Introduction

powdr is a modular compiler stack to build zkVMs. It is ideal for implementing existing VMs and experimenting with new designs with minimal boilerplate.

• Domain specific languages are used to specify the VM and its underlying constraints, not low level Rust code
• Automated witness generation
• Support for multiple provers as well as aggregation schemes
• Support for hand-optimized co-processors when performance is critical
• Built in Rust 🦀

Contributing

powdr is free and open source. You can find the source code on GitHub. Issues and feature requests can be posted on the GitHub issue tracker.

License

The powdr source and documentation are released under the MIT License.

Installation

The only way to install powdr currently is to build it from source. There are two binaries, powdr and powdr-rs.

The former can compile powdr-asm files to powdr-PIL, as well as generate witnesses and proofs.

The latter compiles Rust crates to powdr-asm via RISCV, and can execute powdr-asm code with given inputs.

Prerequisites

You will need the Rust compiler and Cargo, the Rust package manager. The easiest way to install both is with rustup.rs.

On Windows, you will also need a recent version of Visual Studio, installed with the "Desktop Development With C++" Workloads option.

If you want to enable feature estark-polygon, you also need the following runtime dependencies installed on the system:

• gcc
• nlohmann-json3-dev

You will also need the following build time dependencies:

• make
• pkg-config
• libpqxx-dev (Ubuntu) | libpqxx (Arch Linux)
• nasm

Building powdr

Using a single Cargo command (enable the Halo2 backend to use it with the cli):

cargo install --git https://github.com/powdr-labs/powdr --features halo2 powdr-cli

Or, by manually building from a local copy of the powdr repository:

# clone the repository
git clone https://github.com/powdr-labs/powdr.git
cd powdr
# install powdr-cli
cargo install --features halo2 --path ./cli

Building powdr-rs

Using a single Cargo command:

cargo install --git https://github.com/powdr-labs/powdr powdr-rs-cli

Or, by manually building from a local copy of the powdr repository:

# clone the repository
git clone https://github.com/powdr-labs/powdr.git
cd powdr
# install powdr-rs-cli
cargo install --path ./cli-rs

Hello World

Let's write a minimal VM and generate a proof!

machine HelloWorld with degree: 8 {
    // this simple machine does not have submachines

    reg pc[@pc];
    reg X[<=];
    reg Y[<=];
    reg A;

    instr incr X -> Y {
        Y = X + 1
    }

    instr decr X -> Y {
        Y = X - 1
    }

    instr assert_zero X {
        X = 0
    }

    // the main function assigns the first prover input to A, increments it, decrements it, and loops forever
    function main {
        A <=X= ${ std::prover::Query::Input(0) };
        A <== incr(A);
        A <== decr(A);
        assert_zero A;
        return;
    }
}

Let's generate a proof of execution for the valid prover input 0 (since 0 + 1 - 1 == 0)

powdr pil test_data/asm/book/hello_world.asm --field bn254 --inputs 0 --prove-with halo2

We observe that several artifacts are created in the current directory:

• hello_world.pil: the compiled PIL file.
• hello_world_opt.pil: the optimized PIL file.
• hello_world_constants.bin: the computed fixed columns which only have to be computed once per PIL file.
• hello_world_commits.bin: the computed witness which needs to be computed for each proof.
• hello_world_proof.bin: the ZK proof!

Note that the output directory can be specified with option -o|--output, and . is used by default.

Now let's try for the invalid input 1:

powdr pil test_data/asm/book/hello_world.asm --field bn254 --inputs 1 --prove-with halo2

In this case witness generation fails, and no proof is created.

Setup & Verification

The example above omits some important steps in proof generation: Setup and Verification key generation. Some proof systems, such as Halo2 (and other SNARKs), require a Setup to be performed before the proof. Such Setup can be specific to the program or universal, where its artifact is a binary usually called parameters or params. STARKs do not require a previous Setup.

Another step required before the proof is computed is key generation. A proving key and a verification key are generated taking into account the constraints and potentially Setup parameters. The proving key is used by the prover to generate the proof, and the verification key is used by the verifier to verify such proof. A single verification key can be used to verify any number of different proofs for a given program.

Therefore, when computing a proof, it is important that the Setup parameters and the verification key are also available as artifacts.

Below we reproduce the same proof as in the example above, keeping the artifacts needed for verification:

First we run the Setup, where the number given must match the degree of our source file (degree 8). The command below generates file params.bin.

powdr setup 8 --backend halo2 --field bn254

We can now compute the verification key, output in vkey.bin:

powdr verification-key test_data/asm/book/hello_world.asm --field bn254 --backend halo2 --params "params.bin"

The command above can read previously generated constants from the directory specified via -d|--dir, where . is used by default. If the constants are not present it computes them before generating the verification key.

The next command compiles and optimizes the given source, generating the file hello_world_opt.pil. It also computes both the fixed data and the witness needed for the proof, stored respectively in hello_world_constants.bin and hello_world_commits.bin.

powdr pil test_data/asm/book/hello_world.asm --field bn254 --force --inputs 0

We can now generate the proof:

powdr prove test_data/asm/book/hello_world.asm --field bn254 --backend halo2 --params "params.bin" --vkey "vkey.bin"

The proof can be verified by anyone via:

powdr verify test_data/asm/book/hello_world.asm --field bn254 --backend halo2 --vkey "vkey.bin" --params "params.bin" --proof "hello_world_proof.bin"

Note that CLI proof verification works analogously for eSTARK, without the setup step and using the Goldilocks field instead of Bn254.

Another aspect that was omitted in this example is the fact that this proof uses a Poseidon transcript and cannot be verified in a cheap way on Ethereum, even though we can verify it efficiently via powdr. There are two ways to enable verification on Ethereum:

1. Use a different transcript when generating this proof. See section Hello World on Ethereum for the same example targeting EVM verification.
2. Use proof aggregation to compress the proof we just generated using a circuit that can be verified on Ethereum. See section Hello World on Ethereum via proof aggregation to learn how to do that.

Using powdr as a library

Besides the CLI, powdr can also be used as a Rust library. The powdr crate exposes internal crates and data structures needed to compile code, generate witnesses, and compute proofs.

Add powdr to your crate's dependencies:

[dependencies]
powdr = { git = "https://github.com/powdr-labs/powdr", branch = "main" }

The following Rust code has the same workflow as our previous "Hello World" example. The full project can be found here, and as an example in the powdr crate. To run the example in the powdr repository, run:

cargo run --example powdr_crate_usage

You can also enable logs to know what is happening internally:

RUST_LOG=info cargo run --example powdr_crate_usage

use powdr::backend::BackendType;
use powdr::number::buffered_write_file;
use powdr::Bn254Field;
use powdr::Pipeline;

use std::path::Path;

fn main() {
    env_logger::init();

    // Straightforward case
    let _proof = Pipeline::<Bn254Field>::default()
        .from_file("test_data/asm/book/hello_world.asm".into())
        .with_prover_inputs(vec![0.into()])
        .with_backend(BackendType::Halo2, None)
        .compute_proof()
        .unwrap();

    // Step-by-step case

    // First we create the universal setup of size 8
    buffered_write_file(Path::new("params.bin"), |writer| {
        BackendType::Halo2
            .factory::<Bn254Field>()
            .generate_setup(8, writer)
            .unwrap()
    })
    .unwrap();

    // Configure a pipeline
    let mut pipeline = Pipeline::<Bn254Field>::default()
        .from_file("test_data/asm/book/hello_world.asm".into())
        .with_prover_inputs(vec![0.into()])
        .with_backend(BackendType::Halo2, None)
        .with_setup_file(Some("params.bin".into()));

    // Create the verification key
    buffered_write_file(Path::new("vkey.bin"), |w| {
        pipeline.export_verification_key(w).unwrap()
    })
    .unwrap();

    // Add the verification key to a fresh pipeline and create a proof
    let mut pipeline_fresh = pipeline.clone().with_vkey_file(Some("vkey.bin".into()));

    let proof = pipeline_fresh.compute_proof().unwrap();

    // Create yet another fresh pipeline only for proof verification
    let mut pipeline = pipeline
        .with_backend(BackendType::Halo2, None)
        .with_setup_file(Some("params.bin".into()))
        .with_vkey_file(Some("vkey.bin".into()));

    // Verify a proof created by a different Pipeline
    pipeline.verify(proof, &[vec![]]).unwrap();
}

Hello World on Ethereum

This example is a variation of the previous Hello World, targeting verification on Ethereum. In this example we will cover how to generate proofs directly using a Keccak transcript instead of the Poseidon transcript of the previous example, which will enable us to verify proofs onchain. There are almost no differences from the CLI perspective.

machine HelloWorld with degree: 8 {
    // this simple machine does not have submachines

    reg pc[@pc];
    reg X[<=];
    reg Y[<=];
    reg A;

    instr incr X -> Y {
        Y = X + 1
    }

    instr decr X -> Y {
        Y = X - 1
    }

    instr assert_zero X {
        X = 0
    }

    // the main function assigns the first prover input to A, increments it, decrements it, and loops forever
    function main {
        A <=X= ${ std::prover::Query::Input(0) };
        A <== incr(A);
        A <== decr(A);
        assert_zero A;
        return;
    }
}

Since the following command creates a proof, a Solidity verifier, and verifies the proof on the EVM, we need to have solc available in the system. One easy way to install it is by using svm.

powdr pil test_data/asm/book/hello_world.asm --field bn254 --inputs 0 --prove-with halo2 --backend-options "snark_single" -f --params params.bin

The extra parameter --backend-options snark_single tells powdr to produce a single SNARK that uses Keccak. The option -f forces overwriting the compiled files that have been generated before, which is useful if you are running the examples on the same directory as the previous example.

