Through a Fuzzy CTL Logic for Modelling Urban Trajectories

A Framework for Modelling City Evolution from Past to Future

Asma Zoghlami

, Cyril de Runz

1,2

and Herman Akdag

LIASD, University of Paris 8, 2 Rue de la liberté, 93526, Saint-Denis Cedex, France

CReSTIC, University of Reims, Rue des crayères, BP 1035, 51687 Reims cedex 2, France

Keywords: Geographical Information System, Spatiotemporal Modelling, Evolving System, City, Modal Temporal

Logic.

Abstract: A city is by definition a relatively large town of a significant importance. It is a centre of population,

commerce, culture, industry, etc. The city evolves over time and gets morphological, sociological, economic

and political transformations. Geographic Information System (GIS) may be used in spatial analysis of both

the current city and its evolution over time. Based on the past and the present of a city, we are interested in

developing a methodology that goes from the spatiotemporal modelling of its evolution to its prediction in

the future. The motivation behind this research is to create a tool for the decision support at the disposal of

the town hall. This tool aims to help making future decisions about investments, transport networks,

infrastructures, etc. In this paper, we propose a framework that allows defining the possible trajectories of

the city following the spatial, temporal and functional dimensions. The definition of those trajectories will

be attached to a reasoning based on logic according to modalities, time and the imperfect nature of the

information (imprecision, uncertainty, etc.).

1 INTRODUCTION

Geographical Information System (GIS) could be

viewed as a set of tools that allow gathering,

handling, analysing and displaying the data from

various sources (Burrough and McDonnell, 1998).

These data are localized information that contributes

to the space management. Spatial information is

mainly used according to specific objectives. GIS

has an important role in many fields, including

history and urban planning, which involves them in

all phases of data processing starting from the data

collection and ending with the visualization of the

built maps. Historians and geographers may study

the history of cities using GIS for analysing them

over long periods of time. In the urban planning

field, GIS tools enhance urban planners’ analytical,

problem-solving and decision making capabilities.

Urban planning is a future-oriented activity

mainly conditioned by the past and the present.

Therefore, a natural way for modeling the city

evolution is to exploit a logical framework based on

the past of the city that allows modeling rules

according to temporal links using modalities.

Indeed, modal logic and/or temporal logic may help

us for defining consistent processes over time. For

instance, as in the previous example, the emergence

of means of transport (roads, railways) has

contributed very closely to the economic and urban

development (construction, extensions of existing

production sites) and to the sociological

development (populating area, immigration).

To understand past, present and future of the

city, it is important to identify the space in which its

trajectory can be modeled. It is therefore, essential to

establish a list of identified variables as a part of a

city model and to generate a Cartesian logic in the

interaction of these variables in space and time.

Modalities and temporalities will give us the tools to

define this trajectory.

In classic approaches, the imprecision of the

stored information is not considered for modeling

the urban trajectories. In deed, if the uncertainty of

the scenario is studied for the prospective approach

using probabilities, the modelling of the urban

trajectories according to the vagueness of the initial

data is still few studied.

In this paper, section 2 presents the nature of the

studied object and of their possible evolution.

Section 3 introduces the logics we would use.

514

Zoghlami A., de Runz C. and Akdag H..

Through a Fuzzy CTL Logic for Modelling Urban Trajectories - A Framework for Modelling City Evolution from Past to Future.

DOI: 10.5220/0004256305140518

In Proceedings of the 5th International Conference on Agents and Artiﬁcial Intelligence (ICAART-2013), pages 514-518

ISBN: 978-989-8565-39-6

 2013 SCITEPRESS (Science and Technology Publications, Lda.)

Section 4 proposes our framework. Section 5 is

devoted a discussion and the conclusion.

2 URBAN TRAJECTORIES:

OBJECTS AND EVOLUTIONS

Many researches focused on the modeling of the old

cities' history. In particular, several studies targeted

the development of a modeling approach to

represent cities' shapes and their dynamic in the past

time (Güting et al., 2000); (Pumain et al., 2006).

Urban objects are generally represented as the

combination of three features (Peuquet, 2002):

- The function of the object (church, school,

business, etc.),

- The space which is the location of the object,

- The time that corresponds to the existence of the

object over time.

Thus, the trajectory of urban objects is determined

by the different changes occurred in each one of the

previous features. Over the time, the object can

evolve by changing its function, its space or the both

at once. In fact, it may only change its function and

keep its space, change its space and keep its function

or change both the space and the function at once.

It is completely obvious to say that the future of

a city (urban area) is uncertain, and any scenario

should be considered as it is. Nevertheless, we can

study the different possible evolution according to

the knowledge of the past and to the current rules for

the change.

This knowledge is usually vague or imprecise. In

fact, there are many possible sources of information:

such as history studies, maps, city archives, current

urban management laws and directives, etc. Then,

consider the past time to retrace the shape of urban

objects (city, agglomerations, urban areas, etc.)

motivate as to wonder about the geographical

dimensions, and the different changes that the city

would have by evolving in the future.

