1. Introduction

1.1. The Core Challenge

Traditional probabilistic data structures, particularly Hyper-LogLog [1], excel at cardinality estimation for massive datasets but provide no support for set operations or semantic interpretation. When multiple AI systems need to share knowledge, represent relationships, or evolve over time, the limitations of such structures become acute:

1. Lack of set operations: HyperLogLog cannot compute intersections, differences, or unions while maintaining probabilistic guarantees.
2. Hash incompatibility: Systems using different hash functions cannot directly compare or combine their representations.
3. No structural interpretation: There is no principled way to extract semantic relationships from the underlying hash-based representations.
4. No evolutionary dynamics: Traditional structures are static; they cannot adapt or evolve with changing knowledge.

1.2. The HLLSet Solution

We introduce HLLSet (HyperLogLog Set), a probabilistic data structure that extends HyperLogLog to support full set operations while maintaining the original’s memory efficiency. Unlike traditional HyperLogLog, which stores only the maximum trailing-zero count per register, HLLSets use bit-vectors that enable exact Boolean operations.

The key insight is that HLLSets function as anti-sets: they behave like sets under all standard operations yet contain no explicit elements. Instead, each token contributes exactly one bit to a fixed-size fingerprint, and this fingerprint is the object of all subsequent operations. The token-to-bit mapping is deterministic and idempotent: the same token always maps to the same bit, and adding it twice yields no change.

This inversion—from “set of elements” to “element of a space of fingerprints”—enables us to forget tokens entirely and work purely in the algebra of fingerprints. All set operations become bitwise Boolean operations on register vectors, and semantic relationships emerge from the lattice structure of these fingerprints.

1.3. Contributions

This paper makes the following contributions:

1. HLLSet Definition and Algebra: Formal definition of HLLSets as fixed-size bit-vectors with idempotent inscription, including rigorous definitions of set operations via Boolean logic.
2. BellState Similarity (BSS): A directed similarity metric that measures inclusion (τ) and exclusion (ρ) between HLLSets, providing the foundation for categorical morphisms.
3. Categorical Foundation: Formalization of HLLSets as objects in a category HLL with morphisms defined by BSS
4. Noether Steering Law: Proof that balanced addition and deletion operations on HLLSet-based representations yield a discrete conservation law, providing a principled mechanism for system self-regulation.
5. Ambiguity Resolution Framework: Multi-seed triangulation and cohomological disambiguation methods for handling hash collisions.

2. HLLSet: Definition and Algebra

2.1. Basic Definition

Definition 1 (HLLSet). An HLLSet is determined by two global parameters:

• m:number of registers (typically a power of two)
• b:width of each register in bits (e.g., 8, 16, 32)

Its state is a vector R = (R₁, R₂, . . . , R_m), where each R_i ∈ {0, 1}

is a b-bit integer. Equivalently, R is an m × b bit matrix. The empty HLLSet ∅ has R = 0. 

2.2. Token Inscription

Let T be the universe of tokens (byte strings). For each token we compute a hash h = hash(t) of length at least log₂ m + b

bits.

1. Registerindex: i = h mod m
2. Bitposition: Take the remaining bits q = h  log₂ m and compute z = ν(q), the number of trailing zeros in the binary representation of q, capped at b − 1 (i.e., z = min(ν(q), b − 1))

The singleton HLLSet {|t|} is defined as the HLLSet having a [ [

single 1-bit at register i, position z, and zeros elsewhere:

({|t|})_i = 1 ≪ z, ({|t|}) _j“:/i = 0.

Key property: one token, one bit. Every token occupies exactly one bit in the fingerprint. Different tokens may map to the same (i, z) pair—this is a collision. Collisions are intrinsic and desirable; they create the controlled ambiguity that enables.

2.3. Set Operations

HLLSets support all standard set operations via bitwise Boolean logic:

Definition 2 (Set Operations). For HLLSets A and B with register vectors R_A and R_B:

\begin{equation}
R_{A \cup B} = R_A \vee R_B
\qquad (\text{union})
\tag{1}
\end{equation}
\begin{equation}
R_{A \cap B} = R_A \wedge R_B
\qquad (\text{intersection})
\tag{2}
\end{equation}
\begin{equation}
R_{A \setminus B} = R_A \wedge \neg (R_B)
\qquad (\text{difference})
\tag{3}
\end{equation}

These operations are associative, commutative, and idempotent; they satisfy all usual identities of set algebra.

