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An imperfect string matching experience using

deformed fuzzy automata

J.J. Astrain1, J.R. Garitagoitia1, J. Villadangos2,

F. Fari˜ na1, and J.R. Gonz´ alez de Mend´ ıvil1

1Dpt. Matem´ atica e Inform´ atica

2Dpt. Autom´ atica y Computaci´ on

Universidad P´ ublica de Navarra

Campus de Arrosad´ ıa, 31006 Pamplona, Spain

{josej.astrain,joserra,jesusv,fitxi,mendivil}@unavarra.es

Abstract. This paper presents a string matching experience using de-

formed fuzzy automata for the recognition of imperfect strings. We pro-

pose an algorithm based on a deformed fuzzy automaton that calculates

a similarity value between strings having a non-limited number of edi-

tion errors. Different selections of the fuzzy operators for computing the

deformed fuzzy automaton transitions allows to obtain different string

similarity definitions. The selection of the parameters determining the

deformed fuzzy automaton behavior is obtained via genetic algorithms.

1INTRODUCTION

The applications of pattern recognition based on structural and syntactic meth-

ods have to cope with the problem of recognizing strings of symbols that do

not fit with any one of the defined patterns. The problem is usually resolved by

defining a function that allows measuring the similarity (distance or dissimilar-

ity) between pairs of strings [14].

This approach consists basically in the construction of a dictionary by using

the representatives of the classes. For the recognition of an erroneous string, the

string is compared with every sample in the dictionary. A similarity value (or

distance) is then computed with every one of the samples in terms of the edit

operations (normally insertion, deletion and substitution of a symbol) needed for

the transformation of a string into another one. Finally, the string is classified

in the class whose representative obtains greater similarity (less distance). In

the literature different definitions for string distance, as well as algorithms that

calculate them, are proposed [9,10,1,12,15].

Recognition rates of pattern classification systems are sometimes low because

different steps in the recognition process are considered in isolated and sequential

way. Each step makes a decision at its own level, and it could be erroneous.

Later steps have to do their task taking into account erroneous data, which

invariably affects the recognition rate. For instance, text recognizers usually

contain steps for segmentation, isolated character recognition, word recognition,

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2Astrain, Garitagoitia, Villadangos, Fari˜ na and Gonz´ alez de Mend´ ıvil

sentence recognition, and so on, which act sequentially in such a way that for

the classification of a character it is not taken into account the word in which

the character is located. Thus, in the work of a particular step, the information

level pertaining to a later step is not considered.

In order to improve recognition rates and, at the same time, to keep the work

separate in different steps, the use of fuzzy symbols has been proposed [3,2,7,16]

as an adequate way to represent the ambiguity in the classification of previous

steps. A fuzzy symbol is a fuzzy set defined over the total set of symbols, with its

membership function representing the similarity between the observed symbol

and every symbol in the set. Within the concept of string matching, the problem

of classifying strings presents now a higher complexity because it is necessary to

work with strings containing fuzzy symbols which, in addition to edition errors,

present imprecise information in the fuzzy symbols.

We propose a fuzzy automaton for the classification of strings containing any

amount of errors, that can use different fuzzy operations (t-conorms and t-norms)

[7] implementing different distance (in fact, similarity) definitions. Deforming

the fuzzy automaton [11], it can be performed a string matching accepting as

inputs imperfect strings of fuzzy symbols. Given a particular problem we have

to choose adequately the fuzzy values of the transitions as well as the values of

the t-conorm/t-norm parameters. The problem of selecting such parameters is

an optimization problem. In this paper, we propose tuning the parameters of

the fuzzy automaton using a genetic algorithm [5].

