# Turing machine and boolean circuits

As an example of languages that are in P/poly is the UHALT Problem :

UHALT = { 1^n: n's binary expansion encodes a pair such that M halts on input x}

We can create a boolean circuit of just AND gates. However, we can create it only for a fixed 'n', not for all n's ...

My question is : when we consider a language like the UHALD problem, do we consider all of the 'n' possible values ,ie for every 'n' we have a different language? or for every n we have a different language ?

In Turing-machines, i think we can loop, but in boolean circuits.. i dont think we can, thus the impossibility (in my point of view) of making, for example, a circuit for the UHALT problem for every n there is.

The Boolean circuits you are referring to are a non-uniform model. This means that, for every input length, you have a different circuit. When we say that Boolean circuits solve a particular problem what is actually meant is that a family of circuits does, which is an infinite sequence $$(C_n)_{n \in \mathbb{N}_0}$$, $$C_n$$ being the circuit for inputs of length $$n$$.