LLMs and Carbon Chains

Title: A Formal Comparative Model of Finite-State Carbon Chain Programs and Tokenized Sequences in Large Language Models (LLMs)

Abstract: This paper presents a novel formal comparison between finite-state carbon chain programs, derived from constrained covalent bond rotations, and tokenized sequences in large language models (LLMs). By exploring structural, functional, and mathematical parallels, we propose that discrete molecular systems, like carbon chain programs, can serve as analogs to the computational token systems utilized in bio-inspired computation within artificial intelligence, paving the way for advancements in molecular informatics.

1. Introduction: Molecular systems, particularly programmable carbon chain programs, demonstrate discrete state transitions based on chemical rotations and bond conformations. Similarly, large language models (LLMs) operate over sequences of discrete tokens governed by statistical and semantic transformations. Despite their differing domains, both systems can be modeled using formal finite-state architectures, suggesting a profound computational similarity that enhances our understanding of bio-inspired computation. This paper formalizes the comparison and explores its implications.

2. System Definitions

| Concept                    | Carbon Chain Program         | LLM Token Sequence         |

|---------------------------|------------------------------|---------------------------|

| Unit                      | Covalent bond in discrete states | Token (word, subword, character) |

| Alphabet (Σ)             | Set of allowed rotational states (e.g., {0°, 180°}) | Vocabulary of tokens |

| Program                   | Sequence of bond states {s1, s2, ..., sn} | Sequence of tokens {t1, t2, ..., tn} |

| Transition Function (δ)   | Physical rules governing bond-to-bond rotations | Transformer-based attention and embeddings |

| Initial State             | Fixed first bond             | BOS (Beginning of Sentence) token |

| Output                    | 3D spatial trajectory of terminal carbon | Text generation, embeddings, semantic vectors |

3. Mathematical Formalism

3.1 Finite-State Carbon Chain (FSM Representation): Let Σ = {0, 1, ..., m} denote the discrete rotational states. Let Q represent the set of molecular spatial configurations. δ: Q × Σ → Q is the bond transition function. Starting from an initial state q0:

q0 --s1--> q1 --s2--> q2 --s3--> ... --sn--> qn

The final spatial configuration qn is a deterministic function of the sequence {s1, ..., sn}.

3.2 Tokenized LLM Sequence: Let T represent the token vocabulary, and H0 be the initial state (e.g., BOS). At each step i:

Hi = fi(Hi-1, ti)

Where fi denotes the transformer function that computes the hidden state using prior context and the current token.

4. Structural Correspondence

| Feature                   | Carbon Program               | LLM                       |

|---------------------------|------------------------------|---------------------------|

| Initial Condition         | Fixed orientation             | BOS token                 |

| Contextual Logic          | Downstream bonds affected     | Self-attention context     |

| Transformation            | Rotational transition         | Learned embedding dynamics  |

| Output Interpretation      | 3D structure or motif        | Generated or decoded text  |

5. Encoding and Emergence: Carbon chain programs can encode structural motifs such as helices, sheets, or loops, derived from simple rotation rules. In parallel, LLMs produce syntactic or semantic constructs from token sequences. Both systems exhibit emergent complexity: protein-like folds in carbon chains and linguistic intelligence in large language models.

6. Applications and Implications

- Molecular AI: Predict molecular behavior utilizing LLM-style training on bond-state sequences.

- Bio-semantic Embedding: Generate molecular embeddings analogous to word embeddings.

- Shape-to-Text Translation: Map molecular conformation sequences to textual descriptions.

- Programmable Matter: Leverage molecular chains to perform logical operations.

7. Conclusion: Finite-state carbon chain programs and tokenized large language models, while originating from distinctly different scientific realms, can be modeled through shared computational formalisms. This discovery opens up opportunities for bio-inspired computation, applications in synthetic biology, and new hybrid AI-material systems where matter and machine converge through structural, symbolic, and state transitions.

Keywords: carbon chain programs, finite-state machine, large language models, bio-inspired computation.