In order to validate our method, an experimental system for the recognition

of texts has been developed. The character classifiers provide fuzzy characters

that permit to represent the ambiguity of the results obtained by the classifiers

of isolated characters [17,3,2]. The contextual processing step of the system

makes the comparison between the strings of fuzzy characters and the strings of

a dictionary representing the lexicographical context. This step is implemented

by the proposed deformed fuzzy automaton. The experimental results show that

the deformed system has a great capacity for recovering errors. This fact leads

to high word recognition rates even in the presence of a high amount of errors.

The rest of the paper is organized as follows. In Section 2 the algorithm

of the deformed fuzzy automaton that computes similarity between strings is

introduced. Section 3 is devoted to show experimental results obtained when the

deformed fuzzy automaton is applied to a problem of text recognition. Section

4 presents the conclusions. Finally, references end the paper.

2STRING SIMILARITY BASED ON DEFORMED

FUZZY AUTOMATA

This section is devoted to the introduction of a deformed fuzzy automaton that

computes string similarity between two strings, the observed and the pattern

strings respectively.

Initially, a finite deterministic automaton M(ω) which recognizes the pattern

string ω is defined. This automaton will not accept the observed string α unless

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IBERAMIA 20023

α = ω. Second, this automaton is modified as it was introduced in [4], obtaining

a fuzzy automaton MF(ω) in order to accept every observed string α providing

a value for the similarity between α and ω. This modification includes the man-

agement of all the possible edit operations (insertion, deletion, and change of a

symbol).

When the symbols of an observed string are processed by a isolated charac-

ter classifier (ICC), a fuzzy symbol representing the ambiguity of each symbol

classification is obtained. The value of this fuzzy symbol represents the degree of

proximity of the pattern symbol to the observed symbol. Then, it is necessary to

work with imperfect strings of fuzzy symbols, that is, strings which may contain

insertion, deletion, and substitution of fuzzy symbols (edition errors).

Third, in order to work with imperfect fuzzy symbols, the fuzzy automaton

is deformed as it was explained in [4]. Then, an algorithm for the computation

of the deformed fuzzy automaton is introduced (see figure 1).

Input:

M(ω) = (Q,Σ,δ,q0,{qn}), Q = {q0,...,qn}, n is the length of

the string ω

the lists µd

xa

and µi

a

∀j : 0...n

˜ α observed string, length m (˜ yk k-th symbol of ˜ α)

Output: MD(ω, ˜ α)

where MD(ω) = (Q,Σ,µ,σ,η, ˜ µ) is the deformed fuzzy automaton

for M(ω)

qj

x, µc

qi−1qi

qi−1qi

, ∀a,x ∈ Σ, ∀i : 1...n

algorithm computation

initialstate;

∀k : 1..m:

transition(k);

ε-closure;

decision

endalgorithm

procedure initialstate

∀i : 1..n:

Q(qi):= 0;

Q(q0):= 1;

ε-closure

endprocedure

procedure decision

MD(ω, ˜ α):= Q(qn)

endprocedure

procedure transition(k)

∀i : 0..n:

C1:=Q(qi) ⊗ (⊕x∈Σ(µdqi

C2:=Q(qi−1) ⊗ (⊕x∈Σ(µcqi−1qi

where a is that δ(qi−1,qi,a) = 1.

Q?(qi):= C1⊕ C2;

∀i : 0..n: Q(qi) := Q?(qi)

endprocedure

procedure ε-closure

∀i : 1..n:

Q(qi):=max(Q(qi),Q(qi−1) ⊗ µiqi−1qi

where a is that δ(qi−1,qi,a) = 1.

endprocedure

x⊗ µ˜ yk(x)));

xa

⊗ µ˜ yk(x)))

a;

a

);

aQ(q−1) = 0.

Fig.1. Algorithm for the deformed fuzzy automaton.

