On first-order topological queries

On First-Order Topological Queries

Martin Grohe

Institut f¨u r Mathematische Logik

Eckerstr.1,79104Freiburg,Germany

E-mail:grohe@logik.mathematik.uni-freiburg.de

Luc Segou?n

INRIA-Rocquencourt

B.P.105,Le Chesnay Cedex78153,France

E-mail:Luc.Segou?n@inria.fr

Abstract

One important class of spatial database queries is the class of topological queries,i.e.queries invariant under homeomorphisms.

We study topological queries expressible in the standard query language on spatial databases,?rst-order logic with various amounts of arithmetic.Our main technical result is a combinatorial characterization of the expressive power of topological?rst-order logic on regular spatial databases.

1.Introduction

The expressive power of?rst-order logic over?nite re-lational databases is now well understood[AHV95,EF95]. Much less is known in spatial databases(also called con-straint databases),where the relations are no longer?nite but?nitely represented[KLP99].

The notion of genericity(invariance of queries under isomorphisms),fundamental for the relational database model,can be generalized to spatial databases in vari-ous ways[PVV94].Given a group of transformations (translations,af?nities,isometries,similarities,homeomor-phisms,etc.),a query is-generic if for all database instances and each transformation,

.By FO we denote the set of-generic?rst-order queries.The genericity of a?rst-order query is unde-cidable[PVV94],but the expressive power of FO can be understood via sound and complete(decidable)languages.

A language is said to be sound for if it contains only FO queries.It is complete for if it expresses all FO queries.The choice of the group depends on which infor-mation one is interested in.[GVV97]gives sound and com-plete languages for several natural groups of transforma-tions(translations,af?nities,isometries,similarities).The case of the group of homeomorphisms was left open.

Queries invariant under homeomorphisms,which are also called topological queries,are of fundamental impor-tance in various applications of spatial databases.For ex-ample,in geographical databases,queries like“Is region A adjacent to region B?”,“Is there a road from A to B?”,or“Is A an island?”come up very naturally.Therefore,topolog-ical queries have received a lot of attention in the literature (e.g.[KPV97,PSV99,SV98,KV99]).A basic result known about topological queries is that connectivity of a region is not expressible in?rst-order logic[GS99,GS97,BDLW96].

Thinking of geographical databases again,planar(or2-dimensional)database instances,where all relations are em-bedded in the plane,are of particular importance.In [PSV99]it has been proven that all topological properties of a planar spatial database can be represented in a?nite structure called the topological invariant of the instance. In[SV98]it has been shown how this topological invari-ant can be used to answer topological queries.In particular, [SV98]have proven that?rst-order topological queries on a spatial database can be automatically translated into?x-point queries on the topological invariant.The translation of?rst-order topological queries on the spatial database into ?rst-order queries on the topological invariant was proven possible only in the special case of a single relation repre-senting a closed region.It was left open in[SV98]whether this translation could be extended to the case of several re-gions.We answer this question negatively.

The idea of representing the topological information of a spatial database instance by the topological invariant has two important drawbacks:In a sense,the topological in-variant contains too much information;ideally we would just want to store the information that is actually accessible by the query language(which is usually FO).Furthermore, the topological invariant has no straightforward generaliza-tion to higher dimensions.The issue of?nding an invariant more suitable for FO(and computable in any dimension) was raised in[KPV97].

In the special case of one single relation representing a closed planar region,a cone structure was given in[KPV97] capturing precisely the?rst-order topological information. Intuitively,the cone structure is a?nite set containing all the possible small neighborhoods of a point.The results of [KPV97]show that,in this context,?rst-order topological

queries could express only local properties,which is a sit-uation known to be true in the?nite case.[KPV97]asked whether their results generalize to database instances with a region that is not necessarily closed;we give a negative answer to this question.For instances with one closed re-gion that satisfy the additional technical condition of being fully two dimensional,[KV99]introduced a cone logic CL and proved that it is sound and complete for topological FO. They asked if their results generalize to instances with not necessarily closed regions or several regions,again we give a negative answer.

