System Operations

In this section we describe the AIR constraints for Miden VM system operations.

NOOP

The NOOP operation advances the cycle counter but does not change the state of the operand stack (i.e., the depth of the stack and the values on the stack remain the same).

The NOOP operation does not impose any constraints besides the ones needed to ensure that the entire state of the stack is copied over. This constraint looks like so:

$$s_i^{\prime }-s_i=0\text{ for }i\in [0,16)\text{ | degree}=1$$

EMIT

Similarly to NOOP, the EMIT operation advances the cycle counter but does not change the state of the operand stack (i.e., the depth of the stack and the values on the stack remain the same).

The EMIT operation does not impose any constraints besides the ones needed to ensure that the entire state of the stack is copied over. This constraint looks like so:

$$s_i^{\prime }-s_i=0\text{ for }i\in [0,16)\text{ | degree}=1$$

Additionally, the prover puts EMIT's immediate value in the first user op helper register non-deterministically. The Op Group Table is responsible for ensuring that the prover sets the appropriate value.

ASSERT

The ASSERT operation pops an element off the stack and checks if the popped element is equal to $1$. If the element is not equal to $1$, program execution fails.

Stack transition for this operation must satisfy the following constraints:

$$s_0-1=0\text{ | degree}=1$$

The effect on the rest of the stack is:

• Left shift starting from position $1$.

FMPADD

The FMPADD operation pops an element off the stack, adds the current value of the fmp register to it, and pushes the result back onto the stack. The diagram below illustrates this graphically.

Stack transition for this operation must satisfy the following constraints:

$$s_0^{\prime }-(s_0+fmp)=0\text{ | degree}=1$$

The effect on the rest of the stack is:

• No change starting from position $1$.

FMPUPDATE

The FMPUPDATE operation pops an element off the stack and adds it to the current value of the fmp register. The diagram below illustrates this graphically.

The stack transition for this operation must follow the following constraint:

$$fm{p}^{\prime }-(fmp+s_0)=0\text{ | degree}=1$$

The effect on the rest of the stack is:

• Left shift starting from position $1$.

CLK

The CLK operation pushes the current value of the clock cycle onto the stack. The diagram below illustrates this graphically.

The stack transition for this operation must follow the following constraint:

$$s_0^{\prime }-clk=0\text{ | degree}=1$$

The effect on the rest of the stack is:

• Right shift starting from position $0$.