Position-Independent Concepts

• Position-Independent Concepts are principles whose definitions and functional roles remain unchanged regardless of fixed positions in memory, spatial order, or indexing.
• They are applied in diverse fields like computer science, mathematics, and physics to enhance system robustness, flexibility, and semantic clarity.
• Practical implementations include indirect jump instructions, Petri net semantics, and position-independent caching in AI models, leading to modular and efficient designs.

Position-independent concepts are principles, mechanisms, or representations whose definitions, functional roles, or observables do not depend on fixed positions in an indexing scheme, physical memory, spatial order, syntactic structure, or similar positional anchors. Such concepts arise across theoretical computer science, mathematics, physics, and artificial intelligence, where independence from position lends robustness, flexibility, efficiency, and sometimes deeper semantic meaning to models and systems.

1. Foundations and Formal Definitions

A position-independent concept is characterized by the property that its behavior, semantic value, or operational effect remains invariant under positional changes or reordering within an appropriately defined structure. In sequence-based systems (e.g., instruction sets, text strings, or tokenized models), this means the relevant outcome does not rely on the absolute or relative position. In geometry, position independence often refers to combinatorial or incidence properties that do not hinge on the geometric placement of elements. In computational settings, independence may be achieved through abstraction (e.g., using memory indirection), modularization, or statistical representation.

Several formalisms illustrate this:

• Indirect Jump Instructions in Programming: In instruction sequences where jump destinations are stored in registers or stacks rather than coded directly as instruction positions, the control flow dynamically adapts to values computed during execution, enabling program relocation (position independence) without recompilation (0711.0829).
• Petri Net Semantics in Concurrency: An event is defined by its pre- and post-conditions, not by its literal position in the net structure, making process behaviors depend solely on local independence properties; thus, substituting actions or refining their granularity does not affect global correctness (0802.0820).
• General Position in Geometry: A subset of points is in general position if, for example, in ℝ^d, no d + 1 points lie on a hyperplane, regardless of their placement, enabling combinatorial rather than spatial reasoning (Cardinal et al., 2014).
• Density-Based Semantic Representations: In taxonomy learning, concepts are represented as density distributions over context-aware vector embeddings rather than as static word positions in a vocabulary (Schmelzeisen et al., 2019).
• Caching in Machine Learning Models: Position-independent caching decouples key-value (KV) storage from fixed token or content positions, allowing reuse of intermediate activations regardless of their location in the user prompt or document structure (Hu et al., 20 Oct 2024, Zhao et al., 4 Feb 2025).

2. Position Independence in Computation: Methods and Mechanisms

Position independence is realized through several technical strategies across domains:

• Indirection via Registers or Stacks: Indirect absolute, relative, and double indirect jumps draw jump targets from register or stack values. This allows execution to depend on runtime-determined state, so the program code can be loaded at any address without modifying instructions. Translation of such instructions involves replacing the indirect jump with a search or lookup mechanism that interacts with the memory device and behaves identically regardless of position (0711.0829).
• Petri Net Decomposition and Refinement: In concurrent computation models, events are independent when their “neighborhoods”—the set of affected conditions—do not overlap. This independence supports substituting abstract actions with refined implementations without regard to position in the net or ordering in the trace, as shown by the non-interfering substitution theorem (0802.0820).
• Position-Independent Caching in AI Serving: In serving large (and multimodal) LLMs, splitting static context into semantically independent “chunks” allows caching of KV states for portions of input that may reappear at any position in future requests. Systems such as EPIC and MPIC facilitate assembling these chunks dynamically using selective boundary recomputation to restore attention mechanisms’ correctness (Hu et al., 20 Oct 2024, Zhao et al., 4 Feb 2025).

Domain	Mechanism	Position Independence Via
Instruction sequences	Indirect/double jumps	Memory referencing, stack saves
Concurrent process semantics	Petri net neighborhoods	Event independence, substitution
Geometry/combinatorics	Arrangement properties	Hyperplane/point set constraints
ML systems	KV split/cache recombining	Chunked caching, boundary fixing

3. Position-Independent Representations in Mathematics and Geometry

In combinatorial and geometric settings, position independence takes the form of invariance under ordering and placement:

• General Position Subsets: For $n$ points in ℝ^d, a maximal subset is in general position if no hyperplane contains more than $d$ points, reflecting a property that is preserved under reordering or relocation. Results establish that every large enough set gives rise to substantial general position subsets or large cohyperplanar subsets (Ramsey-type dichotomy). The existence of independent hyperplanes—where no cell of the arrangement is bounded by only the chosen hyperplanes—also embodies position independence in cell-complex structures (Cardinal et al., 2014).
• Algorithmic and Ramsey Theory Implications: These invariants allow for the design of coloring, partitioning, or selection algorithms that are robust to geometrical layout, with quantitative bounds on the size of independent subsets.