[KPV97]introduced two local operations on spatial database instances that preserve the equivalence under?rst-order topological queries(called topological elementary equivalence).We call two instances-equivalent if they can be transformed into instances homeomorphic to each other by applying the operations of[KPV97]?nitely often. Our main technical result,from which all the rest easily fol-lows,is that on especially simple instances that we call reg-ular,-equivalence and topological?rst-order equivalence coincide.

The paper is organized as follows:After recalling a few basic de?nitions on spatial databases in Section2,in Sec-tion3we discuss the topology of planar spatial databases and the topological invariant in detail.In Section4we intro-duce topological?rst-order queries and review some results of[KPV97].In Section5,we prove that-equivalence is decidable in PSPACE.Our main result on regular instances is proved in Section6.In Section7,we derive that not all?rst-order topological queries can be translated to?rst-order queries on the topological invariant,and in Section8 we brie?y discuss the problem of?nding a language that is sound and complete for topological FO.

2.Preliminaries

Spatial databases.We?x an underlying structure over the reals;either we let or lin

or poly.1Let be the vocabulary of(i.e.either or

or).

For a point we let

.2For,let

be the open ball with radius around.

A subset,for some,is-de?nable,if there is a?rst-order formula of vocabulary such that.

,and we will assume we have done so in our?gures—it just looks better.Since we are only interested in topological queries,this makes no difference.

A schema is a?nite collection of region names.Let

.An-dimensional spatial database instance over associates an-de?nable set with every

.The sets are called the regions of.Formally,we may interpret the region names as-ary relation symbols and view an instance over as a?rst-order structure of vocabulary obtained by expanding the underlying structure by the relations,for.

In this paper,we only consider2-dimensional(or pla-nar)spatial database instances.For convenience,we also assume that all regions are bounded,i.e.that for every in-stance over a schema and for every there exists a such that for all.The boundedness assumption is inessential and can easily be removed,but it occasionally simpli?es matters.

Queries.An-ary query(for an)of schema is a mapping that associates an-de?nable subset

with every instance over.Here we consider as a one point space and let T RUE,F ALSE;a-ary query is usually called Boolean query.

As our basic query language we take?rst-order logic FO. A?rst-order formula of vocabulary

de?nes the-ary query

of schema.

3.The topology of planar instances

is equipped with the usual topology.

The interior of a set is denoted by int,the closure by cl,and the boundary by bd.We say that a set touches a point if cl.Two sets touch if touches a point or vice versa.

Strati?cations.Strati?cation is the fundamental fact that makes the topology of our instances easy to handle.

A strati?cation of an instance over is a?nite partition of such that

(1)For all,either is a one point set,or is home-

omorphic to the open interval,or is homeo-morphic to the open disk.

(2)For all we either have cl cl or

cl cl is the union of elements of.

(3)For all and we either have or

.

The following lemma follows from the fact that all re-gions of an instance are-de?nable.A proof can be found in[vdD98].

Lemma3.1.For every instance there exists a strati?ca-tion of.

Colors and cones.Let be an instance over.The pre-color of a point is the mapping

int bdi bde ext de?ned by

int if int

bdi if bd

bde if bd

ext if cl

A pre-cell is a maximal connected set of points of the same pre-color.The cone of a point,denoted by cone, is the circular list of the pre-colors of all pre-cells touching .Lemma3.1implies that cones are well-de?ned and?nite.

A point is regular if for every neighborhood of there is a point such that cone cone. Otherwise is singular.It follows from Lemma3.1that an instance has only?nitely many singular points.We call an instance regular if it has no singular points.The

cones

Figure1.Two singular and two regular cones

of regular(singular)points are also called regular(singular, resp.)(cf.Figure1).

