Looking at this again, I’m not sure I understand the two confusions. P vs. NP isn’t about functions that are hard to compute (they’re all polynomially computable), rather functions that are hard to invert, or pairs of easily computable functions that hard to prove are equal/not equal to each other. The main difference between circuits and Turing machines is that circuits are finite and bounded to compute whereas the halting time of general Turing machines is famously impossible to determine. There’s nothing special about Boolean circuits: they’re an essentially complete model of what can be computed in polynomial time (modulo technicalities)
In particular, it’s not hard to produce a computable function that isn’t given by a polynomial-sized circuit (parity doesn’t work as it’s polynomial, but you can write one down using diagonalization—it would be very long to compute, but computable in some suitably exponentially bounded time). But P vs. NP is not about this: it’s a statement that exists fully in the world of polynomially computable functions.
Looking at this again, I’m not sure I understand the two confusions. P vs. NP isn’t about functions that are hard to compute (they’re all polynomially computable), rather functions that are hard to invert, or pairs of easily computable functions that hard to prove are equal/not equal to each other. The main difference between circuits and Turing machines is that circuits are finite and bounded to compute whereas the halting time of general Turing machines is famously impossible to determine. There’s nothing special about Boolean circuits: they’re an essentially complete model of what can be computed in polynomial time (modulo technicalities)
In particular, it’s not hard to produce a computable function that isn’t given by a polynomial-sized circuit (parity doesn’t work as it’s polynomial, but you can write one down using diagonalization—it would be very long to compute, but computable in some suitably exponentially bounded time). But P vs. NP is not about this: it’s a statement that exists fully in the world of polynomially computable functions.