4. Position Independence in AI Model Serving and Representation Learning

Recent advances in large language and multimodal model serving leverage position-independent concepts to enhance efficiency and scalability:

• Position-Independent Caching (PIC): Systems such as EPIC break input content into modular chunks and cache their intermediate KV representations. When the same content reappears (possibly reordered or at different positions within future queries), precomputed caches can be linked and reused. A selective recomputation algorithm (e.g., AttnLink) resolves integration at chunk boundaries by recomputing a small, constant subset of tokens—typically the initial tokens of static segments—to mitigate the “attention sink” problem and restore accuracy (Hu et al., 20 Oct 2024).
• MPIC for Multimodal Input: The MPIC design extends these ideas to multimodal LLMs by decoupling the storage and reuse of image (and other modality) KV caches from fixed positions, employing parallel transfer mechanisms and selective recomputation of tokens sensitive to positional context (Zhao et al., 4 Feb 2025).
• Disentangled and Order-Invariant Visual Concept Learning: Architectures such as Visual Concepts Tokenization (VCT) eschew positional embeddings and enforce cross-attention-only architecture, ensuring that each concept token absorbs one independent visual concept. Explicit disentanglement losses further enforce that tokens correspond to unique factors of variation, making their semantics robust to the spatial location of the source features (Yang et al., 2022).

5. Position-Independent Models in Quantum Physics and Semantics

Physical and semantic systems also exploit position-independent structures:

• Spacetime-Symmetric Quantum Mechanics: Recasting position $q$ as the independent variable with time $t(q)$ as the dependent variable leads to new operator formulations (the “Momentumian” operator) and, under quantization, to evolution equations with half-order (1/2) fractional derivatives with respect to time. These equations and their solutions are position-independent in the sense that the formalism treats time and position on equal footing, often yielding effective potential shifts or coupling structures irrespective of where in configuration space the system is analyzed (Beims et al., 2023).
• Density-Based Semantic Concepts in NLP: In taxonomy learning, a concept is modeled as a probability distribution over high-dimensional contextualized embeddings—reflecting a collection of senses or occurrences—rather than a fixed location in word embedding space. Similarity and hypernymy relations between such concepts are measured by density-based statistics (e.g., inner products, KL divergence), not affected by fixed positional encoding (Schmelzeisen et al., 2019).

6. Applications, Algorithms, and Implications

Position-independent concepts underpin several practical and theoretical advances:

• Relocatable and Modular Code: Indirect jumps and position-independent code are vital for dynamic linking, shared libraries, and secure execution environments (e.g., under address space layout randomization) (0711.0829).
• Concurrent Software Engineering: Modular proofs and refinement in concurrent separation logic depend on Petri net semantics and ownership modeling that are robust to process composition and code rearrangement (0802.0820).
• AI Serving Infrastructure: Position-independent caching dramatically accelerates time-to-first-token and throughput for large models, especially when content is repetitive but its order or dynamic context varies—a typical scenario in retrieval-augmented generation or few-shot prompting. This enables linear scalability with prompt length and more effective handling of multimodal data (Hu et al., 20 Oct 2024, Zhao et al., 4 Feb 2025).
• Interpretability and Editing in Generative Models: Methods such as Head Relevance Vectors (HRVs) in diffusion models pinpoint which cross-attention heads encode specific visual concepts in a way that is spatially and temporally robust. Manipulating these representations facilitates concept strengthening or steers models to resolve misinterpretations, even for polysemous prompts (Park et al., 3 Dec 2024).
• Mechanical and Robotic Design: General force expressions for actuators, cast in terms of geometric variables rather than fixed attachment positions, enable universal application, flexible optimization, and comparative analysis across mechanical configurations (Saxena, 2016).

7. Limitations and Future Directions

While position independence offers efficiency, flexibility, and modularity, several challenges remain:

• Overhead of Indirection: Emulation of indirect jumps or chunk recombination may induce computational or memory overhead from extra operations (e.g., search over registers, selective recomputation in caching) (0711.0829, Hu et al., 20 Oct 2024).
• Clustering Sensitivity and Density Models: In density-based concept representations, results are subject to the choice of clustering or density estimation technique; improper parameterization can yield under- or over-specified concept groups (Schmelzeisen et al., 2019).
• Scalability in Multimodal Systems: As cache sizes grow (e.g., for high-dimensional images), bandwidth and storage can become bottlenecks. Selective recomputation must carefully balance speed and accuracy, especially as the diversity of content and context grows (Zhao et al., 4 Feb 2025).
• Positional Edge Cases in Logic and Geometry: While formal definitions capture position independence, pathological configurations may still challenge separation or independence assumptions (e.g., degenerate geometric arrangements or interference at resource boundaries) (Cardinal et al., 2014, 0802.0820).
• Extensions to Broader Modalities: Future work includes applying these position-independent constructions to new domains, optimizing further for storage and bandwidth, and refining semantic representations to better capture multiword and multimodal concepts (Schmelzeisen et al., 2019, Zhao et al., 4 Feb 2025).

In summary, position-independent concepts provide core framework elements across systems theory, geometry, information representation, AI infrastructure, and physical modeling. Their adoption underpins advances in modularity, efficiency, and semantic clarity, and ongoing research continues to explore their limits and applications in increasingly complex domains.