Coping With Uncertainty in Map Learning

Kennet h Basye Thoma s Dean* Jef f rey Scot t Vi t t er f

Department of Computer Science, Brown Universit y

Box 1910, Providence, RI 02912

Abst r ac t

In many appl i cati ons i n mobi l e roboti cs, i t i s

i mpor t ant for a robot to explore its envi ron

ment i n order to construct a representation of

space useful for gui di ng movement. We refer to

such a representation as a map, and the process

of constructi ng a map f rom a set of measure

ment s as map learning. In thi s paper, we de

velop a framewor k for describing map-l earni ng

problems in which the measurement s taken by

the robot are subject to known errors. We i n

vestigat e two approaches to l earni ng maps un

der such condi ti ons: one based on Val i ant's

probably approximately correct l earni ng model,

and a second based on Rivest Sz Sloan's reli-

able and probably nearly almost always useful

l earni ng model. Bot h methods deal wi t h the

probl em of accumul ated error i n combi ni ng lo

cal measurement s to make global inferences. In

the first approach, the effects of accumulated

error are el i mi nat ed by the use of reliabl e and

probabl y useful methods for discerning the lo

cal properties of space. In the second, the ef

fects of accumul ated error are reduced to ac

ceptabl e levels by repeated expl orati on of the

area to be learned. Fi nal l y, we suggest some i n

sights i nt o why certai n exi sti ng techniques for

map l earni ng perf or m as wel l as they do.

1 I nt r oduct i o n

Many of the problems faced by robot s navi gati ng i n the

envi ronment can be faci l i tated by using expectations in

the f or m of expl i ci t model s of object s and the spaces t hat

they occupy. We use the t er m map to refer to any model

of large-scal e space used for purposes of navi gati on. Map

"Thi s work was supported in part by the National Science

Foundation under grant IRI-8612644 and by the Advanced

Research Projects Agency of the Department of Defense and

was monitored by the Ai r Force Office of Scientific Research

under Contract No. F49620-88-C-0132.

1This work was supported in part by a National Science

Foundation Presidential Young Investigator Award CCR-

8846714 wit h matching funds from I BM, and by National

Science Foundation research grant CCR-8403613.

learning involves expl ori ng the envi ronment, maki ng ob

servations, and then using the observations to construct

a map. The construction of useful maps is compl i cated

by the fact that observations i nvol vi ng the posi ti on, ori

entati on, and i denti fi cati on of spati al l y remot e object s

are i nvari abl y error prone. In thi s paper, we explore a

number of problems involved in constructi ng useful maps

f rom measurement s taken wi t h sensors subject to known

errors.

In previous work [Dean, 1988], we have looked at vari

ous opt i mi zat i on problems related to constructi ng maps

(e.g., construct the most accurate map consistent wi t h

a set of measurements). Even in cases i nvol vi ng onl y a

single di mensi on, such opt i mi zat i on problems can t ur n

out to be NP-har d [Yernini, 1979]. In thi s paper, rather

than look at problems that involve doi ng the best wi t h

what you have, we consider problems t hat involve going

out and getti ng what you need to generate useful repre

sentations. In parti cul ar, we consider a f or m of reliable

and probably almost always useful l earni ng [Rivest and

Sloan, 1988] in which the robot gathers i nf ormat i on to

ensure that it nearl y always (wi t h probabi l i t y 1—6) can

provide a guaranteed perfect pat h f r om one l ocati on to

another. A prerequisit e to thi s sort of l earni ng is t hat

the robot, i n movi ng around i n its envi ronment, can dis

cern the local properties of space wi t h absolut e certaint y

wi t h hi gh probabi l i t y havi ng expended an amount of ef

fort pol ynomi al in 1/2 and n, where n is some measure of

the size of the envi ronment.

By el i mi nat i ng local uncertainty, smal l errors i ncurred

in maki ng local measurement s are not allowed to prop

agate rendering global queries unacceptabl y inaccurate.

