Linear Problems in Valued Fields

A quantifier elimination procedure for a class M over a language L is an algorithm that given an L-formula φ computes a quantifier-free L-formula φ′ such that M |= φ′ ←→ φ. An L-formula is called linear if it contains no products or reciprocals of quantified variables. Quantifier elimination methods for valued fields have been extensively investigated in the past. The existence of a quantifier elimination procedure for the general case including non-linear formulas has been shown independently by Ax and Kochen (1966) and Ershov (1965). The first explicit procedure has been given by Cohen (1969). Considerable progress has been made by Macintyre (1976) turning to a more reasonable language including root predicates in analogy to the reals. This has been made explicit by Weispfenning (1984). The procedures given there are primitive recursive, but far beyond feasibility. Based on ideas of Ferrante and Rackoff (1979) for decision problems, quantifier elimination by elimination sets containing test terms has been introduced for linear formulas by Weispfenning (1988). This technique is very attractive due to its comparatively low complexity a O(c) , where a is the number of atomic formulas, q is the number of quantifiers, and c is the number of quantifier type changes in a prenex input formula. For ordered fields these algorithms have turned out to be feasible, and implementations including powerful simplification algorithms (Dolzmann and Sturm, 1997b) are applicable to a wide range of both academic and non-academic problems, cf. Dolzmann and Sturm (1997a); Dolzmann et al. (1998a,b); Sturm (1996, 1999); Sturm and Weispfenning (1998).