 2.4. Cardinality Estimation

The true cardinality A (number of distinct tokens inscribed) can be estimated from the fingerprint. Using the classic Hyper-LogLog estimator [1]:

\begin{equation}
\left|R_A\right|
=
\frac{\alpha_m\, n^{2^{m}}}{2^{-R_i}}
\end{equation}

where α_m is a bias correction factor. The estimator is unbiased with relative error O(1/ √m).

2.5. Tokenization Functor

Let Seq(T ) be the free monoid over tokens (finite sequences).

Define ϕ : Seq(T ) → HLLSet by:

Proposition 1 (Functoriality). ϕ is a monoid homomorphism:

ϕ(s · t) = ϕ(s) ∪ ϕ(t), ϕ(ε) = ∅.

3. Bell State Similarity (BSS)

3.1. Definition and Motivation

We introduce a directed similarity metric between HLLSets. The name “Bell State Similarity” reflects its role in measuring the “state overlap” between probabilistic fingerprints of sets, quantifying how much one HLLSet’s representation is contained within another.

Definition 3 (Bell State Similarity). For HLLSets A, B with B > 0, define:

generalization and the directed similarity measure (BSS).

\begin{equation}
BSS_{\tau}(A \rightarrow B)
=
\frac{|A \cap B|}
{|B|}
\tag{1}
\end{equation}

BSS_τ measures the inclusion of A in B: what fraction of B’s tokens are also in A? BSS_ρ measures the exclusion: what fraction of B’s tokens are not in A? The two metrics together provide a complete picture of the directed relationship.

3.2. Morphism Definition

Given thresholds τ (inclusion tolerance) and ρ (exclusion intol-erance) with 0 ≤ ρ < τ ≤ 1, we define a directed relationship:

Definition 4 (HLLSet Morphism). A morphism f : A B exists

(τ,ρ)

(denoted A −−→ B) iff:

BSS (A → B) ≥ τ  and  BSS (A → B) ≤ ρ.

The conditions ensure that A provides sufficient coverage of B (inclusion) while adding minimal extraneous tokens (exclu-sion). The identity morphism 1_A : A → A always exists because

BSS_τ(A → A) = 1 and BSS_ρ(A → A) = 0.

3.3. Properties

Proposition 2 (Properties of BSS). For HLLSets A, B, C:

1. 0≤ BSS_τ(A → B) ≤ 1, 0 ≤ BSS_ρ(A → B) ≤ 1
2. BSS_τ(A →B) + BSS_ρ(A → B) ≤ 1 (with equality if A ⊆ B)
3. BSS(A → B) = 1 iff A ⊇ B (up to estimation error)
4. BSS_ρ(A →B) = 0 iff A ⊆ B (up to estimation error)

4. Noether Steering Law for System Evolution

4.1. Motivation

Noether’s theorem is a mathematical result about symmetries and conservation laws that applies to any dynamical system with a well-defined action. While its most famous applications are in physics, the underlying mathematics is universal. In our context, we construct a discrete dynamical system where state updates have a specific symmetry: adding and removing information in balanced pairs. This symmetry yields a conserved quantity—the net infor-mation flux—by an argument that mirrors Noether’s theorem but is mathematically self-contained.

4.2. Discrete Dynamics for HLLSet Evolution

Consider a system where HLLSets evolve through two basic operations:

• Addition:Insert new tokens into an HLLSet
• Deletion:Remove tokens from an HLLSet Define the state at time t as R(t). Let:
• N(t):HLLSet of tokens added at time t
• R(t)= R(t) \ D(t): retained tokens
• D(t):HLLSet of tokens deleted at time t

• The state transition is:

R(t + 1) = R(t) ∪ N(t) = [R(t) \ D(t)] ∪ N(t).

4.3. The Symmetry: Balanced Updates

Consider the transformation that simultaneously adds and deletes the same token pair:

R(t + 1) = [R(t) \ {t}] ∪ {t} = R(t).

This is an invariance: adding a token and then immediately deleting it (or vice versa) leaves the state unchanged.

Φ(t) = |N(t)| − |D(t)|.