The cone-type of,denoted by ct,is a list of all cones appearing in.Furthermore,for every singular cone this list also records how often it occurs.

The color of a point is the pair

cone.

Cells.A cell of color of is a maximal connected set of points of color.The color of a cell is denoted by .Lemma3.1implies that there are only?nitely many cells.Our assumption that all regions are bounded implies that there is precisely one unbounded cell,which we call the exterior of.Lemma3.1implies that every cell has a well de?ned dimension,which is either,,or.The0-dimensional cells are precisely the sets,where is a singular point.

Let be the set of all cells of an instance.We de?ne a binary adjacency relation on by letting two cells be adjacent if,and only if,they touch.We call the graph

the cell graph of.We can partition into three subsets,,and consisting of the,,and-dimensional cells,respectively.Observe that the graph is tri-partite with partition(i.e.

for).Moreover,is planar.

Lemma3.2.Let be an instance and.

(1)If,then either is homeomorphic to the open

disk,or there exists an such that

is homeomorphic to the open disk with holes.

To be de?nite,we let

cl

The topological invariant.Two instances over a schema are homeomorphic if there is a homeomorphism such that for all and we have

.

The topological invariant of an instance over is an expansion of the cell graph that carries enough informa-tion to characterize an instance up to homeomorphism.The vocabulary of is

,where is binary,is8-ary,and

and for are unary.The restriction of to is the cell graph of. consists of the-dimensional cells,for.only contains the exterior of(the unique unbounded cell).For every,the unary relation consists of all cells that are subsets of.

gives the orientation.It is an equivalence relation on the quadruples,where is0-dimensional and are adjacent to.Two such quadruples

,are equivalent if either appears between and in the clockwise order of the cells adjacent to for both or appears between and in the anti-clockwise order of the cells adjacent to for both.3Note that is empty in regular instances.

It is proven in[PSV99]that and are homeomor-phic iff and are isomorphic and that is com-putable from in time polynomial in the size of.Since is a planar graph and canonization of planar graphs is in PTIME,we can actually assume that is canonical in the sense that for homeomorphic instances and we have .4

In[SV98]it is proven that FO-queries over can be translated in linear time into?xpoint+counting queries over .Furthermore,if then FO-queries over can be translated to FO-queries over on instances with just one closed region.(More precisely,this means that there is a recursive mapping that associates with every FO of vocabulary a FO of vocabulary such that for all instances over,where is a closed set,we have.)

The question was left open whether this result extends to instances with one arbitrary region or with several regions. In Section6,we give a negative answer to this question.

ω

2

ω

1

Figure 3.Operations preserving

.

write if and can be transformed into each other

by a ?nite sequence of operations

and .Then the proof of [KPV97]easily yields:

Lemma 4.2.For all instances ,

we have:

.

It is an open question whether the converse of Lemma

4.2holds.In particular,this is interesting because is not known whether is decidable or not,whereas is de-cidable in PSPACE (this is Proposition

5.5of the next sec-tion).[KPV97]have shown that that and coincide on instances with only one closed region.We can extend their result to several regions,but only on instances with only regular cones.

5.Minimal instances

De?nition 5.1.An instance

is minimal if it satis?es the

following two conditions:

(M1)If is homeomorphic to and

are adjacent to and homeomorphic to

,for some

,then .

(M2)If and are adjacent to

and

homeomorphic to

,then

.

Lemma 5.2.There is a PTIME algorithm that associates with every instance a minimal instance such that

.

This can be done in such a way that for homeomorphic

instances

we have .Proof:Suppose ?rst that does not satisfy (M1).We show

that can be transformed to instance with fewer cells violating (M1),by two applications of .

Let be a 1-dimensional cell homeomorphic to

such that both neighbors of have the same color,but neither is homeomorphic to .Then instance locally

looks like Figure 4(1).We apply twice (to the dashed boxes)and obtain an instance that locally looks like Fig-ure 4(3).We have obviously reduced the number of cells violating (M1).