In general, local uncertaint y accumulates as the product

of the distance in generating global estimates. One way

to avoi d thi s sort of accumul ati on is to establish strate

gies such that the robot can discern properties of its

envi ronment wi t h certainty. Most existing map learning

schemes exploi t thi s sort of certaint y in one way or an

other (see Section 4). The rehearsal strategies of Kui per s

[1988] are one exampl e of how a robot mi ght pl an to

el i mi nat e uncertainty. Once we have a method for el i m

i nati ng uncertainty, the probl em then reduces to one of

pl anni ng out and executing the necessary experiment s

to extract certai n i nf ormat i on about the envi ronment.

In si tuati ons i n whi ch i t i s not possibl e to el i mi nat e

local uncertai nt y completely, it is sti l l possibl e to reduce

Basye, Dean and Vitter 663

the effects of accumulated errors to acceptable levels by

performing repeated experiments. To support this claim,

we describe a map-learning technique based on Valiant's

probably approximately correct learning model [Valiant,

1984] that, given small 6 > 0, constructs a map to an

swer global queries such that the answer provided in re

sponse to any given query is correct with probability

1 — 6. The techniques presented apply to a wide range

of map-learning problems of which the specific problems

addressed in this paper are meant to be merely illustra

tive.

2 Spat i al Repr esent at i o n

We model the world, for the purposes of studying map

learning, as a graph with labels on the edges at each

vertex. In practice, a graph will be induced from a set

of measurements by identifying a set of distinctive loca

tions in the world, and by noting their connectivity. For

example, we might model a city by considering intersec

tions of streets to be distinguished locations, and this

will induce a grid-like graph. Kuipers [1988] develops a

mapping based on locations distinguished by sensed fea

tures like those found in buildings (see Figure 1). Fig

ure 2 shows a portion of a building and the graph that

might be induced from i t. Levitt [l987] develops a map

ping based on locations in the world distinguished by the

visibility of landmarks at a distance.

In general, different mappings result in graphs with

different characteristics, but there are some properties

common to most mappings. For example, if the mapping

is built for the purpose of navigating on a surface, the

graph induced wi ll almost certainly be planar and cyclic.

Other properties may include regularity or bounded de

gree. In what follows, we wi ll always assume that the

graphs induced are connected and undirected; any other

properties will be explicitly noted.

Following [Aleliunas et a/., 1979], a graph model con

sists of a graph, G — (V, E), a set L of labels, and a

labeling, Φ : {V x E} —> L, where we may assume that

L has a null element ± which is the label of any pair

(v e V, e e E) where e is not an edge from v. We will

frequently use the word direction to refer to an edge and

its associated label from a given vertex. Wi th this nota

tion, we can describe a path in the graph as a sequence

of labels indicating the edges to be taken at each ver

tex. We can describe a procedure to follow as a function

from V —> L indicating the preferred direction at each

location.

If the graph is a regular tessellation, we may assume

that the labeling of the edges at each vertex is consistent,

i.e., there is a global scheme for labeling the edges and

the labels conform to this scheme at every vertex. For

example, in a grid tessellation, it is natural to label the

edges at each vertex as North, South, East, and West.

In general, we do not require a labeling scheme that is

globally consistent. You can think of the labels on edges

emanating from a given vertex as local directions. Such

local directions might correspond to the robot having

a compass that is locally consistent but globally inac

curate, or local directions might correspond to locally

distinctive features visible from intersections in learning

the map of a city.

In the following, we identify three sources of uncer

tainty in map learning. First, there may be uncertainty

in the movement of the robot. In particular, the robot

may occasionally move in an unintended direction. We

refer to this as directional uncertainty, and we model this

type of uncertainty by introducing a probabilistic move

ment function from {V x L} —► V. The intuition behind

this function is that for any location, one may specify a

desired edge to traverse, and the function gives the loca

tion reached when the move is executed. For example,

if G is a grid with the labeling given above, and we as

sociate the vertices of G with points (i, j) in the plane,

we might define a movement function as follows:

where the ". . ." indicate the distribution governing

movement in the other three directions. The probabili

ties associated with each direction sum to ]. If all direc

tions are equally likely regardless of the intended direc

tion, then the movement function is said to be random.

