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International Journal of Advances in Science and Technology,
Vol. 2, No.5, 2011
Implication of Approximate Reasoning Using Fuzzy Logic
Supriya Raheja1 and Smita Rajpal1
1
Department of Computer Science & Engg., ITM University, Gurgaon, Haryana, India
supriya.raheja@gmail.com, smita_rajpal@yahoo.co.in
Abstract
Fuzzy logic is able to express any natural phenomenon especially related to human decision making
efficiently using linguistic variables. Fuzzy inference is based on the approximate reasoning. In this
paper fuzzy logic and approximate reasoning are presented. Approximate Reasoning with fuzzy logic
proved its various applications in many fields like artificial intelligence, control systems, and in
many real life applications. As we know approximate reasoning with fuzzy logic is an important
framework for understanding and solving various real life applications, so in this paper various
different applications of fuzzy logic with approximate reasoning in computer science & engineering
are describing.
Keywords: Approximate Reasoning (AR), Fuzzy Logic, Linguistic Variable.
1. Introduction
―The power of our thinking and feeling is much higher than the power of living language. If in turn
we compare the power of a living language with the logical language, then we will find that logic is
even poorer.‖ — Zimmermann [9].
Above quotation conveys that any human natural language processes expressing by mathematical
expressions does not always guarantee exact capturing of the process itself. A human tends to use
words rather than numbers to describe how systems behave. Words are a form of imprecise
information appropriate to communication. The conventional approaches to knowledge representation
lack the means for representing the meaning of vague and incompletely understood concepts. So, if any
system, especially related to human decision making expressed linguistically chance of appropriate
capturing of the system increases. Fuzzy logic is such logic, which is based on approximate reasoning
and also capable to express linguistically to capture the vagueness of human mind. Fuzzy logic
simplifies the task of translation between human reasoning and operation of digital computers. Such
translations are made by providing the membership function that defines linguistic values such as
―very", ―highly", ―young", "like", ―healthy". This is particularly important in expert systems, where the
instructions to be programmed are essentially rules of thumb. Fuzzy logic with approximate reasoning
can be applied to the areas which involve human decision making like supervision, monitoring,
planning, scheduling etc.
In this paper author includes the general concept of approximate reasoning and finally author describes
the different applications of approximate reasoning.
2. Preliminaries
Zadeh (1965) introduced fuzzy logic a remarkable venture over the classical logic. Fuzzy
deals with reasoning that is fixed or approximate rather than fixed and exact [10]. In contrast
with "crisp logic", where binary sets have two-valued logic: true or false, fuzzy logic variables
has various ―shades‖ of truth and have a truth value that ranges in degree between 0 and 1.
Fuzzy logic has been extended to handle the concept of partial truth, where the truth value may
range between completely true and completely false. Furthermore, when linguistic variables are
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International Journal of Advances in Science and Technology,
Vol. 2, No.5, 2011
used, these degrees may be managed by specific functions. We can say fuzzy logic provides a
simple way to arrive at a definite conclusion based upon vague, ambiguous, imprecise, noisy, or
missing input information. Fuzzy logic is usually classified into two kinds - one that may be
placed under many-valued logic and the other called ―linguistic logic‖ – that assumes the
possible existence of different kind of truth values e.g. true , more or less true, quite true...
In 1979 Zadeh [5], father of fuzzy logic introduced the theory of approximate reasoning. This
theory provides a powerful framework for reasoning in the face of imprecise and uncertain
information. Prof. Zadeh says, "In its narrow sense, fuzzy logic is logic of approximate
reasoning which may be viewed as a generalization and extension of multi -valued logic‖.
3. Approximate Reasoning
There are two variants of approximate reasoning with the compositional rule of inference:
1) The interpolation method [1]
2) The implication method [2]
Given universes X and Y, a fuzzy set A on X and a fuzzy relation R on X×Y, the result of the
compositional rule of inference is the fuzzy set B on Y defined by
∀ y∈ Y: B(y) = supx min (A(x),R(x,y))
The relation R is determined by a set of N fuzzy rules of the form
IF X = Ai THEN Y = Bi
where Ai and Bi are fuzzy sets on X and Y respectively, for 1 ≤ i ≤ N. Each rule determines a relation
Ri, and the N relations Ri together determine the relation R. In Mamdani's interpolation method, Ri is
given by
∀ x∈ X, ∀ y∈ Y: Ri(x,y) = T(Ai(x),Bi(y))
where T is a t-norm. A t-norm is a function of type [0,1]×[0,1]⇒[0,1] which satisfies T(0,0) = T(0,1) =
T(1,0) = 0, T(1,1) =1. The relation R is defined to be the union of all Ri :∀ x∈ X, ∀ y∈ Y.
