Arithemetic Circuit

exploring how computation could be expressed compuatationally...

It is impossible to talk about Arithmetic circuits without talking about Polynomials.

Polynomials: These are like mathematical expressions that combine numbers (constants like 1, 2, 3) with variables (like x, y, z) using addition, multiplication, and exponents (x², y³). Think of them as building block towers where the blocks are numbers and variables raised to powers.

Arithmetic Circuit: This is a model that represents how you compute these polynomials. It's like a set of instructions with wires and gates. The wires carry the numbers and variables, and the gates perform addition and multiplication to build the final polynomial expression. Similar to how real circuits perform calculations with electronic signals, arithmetic circuits perform computations with these mathematical elements.

Just like Boolean circuits, arithmetic circuits also rely on networks of wires and gates to perform computations. However, unlike Boolean circuits that use logical operations (AND, OR), arithmetic circuits perform calculations using arithmetic operations (addition, multiplication).

Wires carry data to be processed by the gates. In Boolean circuits, wires carry bits while in Arithmetic circuits they carry integers. Wires going into a gate contain the data to be processed by the gate. Wires directed out from a gate contain the result of the operation.

If a gate has no input wires, then it is considered an input gate. An input gate contains the data to be processed by the circuit, which can be either a variable or a number. 1

A very details chapter on Arithmetic circuits has been done by RareSkills here Arithmetic Circuits for ZK (rareskills.io)

Finish up before going to the next chapter :)