Mathematics for Multi-Processor Architecture: First Sketch

This work is supported by a grant from the Mina Foundation

1. Introduction

Modern computation increasingly exhibits a natural bifurcation between high-level program logic and low-level data manipulation. This separation manifests most prominently in the distinction between CPU and GPU processing, where high-level logical operations occur on traditional processors while massive parallel data transformations are offloaded to specialized hardware. This separation is not merely an engineering solution but likely reflects fundamental properties of information processing at scale. This property can be seen in many natural processes, where complex data processing and sequential decision making processes are not naturally split. Think the distinction in the human brain between vision and planning.

2. Evolution of Computational Mathematics

2.1 Traditional CPU-Centric Computation

Historical computational models centered around sequential processing of relatively simple mathematical structures:

• Scalar operations
• Vector computations
• Two-dimensional matrix manipulations

These structures aligned well with CPU architectures optimized for sequential processing and branch prediction.

2.2 Modern Mathematical Structures

Contemporary applications, particularly in cryptography and machine learning, operate on fundamentally different mathematical objects:

• High-dimensional tensors
• Multivariate polynomials over finite fields
• Complex group operations on elliptic curves

These structures exhibit high parallelizability and data homogeneity, naturally lending themselves to GPU computation.

3. The Glue vs. Co-processor Paradigm

Modern computation is being pulled in two directions. On one hand, the fast pace of research and ever growing program complexity demand ever high-levels of computational abstractions. On the other hand, the huge amount of data being processed requires ever more low-level algorithmic and hardware optimizations. This design paradigm forces us to define fundamental computational costs such as the cost of data transfer.

3.1 Data Transfer Costs

A defining characteristic of modern computational systems is the significant cost of data movement between processing layers. This creates a fundamental tension:

• High-level logic requires flexibility and branching
• Low-level operations demand massive parallelism
• Data transfer between layers introduces substantial overhead

3.2 Algorithmic Stability

A crucial observation is the relative stability of low-level algorithmic primitives compared to high-level structures:

• High-level protocols evolve rapidly (e.g., new zero-knowledge proof systems)
• Core mathematical operations remain consistent (FFT, MSM)
• This stability enables focused optimization at the lower level

4. Theoretical Framework for Natural Stratification

This framework is defined by the need to optimize resource allocation, maximize information throughput, and minimize errors. We want to define precise upper and lower bounds for computational stratification, e.g. define exactly when the system splits into two parts. The key variable is the cost of information transfer between layers. And finally we want to define error correction capabilities for the system so that interactions between layers happens in a robust fashion.

4.1 Resource Allocation Optimization

The first fundamental principle governing natural stratification is optimal resource allocation:

1. Computational Resource Partitioning

   • CPUs optimize for branching and sequential logic
   • GPUs optimize for parallel computation
   • Memory hierarchies align with computational patterns

2. Workload Distribution

   • Tasks naturally separate based on their computational characteristics
   • Resource utilization improves through specialization
   • System architecture adapts to computational patterns

4.2 Information Throughput Maximization

The second principle concerns the optimization of information flow through the system:

1. Data Movement Optimization

   • Minimize cross-layer communication
   • Batch similar operations
   • Localize data transformations

2. Processing Alignment

   • Match data structures to processing capabilities
   • Align algorithm structure with hardware architecture
   • Optimize for data locality

4.3 Error Propagation Minimization

The third principle addresses system reliability and error management:

1. Error Isolation

   • Contain computational errors within layers
   • Prevent error cascade across system levels
   • Enable layer-specific error correction strategies

2. Verification Boundaries

   • Create natural checkpoints between layers
   • Enable independent verification of subsystems
   • Facilitate debugging and testing

5. Mathematical Formalization

Computational Systems

We define a $\textbf{computational system}$ as a tuple:

$$\mathcal{C} = (C, \mathcal{L}, \mathcal{P})$$

where:

• $C$ is a finite set of computational resources (e.g., processors, memory units).
• $\mathcal{L}$ is a set of logic operations (sequential computations).
• $\mathcal{P}$ is a set of parallel operations (operations that can be executed concurrently).

Computational Cost and Efficiency

• $\textbf{Computational Cost }(\kappa)$: A function measuring the resource consumption of a computational process, considering time, energy, and hardware utilization.
• $\textbf{Information Processed }(I)$: The amount of information (in bits) that is processed by a computational process.
• $\textbf{Energy or Resource Consumption }(E)$: The total resources expended during computation.

Definition (Computational Stratification) A computational system exhibits $\textbf{natural stratification}$ if there exists a partition function:

$$\phi: \mathcal{C} \rightarrow (\mathcal{C}_h, \mathcal{C}_l)$$

where:

• $\mathcal{C}_h = (C_h, \mathcal{L}_h, \mathcal{P}_h)$ represents the $\textbf{high-level computational layer}$.
• $\mathcal{C}_l = (C_l, \mathcal{L}_l, \mathcal{P}_l)$ represents the $\textbf{low-level computational layer}$.
• $C_h \cup C_l = C$ and $C_h \cap C_l = \emptyset$.

The total computational cost satisfies:

$$\kappa(\phi(\mathcal{C})) < \kappa(\mathcal{C})$$

indicating that stratification leads to a more efficient computation.

In future works, we will further build on this definition to model the computational and communication costs, and define the optimal bounds for when stratification occurs.

6. Conclusion

The natural stratification of computation likely represents a fundamental property of complex information processing systems. Understanding this principle will enable better system design and provide a strong theoretical framework for future computational architectures.

References

Buterin, V. (2024). Glue and coprocessor architectures. https://vitalik.eth.limo/general/2024/09/02/gluecp.html