R(x,y) = maxi T(Ai(x),Bi(y))
The rule structure is based on implications like ―if A is the scenario then B is the action‖. Now, for the
decision maker if the scenario is A, he will take the action B. If the prevailing scenario is in between
any two defined in the rule base, the action will be in between the two corresponding actions [1, 5].The
process of inference which is used by linguistic approach is called fuzzy implication. Fuzzy inference
is based on approximate reasoning. According to Zadeh, fuzzy inference is ―the process or processes
by which a possibly imprecise conclusion is deduced from a collection of imprecise premises‖ [6].The
more prevalent ―fuzzy logic‖ however, is one in which rules of inference such as Modus Ponens (if x
holds and x implies y, then y), Modus Tollens (if not y holds and x implies y,then not x) and
Hypothetical Syllogism (if x implies y, y implies z, then x implies z) are ―fuzzified‖. The intention is to
complete arguments with premises like ―if the voltage is low then the speed is low‖, and ―the voltage is
more or less low‖. The inference process from imprecise or vague premises is becoming more and
more important for knowledge-based systems, especially for fuzzy expert systems. In the implication
method, Ri is given by
∀ x∈ X, ∀ y∈ Y: Ri(x,y) = J(Ai(x),Bi(y))
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International Journal of Advances in Science and Technology,
Vol. 2, No.5, 2011
where J is an implication operator. An implication operator is a function of type [0,1]×[0,1]⇒[0,1]
which satisfies J(0,0) = J(0,1) = J(1,1) = 1, J(1,0) = 0. The relation R is defined to be the intersection of
all Ri:
∀ x∈ X, ∀ y∈ Y: R(x,y) = mini J(Ai(x),Bi(y))
4. Applications of AR
Expressive logical knowledge representation and reasoning tasks are of high computational complexity
and therefore scale badly to large knowledge bases. Since expressive reasoning is a necessity for
advanced artificial intelligence applications, theoreticians and practitioners are therefore faced with a
scalability problem of fundamental nature. Approximate reasoning allows dealing with time critical
reasoning systems in a controlled and logically sound fashion. It is based on controlled alterations of
the inference relation in order to achieve lower reasoning complexities. Consequently, the resulting
systems are much more efficient, but at the price of unsoundness or incompleteness, but in a well
understood manner which allows arriving at correctness estimates, or at algorithms which subsequently
correct initially given answers if more time is available.
3.1 AR in Intelligent systems
AR techniques play an important role in intelligent systems due to its imprecise model of the
environment, presence of stochastic events, limited computational resources and noisy sensing devices.
When a system is composed of various modules that produce approximate results, important
methodological questions arise regarding the management of uncertainty & precision. How can the
performance of approximate components be described? How does the quality of output depend on the
precision of input? How does the execution of system be controlled to maximize the overall
performance?
Approximate reasoning based mechanism is the answer for all the questions mentioned above. Meta-
level control of approximate reasoning is used as a valuable mechanism to trade off decision quality for
deliberation costs. This mechanism allows an intelligent agent to control the level of precision of each
component and maximize the overall performance. In this we mentioned two applications based on
meta-control mechanism: Mobile robot navigation, identification of defective components in a system.
The main aspects of this mechanism include:
1) Simplifying the design and implementation of complex intelligent systems by separating the
design of the performance components from the optimization of the performance.
2) Mechanizing the composition process and monitoring process.
3) Constructing machine independent intelligent systems that can automatically adjust resource
allocation to yield optimal performance.
3.2 AR for the Semantic Web
Scalability is a very important requirement for Semantic Web techniques to be usable in real world
applications. With the increasing use of expressive ontologies in the Semantic Web and enterprise
knowledge management, it is critical to develop scalable and efficient ontology reasoning techniques
that can properly cope with very high data volumes in the Web. Author developed methods and tools to
help overcome the high computational complexity faced by state of the art ontology reasoning tools.
This method meets scalability requirements of real world applications and provides practical reasoning
capability as well as high query performance. More precisely, approximation and discrete-anytime
methods supports the scalability of the ontology reasoning system KAON2 [6]. The scalable
techniques being developed result in a system, called Approximate Reasoning Broker with an
automated benchmarking tool. In short, due to a flexible SOA-based architecture [7], resulting system
shall allow for a practable, scalable reasoning with expressive ontologies in a heterogeneous and
distributed environment, but in a controlled and well-understood way.
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3.3 AR in Transportation Engineering
Transportation engineering, when concentrated on road transportation is concerned about construction
and maintenance of roads and movement of vehicles therein. The second part, i.e., movement of
vehicles requires scheduling and routing of vehicles and a lot of control mechanisms. These all depend
on availability of resources (like, number of vehicles available, whether proper electronic devices are
available for signal controlling at intersection or not etc.) and the demand of people for transportation.
For a particular resource availability condition, the task is to effort for catering the demand efficiently.
For this it is of utmost necessity to have proper control mechanism and scheduling and routing strategy.
Most importantly, it is necessary to understand the motion of vehicles on roads, i.e., the traffic flow
theory. Various models have been proposed to understand traffic flow using various methods till now.