Throughout this paper, we will assume that movement

in the intended direction takes place with probability

better than chance.

A second source of uncertainty involves sensors, and

in particular recognizing locations that have been seen

before. The robot's sensors have some error, and this

664 Machine Learning

can cause error in the recogni ti on of places previousl y

vi si ted; the robot mi ght either f ai l to recognize some

previ ousl y vi si ted l ocat i on, or i t mi ght err by mi st aki ng

some new l ocati on for one seen in the past. We refer to

thi s type of uncertai nt y as recognition uncertai nty, and

model i t by par t i t i oni ng the set of vertices i nt o equiv-

alence classes. We assume t hat the robot is unabl e to

di sti ngui sh between element s of a given class using onl y

its sensors.

A t hi r d source of error involves another mani festati on

of sensor error. In representing the worl d using a graph,

some mappi ng must be established f rom a set of di sti n

guished locations in the worl d to V. Error in the sensors

coul d cause the robot to f ai l to notice a distinguished lo

cati on some of the t i me. For exampl e, a robot t axi mi ght

use intersections as di sti ngui shed locations, leading to a

gri d-l i ke graph. But i f sensor error causes the robot not

to notice that he is passing t hrough an intersection, his

map wi l l become flawed. In expl ori ng an office envi

ronment, the poi nt i n a hal l way i n front of a door may

correspond to a vertex i n the induced graph. If the door

is closed, there is some chance t hat the robot wi l l not

recognize the vertex in traversing the hal l. We model

thi s type of uncertai nt y by i nt roduci ng a probabi l i sti c

movement f unct i on t hat can ski p over vertices. We refer

to thi s type of movement functi on as discontinuous and

to the type of uncertai nt y modeled as continuity uncer

tai nty.

Apparentl y, the three types of uncertai nt y described

above are orthogonal in the sense t hat none i mpl i es or

precludes the others. The issues involved in model i ng

and reasoning about cont i nui t y uncertai nt y are complex

and wi l l not be treated further i n thi s paper. In the

f ol l owi ng, we are concerned wi t h di recti onal and recog

ni t i on uncertai nty.

3 Map Learni ng

For our purposes, a map is a dat a structur e t hat fa

ci l i tates queries concerning connecti vi ty, bot h local and

gl obal. Answers to queries i nvol vi ng global connecti vi t y

wi l l generall y rel y on i nf ormat i on concerning local con

necti vi ty, and hence we regard the fundamental uni t of

i nf ormat i on to be a connection between two nearby loca

tions (i.e., an edge between two vertices in the induced

undirected graph). We say t hat a graph has been learned

completely if for every l ocati on we know al l of its neigh

bors and the di recti ons in whi ch they lie (i.e., we know

every tri pl e of the f or m (u, /, v) where u and v are vertices

and / is the label at u of an edge in G f rom u to v). We

assume t hat the i nf ormat i on used to construct the map

wi l l come f rom expl ori ng the envi ronment, and we iden

t i f y two different procedures involved i n learning maps:

exploration and assimilation. Expl orat i on involves mov

i ng about i n the worl d gatheri ng i nf ormat i on, and as

si mi l at i on involves using t hat i nf ormat i on t o construct

a useful representation of space. Expl orat i on and as

si mi l at i on are generall y handl ed i n paral l el, wi t h assim

i l at i on performed i ncremental l y as new i nf ormat i on be

comes availabl e dur i ng expl orat i on. In thi s section, we

are concerned wi t h the condi ti ons under which a graph

can be compl etel y learned, and how much t i me wi l l be

required for the expl orati on and assi mi l ati on.