The main program is given by the algorithm computation. The algorithm

starts by building the initial fuzzy state which is given by σ (from the defini-

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4Astrain, Garitagoitia, Villadangos, Fari˜ na and Gonz´ alez de Mend´ ıvil

tion of MD(ω)). The body of the algorithm is a loop in which the procedures

transition and ε-closure are executed for every fuzzy symbol ˜ ykof the ob-

served string ˜ α. These two procedures compute the state transition by a fuzzy

symbol and the empty string, respectively. Finally, the procedure decision, com-

putes the final fuzzy state given by η. There are only two possible transitions to

reach state qi: One from state qi−1, representing a substitute operation (C2in

figure 1), the other one from the same state qi, representing a delete operation

(C1in figure 1). The procedure ε-closure represents insert operations.

Operators ⊗ and ⊕ represent the t-norm and the t-conorm respectively. One

can note that for each replacement of a t-conorm and a t-norm, a different

fuzzy automaton is obtained, and thus, a different similarity operator. In [4]

we show that when the operators ⊕ and ⊗ are replaced by the maximum and

the algebraic product operators respectively, a similarity operator that calculates

(transforming the output value as −log(MF(ω,α)) the Generalized Levenshtein

Distance [9] is obtained. The use of different t-conorms and t-norms may be of

interest for practical problems due to different aggregations they provide.

One can note that the string similarity provided by our deformed fuzzy au-

tomaton has a more flexible behavior because it makes use of a parametric

t-norm/ t-conorm for computing the automata transitions. In the following, we

use the Hamacher t-norm/ t-conorm and we denote γ to the parameter that

Hamacher uses to cover the space (γ ≥ 0) of possible t-conorms and t-norms [7].

Fig.2. (Left) Effects of changing the γ parameter of the Hamacher t-norm/t-conorm

when comparing the observed string α with two pattern strings ω1 and ω2. (Right)

Avoiding the effects of the string length.

In the figure 2 (left), we show the effects of changing the γ parameter of

the Hamacher t-norm/t-conorm when comparing the observed string α with two

pattern strings ω1and ω2varying the value of the Hamacher’s parameter. The

figure shows that the final decision depends not only in the number of edit

operations to transform α into ω1or ω2but in the similarity measure obtained

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IBERAMIA 20025

varying the t-norm/t-conorm. In fact, we could select the t-norm/t-conorm that

gives the desired result.

The Levenshtein Edit Distance is dependant of the length of the strings.

This fact has motivated to researches to introduce Normalized Edit Distances

[10], that consider the relation between the number of errors and the length of

the strings when their similarity is analyzed. A similar effect can be obtained

with our deformed fuzzy automaton by using different values of the t-norm/t-

conorm parameter for each target string length. Consider the example depicted

by figure 2 (right). It shows the results obtained for different values of Hamacher

parameters when recognizing a short input string, α1, with one substitution error

and a long input string α2also with one substitution error. If we need that the

errors in short strings be more significant than in long strings, we could assign to

the short string a value of the Hamacher’s parameter higher than the one to the

long string. For example, selecting γ = 1 for computing the fuzzy automaton of

ω1while γ = 0,2 when computing the fuzzy automaton of ω2, we will obtain the

desired result. The similarity between α2and ω2is higher than the similarity

between α1and ω1.

As conclusion, the membership values of the elementary edit operations and

the value of the parameter of the selected parametric t-norm/t-conorm for each

pattern string as the parameters of the system must to be considered. Those

parameters must to be tuned in order to adjust the system to each concrete

problem.

3EXPERIMENTAL RESULTS

In order to evaluate the proposed method, an experimental system for hand-

printed text recognition has been considered. In the experimental system, the

contextual post processing [6] is implemented by the related deformed fuzzy

automaton. The objective is to analyze the capability of the deformed system to

correct the edition errors that are introduced during text creation (typographical

and lexicographical errors) or by the previous stages of the system (segmentation

and individual classification of the characters). In the experiment, the characters

are classified by using a neural network, a perceptron with three layers trained

by using the Back Propagation Training Algorithm [13]. This neural network has

been trained by using 25 alphabets from 25 different authors. When an input

character is obtained from the input text, the neural network produces a fuzzy

symbol, ˜ y = {(x,µ˜ y(x)) | x ∈ Σ}, where µ˜ y(x) is the neural output unit value

associated with the character x of the alphabet Σ (26 letters).