When the a proof is generated, it is verified on a simulated EVM transaction as well!

Verifying SNARK in the EVM...

We can observe again that a proof was created at hello_world_proof.bin.

Now we can generate a Solidity verifier, using the same setup (params) as the previous example:

powdr export-verifier test_data/asm/book/hello_world.asm --field bn254 --backend halo2 --backend-options "snark_single" --params params.bin

A Solidity file verifier.sol is generated. You can verify the same proof that we just generated by passing it to the contract together with the public inputs (which we have not used so far).

Note that the more complex your VM is, the larger the verifier contract will be. In some cases, it might exceed Ethereum's contract size limit. You can mitigate that by using proof recursion on proofs that use Poseidon transcripts. See the next section for details.

Hello World on Ethereum via proof aggregation

This example is yet another variation of the previous Hello World on Ethereum example, still targeting verification on Ethereum but supporting more complex VMs. As noted in the previous section, complex VMs can lead to large Solidity verifiers that exceed the contract size limit on Ethereum. One solution to that problem is to create proofs using the Poseidon transcript, as we did in the first example, and then use proof recursion to create a proof that we know we will be able to verify on Ethereum.

A recursive SNARK works by generating a proof for a circuit that verifies another proof. The circuit we are using here to prove our initial proof recursively is PSE's snark-verifier. This circuit is large enough to be able to prove complex programs that were proven initial with the Poseidon transcript, like our first example. Because of that our aggregation setup params and verification key are going to be larger than before and take longer to compute. The good news are that (i) we can use a pre-computed setup from a previous ceremony, and (ii) the verification key only has to be computed once per program.

First, we need a setup of "size" 2^22. This is the maximum execution trace length of the recursion circuit.

You can generate it using the command line below, or download a pre-computed one here.

This will take a couple minutes if you decide to compute it yourself:

powdr setup 4194304 --backend halo2 --field bn254

We can re-use the new large params.bin for both initial and recursive proofs. Let's start by re-computing our Poseidon proof like in the first example, but using our new setup file:

powdr pil test_data/asm/book/hello_world.asm --field bn254 --inputs 0 --prove-with halo2 --backend-options "poseidon" -f --params params.bin

This generates the initial proof hello_world_proof.bin. We'll also need a verification key:

powdr verification-key test_data/asm/book/hello_world.asm --field bn254 --backend halo2 --backend-options "poseidon" --params params.bin

Let's verify the proof with our fresh verification key to make sure we're on the right track:

powdr verify test_data/asm/book/hello_world.asm --field bn254 --backend halo2 --backend-options "poseidon" --params params.bin --vkey vkey.bin --proof hello_world_proof.bin

In order to avoid confusion between the application's artifacts that we've just generated and the recursion one we're going to generate later, let's rename them:

mv vkey.bin vkey_app.bin
mv hello_world_proof.bin hello_world_proof_app.bin

We can now generate a verification key for the Halo2 circuit that verifies our proofs recursively. This might take up to a minute.

powdr verification-key test_data/asm/book/hello_world.asm --field bn254 --backend halo2 --backend-options "snark_aggr" --params params.bin --vkey-app vkey_app.bin

Note that this verification key can only be used to verify recursive proofs that verify other proofs using the application's key vkey_app.bin.

We can now generate the recursive proof (can take 3 or more minutes and use 28gb RAM):

powdr prove test_data/asm/book/hello_world.asm --field bn254 --backend halo2 --backend-options "snark_aggr" --params params.bin --vkey vkey.bin --vkey-app vkey_app.bin --proof hello_world_proof_app.bin

We have a proof! Note that the proof contents have two fields, proof and publics. The proof object contains the binary encoding of the proof points, and the publics object contains the public accumulator limbs that we need in order to verify the recursive proof.

We can now verify the proof, using the publics object as input (your numbers will likely be different):

powdr verify test_data/asm/book/hello_world.asm --field bn254 --backend halo2 --backend-options "snark_aggr" --params params.bin --vkey vkey.bin --proof hello_world_proof_aggr.bin --publics "269487626280642378794,9378970522278219882,62304027188881225691,811176493438944,234778270138968319485,3212529982775999134,171155758373806079356,207910400337448,188563849779606300850,155626297629081952942,194348356185923309508,433061951018270,34598221006207900280,283775241405787955338,79508596887913496910,354189825580534"

Since the goal of the recursive proof was to be able to verify it on Ethereum, let's do that!

powdr export-verifier test_data/asm/book/hello_world.asm --field bn254 --backend halo2 --backend-options "snark_aggr" --params params.bin --vkey vkey.bin

A Solidity verifier is created in verifier.sol. The contract expects an array of the accumulators' limbs followed by a tightly packed proof, where each accumulator limb uses 32 bytes, and there is no function signature.

Note that this verifier can be used to verify any recursive proof that verifies exactly one Poseidon proof of the given circuit.

Examples

Besides the Hello World examples, you are encouraged to check out these more complex and real-world use cases:

• Ethereum state tests via powdr-RISCV + revm
• Brainfuck VM in 3 different implementations (powdr-asm interpreter, powdr-asm ISA, Rust interpreter + powdr-RISCV)
• Machine Learning zkVM Compiler

Using publics

Public values are a small but important part of verifying ZK proofs. Often, a verifier is interested inputs and/or outputs to a public function.

In the toy example below the prover can show that they know the square root of a public value that is published with the proof.

machine Square with degree: 8 {
    reg pc[@pc];
    reg X[<=];
    reg Y[<=];
    reg A;

    // Expose the register value of A in the last time step
    public N = A(7);

    instr square X -> Y {
        Y = X * X
    }
	
    function main {
        A <=X= ${ std::prover::Query::Input(0) };
        A <== square(A);
    }
}

This example uses a small VM with jump and a square instructions. The program reads the private input from the prover, squares it, and enters an infinite loop to ensure that all the remaining rows are filled with the result of A^2. Since the length of our execution trace is fixed and equals 8, we can tag the 8-th row of A (A[7]) as the publicly exposed number.

Let's run all steps needed to generate and verify a proof that 3² = 9:

1. Setup step:

powdr setup 8 --backend halo2 --field bn254

2. Witness generation:

powdr pil test_data/asm/sqrt_with_public.asm --field bn254 -i 3

3. Verification Key generation:

powdr verification-key test_data/asm/sqrt_with_public.asm --field bn254 --backend halo2 --params params.bin

4. Proof generation:

powdr prove test_data/asm/sqrt_with_public.asm --field bn254 --backend halo2 --params params.bin --vkey vkey.bin

5. Proof verification:

powdr verify test_data/asm/sqrt_with_public.asm --field bn254 --backend halo2 --params params.bin --vkey vkey.bin --proof sqrt_with_public_proof.bin --publics 9

Command-Line Help for powdr

This document contains the help content for the powdr command-line program.

Command Overview:

• powdr↴
• powdr pil↴
• powdr prove↴
• powdr verify↴
• powdr verification-key↴
• powdr export-verifier↴
• powdr setup↴
• powdr reformat↴
• powdr optimize-pil↴

powdr

powdr CLI to compile powdr-asm and powdr-pil programs

Usage: powdr [OPTIONS] [COMMAND]

Subcommands:

• pil — Runs compilation and witness generation for .pil and .asm files. First converts .asm files to .pil, if needed. Then converts the .pil file to json and generates fixed and witness column data files
• prove —
• verify —
• verification-key —
• export-verifier —
• setup —
• reformat — Parses and prints the PIL file on stdout
• optimize-pil — Optimizes the PIL file and outputs it on stdout

Options:

• --log-level <LOG_LEVEL> — Set log filter value [ off, error, warn, info, debug, trace ]

  Default value: INFO

powdr pil

Runs compilation and witness generation for .pil and .asm files. First converts .asm files to .pil, if needed. Then converts the .pil file to json and generates fixed and witness column data files

Usage: powdr pil [OPTIONS] <FILE>

Arguments:

• <FILE> — Input file

Options:

• --field <FIELD> — The field to use

  Default value: gl

  Possible values: gl, bn254

• -o, --output-directory <OUTPUT_DIRECTORY> — Output directory for the PIL file, json file and fixed and witness column data

  Default value: .

• -w, --witness-values <WITNESS_VALUES> — Path to a CSV file containing externally computed witness values

• -i, --inputs <INPUTS> — Comma-separated list of free inputs (numbers)

  Default value: ``

• -f, --force — Force overwriting of PIL output file

  Default value: false

• --pilo — Whether to output the pilo PIL object

  Default value: false

• -p, --prove-with <PROVE_WITH> — Generate a proof with a given backend

  Possible values: estark-starky, estark-starky-composite, estark-dump, estark-dump-composite

• --params <PARAMS> — File containing previously generated setup parameters

• --backend-options <BACKEND_OPTIONS> — Backend options. Halo2: "poseidon", "snark_single" or "snark_aggr". EStark and PilStarkCLI: "stark_gl", "stark_bn" or "snark_bn"

• --export-csv — Generate a CSV file containing the fixed and witness column values. Useful for debugging purposes

  Default value: false

• --csv-mode <CSV_MODE> — How to render field elements in the csv file

  Default value: hex

  Possible values: i, ui, hex

powdr prove

Usage: powdr prove [OPTIONS] --backend <BACKEND> <FILE>

Arguments:

• <FILE> — Input PIL file

Options:

• -d, --dir <DIR> — Directory to find the committed and fixed values

  Default value: .

