THE APPLICATION OF LINGUISTIC FUZZY MODELS
TO THE ANALYSIS OF ARCHITECTURAL WORK OF
Technical University in Prague, Faculty of Architecture, Cabinet of
Architectural Modelling, Prague, Czech Republic
Abstract. The utilization of fuzzy models as a new research tool in
the history of architecture is demonstrated in this paper. Namely, the
models are used to explore Adolf Loos's villas. Fuzzy expert systems
are used to analyse the principles of Adolf Loos's work. Principles
applied by A. Loos in villa design are expressed in the form of fuzzy
rule bases. The formulation of rules results from the analysis of
specific villa designs and draws on theoretical concepts pertaining to
Loos's work -- in particular, on his designing process and the concept
of a Raumplan. Further, the concepts of the theory of architecture
related to a general procedure of construction design are utilized. The
article also gives examples on how to employ fuzzy sets when
describing a building and its characteristics.
Keywords. Fuzzy models; expert systems; architecture
1. Linguistic fuzzy modelling in architecture
The fuzzy set theory allows for mathematical processing of vague
information (imprecise values, inexact relations). In a similar manner, a
natural language fluently handles inexact notions. In proposing the fuzzy
sets theory, Lotfi Zadeh was inspired by usefulness of natural language
(Zadeh, 1965; Zadeh, 1983; Dubois and Prade, 2000). In the history of
architecture, the common rules characterizing works of a famous architect
are often described verbally in literature. If the verbal description is
sufficiently formalized, its meaning can be modelled mathematically by
fuzzy sets and the vague knowledge concerning the architect can be stored in
Fuzzy expert systems that are used in this paper to model the vague
general principles of Adolf Loos’s architectural work are representatives of
2 Z. TALAŠOVÁ
so called linguistic fuzzy models. The linguistic fuzzy models are models
with two levels of description: the first level is linguistic, comprehensible to
a human; the second one, expressed in terms of the fuzzy sets theory, is
mathematical, intelligible to a computer. These models work with linguistic
(fuzzy) variables and linguistically defined functions – fuzzy rule bases. For
a given input, output values are calculated with the help of a rule base
through an approximate reasoning algorithm.
Linguistic variables (e.g. size, shape) are variables whose values are
words (small, roughly squared), and meanings of the words are modelled by
fuzzy sets - usually by fuzzy numbers defined on given closed intervals of
real numbers. In architectural applications, say if a location of an object is to
be described (e.g. that an object is placed in the upper left corner), more
general linguistic variables whose values are modelled by fuzzy vectors
defined on compact sets of real vectors can also be used. The utilization of
linguistic variables for expressing multi-dimensional graphical information
in architecture represents a new development in the linguistic fuzzy
A linguistic function expressed by a fuzzy rule base describes
relationships among linguistic variables in the following way:
If X1 is A1,1 and … and Xn is A1,n, then Y1 is B1,1 and … and Ym is B1,m.
If X1 is Ak,1 and … and Xn is Ak,n, then Y1 is Bk,1 and … and Ym is Bk,m.
Variables on the left-hand side of the formula represent inputs,
conditions, and premises (e.g. desired size of the villa, characteristics of the
plot, location). Variables on the right-hand side express outputs,
consequences, and conclusions (size and shape of the ground plan, shape of
the villa, kind of its facade). The fuzzy rule base is modelled mathematically
by a fuzzy relation.
For the given crisp or fuzzy inputs, the corresponding fuzzy outputs are
calculated through the Mamdani inference algorithm of approximate
reasoning. The output fuzzy data can be defuzzified by the centre-of-gravity
2. Architectural work of Adolf Loos, the Raumplan principle
Adolf Loos, together with Le Corbusier and Miese van der Rohe, ranks
among the most prominent architects of the first half of the 20th century.
From the history of architecture point of view, the most important are his
projects of villas, Müller’s villa in Prague being a masterpiece (Risselada,
THE APPLICATION OF LINGUISTIC FUZZY MODELS... 3
1988). When designing his villas, Adolf Loos applied his Raumplan (i.e.
Space Plan) principle. The principle postulates that no floor plans, no
vertical sections and no facades be designed. According to the principle,
particular spaces, whose sizes and heights have been determined from their
functions, should be designed and located on different height levels in the
The rules of Raumplan can be derived from the actual designs of Loos’s
villas. A villa by Loos is always a building of simple cubic shape with a
vertical structure and four functional levels. The daytime zone is strictly
separated from the night-time zone. There is a difference between the left
and right sides of the building. The front facade is compact and public; the
back facade is terraced and private. The typical feature is centrifugal
structuring of space in the house, with furniture standing at the sides, the
centre being left unobstructed. Various niches and different floor levels are
also typical for these villas.
Adolf Loos himself considered it difficult to photograph his villas, but
verbal description is still possible. Thus, his villas might serve as suitable
objects for an attempt to derive a system of fuzzy rule bases that would
describe the common features of architectural work of this architect.
3. Application of linguistic fuzzy models to the analysis of Adolf Loos’s
In my doctoral thesis a fuzzy model containing principles governing Adolf
Loos’s design of villas will be presented. The fuzzy model consists of
several mutually interconnected fuzzy rule bases and is implemented in
MATLAB. The basic structure of the model (the relationship among
particular rule bases, the choice of input and output variables in each of the
rule bases) was conceived as a sequence of common steps in the design
process (starting with client’s requests and characteristics of the given plot
and location, continuing with decisions about the main parts of the villa, then
about its more detailed parts etc.). The structure of the fuzzy model also
reflects the Raumplan principle – the basic spaces of the house (hall, dining
room, kitchen, and other rooms) are dealt with rather separately. Also
solutions of the exterior and interior of the villa are rather independent. The
particular rules of any fuzzy rule base are derived directly from Loos’s
villas; the designed villas represent fuzzy data from which to determine the
The aim of the fuzzy expert system is not only to provide a practical
support to an architect in his/her design work. The research presented here
aims primarily at the development of new tools and methods for the theory
of architecture. However, it would be possible to organize an experiment
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where a fuzzy model would help an architect in designing a villa in a
Loosian style. Furthermore, the fuzzy model could be tested and tuned by a
similar kind of experiments.
The fuzzy expert system when used to describe the architectural work of
an architect has two main advantages: first, verbal description by fuzzy rule
bases is clear and well-arranged; second, the meaning of the verbally
expressed knowledge can be modelled mathematically, stored in a computer,
and used in experimentation.
To the best of my knowledge, fuzzy logic has not yet been applied in this
manner within the domain of architectural design and theory by other
authors. My previous results were published in Talašová & Achten (2009).
Dubois, D. and Prade, H. (eds.): 2000, Fundamentals of fuzzy sets, Kluwer Academic
Publishers, Boston, London, Dordrecht.
Risselada, M.: 1988, Raumplan versus Plan Libre: Adolf Loos and Le Corbusier, 1919-1930,
Delft University Press, Delft.
Talašová Z., Achten, H.: 2009, Fuzzy rule bases as a tool of the history of architecture:
application to the analysis of villas designed by Adolf Loos. Computation: The New
Realm of Architectural Design - Proceedings of 27th eCAADe Conference, Istanbul,
Zadeh, L. A.: 1965, Fuzzy sets. Inform. Contr. 8(3), 338-353.
Zadeh, L. A.: 1983, Commonsense knowledge representation based on fuzzy logic.
Computer. 16(10), 61-65.