3.1 Tessel l at i o n Gr aph s

I t's not hard to see t hat any connected, undirected graph

can be completel y learned easily if there is no uncer

t ai nt y; [Kuipers and Byun, 1988] describes a way of do

i ng thi s by bui l di ng up an agenda consisting of unex

plored paths leading out of locations and then movi ng

about so as to eventuall y explore al l such paths. Not h

ing about the graph need be known before the explo

rati on begins. Introduci ng the kinds of uncertaint y de

scribed in Section 2 complicates things considerably. If,

however, the graph has addi ti onal structure, then t hat

structur e can often be exploited to el i mi nat e uncertai nty.

In the fol l owi ng, we sketch a proof t hat it is possibl e

to efficientl y learn maps that correspond to regular tes

sellations wi t h boundaries. It turns out t hat the ex

pl orati on component of learning regular tessellations is

qui t e simple; random walks suffice for pol ynomi al -t i me

performance. In the longer version of thi s paper, we

describe an efficient incremental assi mi l ati on procedure

that is called whenever the robot encounters a location

duri ng expl orati on, and then prove the fol l owi ng1.

Lemma 1 The assimilation algorithm, provided will

learn a finite tessellation completely if the exploration

tour traverses every edge in the graph. The overall cost

of assimilation is O(m) where m is the length of the tour.

We now have to ensure that duri ng expl orati on the robot

traverses each edge in the graph at least once wi t h hi gh

probabi l i ty. The fol l owi ng two lemmas establish t hat,

for any connected, regular, undirected graph G and any

b > 0, a random walk of length pol ynomi al in 1/b and the

size of G is sufficient for traversing every edge in G wi t h

probability J - 6.

L e mma 2 For any d > 1, there exists a polynomial

p(d,1/2j) of order O(dlogj) such that with probability 1-S

p visits to a vertex of order d result in traversing all edges

out of the vertex at least once.

Lemma 3 For any connected, regular, undirected graph

G — (V, E) with order d, any S > 0, and any m > I,

there exists a polynomial p( | E|, ra, 1/b) such that with

probability 1 — S, a random tour on G of length p vis-

its every vertex in V at least m times.

In most cases, we can do better than random expl orati on.

If the robot moves i n the di recti on i t i s poi nt i ng wi t h

probabi l i t y better than chance, then the robot can tra

verse every edge in the graph wi t h high probabi l i t y in

t i me linear in the size of the graph. Using the above

three lemmas it is easy to prove the fol l owi ng.

Theor e m 1 Any finite regular tessellation G — (V, E)

can be reliably, probably almost always usefully learned.

The lemmas and for m of the proof described above

provide a framewor k for provi ng that other kinds of

graphs can be reliabl y probabl y almost always usefull y

learned in a pol ynomi al number of steps. In general, al l

1To meet the submission length requirements, all proofs

have been omitted. The longer version of the paper, including

all proofs[Basye et al., 1989], is available upon request.

Basye, Dean and Vitter 665

we require is t hat a pol ynomi al number of visit s to every

vertex provides enough i nf ormat i on to learn the graph.

Perhaps, the most i mpor t ant lesson to extract f rom thi s

exercise is t hat the effects of mul t i pl i cat i v e error in learn

i ng maps of large-scal e space can be el i mi nat ed if there is

a reliabl e met hod for el i mi nat i ng local uncertai nt y t hat

works wi t h hi gh probabi l i t y. The above approach t o

map l earni ng was i nspi red by Rivest's model of learn

i ng [Rivest and Sloan, 1988], i n whi ch compl ex probl ems

are broken down i nt o si mpl e subproblems t hat can be

learned i ndependentl y. In order to learn a useful repre

sentation of the gl obal structur e of i ts envi ronment, i t i s

sufficient t hat a robot have reliabl e and usuall y effective

methods for sensing the l ocal structur e of its envi ron

ment and a met hod for composi ng the local structur e to

generate an accurat e gl obal structure. The sensing met h

ods need not always provi de useful answers; they need

onl y guarantee t hat the answer returned i s not wrong.

