A case study of the difficulty of quantifier elimination in constraint databases: the alibi query in moving object databases

In the constraint database model, spatial and spatio-temporal data are stored by boolean combinations of polynomial equalities and inequalities over the real numbers. The relational calculus augmented with polynomial constraints is the standard first-order query language for constraint databases. Although the expressive power of this query language has been studied extensively, the difficulty of the efficient evaluation of queries, usually involving some form of quantifier elimination, has received considerably less attention. The inefficiency of existing quantifier-elimination software and the intrinsic difficulty of quantifier elimination have proven to be a bottle-neck for for real-world implementations of constraint database systems. In this paper, we focus on a particular query, called the alibi query, that was proposed in the context of moving object databases and that asks whether two moving objects whose positions are known at certain moments in time, could have possibly met, given certain speed constraints. This query can be seen as a constraint database query and its evaluation relies on the elimination of a block of three existential quantifiers. Implementations of general purpose elimination algorithms, such as provided by QEPCAD, Redlog and Mathematica, are in the specific case, for practical purposes, too slow in answering the alibi query and fail completely in the parametric case. The main contribution of this paper is an analytical solution to the parametric alibi query, which can be used to answer this query in the specific case in constant time. We also give an analytic solution to the alibi query at a fixed moment in time, which asks whether two moving objects that are known at discrete moments in time could have met at a given moment in time, given some speed constraints. The solutions we propose are based on geometric argumentation and they illustrate the fact that some practical problems require creative solutions, where at least in theory, existing systems could provide a solution.