• --field <FIELD> — The field to use

  Default value: gl

  Possible values: gl, bn254

• -b, --backend <BACKEND> — Generate a proof with a given backend

  Possible values: estark-starky, estark-starky-composite, estark-dump, estark-dump-composite

• --backend-options <BACKEND_OPTIONS> — Backend options. Halo2: "poseidon", "snark_single" or "snark_aggr". EStark and PilStarkCLI: "stark_gl", "stark_bn" or "snark_bn"

• --proof <PROOF> — File containing previously generated proof for aggregation

• --vkey <VKEY> — File containing previously generated verification key

• --vkey-app <VKEY_APP> — File containing the verification key of a proof to be verified recursively

• --params <PARAMS> — File containing previously generated setup parameters

powdr verify

Usage: powdr verify [OPTIONS] --backend <BACKEND> --proof <PROOF> --vkey <VKEY> <FILE>

Arguments:

• <FILE> — Input PIL file

Options:

• -d, --dir <DIR> — Directory to find the fixed values

  Default value: .

• --field <FIELD> — The field to use

  Default value: gl

  Possible values: gl, bn254

• -b, --backend <BACKEND> — Generate a proof with a given backend

  Possible values: estark-starky, estark-starky-composite, estark-dump, estark-dump-composite

• --backend-options <BACKEND_OPTIONS> — Backend options. Halo2: "poseidon", "snark_single" or "snark_aggr". EStark and PilStarkCLI: "stark_gl", "stark_bn" or "snark_bn"

• --proof <PROOF> — File containing the proof

• --publics <PUBLICS> — Comma-separated list of public inputs (numbers)

  Default value: ``

• --vkey <VKEY> — File containing the verification key

• --params <PARAMS> — File containing the params

powdr verification-key

Usage: powdr verification-key [OPTIONS] --backend <BACKEND> <FILE>

Arguments:

• <FILE> — Input PIL file

Options:

• -d, --dir <DIR> — Directory to find the fixed values

  Default value: .

• --field <FIELD> — The field to use

  Default value: gl

  Possible values: gl, bn254

• -b, --backend <BACKEND> — Chosen backend

  Possible values: estark-starky, estark-starky-composite, estark-dump, estark-dump-composite

• --backend-options <BACKEND_OPTIONS> — Backend options. Halo2: "poseidon", "snark_single" or "snark_aggr". EStark and PilStarkCLI: "stark_gl", "stark_bn" or "snark_bn"

• --params <PARAMS> — File containing previously generated setup parameters. This will be needed for SNARK verification keys but not for STARK

• --vkey-app <VKEY_APP> — File containing the verification key of a proof to be verified recursively

powdr export-verifier

Usage: powdr export-verifier [OPTIONS] --backend <BACKEND> <FILE>

Arguments:

• <FILE> — Input PIL file

Options:

• -d, --dir <DIR> — Directory to find the fixed values

  Default value: .

• --field <FIELD> — The field to use

  Default value: gl

  Possible values: gl, bn254

• -b, --backend <BACKEND> — Chosen backend

  Possible values: estark-starky, estark-starky-composite, estark-dump, estark-dump-composite

• --backend-options <BACKEND_OPTIONS> — Backend options. Halo2: "poseidon", "snark_single" or "snark_aggr". EStark and PilStarkCLI: "stark_gl", "stark_bn" or "snark_bn"

• --params <PARAMS> — File containing previously generated setup parameters. This will be needed for SNARK verification keys but not for STARK

• --vkey <VKEY> — File containing previously generated verification key

powdr setup

Usage: powdr setup [OPTIONS] --backend <BACKEND> <SIZE>

Arguments:

• <SIZE> — Size of the parameters

Options:

• -d, --dir <DIR> — Directory to output the generated parameters

  Default value: .

• --field <FIELD> — The field to use

  Default value: gl

  Possible values: gl, bn254

• -b, --backend <BACKEND> — Generate a proof with a given backend

  Possible values: estark-starky, estark-starky-composite, estark-dump, estark-dump-composite

powdr reformat

Parses and prints the PIL file on stdout

Usage: powdr reformat <FILE>

Arguments:

• <FILE> — Input file

powdr optimize-pil

Optimizes the PIL file and outputs it on stdout

Usage: powdr optimize-pil [OPTIONS] <FILE>

Arguments:

• <FILE> — Input file

Options:

• --field <FIELD> — The field to use

  Default value: gl

  Possible values: gl, bn254

This document was generated automatically by clap-markdown.

asm

powdr-asm is the higher level of abstraction in powdr. It allows defining Instruction Set Architectures (ISA) using virtual and constrained machines.

Modules

powdr exposes a module system to help organise and reuse code.

use my_module::Other as LocalOther;

// we can define a module at `./submodule.asm`
mod submodule;

// we can define a module at `./submodule_in_folder/mod.asm`
mod submodule_in_folder;

use submodule::Other as SubmoduleOther;
use submodule_in_folder::Other as FolderSubmoduleOther;

let zero: int = 0;

// we can also define modules inline
mod utils {
    // Each module has a fresh symbol list. Every external symbol needs to be imported,
    // even from the parent module.
    use super::zero;

    let one = zero + 1;
}

machine Main with degree: 8 {
    // use a machine from another module by relative path
    my_module::Other a;

    // use a machine from another module using a local binding
    LocalOther b;

    // use a machine from another module defined in a different file
    SubmoduleOther c;

    // use a machine from another module defined in a different directory
    FolderSubmoduleOther d;

    reg pc[@pc];

    instr nothing link => a.nothing();
    instr also_nothing link => b.nothing();
    instr still_nothing link => c.nothing();
    instr nothing_again link => d.nothing();

    function main {
        nothing;
        also_nothing;
        still_nothing;
        nothing_again;
        return;
    }
}

mod my_module {
    machine Other with
        latch: latch,
        operation_id: operation_id
    {
        operation nothing<0>;

        col fixed latch = [1]*;
        col fixed operation_id = [0]*;
    }
}

Note that a module can't be called std, as this name is reserved for the powdr standard library.

Similar to Rust, any reference that cannot be resolved is looked up once more in std::prelude. This module exposes basic types and values such as Option, true and false. This means that you can use Option anywhere without prefix.

Declarations

Symbols can be defined via let <name> = <value>;, or via let <name>: <type> = <value>; if you want to specify the type explicitly. The value is an arbitrary PIL-expression. For details, see the Declarations section in the PIL part.

Other symbols available in the current module can be accessed by name, but it is also possible to specify full relative paths in the form of e.g. super::super::module_name::symbol.

Here are some examples of how to define and use symbols:

mod utils {
    // This defines a function by means of a lambda expression that
    // computes the sum of an array of values. We fully specify its type.
    let sum: int, int[] -> int = |len, arr| match len {
        0 => 0,
        _ => arr[len - 1] + sum(len - 1, arr)
    };
    // A simple function that returns the input incremented by one,
    // as an expression.
    let incremented: expr -> expr = |x| x + 1;
    // This is a function that takes an expression as input and returns
    // a constraint enforcing this expression increments by a certain value
    // between rows.
    // The type will be inferred here because `'` is only valid on `expr`.
    let constrain_incremented_by = |x, inc| x' = x + inc;
}

machine Main with degree: 4 {
    // Machines create local scopes in the way functions create local scopes:
    // - all symbols in the machine's module are available without prefix,
    // - new symbols can be defined but are only available inside the machine.
    reg A;
    reg pc[@pc];

    // This defines a witness column,
    let x;
    // and now we force it to stay unchanged.
    utils::constrain_incremented_by(x, 0);

    // We define an instruction that uses a complicated way to increment a register.
    instr incr_a { A = utils::incremented(A) }

    function main {
        return;
    }
}

Machines

Machines are the first main concept in powdr-asm. They can currently be of two types: virtual or constrained.

Virtual machines

Dynamic machines are defined by:

• a degree, indicating the number of execution steps
• a set of registers, including a program counter
• an instruction set
• a set of powdr-pil statements
• a set of functions
• a set of submachines

An example of a simple dynamic machine is the following:

machine HelloWorld with degree: 8 {
    // this simple machine does not have submachines

    reg pc[@pc];
    reg X[<=];
    reg Y[<=];
    reg A;

    instr incr X -> Y {
        Y = X + 1
    }

    instr decr X -> Y {
        Y = X - 1
    }

    instr assert_zero X {
        X = 0
    }

    // the main function assigns the first prover input to A, increments it, decrements it, and loops forever
    function main {
        A <=X= ${ std::prover::Query::Input(0) };
        A <== incr(A);
        A <== decr(A);
        assert_zero A;
        return;
    }
}

Constrained machines

Constrained machines are a lower-level type of machine. They do not have registers, and instead rely on simple committed and fixed columns. They are used to implement hand-optimized computation.

They are defined by:

• a degree, indicating the number of execution steps
• a set of operations
• an operation_identifier column, used to make constraints conditional over which function is called. It can be omitted with _ if the machine has at most one operation.
• a latch column, used to identify rows at which the machine can be accessed from the outside (where the inputs and outputs are passed). It can be omitted if the machine has no operations.
• a set of submachines
• a set of links

An example of a simple constrained machine is the following:

machine SimpleStatic with
    degree: 8,
    latch: latch,
    operation_id: operation_id
{
    operation power_4<0> x -> y;

    col fixed operation_id = [0]*;
    col fixed latch = [0, 0, 0, 1]*;
    col witness x;
    col witness y;

    // initialise y to x at the beginning of each block
    latch * (y' - x') = 0;
    // x is unconstrained at the beginning of the block

    // x is constant within a block
    (1 - latch) * (x' - x) = 0;
    // y is multiplied by x at each row
    (1 - latch) * (y' - x * y) = 0;
}

For more details on the powdr-pil statements, check out the pil section of this book. Note that the parameters of the operation are columns defined in powdr-pil statements.

