A problem in mathematical logic

I am working on a problem that requires eliminating quantifiers from a
logical combination of polynomial equations and inequatlities over the
domain of real numbers, i.e. given a quantified logical expression E, I want
a quantifier-free expression F, such that E and F are equivalent. It has
been shown that the computational complexity of quantifier elimination is
inherently doubly exponential in the number of variables. I want to know
whether quantifier elimination can be done in lesser time if we take the
help of examples.

Suppose I have a quantified logical expression E1 and its equivalent
quantifier-free expression F1. Now I am given another quantified logical
expression E2 and the task is to compute its equivalent quantifier-free
expression, say F2. Instead of spending a lot of time computing F2 from E2
using the quatifier elimination algorithm, I want to know whether there
exists a mapping between E2 and E1, so that F2 can be computed from F1 by
reverse-mapping. This idea will be beneficial only if there exists such a
mapping that can be computed in time less than doubly exponential.

Example of a mapping: If I can show that E1 and E2 are equivalent, then F1
and F2 have to be equivalent. So "equivalence" is an example of such a
mapping. Unfortunately, the general problem of equivalence checking is
NP-hard.

I suspect, there might exist a mapping weaker than equivalence (and
computable in polynomial time) that will suffice for my purposes. Please let
me know if any of you are already aware of any such mapping. Any suggestion
regarding which book/paper to look at would also help.

Thanks much,
Bonny.


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