(1)(2)(3)

Figure 4.

Suppose now that does not satisfy (M2),then it can be transformed to an instance with fewer cells violating

(M2)by an application of

without violating (M1).To see this,let be a 2-dimensional cell in that is ad-jacent to the cells homeomorphic to of the same color.A cell homeomorphic to is adjacent to two 2-dimensional cells.Let be the other neighbors of

,respectively.Then by Lemma 3.4,also have the same color.We have to distinguish between two cases:Case 1:Both and are children of .Then we can reduce the number of cells violating (M2)by an application

of

(cf.Figure

5).

(1)

(2)

Figure 5.

Case 2:

is the parent of and

its child.Figure 6

shows how to

proceed.

(1)

(2)

Figure 6.

A PTIME algorithm transforming a given instance to a minimal instance may proceed as follows:Given ,

the algorithm?rst computes the invariant;this is possi-ble in PTIME[PSV99].The operations and translate to simple local operations on.Our algorithm?rst ap-plies pairs of until(M1)holds(as in?gure4),and then until(M2)holds.This can be done by a simple greedy strategy.The result is a structure that is the toplogical in-variant of a minimal instance.It is shown in[PSV99] that,given an invariant,an instance such that

can be computed in polynomial time.

Because is canonical(cf.Page4),this algorithm also guarantees that for homeomorphic instances we have

.

The language of an instance.A fundamental curve in an instance is an-de?nable continuous mapping

such that and

.If is a fundamental curve in, then for every2-dimensional cell the set is a?-nite union of open intervals(one of which is of the form and one of the form),and for every0-and 1-dimensional cell the set is a?nite union of closed intervals(some of which may just be single points). This follows from the fact that is-de?nable.

We are interested in the?nite sequence of colors appear-ing on a fundamental curve,i.e.in a?nite word

over the alphabet consisting of all colors appearing in. We say that a word is realized in if there is a fun-damental curve such that.The language is the set of all words realized by.

Example5.3.Let and be the instance with

.

has?ve cells,

,,

,and

.Let be the colors of, respectively,and note that has the same color as

.

Figure

7.

Figure8.

Then for example the words and are realized in(cf.Figure7).It is not hard to see that can be described by the?nite automaton displayed in Figure8.

From this example it is easy to see that for every the language is regular;it is accepted by an automaton that is essentially the cell graph.

More formally,let be an instance.We de?ne a?nite automaton as follows:

(where is a new symbol that does not denote a cell).

touches

EXT EXT,where EXT denotes the exterior.

EXT and.

Note that the graph underlying is the cell graph ex-tended by one additional vertex that is only adjacent to the exterior.The proof of the following lemma is straight-forward:

Lemma5.4.For every instance,the automaton ac-cepts.Thus is a regular language.

A walk in a graph is a sequence

,for some,such that

for.For a mapping we let

.Then it is almost im-mediate that for every instance we have

walk in

with EXT(5.1) where as usual EXT denotes the exterior.

Proposition5.5.is decidable in PSPACE.

Proof:

Given two instances and,we want to check in PSPACE whether it is possible to go from to using only homeomorphisms and operations in.Given an in-stance it is possible to compute its invariant in PTIME [PSV99].Thus the problem reduces to checking in PSPACE whether can be derived from using operations from .There are at most polynomially many different ways to apply operators from to(one just has to consider tu-ples of at most5cells and check if an can be applied to this tuple).Let be the set of invariants that can be obtained from by applying one operation from.The previous remarks show that contains at most polyno-mially many elements and can be computed in PTIME.It is therefore possible to enumerate all topological invariants that can be derived from by applying operations in, and check at each step whether it corresponds to or not. The latter can be done in PTIME,because this is deciding whether two planar graphs are isomorphic.

We now give a strategy that ensures that this process will stop.Because and are computable in PTIME (Lemma5.2),we can assume and minimal.

