On lifted problems

Abstract

This study may be viewed from the more general context of a theory of computational problems. An environment E= 〈L,D〉 consists of a class of structures D and a language L for D. A problem in E is a pair of sets of formulas P = 〈Π|Γ〉, with problem predicate Π. Let Ereal = 〈Lreal,{R}〉 and Elin = 〈Llin,Dlin〉 where R are the reals, Dlin is the class of totally ordered structures, Lreal and Llin are the languages of real ordered fields and linear orders, respectively. A problem P = 〈Π|Γ〉 in Ereal is a lifted problem (from Elin) if Π ε Llin. The following interpretes an informal conjecture of Yao: CONJECTURE: Binary comparisons can solve nonredundant, full, lifted problems in Ereal as efficiently as general linear comparisons. The conjecture remains open. We may attack the conjecture by eliminating those comparisons that do not help or by studying those subclass of problems that are not helped by general linear comparisons. Various partial results are obtained, corresponding to these two approaches.