Language Calculus: An Algebraic Framework for LLM Composition

What if we could compose language models the way we compose functions in mathematics? What if there was an algebra of language models?

We present Language Calculus (langcalc)—a comprehensive algebraic framework that transforms how we build and reason about language model systems.

The Problem with Current LLM Composition

Today, combining language models typically means:

• Ad-hoc ensembling techniques
• Manual prompt chaining
• Hardcoded decision trees
• Black-box orchestration layers

There’s no principled way to reason about what these compositions do or how they behave.

The Algebraic Solution

Language Calculus introduces a rich set of operators with well-defined semantics:

Core Operators

M₁ + M₂     Mixture (weighted combination)
k * M       Scaling (temperature/probability adjustment)
M₁ — M₂     Maximum (most confident response)
M₁ & M₂     Minimum (most conservative response)
M₁ ⊕ M₂     Exclusive-or (diverse perspectives)
M ** t      Temperature adjustment
M ¿¿ p      Threshold filtering
M >>> τ     Truncation/limiting
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Why This Matters

These operators satisfy algebraic laws:

(M₁ + M₂) + M₃ = M₁ + (M₂ + M₃)   # Associativity
M₁ + M₂ = M₂ + M₁                  # Commutativity
M + 0 = M                          # Identity
a * (M₁ + M₂) = a*M₁ + a*M₂        # Distributivity
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This means we can transform, optimize, and reason about language model compositions algebraically!

Practical Examples

Ensemble with Confidence Weighting

output = 0.4 * GPT4 + 0.3 * Claude + 0.3 * Llama
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Expert Selection

code_task = (CodeLlama — GPT4) & SafetyModel
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Diverse Brainstorming

ideas = CreativeModel ⊕ ConservativeModel ⊕ TechnicalModel
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Temperature Search

explore = Model ** 1.5
exploit = Model ** 0.2
adaptive = 0.7 * exploit + 0.3 * explore
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Theoretical Foundations

The framework provides:

• Formal semantics for each operator
• Type system ensuring valid compositions
• Equivalence relations for optimization
• Normal forms for canonical representations

This lets us prove properties like:

• Safety preservation under composition
• Bias reduction through specific mixtures
• Computational complexity bounds

Applications

Language Calculus enables:

1. Automatic Optimization: Transform expensive compositions into equivalent cheaper ones
2. Compositional Testing: Verify properties of complex systems from component properties
3. Explainability: Understand what a composition does from its algebraic structure
4. Meta-Learning: Learn optimal compositions for task families

Implementation

The paper includes:

• Reference implementation in Python
• Type checker for valid compositions
• Optimizer for algebraic simplification
• Library of common composition patterns

Read More

For the full mathematical treatment, formal semantics, and experimental validation:

→ View Paper

The paper covers:

• Complete operator semantics
• Algebraic laws and their proofs
• Type system formalization
• Optimization algorithms
• Empirical evaluation on benchmark tasks
• Comparison with existing composition methods

Tags: language model composition, algebraic framework, formal methods, LLM engineering, compositional systems