Multi-Word Extension (MQX)

• MQX is a framework that extends single-word processing to handle multi-word expressions and registers, enhancing expressivity in language, cryptography, and quantum computation.
• It leverages algorithmic innovations such as the Multi-Word Naming Game and multi-word tokenization to boost consensus rates, compress token sequences, and achieve significant speedups.
• Hardware implementations with MQX instructions, like widening multiplication and vectorized addition with carry, enable faster multi-word arithmetic critical for secure and high-performance computing.

The Multi-Word Extension (MQX) refers to a set of methods, algorithms, and hardware enhancements designed to efficiently process, represent, or communicate information units that extend beyond single words—either as multi-word expressions in language or as multi-word registers in hardware. MQX methods emerge in multiple research domains, including NLP, distributed consensus modeling, cryptography, and quantum computation. The term encompasses concrete software and hardware mechanisms for handling multi-word units, with significant implications for efficiency, expressive power, and real-world applicability.

1. Core Definition and Motivation

MQX generally denotes the extension from single-word operations or representations (e.g., single-word naming, scalar arithmetic) to structures, instructions, or communicative patterns involving sequences or bundles of words. Motivation arises in several domains:

• Language and Communication: Naturalistic human communication involves compositional, multi-word expressions; models and algorithms optimized for single words are inadequate for representing semantic, pragmatic, or grammatical nuance (Lou et al., 2015, Poddar, 2016).
• Cryptographic Computations: Large integer arithmetic foundational to many cryptographic and homomorphic encryption protocols necessitates efficient multi-word (e.g., 128-bit or larger) arithmetic. Hardware architectures typically operating on 64-bit words require extensions for proper performance scaling (Zhang et al., 15 Sep 2025).
• Tokenization and Compression in NLP: Multi-word expressions (MWEs) often carry meaning not recoverable by composing the meanings of components; efficient tokenization can fuse frequent MWEs into single tokens, improving model throughput (Gee et al., 15 Feb 2024).
• Quantum Computation: Logical structures with multi-target gates and operations, as in Multi-Target Quantum Computational Logic, reflect domain-general needs to extend beyond the traditional single-word or single-target paradigm (Sergioli, 2018).

2. Formal and Algorithmic Frameworks

2.1 Multi-Word Naming Game (MWNG) Model

The MWNG (Lou et al., 2015) extends classical naming games to simulate agreement on sentences composed of words drawn from fixed grammatical categories (e.g., subject, verb, object). Consensus requires agents to align on both the sentence pattern (ordered combination of categories) and the lexical items per category, yielding a consensus probability:

$$P_{\text{MWNG}} = P_{\text{pattern}} \times \prod_{j \in R} \left( \frac{I_{AB}^j}{L_A^j} \right)$$

where $P_{\text{pattern}}$ is the probability both agents have the same sentence pattern, $I_{AB}^j$ the intersection of words in category $j$, $L_A^j$ the total learned by agent $A$, and $R$ the set of categories in the pattern. Complexity, convergence rates, and memory requirements all scale with the number and overlap of categories.

2.2 Multi-Word Tokenization and Compression

Multi-Word Tokenizer (MWT) strategies (Gee et al., 15 Feb 2024) extend standard subword tokenization by merging statistically frequent $n$-grams (typically bigrams) into single tokens, modifying the vocabulary as $$\mathcal{V} \leftarrow \mathcal{V} \cup \mathcal{G}_K$$ for a set of top-$K$ $n$-grams $\mathcal{G}_K$. Tokenization proceeds via left-to-right maximal-match merging. Fast Vocabulary Transfer assigns embeddings to new tokens as linear combinations of the component word embeddings, followed by masked language modeling fine-tuning.

Sequence length compression, quantified by the ratio of token counts before and after MWT, directly impacts computational cost. Experiments show 20%–50% compression, yielding faster inference and, at fixed model window, higher information density.

2.3 Hardware: Multi-Word SIMD Extension (MQX)

In high-throughput cryptographic kernels, MQX (Zhang et al., 15 Sep 2025) introduces three new AVX-512 instructions:

• Widening multiplication (_mm512_mul_epi64): Multiplies 64-bit lanes to produce full 128-bit products.
• Addition with carry (_mm512_adc_epi64): Adds two 64-bit values per SIMD lane, with input/output carry propagation.
• Subtraction with borrow (_mm512_sbb_epi64): Subtracts with borrow in/out per lane.

These instructions implement "multi-word" arithmetic primitives as single instructions, replacing multi-instruction sequences and significantly lowering latency in polynomial and modular arithmetic required by NTTs, BLAS, and FHE.

3. Empirical Results and Comparative Performance

3.1 Consensus and Efficiency in Multi-Word Models

Simulations of MWNG (Lou et al., 2015) across random, small-world, and scale-free networks show that networks converge to shared sentence patterns and word sets, but with slower rates and increased memory consumption relative to single-word naming games. Agents favor simpler (shorter) patterns if available, as consensus probability decays with the combinatorial product of component match probabilities.