The input texts are obtained from a large corpus of words known as Brown

Corpus [8]. This corpus is formed by 15 different scope texts extracted from

real life documents. We have taken a portion (about 6000 words) of each text.

These texts are referred as Brown-A, Brown-B, ..., Brown-R. Taking the words

of those texts, we have built-up the dictionaries Dic-A, Dic-B, ..., Dic-R (with

an extension that ranges from 1268 to 2213 words).

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6Astrain, Garitagoitia, Villadangos, Fari˜ na and Gonz´ alez de Mend´ ıvil

3.1Parameters of the system

The contextual post processing based on deformed system has been implemented

by using the algorithm introduced in figure 1 modified to work with a set of

words (dictionary), D ⊂ Σ∗. Several parameters of the algorithm must be fixed

for the experiments like the fuzzy operations for computing transitions and the

membership function values associated to transitions. The transitions have been

computed using the Hamacher’s t-conorm and t-norm [7].

We have selected the membership function values associated to transitions

to be independent of symbols and states, so only three values µd, µi, µc are

associated to delete, insert and substitute operation respectively.

In order to determine the values for the three edit operations and the γ

parameter, we use a genetic algorithm [5]. The genetic algorithm can be for-

malized as a function, GA(O,D,MD,{µ},γ), where O is the set of observed

strings of fuzzy symbols; D the set of patterns (dictionary); MD the deformed

fuzzy automaton; {µd, µi, µc} the membership values of the transitions and γ

the parameter of the parametric t-conorm/t-norm. The goal of the genetic algo-

rithm is to determinate the parameter values that minimize the mean quadratic

error of the system: J(O) = (1/2)?

ity of α with one of the patterns. As ˜ α is associated to a unique pattern in the

training set O, the desired output is a vector target(˜ α) = [a1,...,ai,...,an],

such that target(˜ α)[i] = 1 if and only if ˜ α is an element of the class ωi; in other

case, target(˜ α)[i] = 0.

In order to evaluate the error of the fuzzy automaton when classifying a given

observed string ˜ α, we are going to consider as metric the euclidean distance:

||target(˜ α) − out(˜ α)||. The selected optimization function will allow to select

those parameters that maximize the similarity between the observed string of

fuzzy symbols, and its associated pattern also maximize the difference between

the observed string and the other patterns.

We have used a genetic algorithm with 10 populations, evolving during 300

evolution steps. At each step, the best four populations remain in the system for

the next step of evolution. The other six are generated by selecting pairs of pop-

ulations. The crossing and mutation probabilities are 0.25 and 0.1 respectively.

In order to assure that the deformed system does not increase the number of

erroneous words, we have made experiments without introducing errors. In those

tests, we obtained a 100% of recognition rate. We conclude that the deformed

system does not create errors by itself.

˜ α∈O||target(˜ α)−out(˜ α)||. In that equation

out(˜ α) = [MD(ω1, ˜ α),...,MD(ωn, ˜ α)]. Each component represents the similar-

3.2 Experimental results

In a first experiment, we introduce texts with different error rates in order to

examine the capability of the deformed system to correct those errors. Figure

3 presents the results obtained in this experiment (it shows the average values

obtained after 50 experiments). Besides the error reduction rate (with its stan-

dard deviation), the table also shows the percentages of recognition rate (correct

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IBERAMIA 20027

% Error Rate

44.7

95.61 93.26 84.81 66.93

86.084.9

0.723 0.702 0.575 0.550

95.494.6

91.1 90.0

74.173.5

83.7

45

31.371.687.7

% Recognition Rate

% Error Reduction Rate

Deviation (50 iterations)