Submachines

Machines can have submachines which they access by defining external instructions or links. They are declared as follows:

machine MySubmachine {
    ...
}

machine MyMachine {
    MySubmachine my_submachine;
}

Machines can also receive submachines as construction parameters. A machine passed in as an argument can be accessed in the same way as locally declared submachines:

machine MachineWithParam(subm: MySubmachine) {
    // `subm` can be accessed as a submachine
    ...
}

machine MyMachine {
    MySubmachine my_submachine;
    // `my_submachine` is passed to `another_submachine` as a construction argument
    MachineWithParam another_submachine(my_submachine);
}

Registers

Registers are central to a machine. powdr supports a few types of registers:

Program counter

Each machine can have at most one program counter. In the absence of a program counter, the machine is considered static, and no other register can be declared. The program counter is defined as follows:

reg pc[@pc]

At each step execution step, the program counter points to the function line to execute. The program counter behaves like a write register, with the exception that its value is incremented by default after each step.

Write registers

Write registers are the default type for registers. They are declared as follows:

reg A;

They hold a field element, are initialized as 0 at the beginning of a function and keep their value by default. They can be read from and written to.

// write to A
A <=X= 1;
// A is 1

// read from A
B <=X= A;
// A is still 1

Assignment registers

Assignment registers are transient to an execution step: their value is not persisted across steps. They are required in order to pass inputs and receive outputs from instructions, as well as in assignments. For example, if we want to assert that write register A is 0, we can use the following instruction:

reg pc[@pc];
reg A;

instr assert_A_is_zero {
    A = 0
}

function main {
    assert_A_is_zero;
    return;
}

However, if we want the instruction to accept any write register as input, we use an assignment register.

reg pc[@pc];
reg X[<=];
reg A;    

instr assert_zero X {
    X = 0
}

function main {
    assert_zero A;
    return;
}

Read-only registers

Read-only registers are used for function inputs. However, powdr creates them automatically based on functions arguments, so that they do not need to be declared explicitly.

Read-only registers are only mentioned for completeness here and are currently only used inside the compiler. We advise against using them.

Functions

Functions are the entry points to a virtual machine. They can be called from another machine or from the outside.

In this section, we describe functions with this simple virtual machine:

machine Machine with degree: 16 {
    reg pc[@pc];
    reg X[<=];
    reg Y[<=];
    reg Z[<=];
    reg CNT;
    reg A;
    reg B;

    // an instruction to assert that a number is zero
    instr assert_zero X {
        X = 0
    }

    // an instruction to jump to a label
    instr jmp l: label {
        pc' = l 
    }

    // an instruction to jump to a label iff `X` is `0`, otherwise continue
    instr jmpz X, l: label { 
        pc' = XIsZero * l + (1 - XIsZero) * (pc + 1) 
    }
    
    // an instruction to return the square of an input as well as its double
    instr square_and_double X -> Y, Z {
        Y = X * X,
        Z = 2 * X
    }

    function main {
        // initialise `A` to 2
        A <=X= 2;
        // initialise `CNT` to `3`
        CNT <=X= 3;
        start:
        // if `CNT` is `0`, jump to `end`
        jmpz CNT, end;
        // decrement `CNT`
        CNT <=X= CNT - 1;
        // get the square and the double of `A`
        A, B <== square_and_double(A);
        // jump back to `start`
        jmp start;
        end:
        // check that `A == ((2**2)**2)**2`
        assert_zero A - ((2**2)**2)**2;
        // check that `B == ((2**2)**2)*2`
        assert_zero B - ((2**2)**2)*2;
        return;
    }

    // some superpowers on `X` to allow us to check if it's 0
    col witness XInv;
    col witness XIsZero;
    XIsZero  = 1 - X * XInv;
    XIsZero * X = 0;
    XIsZero * (1 - XIsZero) = 0;
}

Function inputs and outputs

Function inputs and outputs are not supported yet

Statements

Labels

Labels allow referring to a location in a function by name.

        start:

Assignments

Assignments allow setting the values of some write registers to the values of some expressions expression using assignment registers.

        CNT <=X= 3;

If the right-hand side of the assignment is an instruction, assignment registers can be inferred and are optional:

        A, B <== square_and_double(A);

This will be inferred to be the same as A, B <=Y, Z= square_and_double(A); from the definition of the instruction:

    instr square_and_double X -> Y, Z {
        Y = X * X,
        Z = 2 * X
    }

Instructions

Instructions which do not return outputs can be used as statements.

        assert_zero A - ((2**2)**2)**2;

Expressions

Field element literals

Field element literals are signed elements of the prime field.

        CNT <=X= 3;

Registers and columns

Registers can be used as expressions, with the exception of assignment registers.

        CNT <=X= CNT - 1;

Instructions

Instructions which return outputs can be used as expressions.

        A, B <== square_and_double(A);

Instructions

Instructions are declared as part of a powdr virtual machine. Once defined, they can be called by any function in this machine. An instruction is composed of:

• a name
• a set of inputs (assignment registers or labels)
• a set of outputs (assignment registers)
• a set of powdr-pil constraints to activate when the instruction is called
• a set of links calling into functions/operations in submachines

Local instructions

A local instruction is the simplest type of instruction. It is called local because its behavior is defined by a set of constraints over registers and columns of the machine it is defined in.

instr add X, Y -> Z {
    X + Y = Z
}

Instructions with links

Instructions may also delegate all or part of their implementation to functions/operations in submachines. Each link in an instruction defines the inputs and outputs of a call to a specific function/operation in a submachine.

Assume we have a submachine with a single operation add:

machine SubMachine with
    latch: latch,
    operation_id: operation_id
{
    col witness operation_id;
    col fixed latch = [1]*;

    operation add<0> x, y -> z;

    col witness x;
    col witness y;
    col witness z;
    z = y + x;
}

An instruction calling into this operation can be declared as follows:

    SubMachine submachine;

    instr add X, Y -> Z link => Z = submachine.add(X, Y); // - trivial usage: only instruction inputs/outputs used in the call

In the previous example, only assignment registers (instruction inputs and outputs) were used to call the submachine. The following example shows more complex usage of link calls:

machine Main with degree: 32 {
    reg pc[@pc];
    reg X[<=];
    reg Y[<=];
    reg Z[<=];
    reg A;
    reg B;
    reg C;

    SubMachine submachine;

    instr add X, Y -> Z link => Z = submachine.add(X, Y); // - trivial usage: only instruction inputs/outputs used in the call
    instr add_to_A X, Y link => A' = submachine.add(X, Y);// - output to a regular register
    instr addAB -> X link => X = submachine.add(A, B);    // - inputs from regular registers
    instr addAB_to_C link => C' = submachine.add(A, B);   // - inputs and output from regular registers
    instr addAB_to_A link => A' = submachine.add(A, B);   // - reusing an input register as output
    instr sub X, Y -> Z link => X = submachine.add(Y, Z); // - swapping input/output
    // expressions can also be used as call parameters
    instr add5 X -> Z link => Z = submachine.add(X, 3+2); // - literal expression as argument
    col fixed STEP(i) { i };
    instr add_current_time_step X -> Z link => Z = submachine.add(X, STEP);// - machine columns can be referenced
    let arr = [1,2,3,4,5];                          // - functions can be used
    instr add_arr_sum X -> Z link => Z = submachine.add(X, std::array::sum(arr));

    instr assert_eq X, Y { X = Y }

    function main {
        A <== add(2, 3);
        assert_eq A, 5;
        add_to_A 6, 7;
        assert_eq A, 13;

        A <== sub(6, 5);
        assert_eq A, 1;
        B <=X= 20;
        C <== addAB();
        assert_eq C, 21;

        A <=X= 2;
        B <=X= 3;
        addAB_to_C;
        assert_eq C, 5;

        A <=X= 33;
        B <=X= 44;
        addAB_to_A;
        assert_eq A, 77;

        A <== add5(2);
        assert_eq A, 7;
        A <== add_arr_sum(3);
        assert_eq A, 18;

        // Note that the result of this operation depends on when it executed (STEP column)
        A <== add_current_time_step(42);
        B <== add_current_time_step(42);
        assert_eq B - A, 1;

        return;
    }
}

A single instruction can activate multiple links, and may also include a set of constraints. Furthermore, each link can be activated conditionally, based on a given boolean flag:

machine Main with degree: 16 {
    reg pc[@pc];
    reg X[<=];
    reg Y[<=];
    reg Z[<=];
    reg W[<=];
    reg A;
    col witness B;
    col witness C;

    SubMachine submachine;

    // multiple links can be activated by a single instruction,
    // witness columns can be used for temporary values,
    // and additional constraints can be used
    instr double_then_mul X, Y -> Z
        link => B = submachine.add(X, X)
        link => C = submachine.add(Y, Y)
    {
        Z = B * C
    }

    // links activated conditional on a boolean flag
    instr add_or_sub W, X, Y -> Z
        link if W => Z = submachine.add(X, Y)
        link if (1 - W) => Z = submachine.sub(X, Y);

    instr assert_eq X, Y { X = Y }

    function main {
        A <== double_then_mul(3, 2);
        assert_eq A, 24;

        A <== add_or_sub(1, 3, 2);
        assert_eq A, 5;
        A <== add_or_sub(0, 3, 2);
        assert_eq A, 1;

        return;
    }
}

Note that links cannot currently call functions from the same machine: they delegate computation to a submachine.

