# Introduction to STARK Recursion

The Polygon zkEVM's rollup strategy is to develop a zero-knowledge Prover (zkProver) that takes a batch of many transactions, proves their validity, and publishes a minimally-sized validity proof for verification.

This document provides the details of how such a validity proof is created. It is a process that involves collating a number of proofs into one, using three methods; recursion, aggregation, and composition.

## Proving Approach​

zkProver is the main component of Polygon zkEVM and solely responsible for proving execution correctness. Instead of using the arithmetic circuit model, the zkProver follows the state machine model.

The approach therefore is to develop a State Machine that allows a Prover to create and submit a verifiable proof of knowledge, and anyone can take such a proof to verify its validity.

The process that leads to achieving such a State Machine-based system takes a few steps;

• Modeling the deterministic computation involved as a State Machine, described in the form of an Execution Trace.
• Stating the equations that fully describe the state transitions of the State Machine, called Arithmetic Constraints.
• Using established and efficient mathematical methods to define the corresponding polynomials.
• Expressing the previously stated Arithmetic Constraints into their equivalent Polynomial Identities.

These Polynomial Identities are equations that can be easily tested in order to verify the Prover's claims.

A Commitment Scheme is required for facilitating the proving and verification. Henceforth, in the zkProver context, a proof/verification scheme called PIL-STARK is used. Check out the documentation here for the Polygon zkEVM's commitment scheme setting.

## Overall Process​

In a nutshell, a state machine's execution trace is expressed in PIL, and this expression is called the PIL Specification of the computation represented by the State Machine.

In the non-recursive case, a PIL specification is transformed into a verifiable STARK proof by using PIL-STARK.

Subsequently, CIRCOM takes the above STARK proof as an input and generates an Arithmetic circuit and its corresponding witness.

The Arithmetic circuit is expressed in terms of its equivalent Rank-1 Constraint System (R1CS), while the witness is actuallly a set of input, intermediate and output values of the circuit wires, satisfying the R1CS.

Finally, Rapid SNARK takes the above Witness together with the STARK Verifier data and generates a SNARK proof corresponding to the previous STARK proof.

The SNARK proof gets published as the validity proof of the original computation.