Nevertheless, every stored data is subject to

imprecision to each component of the information.

Therefore, this component of the information should

be taken into consideration. That is the main goal of

our proposal. In order to present it, we should,

firstly, introduce the different logics we will use.

3 LOGICS

Based on the past and the present of a city, we are

interested in modelling its evolution in the future. To

reach this objective, we have to define its

spatiotemporal trajectory at an instant ti+1. In Figure

2, the city evolves in the space (x, y) during the

period [ti, ti+1]: the points P1’, P2’ and P3’ at ti+1

correspond to the evolution of respectively the

points P1, P2 and P3 observed at the time ti. These

future points are created based on expressions like

“we think that P1 and P1’ should always be

matched”, “it seems that P2 will be P2’ ”, “we know

that P3 will always be P3”, etc.

Figure 1: Illustration of two steps of a possible urban

trajectory.

The elaboration of this trajectory requires the use of

a logic offering ways to reason about expressions

qualified in terms of time called also time modalities

such as “it has been the case that p”, “it has always

been the case that p”, “it will always be the case that

p”, etc.

3.1 Temporal Logic

Classical logic has a static nature that does not allow

handling the concept of properties changing over

time. Temporal logics are consequently, obtained by

extending classical logics with temporal operators

like always, all, some, until and next to express the

evolution of a system over time.

They associate a truth value to a sequence of

states representing the evolution of a system. The

concept of truth in the temporal logic depends on the

world evolution. It means that a proposition may be

false at some time and becomes later true. This

concept may be used to represent the acquired

knowledge. These logics are defined on a set P of

atomic propositions called also proposition

variables. These atomic propositions are combined

through a number of logical connectors, including

the classic connectors (and, or, not, etc) and other

operators called modalities.

Linear logics focus on the executions of the

system without taking into consideration the

interweaving of the different possible futures at a

given point during the execution. In our approach,

we will focus on CTL (Clarke et al., 1986), which

ThroughaFuzzyCTLLogicforModellingUrbanTrajectories-AFrameworkforModellingCityEvolutionfromPastto

Future

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has a representation in the form as a tree of possible

executions. CTL offers the possibility to have

several solutions over time and to model trajectories

due to the branching aspect.

3.2 Modal Logic

In addition to the temporal logic, the elaboration of

the trajectory requires also the use of a modal logic

to express the possibility, prohibition, doubt, etc.

about a logical proposition (c.f. Chellas (1980)).

Modal logic is similar to traditional logic with the

additions of modalities of possibility ("may be p", "it

is possible that p") and modalities of the necessity

("necessarily p", "it is necessary that p"). In addition

to the alethic modalities which are the modalities of

necessity and possibility, modal logic handles also

the modalities of the impossibility ("it is impossible

that p"), the modalities of the belief ("we believe that

p"), the modalities of the knowledge ("it is known

that p") and the deontic modalities such as (" it is

obligatory that p"," it is allowed that p").

3.3 Fuzzy Logic

The fuzzy logic (c.f. Dubois and Prade (2000))

offers the ability to deal with the vagueness through

the representation of each concept by a fuzzy set. A

fuzzy set characterized a concept through its

possible domain values using a membership function

taking values in [0,1]: 0 the confidence in the

domain value for the concept is null, 1 the domain

value is possible, and, in between, the greater the

degree, the higher the confidence. On those sets,

some operators such as AND, OR, NOT, etc. have

been introduced in the literature. Using the three

previous logics, we propose a framework that allows

dealing with soft urban trajectories.

4 OUR FRAMEWORK

In our context, which is the urban objects' evolution

modelling, it has to consider three parameters: time,

space and function of the object (school, business,

etc.). The construction of this trajectory will provide

a global view of the various changes that the city

will undergo. It also enables to predict its functional

and spatial mutations. It is thus possible to answer

questions like "How such a place will evolve?"

"What will be the function of a given place at a

particular time?" etc.

4.1 Fuzzy Object

This work is based on a fuzzy representation

introducing various degrees of membership of an

object to the time, to its function, and to the space as

mentioned previously. Therefore, for each object o

our system may return a triplet of fuzzy sets:

<FTime

ol, FSpaceol, FFunctionol> with respectively

the membership functions <fTimeol, fSpaceol,

fFunction

ol>.

At a state m, one may evaluate the confidence in

the possible presence of the object, called likelihood

coefficient LK

m with the timestamp ti, function Fk,

the shape (geometry) Sj using the Zadeh t-norm:

LKm=

min(fTimeoj(ti),fFunctionoj(Fk),fSpaceoj(Sj))

(1)

4.2 Soft Urban Trajectories

In our context, in place of classic triplet, we have

triplet of fuzzy sets. Therefore for each object ol, we

may obtain at each ti a set of triplet <t

i, Sj, Fk> from

the domain values <FTimeol, FSpaceol, FFunctionol>.