Operations

Operations enable a constrained machine to expose behavior to the outside. If a machine has a single operation, it can simply be declared with its name and parameters:

machine Add with
    latch: latch
{
    // operation name, with column names as inputs and outputs
    operation add a, b -> c;

    col fixed latch = [1]*;

    col witness a;
    col witness b;
    col witness c;

    // constraint "implementing" the operation
    c = a + b;
}

The parameters of the operation (inputs and outputs) must be columns declared in the machine.

If a machine exposes more than one operation, the machine itself needs an operation id column (op_id in the following). Then, each operation needs to be declared with its own unique operation id:

// machine declaration must include an operation id column name
machine AddSub with
    latch: latch,
    operation_id: op_id
{
    // each operation has its own unique operation id
    operation add<0> a, b -> c;
    operation sub<1> a, b -> c;

    col fixed latch = [1]*;
        // it also needs to be declared as a column
    col witness op_id;

    col witness a;
    col witness b;
    col witness c;

    // constraint "implementing" both operations, depending on `op_id`
    c = (1 - op_id) * (a + b) + op_id * (a - b);
}

The actual behavior of an operation is defined by the machine constraints on the columns used as inputs and outputs.

Links

Links enable a constrained machine to call into another machine. They are defined by a call to an operation, where inputs and outputs are expressions. An optional boolean flag restricts the rows in which the link is active. Links without a boolean flag are active in every row.

machine Add4 with
    latch: latch,
    operation_id: operation_id
{
    Add adder;

    operation add4<0> x, y, z, w -> r;

    // Links without a flag are active on every row.
    // - constrain the values of `x`, `y`, and `n` so that `n = adder.add(x, y)`
    link => n = adder.add(x, y);
    // - constrain the values of `z`, `w`, and `m` so that `m = adder.add(z, w)`
    link => m = adder.add(z, w);
    // - constrain the values of `m`, `n` and `r` so that `r = adder.add(m,n)`
    link => r = adder.add(m, n);

    col fixed operation_id = [0]*;
    col fixed latch = [1]*;

    col witness x;
    col witness y;
    col witness z;
    col witness w;
    col witness r;
    col witness m;
    col witness n;
}

machine Add with
    latch: latch
{
    // operation name, with column names as inputs and outputs
    operation add a, b -> c;

    col fixed latch = [1]*;

    col witness a;
    col witness b;
    col witness c;

    // constraint "implementing" the operation
    c = a + b;
}

If a boolean flag is given, the link is only active in rows where the flag evaluates to 1. Whenever a link is active, the columns mapped as inputs and outputs are constrained by the operation implementation. The following example demonstrates how to use links with flags.

col fixed odd_row = [0,1]*;

link if odd_row => z = submachine.foo(x, y); // active on odd rows only
link if (1 - odd_row) => z = submachine.bar(x, y); // active on even rows only

PIL

powdr-pil is the lower level of abstraction in powdr. It is strongly inspired by Polygon zkEVM PIL. We refer to the Polygon zkEVM PIL documentation and document deviations from the original design here.

Declarations

Powdr-pil allows the same syntax to declare various kinds of symbols. This includes constants, fixed columns, witness columns and even higher-order functions. It deduces the symbol kind from the type of the symbol and the way the symbol is used.

Symbols can be declared using let <name>; and they can be declared and defined using let <name> = <value>;, where <value> is an expression. The type of the symbol can be explicitly specified using let <name>: <type>; and let <name>: <type> = <value>;. Symbols with a generic type can be defined using let<TV1, TV1, ..> <name>: <type> = <value>;, where the TV are newly created type variables that can be used in the type.

This syntax can be used for constants, fixed columns, witness columns and even (higher-order) functions that can transform expressions. The kind of symbol is deduced by its type and the way the symbol is used:

• Symbols without a value are witness columns or arrays of witness columns. Their type can be omitted. If it is given, it must be col or col[k].
• Symbols defined with a value and type col (or col[k]) are fixed columns (or arrays of fixed columns).
• Symbols defined with a value and type expr (or expr[k]) are intermediate columns (or arrays of intermediate columns).
• Everything else is a "generic symbol" that is not a column.

Examples:

    // This defines a integer constant. We can omit the type when it is used
    // somewhere that constrains its type. Since it is not used below,
    // we have to specify `: int` (another option would be `fe`, field element).
    let rows: int = 16;
    // This defines a fixed column that contains the row number in each row.
    // Only symbols whose type is "col" are considered fixed columns.
    let step: col = |i| i;
    // Here, we have a witness column, the do not need an explicit `: col`.
    let x;
    // This functions defines a fixed column where each cell contains the
    // square of its row number.
    let square: col = |x| x*x;
    // This is a generic function that computes the sum of an array
    // given its length.
    // It is not stored as a column.
    // If it is used in a constraint, it has to be evaulated, while
    // columns must be used symbolically.
    let<T: Add + FromLiteral> sum: T[], int -> T = |a, len| match len {
        0 => 0,
        _ => sum(a, len - 1) + a[len - 1],
    };
    // This is a constraint that uses the `sum` function:
    sum([x, step], 2) = 0;

Name lookup is performed as follows:

Lookup is performed starting from the current namespace, going up to the root component by component where the first match is used. If all lookups fail, a last attempt is done inside the std::prelude namespace.

Expressions

Depending on the context, powdr allows more or less features for expressions.

Inside values for declarations, you can use a very flexible language which includes many different operators, function calls, lambda functions, tuple types, statement blocks, match statements and others.

In statements and expressions that are required to evaluate to constraints / polynomial identities, only a much more restrictive language can be used. Expressions in that language are called Algebraic Expressions. While you can use the full language everywhere, in the context of a constraint, the result after function evaluation and constant propagation has to be an algebraic expression.

Generic Expressions

The expression language allows the following operators, in order of increased precedence:

• lambda functions: |params| body. Examples: |i| i (the identity), |a, b| a + b (sum)
• || - logical or
• && - logical and
• <, <=, ==, !=, >=, > - comparisons and = - identity operator
• | - bitwise or
• ^ - bitwise xor
• & - bitwise and
• <<, >> - left and right shift
• +, - - addition and subtraction (binary operator)
• *, /, % - multiplication, division and modulo
• ** - exponentiation
• -, ! - numerical and logical negation (unary operators, prefix)
• ' - "next row" operator (suffix)
• [], () - array index access and function calls

Elementary expressions are

• number literals (integers)
• string literals, written in double quotes, e.g. "hello"
• array literals written in square brackets, e.g. [1, 2, 3]
• tuples, having at least two elements, e.g. (1, "abc")
• statement blocks (see below)
• match expressions (see below).
• if expressions (see below).

Parentheses are allowed at any point to force precedence.

Lambda Functions

The only way to declare a function in pil is by assigning a lambda function to a symbol.

Example:

let x = |i| i + 1;

If you want to specify the types of parameters or return values explicitly, you have to do it on the symbol, you cannot do it on the parameters:

let x: int -> int = |i| i + 1;

It is possible to use patterns in the function parameters:

let y: (int, int), int -> int = |(i, j), _| i + j;

If you use patterns, they have to be irrefutable, which means that the pattern has to be able to match any value of the given type.

Statement Blocks

A {-}-delimited block can be used everywhere where an expression is expected.

It has the form { <statement> ; <statement> ; ... ; <expression> }, i.e. a sequence of statements followed by an expression. The statements can either be expressions (f();, only inside constr-functions) or let statements: let x = ...; / let x;

The value of the statement block is the value of the final expression.

Example:

let plus_one_squared = |x| { let y = x + 1; y * y };

Let statements with value can be used everywhere, they just bind an expression to a local variable and allow to avoid repeating the expression. You can use patterns for the left hand side of let statements to destructure values.

Example:

let f = |i| (i / 2, i % 2);
let (quot, rem) = f(7);

The second let statement will create two local variables x and y. You can also ignore values using the _ pattern element. For details, please see the patterns section.

Let statements without value (let x;) create a new witness column and are only allowed inside constr-functions.

Similarly, an expression at statement level (e.g. x * (x - 1) = 0;) can be used to create new constraints that are added to the global constraint set and this can only be done inside a constr-functions.

Note that you can always create constraints and return them from a function, even in pure function.

Example:

let constrain_to_bool: expr -> Constr = |x| x * (x - 1) = 0;

Match Expressions

Match expressions take the form match <value> { <pattern 1> => <value 1>, <pattern 2> => <value 2>, _ => <default value> }, with an arbitrary number of match arms.

The semantics are that the first match arm where the pattern equals the value after the match keyword is evaluated.

Patterns can be used to destructure more complex data types and to capture values inside new local variables. For more details, please see the patterns section.

Example:

let fib = |i| match i {
    0 => 1,
    1 => 1,
    _ => fib(i - 2) + fib(i - 1),
};

If Expressions

If expressions take the form if <condition> { <true value> } else { <false value> }, where the "else" part is not optional.

If the condition evaluates to true, then <true value> is evaluated, otherwise <false value> is.

