Minimal Complete Semantic Units (MCSUs)

• Minimal Complete Semantic Units (MCSUs) are defined as the smallest irreducible semantic units that fulfill both minimality and completeness criteria across linguistic, visual, and graph-based representations.
• They enable effective token-level multi-model collaboration by aligning divergent subword vocabularies and improving performance in tasks like math, commonsense, and symbolic reasoning.
• MCSU concepts extend to various domains including sememe-based language models, sentence simplification, and speech tokenization, demonstrating measurable gains in perplexity, ROUGE scores, and inference efficiency.

Searching arXiv for papers on Minimal Complete Semantic Units and closely related semantic-unit formulations. I’m gathering the relevant arXiv records to ground the article in the cited literature. Minimal Complete Semantic Units (MCSUs) are used in several research traditions to denote units that are simultaneously minimal and complete with respect to meaning. In token-level multi-model collaboration, an MCSU is defined as “the smallest unit of complete meaning in a language”: in alphabetic languages it coincides with whitespace- or punctuation-delimited words or numbers, while in logographic languages each character or word naturally forms an MCSU (Hao et al., 26 Aug 2025). In sememe-based language modeling, the corresponding object is the sememe, “the smallest indivisible unit of meaning” (Gu et al., 2018). In sentence splitting, the analogous object is the “minimal proposition,” a “sound, self-contained utterance presenting a minimal semantic unit that cannot be further decomposed into meaningful propositions” (Niklaus et al., 2019). In knowledge-graph and FAIR work, statement units are the “smallest, independent propositions that are semantically meaningful for a human reader,” and atomic statement units satisfy explicit minimality and completeness conditions under a schema (Vogt et al., 2023, Vogt et al., 30 Sep 2025). This suggests that MCSU is best understood as a family of formally minimal and semantically closed units instantiated at different representational levels rather than as a single universal object.

1. Definitional core and formal criteria

Across the cited literature, the two recurrent constraints are minimality and completeness. In the Dynamic DDS framework, if a conventional tokenizer produces a sequence of subword tokens $T_1, T_2, \ldots, T_n$ that together constitute one semantically whole unit $M$, then $M$ is an MCSU, and its probability is given by the joint product of the autoregressive probabilities of its constituent tokens:

$$P_m = \prod_{i=1}^{n} P(T_i \mid T_1, T_2, \ldots, T_{i-1}) .$$

Tokens that do not themselves carry complete meaning are concatenated until the full unit emerges (Hao et al., 26 Aug 2025).

In sentence splitting, the same logic is expressed propositionally. A minimal proposition has exactly one predicate–argument structure, no subordinate or coordinate clause, and satisfies $|P(s)| = 1$. The minimality condition is that there is no nontrivial split of the sentence into two parts such that both parts still contain propositions. The operational filters are dependency-parse based: the root must be a verb, there must be a nominal subject, and labels such as $acl$, $advcl$, $relcl$, $ccomp$, $xcomp$, $M$0, and $M$1 are excluded (Niklaus et al., 2019).

In FAIR semantics, minimality and completeness are stated as schema-theoretic constraints. A statement unit $M$2 is an MCSU at the atomic level iff $M$3 and $M$4. A compound unit $M$5 is an MCSU at its level iff the union of the child content-graphs satisfies the compound shape and removing any child unit breaks that satisfaction (Vogt et al., 30 Sep 2025). In the knowledge-graph formulation, statement units partition the data graph:

$M$6

so each statement unit is both semantically self-standing and graph-theoretically non-overlapping (Vogt et al., 2023).

2. Token-level MCSUs in dynamic multi-model collaboration

The 2025 Dynamic DDS framework introduces MCSUs to resolve vocabulary misalignment in collaboration among heterogeneous LLMs. The motivating problem is that different LLMs may use divergent subword vocabularies, such as “Lla”+“ma” versus “Ll”+“ama,” making direct vector-space alignment noisy. By elevating the generation unit from raw subword tokens to MCSUs, the same unit “apple” or “2024” is represented and scored across models in a directly comparable way. Appendix E reports that approximately $M$7 of common English words are themselves single tokens in each model, so the MCSU layer introduces minimal overhead while sidestepping subword mismatch (Hao et al., 26 Aug 2025).