Each triplet may represent a possible state m of the

object. We can, thus, compute for each state m the

likelihood coefficient LK

m. Thus each state of the

model will be described by the quad <time, space,

function, likelihood coefficient>, i.e. <t

i, Sj, Fk,

LKm>.

We have valued hypotheses of object presence

over time. Therefore, the use of temporal logic is

important in order to obtain the possible object

mutations and the possible city model evolution. The

main goal is to compute also a confidence index we

may have in the global evolution model.

Figure 2: K transitions of an urban object over time with

likelihood coefficient.

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Figure 2 illustrates an example of the different

possible transitions of an urban object from the

instant t

i+1 to tn. At the instant ti+1, the likelihood

coefficient LKm is assigned to the first state which

has the function Fk+1 and which will take place at the

space S

j+1. At the instant ti+2, two possible transitions

are derived from the previous state leading to two

possible states. LK

m+1 is assigned to the first state

with the parameters <t

i+2, Sj+2, Fk+2> and LKm+2 is

assigned to the second state <ti+2, Sj+3, Fk+3>, and so

on.

4.3 New Operators for Reasoning

in Our Framework

By proposing a modal temporal logic, we aim to

define in space the points that form the trajectory of

the city at an instant t

i+1. In order to deal with the

previous aspect (modality, fuzziness), we define 3

thresholds: the α-mean, the α-min and the α-max of

the set of confidence degree LK

We propose 4 new modal operators that we will

deal with in addition to the classic CTL operators.

At t

∃

<ti, Sj, Fk, LKm> such as LKm < α-mean (2)

∃

<ti, Sj, Fk, LKm> such as LKm ≥ α-mean (3)

∃

<ti, Sj, Fk, LKm> such as LKm = α-min (4)

∃

<ti, Sj, Fk, LKm> such as LKm ≥ α-max (5)

N is true when it exists, at a time t

i, a state <ti, Sj, Fk,

m> where the confidence degree is lower than the

minimum of the confidence degree. The evolution

hypothesis <t

i, Sj, Fk > is then rather non credible. If

P is true, then the hypothesis is rather possible.

When Z is true, the hypothesis is impossible. When

T is true, <t

i, Sj, Fk> is credible.

4.4 Model Checking

Then we may combine the CTL and our operators in

order to define an axiomatic that will be checked

using a model-checking in which we may obtain

both a confidence index in the model and a conflict

index (using spatial constraint and logic). We will

adapt the model-checking (Clarke et al., 1986)

during the analyzing step.

Figure 3 illustrates the operational principle of

the model checker that will be developed for the city

evolution verification. The urban object’s model

evolution will be verified based on a set of

specifications. These specifications are in the form

of rules formulated in the modal temporal logic and

obtained through a learning process carried out on

old maps and plans of the city. The application of

the verification algorithm indicates if the model is

safe or not safe. If the model is safe, the evolution

hypothesis is, then non credible, possible or credible.

The evolution hypothesis is returned with the

confidence degree. If the model is not safe that

means that the evolution hypothesis is impossible.

The latter is returned with the confidence degree and

an example of an unsafe usage.

Figure 3: The model checking adapted to our framework.

5 CONCLUSIONS

In this paper, we introduce a new framework for

modelling and reasoning on urban trajectories. Its

main goal is to take into consideration the whole

complexity of urban objects from their definition

(vagueness, imprecision, space-time-function) to

their exploitation (valued temporal and modal logic)

through a unique and complete framework. In order

to model our data and to produce new evolution

scenarii, our approach uses modal, temporal, and

fuzzy logics in a new kind of GIS.

REFERENCES

Burrough, P. A., McDonnell, R., 1998. Principle of

Geographical Information Systems, Oxford University

Press.

Chellas, B. F., 1980. Modal logic, an introduction,

Cambridge University Press.

Clarke, E. M., Emerson, E. A., Sistla, A. P., 1986.

Automatic Verification of Finite-State Concurrent

Systems Using Temporal Logic Specifications, ACM

Transactions on Programming Languages and

Systems, Vol. 8(2), pp. 244–263.

Dubois, D., Prade, H., 2000. Fundamentals of fuzzy sets.

ThroughaFuzzyCTLLogicforModellingUrbanTrajectories-AFrameworkforModellingCityEvolutionfromPastto

Future

517

The Handbooks of Fuzzy Sets Series. Vol. 7. Springer.

Güttin , R. H., B’hlen, M. H., Erwing, M., Jensen, C. S.,

Lorentzos, N. A., Schneider, M., Vazirgiannis, M.

2000. A foundation for representing and quering

moving objects. ACM Transactions on Database

System, Vol 25(1), pp. 1–42.

Peuquet, D. J., 2002. Representations of Space and Time,

The Guilford Press, N.Y.

Pumain, D., Bretagnolle, A. Glisse, B. 2006. Modeling the

future of cities. ECCS’06, In Proceedings of the

European Conference of Complex systems, Oxford,

pp. 25–29.

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