It is easy to see that operations in do not change the cones of the instances involved.Therefore implies that and have the same cone-type(cf.Page3)and the same orientation relation.This can be checked in PTIME. As the cones determine the number of1-dimensional cells homeomorphic to,this implies that all instances such that are such that their respective skeletons verify.

We may view the skeletons and as embedded pla-nar graphs with the0-dimensional cells as vertices and the 1-dimensional cells homeomorphic to as edges.Let and,resepectively,denote the faces of these em-bedded graphs.Note that these faces correspond precisely to the connected components of and,re-spectively.The number of connected components of is bounded by,the number of cones of.Therefore the number of faces of is given by the Euler formula, we have,where is the number of connected components of,and its number of edges and vertices.As is bounded by,so is.This gives a linear(in)bound on the number of connected compo-nents of.We would also like to bound the size of each connected component of(or equivalently the number of1-dimensional cells homeomorphic to).Un-fortunately,repeated application of the operations and may produce arbitrary large such components.

Nevertheless,it is possible to decide in PSPACE whether for an instance such that there is an isomor-phism from to that preserves the coloring and ori-entation.Furthermore,if there is such a,it can be computed in PSPACE.The complexity is PSPACE because there is no need to produce extra1-dimensional cells home-omorphic to during this process.This would only restrict the number of possible connections between the connected components of the skeleton.

So without loss of generality we can now assume that and are minimal and have isomorphic skeletons.Recall that every face()corresponds to a con-nected component of(,respectively). By Lemma3.3,is a tree with a canonical root,its exterior cell.

To?nd out whether,for each isomorphism

that preserves the coloring and orientation we check whether there are instances and

such that,,and for every face

we have.(Here denotes the face corresponding to under the isomorphism).

It suf?ces to prove the following for every: Claim1.Either,or and can be transformed to instances and,respectively,with,

such that in the component is isomorphic to in ,and for all,the component in has remained the same as in,and for all,the component in has remained the same as in.

It can be decided in PSPACE which of the two alterna-tives holds,for the latter can also be computed in PSPACE.

We can go through all isomorphism in PSPACE,so we ?x one.We also?x a face.

Before going on we need the following de?nitions.A branch of a tree is a minimal walk in going from the root of to one of its leaves.A walk in is said to be regular if it never goes through a singular cell.Let be the set of colors of and.For a word,we de?ne as the word of computed using rules in the same spirit as in Lemma5.2.More precisely this means rewriting the word using the rules for all colors. Fact1.Let be a regular walk in which starts in .Then it is possible to transform in such a way that contains a branch,such that.

This can be proved by a single loop on the length of

starting from the end.Without loss of gen-erality we can assume that is a2-dimensional cell.We start by constructing in a new ball of color .This is easily done by applying or once.Assume next that we have constructed a subinstance whose cell graph is a path attached to.Again we apply or once in order to get in sur-rounding.

The same kind of induction(but starting from the be-ginning of the word this time)shows that the converse also holds:

Fact2.If it is possible to construct using operations in a new branch in then there exists a walk in starting from such that.

We say that a walk realizes a word in if

.It can be checked in PSPACE whether a given word is realized by a walk in starting in;one way to see this is to reduce the problem to the question of whether two regular languages have a non-empty intersec-tion.

Now we can prove Claim1.We?rst transform to. For each branch of starting from,let and check whether there is a walk that realizes. If this is not the case,Fact2shows that.If it is the case,use Fact1to construct the corresponding branch in. To construct,do the same after reversing the role of and.It is clear that if the algorithm does not?nd out that

on its way,after minimizing the resulting instances

they satisfy the claim.