The probl em then becomes largel y one of det ermi ni ng a

sequence of sensing and movement tasks t hat wi l l pro

vide useful answers wi t h hi gh probabi l i t y. There are sit

uati ons, however, i n whi ch reliabl e sensing methods are

not available, and i t i s st i l l possibl e to learn useful maps

of large-scal e space.

3.2 Gener a l Gr aph s

The next probl em we l ook at involves bot h recogni

ti on and di recti onal uncertai nt y wi t h general undi rected

graphs. We show t hat a f or m of Val i ant's probabl y ap

proxi mat el y correct l earni ng i s possibl e when appl i ed to

l earni ng maps. In thi s section, we consider the case in

which movement i n the i ntended di recti on takes place

wi t h probabi l i t y better t han chance, and t hat, upon en

teri ng a vertex, the robot knows wi t h certai nt y the local

name of the edge upon whi ch i t entered. We cal l the

l atter requi rement reverse movement certainty. Result s

for related model s are summari zed in the next section.

At any poi nt i n t i me, the robot i s faci ng i n a di recti on

defined by the label of a part i cul ar edge/vertex pai r—t he

vertex being the l ocati on of the robot and the edge being

one of the edges emanat i ng f r om t hat vertex. We assume

that the robot can t ur n to face i n the di recti on of any of

the edges emanat i ng f rom the robot's l ocati on. We also

assume t hat upon enteri ng a vertex the robot can de

termi ne wi t h certai nt y the di recti on i n whi ch i t entered.

Di recti onal uncertai nt y arises when the robot at t empt s

to move in the di recti on it is poi nt i ng. Let 7 > 0.5 be

the probabi l i t y t hat the robot moves i n the di recti on i t

i s currentl y poi nt i ng. Mor e t han 50% of the t i me, the

robot ends up at the other end of the edge defi ni ng i ts

current di rect i on, but some percentage of the t i me i t ends

up at the other end of some other edge emanat i ng f r om

its st art i ng vertex. Whi l e the robot won't know t hat i t

has ended up at some uni ntended l ocat i on, i t wi l l know

the di recti on t o fol l ow i n t r yi ng t o ret ur n t o its previous

l ocati on.

To model recogni ti on uncertai nty, we assume t hat the

vertices V are part i t i oned i nt o two sets, the di sti ngui sh

able vertices D and the i ndi sti ngui shabl e vertices I. We

are able to di sti ngui sh onl y vertices in D. We refer to

the vertices in D as landmarks and to the graph as a

landmark graph. We define the landmark distribution

parameter', r, to be the maxi mu m distance f r om any ver

tex in I to i ts nearest l andmar k (i f r = 0, then I is

empt y and al l vertices are l andmarks). We say t hat a

procedure learns the local connectivity within radius r of

some v E D if it can provi de the shortest pat h between

v and any other vertex in D wi t hi n a radius r of v. We

say t hat a procedur e learns the global connectivity of a

graph G within a constant factor if, for any t wo vertices

u and v in D, it can provi de a pat h between u and v

whose l engt h is wi t hi n a constant factor of the l engt h of

the shortest pat h between u and v in G.

We begi n by showi ng t hat the mul t i pl i cat i v e error i n

curred i n t r yi ng to answer gl obal pat h queries can be

kept low i f the l ocal error can be kept l ow, t hat the t ran

si ti on f r om a local uncertai nt y measure to a gl obal un

certai nt y measure does not increase the compl exi t y by

more t han a pol ynomi al factor, and t hat i t i s possibl e

to bui l d a procedur e t hat direct s expl orati on and map

bui l di ng so as to answer gl obal pat h queries t hat are ac

curat e and wi t hi n a smal l constant factor of opt i mal wi t h

high probability.

