One-dimensional and n-dimensional Usage of the Finite State Automata

In this paper we describe teoretical solution and practical usage of a Finite State Automata (FSA) in one and n-dimensional space. We classify approximate pattern matching problems using algebraic structures. Construction of algebra for pattern matching and operations applied to FSA corresponds to the creation of new pattern matching problems. We present several operations and describe some identified properties of the algebra defined this way. We propose a formal description useful for all similar problems to use better the known algorithms. Finally we describe a basic idea for using FSA on ndimensional space. Here we consider transformations of the input strings for weighted finite automata (WFA) in order to obtain dimension transformations which preserve multidimensional similarities. 1 Motivation and Introduction FSA is useful tool for compression, storage and manipulation with data and for (approximate) string matching [4][13][17][18][25]. In this paper we want to describe both common mathematical background and practical usage of FSA in various areas. Described backgrounds and solutions are beneficial in the creation a solid algorithm for indexing several data and for creation of new database storage solutions. 1.1 Motivation and background We start a motivation in one-dimensionally space to understanding better a problem. We present on of simplest applications of FSA. This approach can be liable to unified and later expanded for 2D, 3D and ndimensionally approach. Researchers within the fields of Genetics and Biochemistry have become reliant on computers in their work. The ability to match a deoxyribonucleic acid (DNA) pattern into a long DNA sequence is something for which computers have been used and is an essential part of current research. However, many of the resources used in this process are too expensive and generally use a brute force approach to pattern matching. DNA is represented as being made up of four letters or base pairs, A, C, G and T. A DNA sequence is a onedimensional sequence of these four letters. A DNA pattern is also made up of the four base pairs and is generally between four and sixty base pairs in length. The problem is to find an efficient method to match specific patterns in a DNA sequence, i.e. to find a DNA pattern in a DNA sequence. One of the complications that are faced is that DNA patterns are not always found perfectly in a DNA sequence [1]. The pattern could be split in the sequence by irrelevant and unimportant or mutated data. Biotechnology and pharmaceutical companies are find new DNA patterns and then need to be able to trace this pattern to other humans and possibly animals. There is a need to match the DNA pattern that has been found to the DNA sequences of other people or animals, discovering whether the pattern exists in these other sequences. The pattern matching is the first part of an essential process. This would involve a person searching through a sequence and trying to identify a pattern using the human eye. If the sequence contained ten thousand base pairs, the entire sequence would need to be looked at. If multiple patterns were being searched for, each sequence would have to be scanned in its entirety for each pattern. Thus this procedure meant that only a single DNA pattern could be searched for in a single sequence at any time. This process was very time-consuming and could result in low accuracy because of missed and incorrectly identified patterns. There are a number of conventional string matching algorithms. These algorithms are used to find a certain string in a large body of text. Two of the better known algorithms are the Boyer Moore and the Knuth-MorrisPratt algorithms [2]. These algorithms are used in the problem of detecting the occurrence of a particular substring, called the pattern, in another string, called the text. The conventional string matching algorithms are only able to match patterns, where they exist Proceedings of the Spring Young Researcher's Colloquium On Database and Information Systems SYRCoDIS, Russia, 2006 completely, that is unbroken and with no irrelevant data in between any letters. Approximate string matching extends the conventional string matching [22] ideas described before, but there exists some problems [7]. Here the algorithms need to be able to cope with errors in the strings. This is thus more relevant to the research than conventional matching algorithms. Navarro [22] describes the problem as find a pattern in a text, allowing a limited number of errors in the matching. Where m is the length of the pattern, n is the length of the text and k is the maximum number of errors allowed. Navarro also describes the approximate string matching problem as being applicable to computational biology, signal processing, text retrieval, hand writing recognition, image detection and virus and intrusion detection. But there are a number of different methods that have been developed in order to handle errors and we propose a formal description useful for all these problems for better using of known algorithm.