Solving efficiently NP problems with infinite precision

One such model of computation is explained in this note. In this model of computation, we are allowed to perform arithmetic operations, logical operations, and comparisons on integers of arbitrary precision. We encode Boolean functions $f\colon \{0,1\}^n \to \{0,1\}$ as bitstrings of length $2^n$, thinking of the index of a bit as encoding a truth assignment. Explicitly, if $i = i_{n-1} 2^{n-1} + \cdots + i_0$, where $i_0,\ldots,i_{n-1} \in \{0,1\}$, then the $i$'th bit of the bitstring (where the $0$th bit is the LSB) is $f(i_0,\ldots,i_{n-1})$.

Given functions $f,g$, we can compute $f \land g, f \lor g, \lnot f$ using bitwise operations. The projection function $(x_1,\ldots,x_n) \mapsto x_i$ corresponds to the bitstring $$ 2^{2^{i-1}} \prod_{I \neq i-1} (1 + 2^{2^I}) $$ which can be computed efficiently (in this model) using repeated squaring.

Given a Boolean formula (or even a Boolean circuit) on $n$ inputs, we can efficiently compute the bitstring corresponding to the function computed by the formula. Comparing the result to zero, we can solve SAT.

In fact, we can even solve QBF. Roughly speaking, to implement a quantifier, we decompose the bitstring into two halves, and then either AND them (in the case of $\forall$) or OR them (in the case of $\exists$). This is possible if we also allow the shift operation.