OBJECTIVE OF THE RESEARCH The aim of this thesis is as follows: i. We propose to present a critical study of the existing multiset models for DNA and membrane computing. ii. We propose to study membrane computing specifically by way of providing a multiset–based tree model. iii. We also wish to outline constructions of multiset-based biological simulators.

A Study of the Application of Multiset to Membrane Computing

Chapter One

OBJECTIVE OF THE RESEARCH

The aim of this thesis is as follows:
i. We propose to present a critical study of the existing multiset models for DNA and membrane computing.
ii. We propose to study membrane computing specifically by way of providing a multiset–based tree model.
iii. We also wish to outline constructions of multiset-based biological simulators.

CHAPTER TWO
REVIEW OF LITERATURE
In this chapter, we look at some of the various efforts in developing multiset and its application to mathematics, linguistics, statistics, computer science, membrane and molecular (or DNA) computing. Details of the study of some significant applications appear in chapter four; of particular interest being its application to membrane computing.
CELL BIOLOGY
Throughout time, thoughts and ideas of life have evolved, stretching from biogenesis and spontaneous generation to the modern cell theory. In 1824, Rene Dutrochet discovered that “the cell is the fundamental element in the structure of living bodies, forming both animals and plants through juxtaposition.” However, the first sightings of the internal
action of the cell were made by Robert Brown. Schwann created the term “cell theory” and declared that plants consisted of cells. This declaration was made after that of Mathias Schlieden (1804 – 1881) that animals are composed of cells and that living organisms are made up of basic microscopic units called cells. All cells fall into one of the two major classifications: prokaryotes and eukaryotes. The later is more complex and contains nuclear materials. Plants and animals are made of cells and that most of the cell organelles are considered identical, yet they do have their differences (Harrison, 1994).

 SOME HISTORY OF THE DEVELOPMENT OF MULTISETS
The idea of having a repeated element in a set dates back to as far as numbers itself. For example, evidences of representing a number by a collection of tally marks or units are found in the work of the Babylonians in 200 B.C., Egyptians and Greek in 3500 – 1700 B.C. Knuth (1981) notes that enumeration of permutations of a set was known in ancient times and historically the first known document is the Hebrew Book of Creation in 100 A.D., followed by the Indian Classic Anuyogadvarā-sutra in 500 A.D.; and the corresponding result for multisets seems to have appeared first in another Indian Classic
He further notes that Kircher (C. 1650, pp. correctly gave the number of permutations of multiset {m.C, n.D} for several values of m and n. A generalization of the rule for enumerating the permutations of multiset appeared
in Prestet’s Elémens de mathématiques (Paris 1675, 351 – 352), and later in John Walli’s Treatise of Algebra 2 (Oxford 1685, pp. 117 – 118). By exploiting Dominique Floata’s work done in 1965, Knuth present a good number of significant results on multiset permutations (Singh, 2006).

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CHAPTER THREE
FUNDAMENTALS OF MULTISET
In this chapter, we intend to explicate the meaning of multiset, show various ways in which a multiset can be represented. We define operations on multiset, submultisets, similar multisets, ordered pair of two multiset terms, power multists, union intersections and sum of multisets, difference and complementation of multisets and function between multisets. We show that Cantor’s theorem and Schröder–Bernstein’s theorem fail with multisets. Finally, we talk about the Darshowitz–Manna ordering on multisets.
PRELIMINARY
A multiset is a collection of elements in which repetition of elements is allowed. A set is a multiset in which distinct elements occur only once. The copies of an element in a multiset are called indistinguishables. The number of occurrences of an element in a multiset is called its multiplicity. The multiplicity of an element in a multiset contributes to the cardinality of the multiset. That is, the cardinality of a multiset is the sum of the multiplicities of the elements in the multiset. A multiset is finite if the distinct elements of the multiset are finite and every element has finite multiplicity. A multiset is therefore infinite if the distinct elements of the multiset are infinite or some elements have infinite multiplicities (Blizard, 1991).

CHAPTER FOUR
APPLICATION OF MULTISET TO MEMBRANE COMPUTING
In this chapter, we highlight some biological concepts as relating to cells and membrane stru