Cook-Levin reduction is both deterministic polynomial time and parsimonious and that's mean that from every non deterministic Turing machine $M$ and string $w$ it is possible in polynomial time deterministically to create the boolean formula $\alpha_{M,w}$ so that every satisfying assignment of $\alpha_{M,w}$ is accepting path/route of $M$ with input $w$ and every falsifying assignment of $\alpha_{M,w}$ is rejecting path/route of $M$ with input $w$, and also $\alpha_{M,w}$ is in conjunctive normal form.
Therefore $\alpha_{M,w}$ is satisfiable if and only if $M$ with input $w$ has accept path/route and $\alpha_{M,w}$ is falsifiable if and only if $M$ with input $w$ has reject path/route.
I think that it is also possible from every non deterministic Turing machine $M$ and string $w$ it is also possible in polynomial time deterministically to create the boolean formula $\beta_{M,w}$ so that every satisfying assignment of $\beta_{M,w}$ is rejecting path/route of $M$ with input $w$ and every falsifying assignment of $\beta_{M,w}$ is accepting path/route of $M$ with input $w$ and $\beta_{M,w}$ will be also in conjunctive normal form.
Therefore $\beta_{M,w}$ is satisfiable if and only if $M$ with input $w$ has reject path/route and $\beta_{M,w}$ is falsifiable if and only if $M$ with input $w$ has accept path/route.
Am I right?
