Abstract. It has been argued that the linear database model, in which semi-linear sets are the only geometric objects, is very suitable for most spatial database applications. For querying linear databases, the language FO + linear has been proposed. We present both negative a n d positive results regarding the expressiveness of FO+linear. First, we s h o w that the dimension query is de nable in FO + linear, which allows us to solve several interesting queries. Next, we s h o w the non-de nability o f a whole class of queries that are related to sets not de nable in FO+linear. This result both sharpens and generalizes earlier results independently found by Afrati et al. and the present authors, and demonstrates the need for more expressive linear query languages if we w ant to sustain the desirability of the linear database model. In this paper, we s h o w h o w FO + linear can be strictly extended within FO + poly in a safe way. Whether any of the proposed extensions is complete for the linear queries de nable in FO + poly remains open. We d o s h o w, however, that it is undecidable whether an expression in FO + poly induces a linear query.
1 IntroductionSeveral authors have suggested to use first-order logic over the real numbers to describe spatial database applications. Geometric objects are then described by polynomial inequalities with integer coefficients involving the coordinates of the objects. Such geometric objects are called semi-algebraic sets. Similarly, queries are expressed by polynomial inequalities. The query language thus obtained is usually referred to as FO + poly. From a practical point of view, it has been argued that a linear restriction of this so-called polynomial model is more desirable. In the so-called Zinear model, geometric objects are described by linear inequalities, and are called semilinear sets. The language of the queries expressible by linear inequalities is usually referred to as FO + linear.As part of a general study of the feasibility of the linear model, we show in this paper that semi-linearity is decidable for semi-algebraic sets. In doing so, we point out important subtleties related to the type of the coefficients in the linear inequalities used to describe semi-linear sets. An important concept in the development of the paper is regularity, of which we point out the geometric sign&axe.We show that the regular points of a semi-linear set can be computed in FO + linear.Following the seminal work by Kanellakis, Kuper, and Revesz [12] on constraint query languages with polynomial constraints, various researchers have introduced geometric database models and query languages within this framcwork [lo, 171. These researchers have studied the desirability of their models for database applications involving geometric data objects, as well as the expressiveness of the proposed geometric query languages. We adopt the formalism of [17], which we shall call the polynomial spatial database model, in which both geometric objects and queries are exprcsscd using polynomial inequalities. Geometric objects described by polynomial inequalities are called semi-algebraic sets, and the query language using polynomial inequalities is referred to as FO I-poly.The decidability of semi-linearity of semi-algebraic sets has an important consequence. It has been shown that it is undecidable whether a query expressible in FO + poly is linear, i.e., maps spatial databases of the linear model into spatial databases of the linear model. It follows now that, despite this negative result, there exists a syntactically definable language precisely expressing the linear queries expressible in FO f poly.Recently, several authors [l, 2, 4, 10, 12, 13, 23, 241 discussed linear spatial database models which can be seen ~9 linear restrictions of the polynomial database model. These linear models allow users to define relational databases, which may, besides conventional data, contain linear gcometric data objects, which suffice for the majority of applications encountered in GIS, geometric modeling, and spatial and temporal databases [16,18). Furthermore, data structures and algorithms have been developed to efficiently implement a wid...
Abstract. The research presented in this paper is situated in the framework of constraint databases introduced by Kanellakis, Kuper, and Revesz in their seminal paper of 1990, specifically, the language with real polynomial constraints (FO + poly). For reasons of efficiency, this model is implemented with only linear polynomial constraints, but this limitation to linear polynomial constraints has severe implications on the expressive power of the query language. In particular, when used for modeling spatial data, important queries that involve Euclidean distance are not expressible. The aim of this paper is to identify a class of two-dimensional constraint databases and a query language within the constraint model that go beyond the linear model and allow the expression of queries concerning distance. We seek inspiration in the Euclidean constructions, i.e., constructions by ruler and compass. We first present a programming language that captures exactly the first-order ruler-and-compass constructions that are expressible in a first-order language with real polynomial constraints. If this language is extended with a while operator, we obtain a language that is complete for all ruler-and-compass constructions in the plane. We then transform this language in a natural way into a query language on finite point databases, but this language turns out to have the same expressive power as FO + poly and is therefore too powerful for our purposes. We then consider a safe fragment of this language and use this to construct a query language that allows the expression of Euclidean distance without having the full power of FO + poly.
Abstract. A general linear spatial database model is presented in which both the representation and the manipulation of non-spatial data is based on rst-order logic over the real numbers with addition. We rst argue the naturalness of our model and propose it as a general framework to study and compare linear spatial database models. However, we also establish that no reasonable safe extension of our data manipulation language can be complete for the linear spatial queries in that even very simple queries such as deciding colinearity or computing convex hull of a nite set of points cannot be expressed. We s h o w that this fundamental result has serious rami cations for the way i n w h i c h query languages for linear spatial database models have to be designed.
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