Example:

let is_seven = |i| if i == 7 { 1 } else { 0 };

Algebraic Expressions

For constraints (or functions called at a place where a constraint is expected), the expression syntax is limited: After evaluating function calls and performing constant propagation, the resulting expression has to be an "algebraic expression". These are restricted in the following way:

• You can freely use the operators +, -,*.
• The operator ** must have a number as exponent.
• The operator [i] must have a column name on the left-hand side and the index must be a number.
• The operator ' must have a column or [i] on the left-hand-side.
• No other operators are allowed.

Arbitrary parentheses are allowed.

The following example illustrates how you can still use the generic language:

namespace Main(16);
    // Returns folder(...folder(folder(0, f(0)), f(1)) ..., f(length - 1))
    // This is a generic function.
    let<T1, T2> fold: int, (int -> T1), T2, (T2, T1 -> T2) -> T2 =
        |length, f, initial, folder| match length {
            0 => initial,
            _ => folder(fold(length - 1, f, initial, folder), f(length - 1))
        };
    // returns f(0) + f(1) + ... + f(length - 1)
    let sum = |length, f| fold(length, f, 0, |acc, e| acc + e);
    // This function takes an algebraic expression (a column or expression
    // involving columns) and returns an identity that forces this expression
    // to equal 20. Note that `=` is not an assignment but creates an identity constraint.
    let equals_twenty: expr -> Constr = |x| x = 20;
    // This declares an array of 16 witness columns.
    col witness wit[16];
    // This expression has to evaluate to an identity, but we can still use
    // higher order functions and all the flexibility of the language.
    // The sub-expression `sum(16, |i| wit[i])` evaluates to the algebraic
    // expression "wit[0] + wit[1] + ... + wit[15]", which is then
    // turned into the identity by `equals_twenty`
    // wit[0] + wit[1] + ... + wit[15] = 20.
    equals_twenty(sum(16, |i| wit[i]));

    // We constrained the sum to equal twenty, but there is no unique solution
    // to that constraint. In order to fully constrain the system, we need to
    // add something more: The first fifteen columns should all be one.

    // returns [f(0), f(1), ..., f(length - 1)]
    let make_array = |length, f| fold(length, f, [], |acc, e| acc + [e]);
    // If an expression evaluates to an array of constraints, all the
    // constraints in the array are added to the system.
    make_array(15, |i| wit[i] = 1);

Constr and Query Functions

Every function in PIL is either a pure, a constr or a query function. They are denoted by

• |...| ...
• constr |...| ...
• query |...| ...

Inside constr functions, it is possible to create new witness columns and add constraints to the set of constraints (see the Statement Blocks section for details).

Inside query functions, it is possible to evaluate the value of a column on the "current" row using the std::prover::eval function.

Both actions require a certain context to be available, which is not the case for example when the values of a fixed column are computed.

A query function can only be used in the query or hint part of a witness column while constr functions can only be evaluated in the constraint part of a namespace or machine.

You can define and call new constr functions inside a constr function and you can call and define new query functions inside query functions, but as soon as you enter a pure function, this is not possible any more.

Examples:

// This function creates and returns a new witness column.
let new_wit = constr || { let x; x };
// Queries the current value of a column and returns its square.
let square_of = query |x| { let v = std::prover::eval(x); v * v };
// Creates a new witness column, constrains it to be boolean and returns it.
let new_bool = constr |x| { let x = new_wit(); x * (x - 1) = 0; x };
// This is a pure function that only returns a constraint, but does not add it
// to the global set of constraints.
let bool_constraint: expr -> Constr = |x| x * (x - 1) = 0;

Patterns

Patterns are a way to destructure or match certain values. They are valid in match arms, function parameters or left hand sides of let statements in blocks.

A pattern is built up in from the following components:

• _ - the "catch all" pattern that matches anything
• x - for an identifier x, matches anything and assigns the value to the new local variable of that name
• k - for a literal number k, matches the exact number, either as an int or a fe
• -k - for a literal number k, matches the exact negated number, either as an int or a fe
• "text" - for a string literal, matches the exact string literal as a string
• (a, b, c) - for a tuple, matches a tuple-typed value if all the components match
• [a, b, c] - for an array, matches array values of exactly the same length if all the components match
• [a, .., b, c] - matches an array that has an initial segment of a and ends in b, c. The omitted part can be empty.
• X::Y(a, b) - for an enum variant X::Y, matches that enum variant if all the enum fields match.

Patterns can be nested, which means that the components of tuple and array patterns are themselves patterns.

Some examples:

// This pattern de-structures the first function parameter.
let f: (int, int), int -> int = |(a, b), c| (a + c, b);
// Matches a tuple, ignores the second component.
let (x, _) = f((6, 7), 3);
// The match statement typically uses patterns to check for certain values
// but it can also destructure and create new local variables valid inside
// the match arm.
let t = match (x, f((1, x), 2)) {
    (0, _) => 0,
    (1, _) => 7,
    (_, y) => y,
    _ => 9
};
let head: int[] -> int = |x| match x {
    // Matches the first element of a non-empty array and binds it to a local variable.
    [a, ..] => a,
    [] => std::check::panic("Called 'head' on empty array."),
};

Note that PIL does not check that patterns in a match expression are exhaustive.

(Ir-)refutability

A pattern is refutable if there is a value of the correct type that the pattern does not match. An example is the pattern 7 since it does not match all integers, or the patten [x, ..], because it does not match the empty array.

Refutable patterns are fine in match arms, because if the pattern does not match, the evaluator will just continue trying the next match arm, but they are disallowed in let statements and in function parameters, because there we do not have the option of "trying the next arm".

Example:

let f: int -> int[] = |i| match i {
    // This is a refutable pattern, but it is fine
    // because we will try the next match arm.
    0 => [],
    _ => f(i - 1) + [i],
};
// This pattern does not match all `int[]`, because it requires a length
// of at least one.
let [x, ..] = f(8);

The following patterns are refutable:

• all integer literal patterns
• all string literal patterns
• enum variant patterns
• tuple patterns that have refutable components
• array patterns that are not [..].

Variable patterns and _ are always irrefutable.

Types

The powdr-pil language has the following types:

• bool
• int (integer)
• fe (field element)
• string
• tuple
• array
• function type
• expr (expression)
• ! ("bottom" or "unreachable" type)
• enum types

The col type is special in that it is only used for declaring columns, but cannot appear as the type of an expression. See Declaring and Referencing Columns for details.

Powdr-pil performs Hindley-Milner type inference. This means that, similar to Rust, the type of a symbol does not always have to be specified. The compiler will try to find a type for every symbol depending both on the value assigned to the symbol and on the context the symbol is used in. It is an error if the type is not uniquely determined.

Symbols can have a generic type, but in those cases, you have to explicitly specify the generic type. Such declarations can require type variables to satisfy certain trait bounds. Currently, only built-in traits are supported (see the next section).

Literal numbers do not have a specific type, they can be either int, fe or expr (the types that implement the FromLiteral trait), and their type can also stay generic until evaluation.

Example

The following snippet defines a function that takes a value of a generic type and returns the value incremented by one. The type bounds on the generic type are FromLiteral and Add. The type checker will complain if we do not specify the type bounds. The bound Add is required because we use the + operator in the function and FromLiteral is needed because we use the literal 1 as a value of that type.

let<T: FromLiteral + Add> add_one: T -> T = |i| i + 1;

Declaring and Referencing Columns

A symbol declared to have type col (or col[k]) is a bit special:

If you assign it a value, that value is expected to have type int -> fe or int -> int (or an array thereof). This allows the simple declaration of a column let byte: col = |i| i & 0xff; without complicated conversions. The integer value is converted to a field element during evaluation, but it has to be non-negative and less than the field modulus.

If you reference such a symbol, the type of the reference is expr. A byte constraint is as easy as [ X ] in [ byte ], since the expected types in plookup columns is expr. The downside is that you cannot evaluate columns as functions. If you want to do that, you either have to assign a copy to an int -> int symbol: let byte_f: int -> int = |i| i & 0xff; let byte: col = byte_f;. Or you can use the built-in function std::prover::eval if you want to do that inside a prover query or hint.

All other symbols use their declared type both for their value and for references to these symbols.

Built-in Traits

FromLiteral: Implemented by int, fe, expr. The type of a number literal needs to implement FromLiteral.

Add: Implemented by int, fe, expr, T[], string. Used by <T: Add> +: T, T -> T (binary plus).

Sub: Implemented by int, fe, expr. Used by <T: Sub> -: T, T -> T (binary minus).

Neg: Implemented by int, fe, expr. Used by <T: Neg> -: T -> T (unary minus).

Mul: Implemented by int, fe, expr. Used by <T: Mul> *: T, T -> T (binary multiplication).

Pow: Implemented by int, fe, expr, Used by <T: Pow> **: T, int -> T (exponentiation).

Ord: Implemented by int. Used by <T: Ord> op: T, T, -> bool for op being one of <, >, <=, >=.

Eq: Implemented by int, fe, expr. Used by <T: Eq> op: T, T -> bool for op being one of ==, !=.

List of Types

Bool

Type name: bool

Booleans are the results of comparisons. They allow the following operators:

• &&: logical conjunction
• ||: logical disjunction
• !: logical negation

Short-circuiting is not performed when evaluating boolean operators. This means that (1 == 1) || std::check::panic("reason") will cause a panic abort.

Integer

Type name: int

Integers in powdr-pil have unlimited size. Array index requires an integer and row indices (for example the input to a fixed column defined through a function) are also integers.