Title: A Formal Comparative Model of Finite-State Carbon Chain Programs and Tokenized Sequences in Large Language Models (LLMs)

Abstract: This paper presents a novel formal comparison between finite-state carbon chain programs, derived from constrained covalent bond rotations, and tokenized sequences in large language models (LLMs). By exploring structural, functional, and mathematical parallels, we propose that discrete molecular systems, like carbon chain programs, can serve as analogs to the computational token systems utilized in bio-inspired computation within artificial intelligence, paving the way for advancements in molecular informatics.

1. Introduction: Molecular systems, particularly programmable carbon chain programs, demonstrate discrete state transitions based on chemical rotations and bond conformations. Similarly, large language models (LLMs) operate over sequences of discrete tokens governed by statistical and semantic transformations. Despite their differing domains, both systems can be modeled using formal finite-state architectures, suggesting a profound computational similarity that enhances our understanding of bio-inspired computation. This paper formalizes the comparison and explores its implications.

2. System Definitions

| Concept                    | Carbon Chain Program         | LLM Token Sequence         |

|---------------------------|------------------------------|---------------------------|

| Unit                      | Covalent bond in discrete states | Token (word, subword, character) |

| Alphabet (Σ)             | Set of allowed rotational states (e.g., {0°, 180°}) | Vocabulary of tokens |

| Program                   | Sequence of bond states {s1, s2, ..., sn} | Sequence of tokens {t1, t2, ..., tn} |

| Transition Function (δ)   | Physical rules governing bond-to-bond rotations | Transformer-based attention and embeddings |

| Initial State             | Fixed first bond             | BOS (Beginning of Sentence) token |

| Output                    | 3D spatial trajectory of terminal carbon | Text generation, embeddings, semantic vectors |

3. Mathematical Formalism

3.1 Finite-State Carbon Chain (FSM Representation): Let Σ = {0, 1, ..., m} denote the discrete rotational states. Let Q represent the set of molecular spatial configurations. δ: Q × Σ → Q is the bond transition function. Starting from an initial state q0:

q0 --s1--> q1 --s2--> q2 --s3--> ... --sn--> qn

The final spatial configuration qn is a deterministic function of the sequence {s1, ..., sn}.

3.2 Tokenized LLM Sequence: Let T represent the token vocabulary, and H0 be the initial state (e.g., BOS). At each step i:

Hi = fi(Hi-1, ti)

Where fi denotes the transformer function that computes the hidden state using prior context and the current token.

4. Structural Correspondence

| Feature                   | Carbon Program               | LLM                       |

|---------------------------|------------------------------|---------------------------|

| Initial Condition         | Fixed orientation             | BOS token                 |

| Contextual Logic          | Downstream bonds affected     | Self-attention context     |

| Transformation            | Rotational transition         | Learned embedding dynamics  |

| Output Interpretation      | 3D structure or motif        | Generated or decoded text  |

5. Encoding and Emergence: Carbon chain programs can encode structural motifs such as helices, sheets, or loops, derived from simple rotation rules. In parallel, LLMs produce syntactic or semantic constructs from token sequences. Both systems exhibit emergent complexity: protein-like folds in carbon chains and linguistic intelligence in large language models.

6. Applications and Implications

- Molecular AI: Predict molecular behavior utilizing LLM-style training on bond-state sequences.

- Bio-semantic Embedding: Generate molecular embeddings analogous to word embeddings.

- Shape-to-Text Translation: Map molecular conformation sequences to textual descriptions.

- Programmable Matter: Leverage molecular chains to perform logical operations.

7. Conclusion: Finite-state carbon chain programs and tokenized large language models, while originating from distinctly different scientific realms, can be modeled through shared computational formalisms. This discovery opens up opportunities for bio-inspired computation, applications in synthetic biology, and new hybrid AI-material systems where matter and machine converge through structural, symbolic, and state transitions.

Keywords: carbon chain programs, finite-state machine, large language models, bio-inspired computation.