 2013).
• Algorithmics (Graph and Distributed Computation): In distributed, online, sequential, and dynamic models, locality quantifies the radius-$T$ neighborhood upon which a node's computations can depend before output is determined. This formalizes the minimal neighborhood size needed for local decisions in problems like coloring or labeling, with explicit complexity trichotomies for Locally Checkable Labeling (LCL) problems (Akbari et al., 2021).
• Error Correction and Codes: Code locality traditionally refers to the minimal set of code symbols required to recover a specific symbol. For a code of length $n$, locality $r$ means each coordinate can be reconstructed from a subset of no more than $r$ others (Krishnan et al., 2017, Pacifico, 2024). In storage and coded computation, this is further abstracted via computational locality, determining the minimal worker or server redundancy required to mitigate stragglers for nonlinear computation tasks (Rudow et al., 2020).
• Spectral and Geospatial Analysis: In spatial statistics, locality is operationalized via localized association measures such as the Local Indicator of Colocation Quotient (LCLQ), which computes property associations using weighted neighbor aggregations in physical metric or network space (Wang et al., 2020).
• Formal Topology and Category Theory: Locales (Loc) and their point-free generalizations (positive topologies, PTop) provide categorical frameworks where locality is reflected in morphisms preserving open-set structure and continuity. The Top $\dashv$ Loc adjunction and its factorization through positive topologies illustrate deep connections between classical, point-based, and constructive notions of space (Ciraulo et al., 2018).

2. Locality as a Quantitative and Structural Constraint

Locality is often operationalized or measured via explicit mathematical quantities:

• Interval Abundance in Causal Sets: For a Poisson-sprinkled causal set in $d$-dimensional Minkowski space,

$$\big\langle N_m^{(d)}(\rho, V)\big\rangle = \rho^2 \int_{I[p,q]} dV_x \int_{I[x,q]} dV_y \, \frac{(\rho V_{xy})^m}{m!} e^{-\rho V_{xy}}$$

This can be further normalized to $$S_m^{(d)} = \langle N_m^{(d)}\rangle/\langle N_0^{(d)}\rangle$$ which is scale-invariant in the large $N$ limit and uniquely characterizes flat, local regions versus nonlocal or curved structure (Glaser et al., 2013).

• Neighborhood Radius in Graph Models: Local computation models, such as LOCAL, SLOCAL, dynamic-LOCAL, and online-LOCAL, define complexity and feasibility based on the minimal radius $T$ that forms the dependency horizon for nodes. The separation of models and the trichotomy for LCLs ($O(\log^* n)$, $\Theta(\log n)$, $n^{\Theta(1)}$) is tightly linked to how locality constraints propagate (Akbari et al., 2021).
• Block-Sparsity Penalties in Transformers: In recent LLM architectures, such as localist LLM frameworks (Diederich, 10 Oct 2025) and AILA (Diederich, 5 Nov 2025), locality is explicitly enforced via group-sparsity penalties:

$$L_\text{sparsity}(W_Q, W_K; \alpha) = \sum_{h=1}^H \sum_{i=1}^p \alpha_i^{(h)} \left(\|W_{Q,i}^{(h)}\|_F + \|W_{K,i}^{(h)}\|_F\right)$$

The corresponding locality dial parameter $\lambda$ or vector $\{\alpha_i^{(h)}\}$ enables a continuous trade-off between localist and distributed internal encodings, with precise control over the concentration of attention (as measured by entropy and pointer fidelity).

• Colocation Quotients in Spatial Analysis: The LCLQ at location $i$, using $k$ nearest-neighbors (under Euclidean or network metrics), provides a normalized measure of local association and supports hypothesis testing via conditional permutation (Wang et al., 2020).
• Residuated-Lattice Model Theory: In many-valued logic and substructural logics on residuated lattices, locality theorems (Hanf, Gaifman) impose constraints on the local neighborhood structure necessary for the satisfaction of formulas, with locality determined by algebraic thresholds in the interpretation of quantifiers and connectives (Carr, 19 Jun 2025).