% Class. Error Reduction Rate

% Insertion Error Reduction Rate

% Deletion Error Reduction Rate

% Substitution Error Reduction Rate 84.5

Poss (Recognition Rate 99%)

78.862.4

90.7

82.2

67.9

78.2

15

74.7

57.4

51.4

58.7

15

Fig.3. Results obtained for different error rate. Input text: Brown-A. Dictionary: Dic-

A (1725 words).

words), corrected classification errors, corrected insertion errors, corrected dele-

tion errors and corrected substitution errors. The row labelled ‘Poss’ indicates

the interval (when the output of the deformed system is given in the rank level

[17]) where, more than 99% of the corrected words fall into. For example, if

Poss= 4 we assure that more than the 99% of corrected words are between the

first and the fourth position when the output of the automaton is given in the

rank level. Such a rank output becomes useful when a higher level of context

(i.e., syntactic or semantic analysis) is wanted to be used.

In a second experiment we have evaluated the influence of the text domain.

Figure 4 shows the results obtained with the texts Brown-B (Dic-B) to Brown-R

(Dic-R). Again, we introduce an error every 15 characters (segment length). This

corresponds to an error rate from 25.4% to 34.3% depending on the text. We

can see how the best correction rate is obtained with the Brown-E text. This is

so because such a text has the lowest percentage (58.56%) of short words (less

than six characters). Note that short words are difficult to correct because they

have less context. On the other hand, the worst correction rate is obtained with

the Brown-N text. This text has the highest percentage of short words (79.68%).

Finally, we made an experiment intended to evaluate the influence of fuzzy

symbols and the fuzzy operations chosen for the transitions of the automata

(see figure 5). In the first line the results correspond to the use of Levensthein

Distance [9] in order to compare the observed strings with the words in the dic-

tionary. In this case the observed strings are composed of individual characters,

being those that present greater value in the membership function in the cor-

responding fuzzy symbols. The edit operation costs used by the algorithm [15]

implementing the Levensthein Distance have unitary values. Therefore, an ob-

served string will be classified as the word in the dictionary having the smallest

number of edit operations.

As it was proved in [4], when the fuzzy automaton uses as t-conorm the

maximum and as t-norm the algebraic product, it permits to calculate the Lev-

ensthein Distance giving appropriate values to the transitions modelling the edit

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8Astrain, Garitagoitia, Villadangos, Fari˜ na and Gonz´ alez de Mend´ ıvil

ER RRDRR ERR CLerrIerrDerr Cerr Poss

Dic-B (2087 p.) 30.223 95.100 0.249 83.787 95.237 88.435 70.806 81.987

Dic-C (2213 p.) 30.503 95.181 0.241 84.203 94.623 89.455 72.071 81.996

Dic-D (1658 p.) 30.345 95.338 0.288 84.636 95.255 89.033 73.199 82.591

Dic-E (2063 p.) 34.285 96.149 0.245 88.768 96.295 91.601 80.532 88.114

Dic-F (1918 p.) 30.347 95.361 0.227 84.175 94.686 89.168 73.147 83.101

Dic-G (1899 p.) 31.195 95.726 0.261 86.300 96.015 90.564 74.835 84.940

Dic-H (1571 p.) 32.139 96.031 0.212 87.649 96.000 91.890 77.714 86.120

Dic-J (1480 p.) 31.706 95.969 0.237 87.285 96.448 90.039 77.622 86.152

Dic-K (1268 p.) 30.790 96.132 0.253 87.439 96.124 91.055 78.311 85.961

Dic-L (1449 p.) 25.376 94.627 0.281 78.828 92.865 84.465 63.290 75.383

Dic-M (1725 p.) 28.181 94.908 0.243 81.933 93.957 87.968 67.204 79.543

Dic-N (1447 p.) 25.880 94.454 0.246 78.568 92.942 85.939 61.122 74.938

Dic-P (1720 p.) 26.943 94.478 0.235 79.504 93.219 85.723 63.619 76.255

Dic-R (2078 p.) 28.717 95.089 0.265 82.900 94.173 87.971 69.918 80.386

5

4

4

4

5

4

4

4

4

5

5

5

5

5

Fig.4. Results (in percentage) obtained for different text domains. ER: Error Rate;