At each autoregressive step, each model computes its next-token distribution, performs Top-$M$8 sampling, and converts token candidates into candidate MCSUs. If a token is already a complete semantic unit, it is accepted directly; otherwise the model continues generating until the concatenation forms a valid MCSU. Each model then yields a Top-$M$9 distribution over MCSUs rather than over raw subword tokens. DDS forms the union $M$0 of all proposed MCSUs, renormalizes each model’s distribution over $M$1, assigns negligible mass $M$2 to non-$M$3 items, computes pairwise $M$4, retains only those distributions whose divergence to at least one other distribution is below $M$5, averages the retained distributions, and selects the MCSU with the highest probability (Hao et al., 26 Aug 2025).

The reported gains are end-to-end gains of DDS, whose operation relies crucially on MCSUs. On math reasoning, DEEPEN average is $M$6 and DDS is $M$7. On commonsense reasoning, DEEPEN average is $M$8 and DDS is $M$9. On symbolic reasoning, DEEPEN average is $$P_m = \prod_{i=1}^{n} P(T_i \mid T_1, T_2, \ldots, T_{i-1}) .$$0 and DDS is $$P_m = \prod_{i=1}^{n} P(T_i \mid T_1, T_2, \ldots, T_{i-1}) .$$1. On HumanEval / C-Eval, DEEPEN is $$P_m = \prod_{i=1}^{n} P(T_i \mid T_1, T_2, \ldots, T_{i-1}) .$$2 and DDS is $$P_m = \prod_{i=1}^{n} P(T_i \mid T_1, T_2, \ldots, T_{i-1}) .$$3. On TruthfulQA, single Qwen ROUGE is $$P_m = \prod_{i=1}^{n} P(T_i \mid T_1, T_2, \ldots, T_{i-1}) .$$4 and DDS is $$P_m = \prod_{i=1}^{n} P(T_i \mid T_1, T_2, \ldots, T_{i-1}) .$$5 (Hao et al., 26 Aug 2025).

The limitations are also explicit. DDS incurs linear compute and memory overhead in the number of models; loading multiple large models may be infeasible on resource-constrained devices; DDS may sometimes retain an incorrect high-confidence MCSU if all distributions diverge; and while MCSUs work well for languages with clear word or character boundaries, more work is needed for highly agglutinative or morphologically rich languages (Hao et al., 26 Aug 2025).

3. Sememes, sense prediction, and lexical grounding

In the Sememe-Driven LLM, the semantic atom is the sememe. A sememe is “the smallest indivisible unit of meaning,” and each word sense $$P_m = \prod_{i=1}^{n} P(T_i \mid T_1, T_2, \ldots, T_{i-1}) .$$6 is annotated with a set of sememes $$P_m = \prod_{i=1}^{n} P(T_i \mid T_1, T_2, \ldots, T_{i-1}) .$$7 drawn from a global inventory $$P_m = \prod_{i=1}^{n} P(T_i \mid T_1, T_2, \ldots, T_{i-1}) .$$8. The model replaces the usual word-only softmax decoder with a three-stage sememe $$P_m = \prod_{i=1}^{n} P(T_i \mid T_1, T_2, \ldots, T_{i-1}) .$$9 sense $|P(s)| = 1$0 word decoding, where the sememe predictor estimates

$|P(s)| = 1$1

the sense predictor combines sparse sememe experts through a product-of-experts mechanism, and the word predictor marginalizes over senses:

$|P(s)| = 1$2

Sparsity arises because each expert only controls senses containing that sememe, and on average each word has approximately $|P(s)| = 1$3 sememes (Gu et al., 2018).

The reported empirical effects are specific. On Chinese news language modeling on People’s Daily, replacing the standard softmax with SDLM reduces perplexity by $|P(s)| = 1$4–$|P(s)| = 1$5 points across LSTM, Tied LSTM and AWD-LSTM backbones. Multi-sense words see a $|P(s)| = 1$6 perplexity gain versus $|P(s)| = 1$7 for single-sense words. Words with more sememes, such as $|P(s)| = 1$8–$|P(s)| = 1$9, benefit up to $acl$0 lower perplexity. On headline generation, SDLM raises ROUGE-1/2/L by approximately $acl$1–$acl$2 F1 points. Removing $acl$3 of HowNet connections only slightly degrades performance (Gu et al., 2018).