Integer implements FromLiteral, which means that literal numbers can be used in contexts where int is expected.

Integers allow the following operators, whose result is always an integer:

• +: addition
• -: subtraction (also unary negation)
• *: multiplication
• /: integer division rounding towards zero, division by zero results in a runtime error
• **: exponentiation, the exponent needs to be non-negative and fit 32 bits, otherwise a runtime error is triggered
• %: remainder after division, (for signed arguments, p % q == sgn(p) * abs(p) % abs(q)), remainder by zero results in a runtime error
• &: bit-wise conjunction
• |: bit-wise disjunction
• ^: bit-wise exclusive or
• <<: bit-wise shift left, the shift amount needs to be non-negative and fit 32 bits, otherwise a runtime error is triggered
• >>: bit-wise shift right, the shift amount needs to be non-negative and fit 32 bits, otherwise a runtime error is triggered

The exponentiation operator on field elements requires a non-negative integer as exponent. It has the signature **: fe, int -> fe.

In addition, the following comparison operators are allowed, the result is a boolean:

• <: less than
• <=: less or equal
• ==: equal
• !=: not equal
• >=: greater or equal
• >: greater than

Field Element

Type name: fe

Field elements are elements of a particular but unspecified prime field. The exact field is chosen when powdr is run. The modulus of that field can be accessed via std::field::modulus().

Field elements are the values stored in (fixed and witness) columns. Arithmetic inside constraints (algebraic expressions) is also always finite field arithmetic.

The type fe implements FromLiteral, which means that literal numbers can be used in contexts where fe is expected. If the literal number is not less than the field modulus, a runtime error is caused.

Field elements allow the following operators, where the result is always a field element:

• +: finite field addition
• -: finite field subtraction (also unary negation)
• *: finite field multiplication

There is also an exponentiation operator ** on field elements. It requires the exponent to be a non-negative integer and thus has the signature **: fe, int -> fe. If the exponent is negative, a runtime error is triggered. 0**0 is defined as 1.

The following comparison operators exist for field elements, whose result is a boolean:

• ==: equality comparison
• !=: inequality comparison

Since finite fields do not have an inherent order as integers do, if you want to compare them using <, you have to first convert them to integers.

String

Type name: string

String literals are written as "string content". They are mainly used for debugging or documentation purposes, since they cannot occur in constraints.

They allow the following operators:

• +: string concatenation

Tuple

Type name: (..., ..., ...)

Tuples are complex types that are composed from other types, either zero or two or more. There is no tuple type with a single element ((int) is the same as int). The empty tuple type is written as ().

Examples include (int, int) (a pair of integers) and ((fe[], int), ()) (a tuple consisting of a tuple that contains an array of field elements and an integer and an empty tuple).

Tuples values are constructed using parentheses: (1, 2) constructs the tuple that consists of a one and a two.

Tuples do not allow any operators.

Array

Type name: _[]

Arrays are statically or dynamically-sized collections of elements each of the same type denoted for example as int[] (a dynamically-sized array of integers) or int[2] (an array of integers with static size two). Array values can be constructed inline using [1, 2] (the array containing the two elements one and two).

The built-in function std::array::len can be used to retrieve the length of an array (statically or dynamically sized) and the elements of an array a can be accessed using a[0], a[1], etc.

The type checker currently only knows dynamically-sized arrays, which means that it does not compare the sizes of statically-sized array types.

Arrays allow the following operators:

• +: array concatenation
• _[]: array index access, the index needs to be a non-negative integer that is less than the length of the array, otherwise a runtime error is triggered

Function

Type name: T1, T2, ..., Tn -> T0

Function type names are for example denoted as int, fe -> int or -> int. Note that (int, fe) -> int is a function that takes a single tuple as parameter while int, fe -> int takes two parameters of type integer and field element.

Functions can be constructed using the lambda expression notation. For example |x, y| x + y returns a function that performs addition. The lambda expression || 7 is a function that returns a constant (has no parameters). Lambda functions can capture anything in their environment and thus form closures.

Functions allow the following operators:

• _(...): function evaluation

Powdr-pil is usually side-effect free, but there are some built-in functions that have side-effects: These are std::debug::print and std::check::panic and all functions that call them. Expressions are eagerly evaluated from left to right.

Expression

Type name: expr

Expressions are the elements of the algebraic expressions used in constraints.

References to columns have type expr and expr also implements FromLiteral, which means that literal numbers can be used in contexts where expr is expected.

Example:

let x: col;
let y: col;
let f: -> expr = || x + y;
let g = || 7;
f() = g();

The first two lines define the witness columns x and y. The next two lines define the utility functions f and g. The function f adds the two columns x and y symbolically - it essentially returns the expression x + y. The last line is at statement level and it is expected that it evaluates to a constraint, in this case, a polynomial identity. Because of that, g is inferred to have type -> expr, which is compatible with the literal 7.

Since expressions are built from abstract column references, applying operators does not perform any operations but instead constructs an abstract expression structure / syntax tree.

Expressions allow the following operators, which always construct new expressions:

• +: additive combination of expressions
• -: subtractive combination of expressions (also unary negation)
• *: multiplicative combination of expressions
• **: exponential combination of an expression with an integer constant
• ': reference to the next row of a column, can only be applied directly to columns and only once

The operator = on expressions constructs a constraint (see [../builtins#Constr]).

Bottom Type

Type name: !

The bottom type essentially is the return type of a function that never returns, which currently only happens if you call the panic function. The bottom type is compatible with any other type, which means that you can call the panic function in any context.

Enum Types

Enums are user-defined types that can hold different named alternatives plus data. An enum type has a (namespaced) name that uniquely identifies it and is also used to reference the type.

Enums are declared in the following way:

enum EnumName {
    Variant1,
    Variant2(),
    Variant3(int),
    Variant4(int, int[], EnumName),
}

The variants must have unique names inside the enum and they can optionally take additional data. Each variant declares a type constructor function that can be used to create a value of the enum:

let a = EnumName::Variant1;
let b = EnumName::Variant2();
let c = EnumName::Variant3(3);
let d = EnumName::Variant4(1, [2, 3], EnumName::Variant1);

Recursive enums are allowed.

Enums do not allow any operators.

Fixed columns

powdr-pil requires the definition of fixed columns at the time of declaration.

For example:

col fixed ONES = [1]*; // this is valid
// col fixed ONES; // this is invalid

A number of mechanisms are supported to declare fixed columns. Let N be the total length of the column we're defining.

Values with repetitions

powdr-pil supports a basic language to define the value of constant columns using:

• arrays, for example [1, 2, 3]
• repetition, for example [1, 2]*
• concatenation, for example [1, 2] + [3, 4]

These mechanisms can be combined, as long as a single repetition is used per column definition.

// valid, as for a given total length, only one column fits this definition for a given `N`
col fixed A = [1, 2] + [3, 4]* + [5];

// invalid, as many columns fit this definition
// col fixed A = [1, 2]* + [3, 4]*

Mappings

A column can be seen as a mapping from integers to field elements. In this context, different functions are supported:

col fixed B(i) { i + 1 };

col fixed C(i) {match i {
    0 => 1,
    _ => 0
}};

Built-ins

Functions

The following functions are built into the compiler. They need to be defined to be accessible, but their assigned value is ignored and the compiler replaces it with the following.

Array length

let<T> std::array::len: T[] -> int

Returns the length of an array as an integer.

Example:

let x = [1, 2, 3];
let l = std::array::len(x); // returns 3

Panic

let std::check::panic: string -> !

Aborts evaluation and prints its argument as error message as a side-effect of its evaluation.

Since panic does not generate a constraint, it cannot be used for correctness checks. The verifier only checks constraints / identities and thus ignores anything that could lead to a panic. Panic should only be used to check prover-internal consistency.

Example:

let secp256k1_inverse = |x|
    if x == std::convert::fe(0) {
        panic!("Tried to compute the inverse of zero.")
    } else {
        std::math::ff::inverse(x, secp256k1_modulus);
    };

Conversions

let<T: FromLiteral> std::convert::fe: T -> fe

This function is meant to be used on int, but also works on fe for convenience.

It converts a non-negative integer less than the field modulus to a field element. Causes a type error in all other cases.

If the argument is already a field element, it is returned without modification.

let<T: FromLiteral> std::convert::int: T -> int

This function is meant to be used on fe, but also works on int for convenience.

It converts a field element to an integer.

If the argument is already an integer, it is returned without modification.

let<T: FromLiteral> std::convert::expr: T -> expr

This function is meant to be used on int, but also works on fe and expr for convenience.

It converts an integer to an expr.

If the argument is already an expr, it is returned without modification.

Printing

let std::debug::print: string -> Constr[]

This function takes a string and prints it on the standard output during evaluation, as a side-effect of its evaluation.

This function should only be used for debugging purposes.

Note that the function does not append a newline at the end.

It returns an empty Constr array so that it can be used at statement level where constraints are expected.

Modulus

let std::field::modulus: -> int

Returns the current field's modulus as an integer.

Example:

// Inside a machine
if std::field::modulus() != 2**64 - 2**32 + 1 {
    panic!("This machine can only be used with the Goldilocks field.")
} else {
    []
};

Evaluate

let std::prover::eval: expr -> fe

Evaluates a column (potentially with ' applied) on the current row.

This function can only be used for prover queries or hints and it only works on columns (and those with ' applied). This means you cannot use std::prover::eval(x + 1).