In the next section,it will be convenient to work with

a slight simpli?cation of the cell graph.We call a2-

dimensional cell EXT of an instance inessential if

is homeomorphic to a disk(and thus has precisely one

neighbor in),and the neighbor of in has another

neighbor with.Let be the graph

obtained from by deleting all vertices that are inessential

cells.We call the reduced cell graph of.Then(5.1)

actually holds with instead of:

walk in

with EXT(5.2)

6.Regular instances

Recall that an instance is regular if all points

are regular.The main result of this section is that and

coincide on regular instances.As a corollary,we will see

that the equivalence(4.1)does not extend beyond instances

with one closed region.

To illustrate where the problems are,let us start with a

simple example:

Example6.1.Let and consider the two in-

stances with,

5Furthermore,these are precisely the star-free regular languages.

Lemma6.4.Let be a minimal regular instance.Then is aperiodic.

Proof:

Let be a minimal regular instance.

The crucial step is to prove the following claim:

Let,,and

a walk in such that and

for.

Then.

(6.3)

Suppose for contradiction that(6.3)is wrong.Choose minimal such that there is a,an, and a walk in with and

for,but.

Since adjacent vertices in a cell graph(and thus in a reduced cell graph)have different colors and

,we have.

For notational convenience,we let,

,,and.We choose an

such that there is no such that is in the subtree below.Then is the parent of. Let,,and. Then.Let. Then for we have

.By(6.2),this implies

.If,this implies,a contradic-tion.If we can de?ne a walk from by deleting and,for.Then(6.3)holds with,,,and,in contradiction to the minimality of.

This proves(6.3).

Now let and such that

.We shall prove that. Let be the length of,respectively,and

a walk in with. Since there exist with

such that.Applying(6.3)to the walk

we infer. But then

is a walk with.

Lemma6.5.Let be a minimal regular instance and instances such that and. Then.

Proof:Recall the de?nitions of the operations and from Figure3.It suf?ces to prove the statement for in-stances with or.

Let.Note that there is a word

obtained from by replacing some let-ters by subwords(if)or(if ).

Suppose for contradiction that.Then there is a walk in such that.Let

and.By(6.2),when-ever,we have.Hence

is a walk in with

.Similarly,if

,we have,which implies that.Thus, a contradiction.

This shows that.

Lemma6.6.Let be a minimal regular instance and a regular instance with.Then there is a

such that for the curve we have

.

Proof:We assume that and are instances over the same schema.Then,denoting by EXT and EXT the exte-rior of,,respectively,we have EXT EXT. For every walk in with EXT we de?ne a sequence as follows:We let EXT.For we let be the(by(6.2)unique) neighbor of that has the same color as,if

and such a neighbor exists,and otherwise.

Since is a tree and satis?es(6.2),im-plies or(for).It fol-lows that for any walk with EXT and

with we either have

or or.

Note that actually the sequence only depends on the word.This if EXT(and thus), then if,and only if,.

Now suppose that.Let be max-imal with.Then.Observe that must be a child of in the tree.

Choose such that the curve intersects the cell .Let be a walk in that corresponds to the curve and let be minimal such that .Then,because is a child of. Then either,or.In both case we have .Thus.

Theorem6.7.Let be a regular instance of schema. Then there is a sentence FO top of vocabulary

such that an instance of the same schema satis?es if,and only if,.

Proof:Let be the set of colors appearing in.For every aperiodic language there is a formula

FO of vocabulary such that a regular instance

satis?es for a if,any only if,. This is an easy consequence of the theorem of McNaughton, Papert and Sch¨u tzenberger that the aperiodic languages are precisely the?rst-order de?nable languages.

Let and.By Lemma6.4,is aperiodic.Let

Clearly,satis?es.

We claim that for all instances we have:

(6.4) To prove this claim,note?rst that every instance satisfying realizes the same cones as and thus is regular.

Let be a regular instance.Assume?rst that. Then by Lemma6.6,.Because ,by Lemma6.5we have.Thus .

Conversely,if,then by Lemma6.6,

.By(6.1),it follows

and thus.

This proves(6.4).