3.2 NLP Acceleration and Robustness via MWT

MWT (Gee et al., 15 Feb 2024) consistently yields boosted performance and robustness to sequence truncation across text classification and compression benchmarks. Even with massive truncation (sequence length reduced by a factor of 4), models retain high macro-F1. On DistilBERT, using MWT yields up to 18× inference speed improvements, with negligible to moderate ($\leq 2.7\%$ absolute) drops in accuracy, as compared to BPE tokenization.

3.3 Hardware Speedups for Cryptography

Parallelizing MQX-enabled kernels (Zhang et al., 15 Sep 2025) on commodity server CPUs results in 2.1×–2.7× speedup over plain AVX-512, 38×–62× over scalar baselines for BLAS/NTT, and brings CPU kernel performance within $35\times$ of state-of-the-art ASIC accelerators. Roofline analysis demonstrates that, with sufficient core scaling and cache bandwidth, CPU performance can approach ASIC throughput for large-integer cryptographic operations.

MQX Instruction Examples

Instruction	Operation	Purpose
_mm512_mul_epi64	64-bit × 64-bit $\rightarrow$ 128-bit	Wide prod. in modular mult.
_mm512_adc_epi64	64-bit add with carry in/out	Vectorized carry addition
_mm512_sbb_epi64	64-bit subt. with borrow in/out	Vectorized borrow subtr.

4. Applications and Impact across Domains

• Distributed Consensus and Language Evolution: MWX frameworks elucidate how syntactic structure and lexical complexity influence consensus in agent-based simulations, with potential implications for models of language evolution and cultural transmission (Lou et al., 2015).
• Model Compression and Industrial NLP: MWT and analogous MQX-driven techniques interface seamlessly with model distillation, quantization, and domain-adaptation regimes, supporting scalable, efficient deployment in computationally constrained environments (Gee et al., 15 Feb 2024).
• Cryptography and Secure Computation: MQX instruction sets serve as a crucial architectural enabler for efficient homomorphic encryption and lattice-based cryptography, closing the gap with custom hardware and making cryptographic workloads more economically viable on general-purpose systems (Zhang et al., 15 Sep 2025).
• Quantum Logic and Cognition: In computational logic, multi-target extensions such as MT-QCL allow logical operations over registers carrying multiple semantic units—paving the way for models with richer, non-compositional semantic representations (Sergioli, 2018).

5. Scalability, Limitations, and Engineering Considerations

5.1 Scalability

• Communication and Language Games: Convergence rates and memory usage in MWNG models increase with the number of categories and complexity of sentence patterns; overlapping patterns bias agents to select the shortest available forms (Lou et al., 2015).
• Tokenization-Based Approaches: Sequence compression and speedups from MWT scale with the number and frequency mass of multi-word tokens incorporated. Speedups are upper-bounded by the proportion of input reducible via pattern merging (Gee et al., 15 Feb 2024).
• Hardware Implementations: MQX benefits are realized when computation is compute-bound (fit into cache); for working sets exceeding L2, performance becomes memory-bound (Zhang et al., 15 Sep 2025).

5.2 Engineering and Adoption

• MQX is designed for minimal disruption to AVX-512 microarchitectures but still entails nontrivial ALU changes and microcode updates. Actual silicon implementation may expose additional constraints not captured in the Proxy ISA modeling adopted by the authors (Zhang et al., 15 Sep 2025).

5.3 Theoretical and Empirical Boundaries

• MWX models highlight a trade-off between expressivity and consensus probability: increasing the granularity of representation (e.g., more sentence roles, more word categories per unit) lowers the probability of successful local agreement and lengthens convergence (Lou et al., 2015).
• In hardware, MQX offers diminishing returns for arithmetic larger than a few words wide, as instructions remain fixed at 64- or 128-bit atomicity and higher word widths would require further architectural changes (Zhang et al., 15 Sep 2025).

6. Implications and Future Directions

MQX methods demonstrate that extending single-word models—whether in information representation, communication, or computation—to multi-word units can yield substantial benefits in expressivity, efficiency, and real-world relevance. These extensions underpin advances in parallel cryptographic computation on general-purpose CPUs, movement towards more realistic models of natural language use and evolution, and improved compression and inference speed for neural models in NLP.

Anticipated directions include:

• Extension of token merging schemes to more complex or adaptive compositionality (e.g., phrases with syntactic variation) in NLP tokenizers (Gee et al., 15 Feb 2024).
• Generalization of MWNG to communications involving higher-order syntactic and pragmatic cues, or to hierarchical language production (Lou et al., 2015).
• Expansion of hardware MQX instructions to variable word sizes and advanced modular arithmetic forms, informed by anticipated cryptographic protocol requirements (Zhang et al., 15 Sep 2025).
• Integration with models of multi-word entity recognition, multi-sense embeddings, and systematic semantic extension for better handling of figurative, idiomatic, and domain-specific language.

MQX thus serves as a key conceptual and technical axis for advancing both the efficiency and fidelity of language and computation in systems spanning agent-based models, NLP, and hardware for secure computation.