A different lexical formulation comes from graph-theoretic analysis of dictionaries. A dictionary is represented as a directed graph $acl$4, with an arc $acl$5 when $acl$6 appears in the definition of $acl$7. The Kernel is extracted by recursively deleting vertices with out-degree $acl$8, and empirical dictionaries exhibit $acl$9–$advcl$0 of $advcl$1, Core $advcl$2–$advcl$3 of $advcl$4, Satellites $advcl$5–$advcl$6 of $advcl$7, and $advcl$8–$advcl$9 of $relcl$0. MGS words are learned earlier, more concrete, more frequent, and more imageable than non-MGS words (Picard et al., 2013).

Within that exposition, an MCSU is defined as a special kind of Minimal Grounding Set $relcl$1 such that $relcl$2 is an MGS of $relcl$3, the induced subgraph $relcl$4 is strongly connected, and no proper subset satisfies both properties (Picard et al., 2013). This suggests an interpretation of MCSUs as minimal mutually interdefinable lexical backbones rather than merely local semantic atoms.

4. Minimal propositions in sentence splitting

Niklaus et al. operationalize MCSUs as “minimal propositions” in the MinWikiSplit corpus. A minimal proposition is “a sound, self-contained utterance presenting a minimal semantic unit that cannot be further decomposed into meaningful propositions.” Formally, the output sentence has exactly one predicate–argument structure, the dependency root is a verb, there is at least one nominal subject, and subordinate or coordinate clause labels are excluded. Equivalently, $relcl$5, and no further decomposition preserves nonempty propositions on both sides (Niklaus et al., 2019).

The splitting engine is DisSim, which applies a set $relcl$6 of hand-written transformation rules over dependency trees. Relative-clause reduction and coordination splitting are representative examples. If $relcl$7 denotes the transformation induced by exhaustive application of the rules, then

$relcl$8

such that each $relcl$9 fulfills the MCSU criteria and no further rule applies (Niklaus et al., 2019).

The corpus contains $ccomp$0 complex–simplified pairs. On a held-out sample of $ccomp$1 complex inputs, the reported statistics are: average number of tokens per simplified sentence $ccomp$2 $ccomp$3; average number of simplified sentences per complex input $ccomp$4 $ccomp$5; percentage of outputs identical to inputs $ccomp$6 $ccomp$7; average word-level Levenshtein distance from input $ccomp$8; SAMSA $ccomp$9; and $xcomp$0 (Niklaus et al., 2019).

The manual evaluation uses a random sample of $xcomp$1 outputs and three criteria: grammaticality, meaning preservation, and structural simplicity on a $xcomp$2–$xcomp$3 scale. Inter-annotator agreement is $xcomp$4, $xcomp$5, and $xcomp$6. Mean scores are $xcomp$7, $xcomp$8, and $xcomp$9 (Niklaus et al., 2019). In this line of work, an MCSU is therefore a mono-propositional sentence unit designed for simplification, parsing, and downstream processing.

5. Statement units, compound units, and FAIR semantics

In knowledge-graph work, semantic units are identifiable, semantically meaningful subgraphs. Statement units represent “smallest, independent propositions that are semantically meaningful for a human reader,” while compound units represent semantically meaningful collections of semantic units. The collection of all statement-unit data-graphs forms a partition of the data layer, which supports graph-alignment, subgraph-matching, knowledge-graph profiling, and management of access restrictions to sensitive data (Vogt et al., 2023).

The framework distinguishes assertional, contingent, and universal statement units. Assertional units express ABox-like facts about named individuals. Contingent units use some-instance resources to express existential readings. Universal units use every-instance resources to express universal readings. A later extension introduces four representational resource types—some-instance, most-instances, every-instance, and all-instances—to model assertional, contingent, prototypical, and universal statements without relying on blank nodes or OWL class expressions. Internal constraints include basic consistency, schema conformance, unique subject partitioning, and meta-layer separation (Vogt, 2024).