It follows easily that FO top.Indeed,assume that satis?es.For every homeomorphic to we have

.Thus iff iff .

We let

minimal regular instance

with

Then FO top.

We shall prove that for any instance of schema we have

Suppose that.Then in by(6.4). By Lemma6.6,this implies and

for any minimal regular with .By Lemma6.3,this implies.By Lemma6.2,it follows that and are homeomor-phic.Thus.

Conversely,suppose that.Then by Lemma4.2, .Thus,because FO top and

(the later follows easily from the previous paragraph and (6.4)).

Corollary6.8.For all regular instances we have

.

Finally,we are ready to prove that the equivalence(4.1)

does not extend beyond instances with one closed region. Corollary6.9.The two instances of Example6.1are not elementary equivalent.Neither are the instances over de?ned by:

7.Translating sentences to the topological in-

variant

Recall that it is proven in[SV98]that there is a recur-

sive mapping that associates with every FO of vocab-ulary a FO of vocabulary such that for all instances over,where is a closed set,we have

.

The purpose of this section is to prove that this does not extend to arbitrary instances.

Proposition7.1.There is a sentence FO top of vocab-ulary such that for every sentence FO of vo-cabulary there is an instance such that and .

Proof:Let be the instances de?ned by

For,let be de?ned by

where for all

(cf.Figure

11).

Figure11.The instances and.

Note that for all we have and

,but.Corollary6.8implies that there is a

sentence FO top such that for all we have and.

Let.The graph is just a path with ver-tices,say,.Denoting the colors by, the colors on this path form the following sequence:

times times

is the same,except that and

.The other relations of the topological in-variant are identical in and.is empty, consists of all with even,and consists of the with odd.The orientation is empty.Finally,.

Standard Ehrenfeucht-Fra¨?ss′e techniques show that for every sentence FO there is an such that

if,and only if,.

The statement of the proposition follows.

8.On completeness of languages

An open problem that we have not considered so far is to?nd a(recursive)language that expresses precisely the ?rst-order topological queries.Although this is certainly an interesting question,we doubt that,even if theoretically such a language exists,it would be a natural language that may serve as a practical query language.Our results show that?rst-order topological queries are not local;on the other hand,it is known that?rst-order logic fails to express the most natural non-local topological queries such as connec-tivity of a region.

[KV99]have introduced a topological query language, the cone-logic CL,only expressing local properties.This language is a two tier language that allows to build?rst-order expressions whose atoms are again?rst-order expres-sions talking about the cones and colors of points.[KV99] have proven that CL captures precisely the?rst-order topo-logical properties of instances with only one closed region that is“fully2-dimensional”.If the underlying structure is ,it follows from[SV98]that the condition of being full 2-dimensional is not needed.Corollary6.9shows that this result does not extend to instances with several closed re-gions or one arbitrary region.

We propose to extend CL by a path operator,as it has been introduced in[BGLS99].Let us call the resulting topological query language PCL.The results of[BGLS99] show that this language has the basic properties expected from a reasonable spatial database query language.In addi-tion,it admits ef?cient query evaluation(the cost of evalu-ating a PCL-query is not substantially higher than the cost of evaluating a?rst-order query).

An example of a PCL-query not expressible in FO is connectivity of regions.We conjecture that conversely ev-ery FO top-query is expressible in PCL.The idea behind this conjecture is that local properties(expressible in CL)to-gether with the language of an instance,which can be de-scribed by the path-operator,seem to capture the?rst-order topological properties of an instance.As a?rst step to-wards proving this conjecture,let us remark that Corollary 6.8implies that on regular instances every FO top-sentence is equivalent to a set of PCL-sentences.

9.Conclusions

The results of this paper give a good understanding of ?rst-order topological queries on regular instances.Of course one may argue that regular instances are completely irrelevant—look at any map and you will?nd singular points.However,we could use our results to answer several open questions concerning topological queries on arbitrary instances.