In the following example, the column x is evaluated in a prover hint that returns the square root of a number. Example:

machine Sqrt {
    let sqrt_hint: fe -> fe = |x| match x {
        // Code to compute the square root of x goes here.
    };

    col witness x;
    col witness y(i) query std::prover::Query::Hint(sqrt_hint(std::prover::eval(x)));

    y * y = x;

}}

Challenges

let std::prelude::challenge: int, int -> expr

Constructs a challenge object, essentially asking the verifier for a random number.

The first argument is the proof stage and the second is the identifier of the challenge.

If you want two challenges to be different, you have to choose different IDs.

Degree

let std::prover::degree: -> int

Returns the current number of rows / the length of the witness columns, also known as the degree.

Types

There are some types that are not proper built-in types (in the sense that they are not treated specially in the type system), but they are defined in the standard library and are referenced by built-in functions.

Constr

The type Constr or more specifically, std::prelude::Constr is the type of a constraint. Expressions at statement level are required to evaluate either to Constr or Constr[].

It is defined as follows:

enum Constr {
    /// A polynomial identity.
    Identity(expr, expr),
    /// A lookup constraint with selectors.
    Lookup((Option<expr>, Option<expr>), (expr, expr)[]),
    /// A permutation constraint with selectors.
    Permutation((Option<expr>, Option<expr>), (expr, expr)[]),
    /// A connection constraint (copy constraint).
    Connection((expr, expr)[])
}

The operator = can be applied on two expr values and results in a Constr::Identity.

The type implements no traits and allows no operators.

Operators

The following operators are supported by powdr-pil with their respective signatures.

let<T: Add> +: T, T -> T
let<T: Sub> -: T, T -> T
let<T: Neg> -: T -> T

let<T: Mul> *: T, T -> T
let /: int, int -> int
let %: int, int -> int
let<T: Pow> **: T, int -> T

let <<: int, int -> int
let >>: int, int -> int
let &: int, int -> int
let |: int, int -> int
let ^: int, int -> int

let<T: Ord> <: T, T -> bool
let<T: Ord> <=: T, T -> bool
let<T: Ord> >: T, T -> bool
let<T: Ord> >=: T, T -> bool
let<T: Ord> <: T, T -> bool

let<T: Eq> ==: T, T -> bool
let<T: Eq> !=: T, T -> bool

let =: expr, expr -> Constr
let ': expr -> expr

let ||: bool, bool -> bool
let &&: bool, bool -> bool
let !: bool -> bool

Frontends

While any frontend VM can be implemented in powdr-asm, powdr comes with several frontends for popular instruction set architectures.

RISCV

A RISCV frontend for powdr is already available.

How to run the Rust-RISCV example

# Install the riscv target for the rust compiler
rustup target add riscv32imac-unknown-none-elf
# Run the powdr-rs compiler. It will generate files in ./output/
powdr-rs compile riscv/tests/riscv_data/sum -o output
# Run powdr to compile powdr-asm to powdr-PIL and generate the witness
# -i specifies the prover witness input (see below)
powdr pil output/sum.asm -o output -f -i 10,2,4,6

The example Rust code verifies that a supplied list of integers sums up to a specified value.

{{#include ../../../riscv/tests/riscv_data/sum/src/lib.rs}}

The function get_prover_input reads a number from the list supplied with -i.

This is just a first mechanism to provide access to the outside world. The plan is to be able to call arbitrary user-defined ffi functions that will translate to prover queries, and can then ask for e.g. the value of a storage slot at a certain address or the root hash of a Merkle tree.

RISCV and ZK-Continuations

ZK-continuations can be used to make proofs for unbounded execution traces of Rust programs, where the trace is split into many different chunks. A proof is computed for each chunk, and all proofs can be combined with recursion/aggregation until a single proof remains.

For the details of how memory is handled in the ZK-continuations case please see this.

powdr-rs has experimental support to ZK-continuations, which can be used as follows.

Let's use as example test many chunks from the riscv crate:

{{#include ../../../riscv/tests/riscv_data/many_chunks/src/lib.rs}}

First we need to compile the Rust code to powdr-asm:

powdr-rs compile riscv/tests/riscv_data/many_chunks

Now we can use podwr's RISCV executor to estimate how many cycles are needed:

powdr-rs execute many_chunks.asm

...
Execution trace length: 750329

By default, powdr-RISCV uses chunks of length 2^18. That means we will need at least 3 chunks.

For the continuations case, the compiled assembly code looks different because of the external memory commitments, so we need to recompile using the --continuations flag:

powdr-rs compile riscv/tests/riscv_data/many_chunks --continuations

We can now execute the program with continuations enabled:

powdr-rs -- execute many_chunks.asm --continuations

The output now is longer:

Running chunk 0...
Building bootloader inputs for chunk 0...
26 unique memory accesses over 2 accessed pages: {31, 32}
Estimating the shutdown routine to use 1362 rows.
Bootloader inputs length: 1285
Simulating chunk execution...
Initial memory root hash: 44cd91c12033ad4c6a6b19793b73f1a66d99a0e0bf63494c12ceb1f451ec9452
Final memory root hash: e6a6fd63d864c734d6a4df4988f733738790262af9c7b014a1c036679baf1125
Chunk trace length: 260782
Validating chunk...
Bootloader used 3134 rows.
  => 257648 / 262144 (98%) of rows are used for the actual computation!
Proved 257647 rows.

Running chunk 1...
Building bootloader inputs for chunk 1...
0 unique memory accesses over 0 accessed pages: {}
Estimating the shutdown routine to use 42 rows.
Bootloader inputs length: 83
Simulating chunk execution...
Initial memory root hash: e6a6fd63d864c734d6a4df4988f733738790262af9c7b014a1c036679baf1125
Final memory root hash: e6a6fd63d864c734d6a4df4988f733738790262af9c7b014a1c036679baf1125
Chunk trace length: 262102
Validating chunk...
Bootloader used 62 rows.
  => 262040 / 262144 (99%) of rows are used for the actual computation!
Proved 262039 rows.

Running chunk 2...
Building bootloader inputs for chunk 2...
2 unique memory accesses over 1 accessed pages: {31}
Estimating the shutdown routine to use 702 rows.
Bootloader inputs length: 684
Simulating chunk execution...
Initial memory root hash: e6a6fd63d864c734d6a4df4988f733738790262af9c7b014a1c036679baf1125
Final memory root hash: e6a6fd63d864c734d6a4df4988f733738790262af9c7b014a1c036679baf1125
Chunk trace length: 232057
Validating chunk...
Bootloader used 1610 rows.
  => 259832 / 262144 (99%) of rows are used for the actual computation!
Done!

The step above is informational, but for the proofs we also need the full witness:

powdr-rs execute many_chunks.asm --witness

The witnesses are written in ./chunk_0/commits.bin, ./chunk_1/commits.bin and ./chunk_2/commits.bin.

Now that we have the witnesses for all chunks we can use powdr (instead of powdr-rs) to compute proofs for each chunk:

powdr prove many_chunks.asm -d chunk_0 --backend estark-dump
powdr prove many_chunks.asm -d chunk_1 --backend estark-dump
powdr prove many_chunks.asm -d chunk_2 --backend estark-dump

These proofs are mock proofs for the sake of the example, but any backend should work here.

After generating real proofs, each specific proof system can be used for the recursion/aggregation parts. A follow-up tutorial on that is coming soon.

Valida

A Valida front end for powdr is under development. If you are interested, feel free to reach out!

EVM

An EVM frontend for powdr is under development. If you are interested, feel free to reach out!

Backends

powdr aims to have full flexibility when it comes to generating proofs and comes with a few built-in backends to get started with zkVMs.

Halo2

powdr supports the PSE fork of halo2 with the bn254 field.

eSTARK

powdr supports the eSTARK proof system with the Goldilocks field, implemented by the starky library from eigen-zkvm.

Architecture

powdr applies a number of steps in order to reduce a powdr-asm program into PIL.

We provide a high level overview of these steps.

            ┌────────────┐           ┌──────────┐
            │            │           │          │
 powdr-asm  │            │ AIR graph │          │  PIL
───────────►│  compiler  ├───────────┤  linker  ├──────►
            │            │           │          │
            │            │           │          │
            └────────────┘           └──────────┘

Compiler

In this section, we explain how the powdr compiler reduces a program made of virtual and constrained machines to a set of AIRs.

Virtual machine reduction

The first step is to reduce virtual machines to constrained machines. This step is run on all machines and does not affect constrained machines. As a result of this step, for each machine:

• Local instructions are reduced to constraints
• External instructions are reduced to links
• Functions are reduced to operations

Block enforcement

Block enforcement applies on constrained machines. It makes sure that the operation_id is constant within each machine block.

AIR generation

At this point, all machines contain only:

• an optional degree
• constraints
• links to other machines
• operations

Let's define AIR as a data structure with only these elements.

Starting from the main machine's type, we create a tree of AIR objects by traversing its submachines, recursively instantiating each machine as an AIR. Let's define the AIR tree as the resulting tree.

Linker

A linker is used to turn an AIR tree into a single PIL file. The linking process operates in the following way:

1. Create an empty PIL file
2. Start from the main AIR. If it defines a degree, let main_degree be that value. If it does not, let main_degree be 1024.
3. For each AIR
   1. Create a new namespace in the PIL file
   2. If a degree is defined, set it as the namespace degree. If no degree is defined, set the namespace degree to main_degree
   3. Add the constraints to the namespace
   4. Turn the links into lookups and add them to the namespace

The result is a monolithic AIR where:

• each machine instance is a namespace
• each namespace defines its own degree