The Grammar of FAIR generalizes this architecture to atomic and composite units. Let $M$00 be the universe of all semantic units, partitioned into $M$01 and $M$02, with $M$03 and $M$04. Each semantic unit carries a globally unique persistent identifier $M$05, a semantic-unit class $M$06, a content-graph $M$07 for atomic units, and a meta-graph $M$08 for all units. An atomic statement unit is an MCSU iff there is no proper subgraph that still satisfies the schema of its class; a compound unit is an MCSU at its level iff the union of child content-graphs satisfies the compound shape and removing any child unit breaks semantic closure (Vogt et al., 30 Sep 2025).

The semantic-modularisation principle states that any dataset $M$09 is partitioned into disjoint MCSUs $M$10 such that $M$11 and $M$12 for $M$13, that composition is defined whenever a subset matches a compound-unit class, that there is a unique neutral unit $M$14, and that every MCSU has full provenance, license, and schema-metamodel information (Vogt et al., 30 Sep 2025). The result is an architecture in which each unit is independently addressable, machine-actionable, and compositionally predictable.

A representative workflow starts from the sentence “Parasite X has a mass of 24.76 grams.” After tokenization and normalization, the system constructs a content-graph containing triples for the statement-unit type, subject, predicate, value, and unit, attaches provenance and shape metadata, checks $M$15 together with $M$16, and publishes the result as a Nanopublication or Statement FDO (Vogt et al., 30 Sep 2025). In this setting, MCSUs are not merely linguistic segments; they are minimal schema-satisfying semantic objects for FAIR data infrastructures.

6. Speech-scale units, recurring patterns, and open problems

SyllableLM transfers the MCSU idea to speech. Here an MCSU is a “syllable-like” token with two key properties: minimality, because it cannot be further subdivided without losing cohesive semantic content, and completeness, because it carries a self-contained chunk of meaning roughly equivalent to a spoken syllable. Each unit corresponds to a contiguous span of HuBERT/Data2Vec2 frame-level features (Baade et al., 2024).

The first stage, LossPred, extracts noisy boundaries from a frozen pretrained speech encoder using masked prediction behavior. With masking span length $M$17 frames, approximately $M$18 second of audio, the method constructs a $M$19 loss-prediction matrix and segments it by normalized cut; a fast dynamic-programming variant minimizes the sum of squared distances to segment means while respecting a maximum group length $M$20. The second stage, SylBoost, refines boundaries by minimizing

$M$21

re-extracting boundaries after each iteration and finally quantizing mean-pooled segment features by $M$22-means plus agglomerative clustering (Baade et al., 2024).

The method provides controllable token rates. Reported operating points include $M$23 Hz $M$24 bps after deduplication with $M$25, $M$26 Hz with $M$27 units $M$28 bps, and $M$29 Hz $M$30 bps. On LibriSpeech test-clean, unsupervised syllable boundary detection reaches $M$31 for SD-HuBERT at $M$32 Hz, $M$33 for SylBoost (HuBERT), and $M$34 for SylBoost (Data2Vec2). On unit-to-speech resynthesis, WER/CER improve from $M$35 for SD-HuBERT $M$36 Hz $M$37 units to $M$38 for SylBoost (Data2Vec2) at $M$39 Hz $M$40, and to $M$41 for SylBoost (Data2Vec2) at $M$42 Hz $M$43. On zero-shot spoken language modeling, SyllableLM with $M$44 units at $M$45 Hz and $M$46 bps yields $M$47, $M$48, and $M$49; with $M$50 units at $M$51 Hz and $M$52 bps it yields $M$53, $M$54, and $M$55. The system is reported to match or exceed AudioLM and TWIST with $M$56 less compute and $M$57 faster inference (Baade et al., 2024).

Taken together, these works show that the MCSU idea recurs whenever a representation must be both irreducible and self-sufficient: a word-or-number unit for multi-model token fusion, a sememe for sense prediction, a mono-propositional sentence for simplification, a schema-closed statement unit for FAIR knowledge graphs, or a syllable-like segment for speech modeling. This suggests that the central research question is not whether there is one canonical MCSU, but how minimality and completeness should be formalized at a chosen representational level. The open problems stated in the cited work are correspondingly level-specific: vocabulary alignment across model families, typological coverage beyond languages with clear word or character boundaries, efficient inference under multi-model lockstep execution, and semantic modularization that remains interoperable across logical frameworks and data infrastructures (Hao et al., 26 Aug 2025, Vogt et al., 30 Sep 2025).