As a matter of fact,we have shown that the previous un-derstanding of?rst-order topological queries,viewing them as”local”in the sense that they can only speak about the colors of points,is insuf?cient;this may be our main con-tribution.

The problem of a characterization of topological elemen-tary equivalence on arbitrary planar instances remains open. We conjecture that Corollary6.8generalizes,i.e.that is the same as on arbitrary instances.If this was true,by Proposition5.5topological elementary equivalence would be decidable in PSPACE.Let us remark that we do not be-lieve that the PSPACE-bound of Proposition5.5is optimal, we see no reason why should not be decidable in NP or even PTIME.

References

[AHV95]S.Abiteboul,R.Hull,and V.Vianu.Foundations of Databases.Addison Wesley,1995.

[BDLW96]M.Benedikt,G.Dong,L.Libkin,and L.Wong.

Relational expressive power of constraint query lan-

guages.In Proc.ACM Symp.on Principles of

Database Systems,1996.

[BGLS99]M.Benedikt,M.Grohe,L.Libkin,L.Segou?n.

Reachability and Connectivity Queries in Constraint

Databases.In Proc.ACM Symp.on Principles of

Database Systems,2000.

[BL97]M.Benedikt and https://www.360docs.net/doc/d1107166.html,nguages for relational databases over interpreted structures.In Proc.ACM

Symp.on Principles of Database Systems,1997. [BL98]M.Benedikt and L.Libkin.Safe constraint queries.

In Proc.ACM Symp.on Principles of Database Sys-

tems,1998.

[EF95]H-D Ebbinghaus and J.Flum.Finite Model Theory.

Springer,1995.

[GS97]S.Grumbach and J.Su.Queries with arithmeti-cal constraints.Theoretical Computer Science,

173(1):151–181,1997.

[GS99]S.Grumbach and J.Su.Finitely representable databases.Journal of Computer and System Sciences,

55(2):273–298,1999.

[GVV97]M.Gyssens,J.Van den Bussche,and D.Van Gucht.

Complete geometrical query languages.In Proc.

ACM Symp.on Principles of Database Systems,pages

62–67,New York,1997.ACM.

[JPV95]J.Van Den Bussche J.Paredaens and D.Van Gucht.

First-order queries on?nite structures over the reals.

In Proc.IEEE Conf.on Logic in Computer Science,

pages79–87,1995.

[KLP99]G.Kuper,L.Libkin,and J.Paredaens,editors.Con-straint Databases.Springer Verlag,2000.

[KPV97] B.Kuijpers,J.Paradaens,and J.Van den Buss-che.On topological elementary equivalence of spatial

databases.In Proc.of Intl.Conf.on Database Theory,

pages432–446,1997.

[KV99]Bart Kuijpers and Jan Van den Bussche.On captur-ing?rst-order topological properties of planar spatial

databases.In Proc.of Intl.Conf.on Database Theory,

pages187–198,1999.

[MP71]R.McNaughton and S.Papert.Counter-free Au-tomata.MIT Press,1971.

[PSV99] C.H Papadimitriou,D.Suciu,and V.Vianu.Topolog-ical queries in spatial databases.Journal of Computer

and System Sciences,58(1):29–53,1999.

[PVV94]J.Paredaens,J.Van Den Bussche,and D.Van Gucht.

Towards a theory of spatial database queries.In Proc.

ACM Symp.on Principles of Database Systems,pages

279–288,1994.

[Sch65]M.P.Sch¨u tzenberger.On?nite monoids having only trivial https://www.360docs.net/doc/d1107166.html,rmation and Control,8:190–

194,1965.

[SV98]L.Segou?n and V.Vianu.Querying spatial databases via topological invariants.In Proc.ACM Symp.on

Principles of Database Systems,1998.

[vdD98]L.van den Dries.Tame Topology and O-minimal Structures.Cambridge,1998.