In view of the paper "Forcing As A Computational Process" by J. Hamkins, R. Miller and K. Williams, I have revised my original question, part (a) about the computability of the Forcing Truth Definition, Looking into the Future within Forcing as follows (instead of Cohen's book, using Shoenfield paper :
https://mathweb.ucsd.edu/~sbuss/CourseWeb/Math260_2012F2013W/Shoenfield_UnramifiedForcing.pdf)
In demonstrating the definability of forcing in the base model M, the definition of $\Vdash$* includes (page 362 definition (c)):
p $\Vdash$* $\neg$ $\Phi$ $\;$ iff $\;$ For all q $\leqq$ p $\neg$ (q $\Vdash$* $\Phi$ ) …………………(1)
Although the complexity of the construction of a formula $\Phi$ is finite, there is an infinite number of finite q $\leqq$ p to test for q $\Vdash$* $\Phi$. If none of the q lead to q $\Vdash$* $\Phi$, this would mean that a completed infinity of tests may be required to establish (1) making the overall calculation not computable.
However "Forcing As A Computational Process" in Main Theorem 1 on page 1 says that : G is computable from the Atomic Diagram of M.
So I was wondering whether anyone could help to identify the error in thinking that "determination of expressions like (1) could require a completed infinity of calculations."
Best Answer
Note that in the Forcing as a computational process paper, the theorem merely states that some generic is computable from (the atomic diagram of) $M$, not that every generic is.
Proof:
Now what does this tell us about the computability of the forcing relation? Not much. The paper goes into this a bit, talking about how with only access to the membership relation in $M$, we can't compute much about the theory of $M[G]$ (where $G$ is computable from $M$ as above). With access to the entire theory of $M$, we can compute more of $M[G]$, as expected, with negation increasing the complexity needed to compute more about $M[G]$.