Recursion

Understanding Recursion

Recursion is a concept commonly used in mathematics, statistics, and computer science, as well as other fields such as linguistics, logic, artificial intelligence, and gaming.

It involves a function calling itself directly or indirectly in a circular loop.

Divide, Conquer, and Refine

Recursion allows for the refinement of computation accuracy and reduces the overall computational load required.

By breaking down a larger and more complex problem into smaller pieces, often called “divide and conquer,” recursion enables quicker and more feasible problem-solving.

It relies on the repeated calling of a function, following base cases and recursive steps.

Sequencing and Flexibility in Problem-Solving

Base cases represent the most straightforward instances of a problem with inputs that can compute the output.

Recursive steps involve calling the same function with decreasing input sizes or complexities.

Both iterative and recursive algorithms break down problems into smaller components, but iterative algorithms require explicit control logic to manage the sequencing of the algorithms.

Recursive functions, on the other hand, determine sequencing through the interaction of vector data and the recursive function itself.

This flexibility is beneficial when dealing with data structures that cannot be predefined.

Application of Recursion in Blockchain Technology

Recursion generates proofs on the blockchain and has found applications in various production systems on the Ethereum mainnet.

Recursion technology overcomes the limitation of fitting a limited number of transactions into a proof determined by a single block‘s computation capacity.

Recursion can generate a single proof from multiple verified proofs containing many underlying transactions.

This approach, known as recursive scaling, allows the processing of numerous transactions within a single proof.

Enhancing Layer-2 Scaling

With the addition of recursion, multiple transactions or statements can be sent to SHARP and proven in parallel.

Each proof is then validated by a STARK verifier and merged again through a Recursive Verifier statement.

This recursive loop can repeat until a final proof is submitted to Layer 1 for validation by a Solidity verifier smart contract.

This recursive approach attests to all original statements, enabling the processing of multiple on-chain transactions within a single proof.

The potential for indefinite recursion offers “hyper-scaling” capability.

Recursion further enhances roll-up technology and layer-2 scaling in blockchain systems.

Benefits of Recursion

Recursion provides several benefits for validity proofs, including reduced gas costs resulting from the compression of multiple proofs into one.

This allows for more transactions to be included in a single proof, effectively reducing the gas cost per transaction.

The computational resource barrier for proof size is no longer a constraint since extensive statements no longer need to be proven in one go.

Enhanced Scalability and Participation

Recursion also reduces latency as statements containing smaller transactions can be proven in parallel without waiting for other transactions.

Provers no longer need to perform extensive off-chain computations, lowering the barriers to becoming a prover and encouraging a decentralized network of provers to increase the network’s processing capacity.

Recursion Beyond Layer 2

Recursion sets the stage for layer 3 use cases in blockchain technology.

While recursion has primarily been used to generate proofs on layer 2 that settle on layer 1, a new layer opens up the possibility of submitting transaction proofs from layer 3 to layer 2.

Execution is performed on the top layer, and the proof is verified on layer 2. This further unlocks performance optimization and cost benefits, preserving transaction integrity and security.

Layer 3 Networks with Recursion for Improved Privacy

Private layer 3 networks offer high customization, allowing protocols to set their own operational parameters and implement privacy-preserving features.

The tailored nature of layer 3 networks, combined with improved processing capabilities from recursion, enables customized chain experiences while providing tools for performance and cost optimization.

The benefits of layer 3 include hyper scalability through the multiplicative effect of recursive proving, privacy features, and improved interoperability between layer 2 and layer 3.

Over time, additional use cases and benefits of recursion in blockchain development will continue to emerge.

By unlocking parallelization, recursion makes hyper-scaling possible, improving latency and reducing gas fees simultaneously.