3. Locality in Theoretical and Practical Computation

• Hardware and Memory: The locality of reference is a key principle guiding the design and analysis of hardware-agnostic algorithms. The Locality-of-Reference (LoR) model assigns cost functions $\ell(d, \delta)$ to memory jumps, parameterized by spatial and temporal locality. Afshani et al. show that cache-oblivious algorithms that are optimal in the classic ideal-cache model remain optimal under any LoR model satisfying reasonable concavity and tall-cache conditions (Afshani et al., 2019).
• Coded Computation and Straggler Mitigation: Computational locality dictates the minimal number of needed workers for robust computation in the face of straggler nodes. For function classes like Reed–Muller codes, this locality is explicitly bounded, and exploitation of symmetry or linear dependencies among inputs results in schemes that dramatically reduce redundancy compared to naive replication—these adaptive approaches strictly beat the lower bounds for "input-oblivious" coded computation (Rudow et al., 2020).
• Error Control and Recovery: In coding theory, locality is crucial for efficient error recovery. LRC codes provide explicit upper bounds on code distance given locality $r$, via the bound $d \le n - k - \lceil k/r \rceil + 2$ (Pacifico, 2024). For cyclic codes, the addition of locality "trains" (specific sets of zeros) reduces trellis state complexity, directly translating into lower decoding complexity and increased throughput. Locality-aware decoders (e.g., quick-look ML stage augmented by local parity-checks) yield SNR gains with negligible cost (Krishnan et al., 2017).
• Multi-Agent Systems and RL: In multi-agent reinforcement learning, locality is encoded via dependency graphs and partitioned critic structures, e.g., in Loc-FACMAC, which assigns local critics to partitions defined by induced subgraphs of agent interaction, thereby reducing sample complexity and enabling scalable training (Shek et al., 24 Mar 2025).

4. Locality as a Tool for Interpretability and Modularity

Controlling locality is central to the interpretability and modularity of complex systems:

• Neural Network Internal Structure: In transformer-based models, the locality dial architecture enables explicit interpolation between highly interpretable, rule-based (localist) modes, and generic, distributed representations. This mechanism affords predictable control of properties such as attention entropy and pointer fidelity, with theoretical bounds linking margin parameters and block-sparsity penalties to the degree of achieved locality (Diederich, 10 Oct 2025, Diederich, 5 Nov 2025).
• Rule Injection and Auditable AI: The dynamic rule injection and locality dial allow for the integration of symbolic rules and constraint verification in LLMs without retraining, supporting regulatory requirements and transparent auditing by precisely restricting attention pathways (Diederich, 10 Oct 2025).
• Spatial Analytics: Locality-aware spatial indicators enable the detection of local "hotspots" or associations that would be masked by global correlations, with the LCLQ facilitating mapping and localized policy interventions (Wang et al., 2020).

5. Locality and Foundations: Physics, Topology, and Logic

• Quantum Gravity and Causal Sets: The interval-abundance criterion in causal set theory formalizes locality without reference to continuum geometry, providing both a test for manifoldlikeness and a covariant estimator of continuum dimension. This approach distinguishes local (flat) from nonlocal (curved, topological, or dynamically generated) structures by the empirical $N_m$ profile (Glaser et al., 2013).
• Higher-Spin Field Theory: In AdS higher-spin theories, Skvortsov and Taronna provide a criterion for admissible pseudo-local field redefinitions: the infinite-derivative tails of interaction vertices must have sufficient decay for the sum of overlap coefficients to converge. Only such pseudo-local terms can be reorganized into standard local field theory observables that match CFT correlators. Standard Vasiliev constructions may fall outside this class, requiring field redefinitions constrained by the locality criterion (Skvortsov et al., 2015).
• Locales, Formal Topologies, and Category Theory: Locality is structurally embedded in the morphism spaces and adjunctions between Top (topological spaces), PTop (positive topologies), and Loc (locales). The factorization of Top $\dashv$ Loc via positive topologies demonstrates that point-free and predicative notions of space retain the essentials of local structure while supporting constructive and fibrational extensions (Ciraulo et al., 2018).

6. Open Problems and Broader Implications

• Sufficiency of interval-abundance locality for manifoldlikeness in causal sets remains conjectural, with a rigorous classification still open (Glaser et al., 2013).
• For multi-agent systems, the optimal construction of locality partitions relative to environment structure and agent dependency graphs is an active research area (Shek et al., 24 Mar 2025).
• Nontrivial extensions of classical model-theoretic locality theorems (Hanf, Gaifman) to many-valued and substructural logics depend delicately on algebraic properties (well-connectedness, presence of co-atoms, etc.), with failures of locality occurring outside these settings (Carr, 19 Jun 2025).
• In computational models with locality-of-reference cost, future directions include formalizing universality classes of hardware or memory architectures for which cache-oblivious algorithms remain optimal (Afshani et al., 2019).

Research across disciplines continues to illuminate the structural, computational, and operational roles of locality: as a means of enforcing efficient computation, ensuring safety and recoverability, enabling transparency, and providing the mathematical backbone for the emergence and analysis of complex systems.