RR: Recognition Rate; DRR: Standard Deviation of RR for 50 experiences; ERR: Error

Reduction Rate; CLerr: Classification Error Reduction Rate; Ierr: Insertion Error Re-

duction Rate; Derr: Deletion Error Reduction Rate; Cerr: Substitution Error Reduction

Rate; Poss: number of words needed for 99% recognition rate.

operations. In this way, giving values µi= 0.1, µd= 0.1, µc= 0.1 one would

obtain the same results as obtained by using the Levensthein Distance. However,

we can provide a deformed system for this fuzzy automaton to work with strings

of fuzzy characters. Therefore, it is possible to evaluate the influence by taking

into account all the information contained in the fuzzy symbols. These results

are shown in the second line of figure 5. The third line shows the results obtained

with the deformed system using Hamacher’s t-conorm and t-norm together with

the used values for the transitions in previous experiments. This enables us to

compare the influence of the election of fuzzy operations.

In figure 5 the average values obtained through a simulation with one hun-

dred texts are shown. The best results are obtained with the deformed system

method. Differences between Levensthein, max-prod and Hamacher are minor if

the number of errors per word is low (error rate 31.3%), and major if the number

of errors per word is high (error rate 71.6%). Differences between max-prod and

Hamacher are relevant in terms of error reduction rate (near eight points). When

a deformed system is used with a high number of errors per word, RR and ERR

become higher.

4 CONCLUSIONS

In this work we have introduced a fuzzy method that allows recognizing imper-

fect strings of fuzzy symbols. The basic idea is to use a deformed system built

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IBERAMIA 20029

ER = 31.3% ER = 44.7% ER = 71.6%

RRERRRRERR RR ERR

Levenshtein distance 89.811 67.451 84.309 65.742 65.296 51.540

Max-prod (deformed) 93.014 77.516 90.049 77.650 80.597 72.833

Hamacher (deformed) 95.677 86.154 93.449 85.340 86.011 80.463

Fig.5. Influence of fuzzy symbols and the t-conorm/t-norm in recognition rate. Input

text: Brown-A. Dictionary: Dic-A (1725 words). ER: Error Rate; RR: Recognition

Rate; ERR: Error Reduction Rate.

from a fuzzy automaton. The fuzzy automaton is defined in such a way that

its transitions model every possible edit operation. Thus, it allows to compute

the similarity between an observed string and a pattern string. When this fuzzy

automaton is extended by means of a deformed system, the same computation

can be performed over strings of fuzzy symbols.

Our method is quite general. It does not assume a limit in the number of

errors to be handled. Moreover, the deformed system allows to adjust the values

of transition fuzziness and to select different fuzzy operations for computing

the transitions. We wish to emphasize the influence that the fuzzy operations

used when computing the transitions has on the obtained results. We have used

the parametric t-norm/t-conorm proposed by Hamacher [7]. Variations of the

t-norm/t-conorm parameter lead to very different results.

Due to the difficulty to adjust the values of transition fuzziness once selected

the fuzzy operators, we have used a genetic algorithm that obtains good results.

The deformed system has been used as the layer of contextual post processing

in a system of text recognition. Section 3 shows that the deformed system gives

high ratios of error recovery even when the number of edition errors considered

is high.

It has been clearly established that, although usually the string matching

methods are based on discrete mathematics and probabilistic techniques, fuzzy

techniques can also be useful within this area.

Acknowledgments

The authors wish to thank to the enterprise Investigaci´ on y Programas, S.A.

(IPSA) for the contribution to the experimental part and for the financial sup-

port through research grant OTRI-2001,4059 (Universidad P´ ublica de Navarra).

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