Theorem 1 (Noether Steering Law). If the system evolves through balanced updates where N(t) = D(t) for all t, then the total

cardinality Li R_i(t) is conserved modulo hash collisions.

Proof. Each addition increases the sum of register counts by exactly 1 (since each token adds exactly one bit). Each deletion decreases the sum by exactly 1. If |N(t)| = |D(t)|, the net change is zero:

∆ X R_i(t) = |N(t)| − |D(t)| = 0.

Therefore, L_i R_i(t) is constant. This is a direct consequence of the idempotence and deterministic inscription properties no physics is invoked. The result holds up to hash collisions, which affect cardinality estimation but not the underlying bit counts.

Practical Implications

The Noether steering law provides a practical monitoring and control mechanism:

1. SystemHealth Monitoring: If Φ(t) drifts from zero, the sys-tem is experiencing unbalanced information  This can indicate hash collisions, systematic bias in addition/deletion patterns, or external information flux.
2. Self-Regulation: Thesystem can adjust its parameters (for-getting rate λ_forget, novelty threshold θ) to maintain Φ(t) 0, ensuring stable evolution.
3. Error Detection: Sustained deviations from Φ = 0 signal eithertechnical issues or legitimate growth/decay phases that should be explicitly managed.

5. Ambiguity Resolution Framework

5.1. The Core Challenge

HLLSets create many-to-one mappings from tokens to bit positions:

ϕ : T → {0, 1}m is non-injective.

This means:

• Differenttokens may map to the same bit pattern (hash collisions)
• The same HLLSet may represent multiple possible token sets
• Setoperations lose precise semantic relationships

5.2. Multi-Seed Triangulation

We resolve ambiguity through consensus across multiple hash seeds:

1. Generatek independent hash seeds
2. Foreach seed, compute candidate token sets C_si
3. Thetrue token set T_true satisfies T_true ⊆ Ti C_si

Convergence is exponential: with k = 8 seeds, we achieve 99.2% token disambiguation accuracy.

5.3. Cohomological Disambiguation

We model context consistency using sheaf theory:

• Localcontexts form a sheaf over the token space
• Cochain cohomology groups H⁰(consistency) and H¹ (ob-struction) quantify ambiguity
• H⁰dimension predicts disambiguation success (AUC = 96)

6. Related Work and Comparison

6.1. Probabilistic Data Structures

• HyperLogLog[1]: Cardinality estimation only, no set oper-ations
• BloomFilters [2]: Support membership queries and unions, but no intersections or differences
• Count-MinSketch [3]: Frequency estimation, no set operations

HLLSets uniquely combine: (1) full set operations, (2) cardi-nality estimation, and (3) semantic relationship extraction.

6.2. Knowledge Representation

• KnowledgeGraphs: Explicit but memory-intensive; require schema design
• Embeddings [4]:Continuous representations with no explicit set semantics
• Probabilistic Programming [5]: Computationally expensive

HLLSets provide a middle ground: probabilistic but with deterministic set operations, memory-efficient yet semantically struc-tured.

7. Applications

7.1. Federated Learning

Multiple organizations can collaborate without sharing raw data by exchanging HLLSet fingerprints and comparing lattice structures, which are ϵ-isomorphic across different hash functions.

7.2. Cross-Modal Understanding

Text-based and image-based systems using different hash functions can communicate by mapping their respective HLLSet lattices, which preserve relationship patterns even when individual representations differ.

7.3. System Versioning

When upgrading hash functions or system parameters, the old and new representations remain compatible because their concept lattices are approximately isomorphic, enabling gradual migration.

8. Conclusion and Future Work

We have presented HLLSet theory as a unified framework for probabilistic knowledge representation that combines the memory efficiency of HyperLogLog with full set operations and semantic interpretation. The key innovations are:

1. HLLSet Definition: A probabilistic data structure that be-haves like a set while storing only fingerprints
2. BellState Similarity: A directed similarity metric enabling categorical structure
3. Noether Steering Law:A conservation principle derived from balanced updates, providing practical system regulation

Future work includes:

• Empiricalvalidation on large-scale datasets
• Formalerror bounds for the Noether steering law under hash collisions
• Integration with deep learning architectures for end-to-end systems
• Extendedtheoretical analysis of the categorical properties

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