L e mma 4 Let G be a landmark graph with distribution

parameter r, and let c be some integer > 2. Given a pro-

cedure that, for any S1 > 0, learns the local connectivity

within cr of any landmark in G in time polynomial in

1/2- with probability 1 - S1, there is a procedure that learns

the global connectivity of G with probability 1 — Sg for

any Sg > 0 in time polynomial in 1/S- and the size of the

graph. Any global path returned as a result will be at

most c/c-2 times the length of the optimal path.

The procedur e presented i n the proof of Lemma 4

searches out war d f r om a vertex v E D to a distance cr,

and then uses the edges f ound whi l e enteri ng vertices on

the out war d pat h to at t empt to ret ur n to v. The direc

ti ons used on the way out f or m an expectati on for the

label s observed on the way back. When these expecta

ti ons are not met, the traversal is said to have fai l ed, and

the procedur e tries agai n. The procedur e keeps track of

the edge/vertex label s associated wi t h vertices visited

duri ng expl orati on i n order to ensure t hat i t explores al l

paths of l engt h cr or less emanat i ng f r om each vertex

i n D wi t h hi gh probabi l i t y.

There is a possi bi l i t y t hat some combi nat i on of move

ment errors coul d resul t in false posi ti ve or false nega

ti ve tests. But we show by expl oi t i ng reverse certai nt y

t hat we can stati sti cal l y di sti ngui sh between the true

and false test results. By at t empt i ng enough traversals,

the procedur e can ensure wi t h hi gh probabi l i t y t hat the

most frequentl y occurri ng sets of directions correspond

i ng to perceived traversal s actual l y correspond to paths

in G. What is requi red, then, is for the l earni ng pro

cedure to do enough expl orati on to i denti f y al l paths of

l engt h cr or less in G wi t h hi gh probabi l i t y.

L e mma 5 There exists a procedure that, for any S1 > 0,

learns the local connectivity within cr of a vertex in any

landmark graph with probability l — 6\ in time polynomial

in 1/S1, 1/1-2r and the size of G, and exponential in r.

666 Machine Learning

3.3 Rel at e d Model s

We can get the same results as in the last section if we

allow movement uncertai nt y i n the reverse di recti on, but

demand forwar d movement certainty. The al gori thms

are si mi l ar, the j usti fi cati ons different. In thi s case, the

graph can be reliabl y navigated by the same agent t hat

di d the map learning.

We are also investigating ways to remove the require

ment of either reverse certaint y or forwar d certainty. Re

verse certaint y is used in the last section to hel p di sti n

guish probabi l i sti cal l y between true and false results in

our testing procedures. We can show, for example, that

if r ( l —7) is bounded by a smal l constant, then efficient

map learning is possibl e wi t hout either the reverse cer

t ai nt y or forwar d certaint y requirement. Another way

around thi s restriction is to allow the expl ori ng agent to

drop pebbles or beacons to remember where it has been.

4 Rel at ed Wor k

There have been many approaches to dealing wi t h un

certaint y in spatial reasoning [Brooks, 1984, Davi s, 1986,

Durrant -Whyt e, 1988, Kuipers, 1978, Lozano-Perez,

1983, McDermot t and Davis, 1982, Moravec and Elfes,

1985, Smi t h and Cheeseman, 1986], but most of these

methods suffer f rom the effects of mul t i pl i cat i ve error in

esti mati ng relative position and ori entati on. Thi s paper

is concerned wi t h el i mi nati ng the effects of mul t i pl i cat i ve

error by either el i mi nati ng local uncertai nt y altogether

or by taki ng enough measurement s to ensure t hat such

effects are reduced to tolerabl e levels. In thi s section, we

consider two related approaches.

