A scheme $\system{} = (\setup{}, \prove{}, \verify{})$ is defined as follows:

$\setup{}(ecs, src) \to (pk, vk)$ takes an executable constraint system $ecs$ encoding a program’s logic, and a circuit independent stuctured reference string $src$, outputs proving key $pk$ and a verification key $vk$.

$\prove{}(ecs, x, x', w, pk) \to \pi_{ecs}$ takes the program encoding $ecs$, public inputs $x$, private inputs $x'$, and witness $w$, outputs proof $\pi_{ecs}$ if $w$ is a valid variable assignment for $ecs$ with proof arguments $x, x'$.

$\verify{}(\pi_{ecs}, x, vk) \to \{0, 1\}$ outputs $1$ if $\pi_{ecs}$ is valid with respect to public inputs $x$ and verification key $vk$.