On the operation side, lots of problems are encountered every day, namely, delay, traffic jam, accidents
and so on. Studies have been conducted on analysis of delay and queue on roads and also on reasons
and mitigations of accidents.
There are some areas in transportation engineering where imprecise perception involves. One of them
is modeling the behavior of drivers (which is a human thought process). Modeling driver behavior
involves to predict how a driver of a particular vehicle responds (in terms acceleration) according to the
behavior of other drivers (in terms acceleration) and the environment. This environment may be static
(roadway features) as well as dynamic (other vehicles). So, fuzzy logic has been applied into modeling
driver behavior, which is imprecise in nature. Fuzzy logic has also been applied into transportation
planning for deciding mode choice and route choice. Control devices (where the perception of the
condition is not clear and the consequence of decision is not known) are suitable places for
applicability of fuzzy logic. Fuzzy logic has been applied into traffic modeling, transportation
planning, traffic control and some more areas in transportation engineering. In the modeling and
planning side fuzzy inference system has been developed to capture the imprecise decision making of
drivers; whereas, in the control and application side, the concept of fuzzy logic has been implemented
into machines.
3.4 AR to Object-Oriented design methods
In applications of approximate reasoning, one usually encodes fuzzy information into a set of fuzzy
inference rules, and performs fuzzy inferences with either fuzzy or crisp input. This is the case where
the rules are crisp, and the input consists of uncertain decisions. In this author introduces the concept of
fuzzy Booleans, and show that the use of fuzzy Booleans in Zadeh's compositional rule of inference
approach to approximate reasoning [4] is particularly suited to this kind of application can be derived
from the original set of rules. Approximate reasoning using a set of fuzzy rules with crisp antecedents
and crisp consequents is equivalent with application of Zadeh's extension principle [3, 5], irrespective
whether the interpolation [1] or the implication method [2] is adopted, and irrespective of the particular
t-norm or implication operator. As a consequence, in some cases the approximate reasoning process
can be replaced by application of Zadeh's extension principle, leading to a significant increase in
efficiency. Approximated reasoning with fuzzy Booleans can be applied in the field of object-oriented
design methods.
5. Conclusion
Various real life applications of computer science and engineering like intelligent systems, semantic
web, object oriented designing using approximate reasoning has been studied in this paper. In all of
these applications approximate reasoning has proved to be more successful or beneficial in terms of
overall performance, scalability and efficiency over the other conventional reasoning methods used in
the computer science and engineering. In future, I will implement approximate reasoning approach to
various other applications of computer science & engineering.
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Vol. 2, No.5, 2011
References
[1] E.H. Mamdani and S. Assilian, ―An experiment in linguistic synthesis with a fuzzy logic
controller‖, Intern. Journal of Man- Machine Studies 7, 1–13, 1975.
[2] G.J. Klir and B. Yuan, ―Fuzzy sets and fuzzy logic‖, theory and applications, Prentice Hall, 1995.
[3] R.R. Yager, ―A characterization of the extension principle‖, Fuzzy Sets and Systems 18, 205-217,
1986.
[4] L. Zadeh, ―Outline of a New Approach to the Analysis of Complex Systems and Decision
Processes‖, IEEE Trans. on Systems, Man and Cybernetics 3, 28-44, 1973.
[5] L. Zadeh, ―The concept of a linguistic variable and its application to approximate reasoning‖.
Information Sciences 8, 199-251, 1975.
[6] Motik, B., ―Reasoning in Description Logics using Resolution and Deductive Databases‖, PhD
thesis, University at Karlsruhe, 2006.
[7] Tuvshintur Tserendorj, Sebastian Rudolph, M.K., Hitzler, ―Approximate owl-reasoning with
screech‖ Technical Report, http://logic.aifb. uni-karlsruhe.de/screech, 2007.
[8] Christer Carlsson, ―Uncertainty modelling and Approximate reasoning‖, IEEE, 1060-3425/98,
1998.
[9] Zimmermann, H.J., ―Fuzzy set theory and its applications‖, Fourth edition, Kluwer Academic
Publishers, Boston/Dordrecht/London, 2001.
[10] Zadeh, L.A., ―Fuzzy sets‖, Information and Control, vol. 8, pp 338–353, 1965.
Authors Profile
Supriya Raheja, ITM University, pursuing her PhD in Computer Science from Banasthali University. She had
done her engineering from Hindu college of Engineering, Sonepat and masters from Guru Jambeshwar University
of Science and Technology, Hisar. She is working as a Reviewer/Committee member of various International
Journals and Conferences. Her total Research publications are seven.
Dr. Smita Rajpal, ITM University, completed her PhD in Computer Engineering. She has a total work experience
of 11 years. She is specialized in TOC, Compiler Design, Soft Computing and RDBMS. She is a Java certified
professional. She is working as an Editorial Board Member / Reviewer/Committee member of various
International Journals and Conferences. She is an active member of IEEE. Her biography is a part of Marquis
who’s who in the world, 2010.Her total Research publications are 17 and book chapter’s-5.She has published three
books.
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