Kuipers defines the noti on of "pl ace" in terms of a set

of related visual events [Kuipers, 1978]. Thi s noti on pro

vides a basis for i nduci ng graphs f rom measurements. In

Kui pers' framewor k [1988], locations are arranged in an

unrestricted planar graph. There is recognition uncer

tai nty, but there is no directional uncertai nt y (i f a robot

tries to traverse a parti cul ar hal l, then i t wi l l actual l y

traverse that hal l; it may not be able to measure exactl y

how long the hal l is, but i t wi l l not mi stakenl y move

down the wrong hal l ). Kuiper s goes to some l engt h to

deal wi t h recognition uncertainty. To ensure correctness,

he has to assume t hat there is some reference l ocati on

that is distinguishabl e from al l other locations. Since

there is no di recti onal uncertainty, any two locations can

be distinguished by traversing paths to the reference lo

cati on. Given a procedure that is guaranteed to uniquel y

i denti f y a location if it succeeds, and succeeds wi t h hi gh

probabi l i ty, we can show that a Kuipers-styl e map can

be reliabl y probabl y almost always usefull y learned using

an analysi s si mi l ar to t hat of Section 3. In fact, we do

not require that the edges emanati ng f rom each vertex

be labeled, j ust that they are cyclicall y ordered.

Dudek et al [1988] consider the probl em of learning

a graph in which al l vertices are i ndi sti ngui shabl e and

upon entering a vertex the robot can leave by any arc

indexed f rom the one i t entered on. The robot can always

Basye, Dean and Vitter 667

retrace its steps i f i t remember s the di recti ons i t took at

each poi nt duri ng expl orat i on. The author s show t hat

the probl em i s unsolvabl e i n general, but t hat by provi d

i ng the robot wi t h a number of di sti nct marker s (k; > 1)

the robot can learn the graph i n t i me pol ynomi al i n the

gr aphs size. In order to place a marker on a part i cu

lar vertex, the robot must vi si t t hat vertex; i n order to

recover the marker at later t i me, the robot must ret ur n

to the vertex. A vertex wi t h a marker on it acts as a

t emporar y l andmark. No assumpti on i s made regardi ng

the pl anari t y of the graph. The probl em wi t h a singl e

marker t hat can be placed once but not recovered is also

unsolvable, but, i f you allow a compass i n addi t i on, the

probl em can be solved i n pol ynomi al t i me.

Levi t t et a/[1987] describe an approach to spati al rea

soning t hat avoids mul t i pl i cat i v e error by i nt roduci ng lo

cal coordi nat e systems based on l andmarks. Landmarks

correspond to envi ronmental features t hat can be ac

qui red and, more i mpor t ant l y, reacquired i n expl ori ng

the envi ronment. Gi ven t hat l andmarks can be uni quel y

i denti fi ed, one can induce a graph whose vertices corre

spond to regions of space defined by the l andmarks vis

ibl e i n t hat region. The resul ti ng probl em involves nei

ther recogni ti on nor movement uncertai nty. Our results

in Section 3 bear di rectl y on any extension of Levi t t's

work t hat involves either recogni ti on or movement un

certainty.

5 Concl usi on

Thi s paper examines the rol e of uncertai nt y i n map

l earni ng. We assume an envi ronmental model t hat pro

vides for a fi ni t e set of di sti ncti ve locations t hat can be

rel i abl y detected and repeatedl y f ound. Under thi s as-

sumpt i on, the probl em of map l earni ng reduces to one

of ext ract i ng the structur e of a graph t hrough a process

of expl orati on i n whi ch onl y smal l part s of the struc

ture can be sensed at a t i me and sensing is subject to

error. We are part i cul arl y interested i n showing t hat cu

mul at i ve errors i n reasoning about the gl obal properties

of the envi ronment based on local measurement s can be

reduced to acceptabl e levels using a pol ynomi al (i n the

size of the graph) amount of expl orat i on. The results i n

thi s paper shed l i ght on several exi sti ng approaches to

map l earni ng by showing how they mi ght be extended

to handl e various types of uncertai nty. Our basic frame-

work is general enough to be appl i ed to a wide vari et y of

map l earni ng probl ems. We have i denti fi ed one part i cu

lar source of uncertai nty, namel y cont i nui t y uncertai nt y

(see Section 2), t hat we believe of parti cul ar interest in

l earni ng maps of bui l di ngs and other envi ronment s pos

sessing an easil y discernabl e structure.

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