Relation between quantifier elimination, elimination theory and model theory

Question

I am interested in quantifier elimination over the field of real numbers (as in Tarski-Seidenberg theorem), particularly from an algorithmic approach (e.g. Cylindrical Algebraic Decomposition). While looking for introductory references, I have stumbled upon "Elimination Theory" (e.g. Chap. 3 of Cox et al. "Ideals, Varieties and Algorithms") and "Model Theory", which appear to have quantifier elimination as a subdiscipline.

What are exactly the differences between quantifier elimination, elimination theory, and model theory?

Moreover, given that I am interested in an overview of quantifier elimination from an algorithmic perspective (ultimately, I need to apply it in practice), what are good, preferably undergraduate-/beginning graduate-level references?

Quantifier Elimination is a property of formal theory and is usually "used" in Model Theory. — Mauro ALLEGRANZA, Jan 28 '22 at 11:55
I think that if you are interested in an algorithmic perspective, you should probably look at computability/recursion theory (or maybe proof theory), rather than model theory. — tomasz, Jan 29 '22 at 00:22

score 1 · Answer 1 · answered Jan 28 '22 at 12:11

Not an answer, just a long comment

Time ago, there has been some interest in quantifier elimination from an algorithmic perspective. I remember this book

Poizat, B.: Les petits cailloux. ALEAS, Lyon (1995)

and this paper

Poizat, B.: Une tentative malheureuse de construire une structure éliminant rapidement les quanteurs. In: Clote, P.G., Schwichtenberg, H. (eds.) CSL 2000. LNCS, vol. 1862, pp. 61–70. Springer, Heidelberg (2000)

Relation between quantifier elimination, elimination theory and model theory

1 Answers1