The Importance of discrete mathematics in computer science


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1 The Importance of discrete mathematics in computer science Exemplified by deductive logic and Boolean algebra By Mathias Schilling BSc Computing, Birkbeck, University of London
2 Preface Discrete mathematics (DM) is a branch of mathematics which deals with mathematical operations within a subset of the real numbers where data is considered more as objects rather than numbers. Characteristically DM provides us problem solving solutions for only distinct and countable quantities which stands clearly in contrast to e.g. the mathematical analysis which encompasses continuous functions applied on uncountable and infinitive quantities. The term discrete (Latin discretum) highlights exactly this difference. Even though some sections of DM like e.g. number or graph theory are relatively old, discrete mathematics has been overshadowed for centuries by the continuous mathematics due to the invention of infinitesimal calculus and its multifaceted applications within natural sciences (particularly in physics). It was only during the 20 century that discrete mathematics has become again more and more important. The new computer based possibilities of data processing have opened a new chapter and pushed the development into the field of discrete mathematics as well as into the computer sciences (CS) itself. But what is that exactly gives discrete mathematics such an importance within the CS? To get a better understanding on how discrete maths concepts are related to computer sciences and why it is relevant for computer engineers to have a solid knowledge in this area we will take a look into mathematical logic as one of major DM topics. Page 1 / 9
3 Mathematical logic in computer sciences History Mathematical logic (ML) emphasises the principles of valid reasoning, inference, validity and soundness and, therefore finds applications in mostly all areas of computing. The study of mathematical proof is specifically important within logic, and has applications to automated theorem proofing and formal verification of software. Before diving deeper into the actual context of ML, we will have a brief look into the historical background of ML, in general and deductive reasoning in detail. Logic is a science of the necessary laws of thought, without which no employment of the understanding and the reason takes place. Immanuel Kant, 1785 The origins of logic can be traced back to the works of the great Greek philosopher Aristotle (384 B.C B.C.) who lived during the golden age of Greek culture. He was one of the first denoted thinkers who believed in the concept of logic with reasoning and arguments as fundamentals for knowledge and covered this thesis in his monumental masterpiece Organon 1. One of his main aims was to establish rules that would enable Greek citizens to distinguish arguments that would formally be valid and correct from those which would be invalid and, therefore wrong. He developed a collection of logical rules for what we currently now as deductive reasoning (syllogismos). Two fundamental laws of logic were established by Aristotle. The Law of NonContradiction and the Law of Excluded Middle. The Law of Noncontradiction 2 establishes that 1 (Epp, 2010) 2 (Gottlieb, 2007) Page 2 / 9
4 no proposition can be true and false at the same time. The Law of the Excluded Middle 3 defines that a proposition must be either true or false. Aristotle s logic is well designed in a way that it is possible to determine the validity of an argument regardless of the matter that is being argued. The engine behind his logic is the syllogism with two premises and one conclusion such as for example: All man are mortal. (Premises 1) Socrates is a man. (Premises 2)  Therefore, Socrates is mortal. (Conclusion) In syllogisms the conclusion is deduced from the premises. If we do not know the conclusion Socrates is mortal than the fact that All man are mortal and Socrates is a man can be used as a proof that indeed Socrates is mortal. This is what is known as deductive argument, which means that the conclusion is a necessary result of the premises. The very beauty of a wellexecuted syllogism is that it cannot guide from true premises to a false conclusion. For instance, in the following argument, formalize it by expressing it as All trees are plants. (True) All willows are trees. (True)  Therefore, all willows are trees. (True) All P s are R s. (True) All Q s are P s. (True)  Therefore, all Q s are R s. (True) 3 (Horn, 2006) Page 3 / 9
5 and notice that it is formally valid and that its validity is completely independent of its statements. Let us have a look onto another, more discussable example to illustrate this validity independency. All trees are rectangular. (False) All cats are trees. (False)  Therefore, all cats are rectangular. (False) Of course the argument is highly questionable, but it is formally valid even its premises and conclusions are false. In the same manner a statement which premises are false and conclusion is true can also be formally valid like for example All trees are flowers. (False) All roses are trees. (False)  Therefore, all roses are flowers. (True) Gottfried Leibniz (1646 A.C A.C.), a german philosopher and mathematician, who was intent upon logic, interpreted Aristotle s thesis by using symbols to mechanize deductive reasoning processes in the same way like reasoning process about numbers and their relationships, was mechanized by algebraic notation 4. Boolean algebra In 1847, over hundred years later, Leibniz idea was finally realized by an english, selftaught mathematicians named George Boole ( ). Boole extended the concept from Leibniz. He created the first algebra of logic and published his thoughts on this topic in his works The Mathematical Analysis of Logic (1847) and An Investigation of the Laws of Thought (1854). He found out that the symbols of logic (today known as the subject of symbolic logic) behave exactly like those in algebra and he used algebraic symbols to express 4 (Epp, 2010) Page 4 / 9
6 logical relations. He claimed that only three operations (AND, OR and NOT) are needed to perform all other logical functions. The terminology Boolean algebra was later suggested by Sheffer in Today George Boole (along with Charles Babbage, who designed the first mechanical calculator, confer Difference Engine) is considered as one of the grandfathers of computing. Applying Boolean algebra But George Boole s thoughts seemed to have been sleeping in a drawer until Claude Shannon ( ), an american mathematician and electric engineer, applied them in While Shannon was studying for his master degree at the Institute of Technology in Massachusetts, he discovered an analogy between the operations of switching devices, as in switching circuits, and the operations of logical connectives. Shannon used Boolean algebra in his thesis (A Symbolic Analysis of Relay and Switching Circuits) to design logic circuits using electromechanical relays. Comparable ideas for application came from the russian logician and theoretician Victor Shestakov in 1935, but unfortunately he did not publish his thoughts until 1941 (Shestakov, V.I. Algebra of Two Poles Schemata). Because Shannon s discovery led us more to our actual question on how Boolean algebra as part of discrete mathematics is applied in modern computer we will have a little deeper look into it. 5 (Huntington, 1933) Page 5 / 9
7 (Figure 1) 6 Imagine figure 1 (a) which shows a simple switch. It is obvious, that current can only flow between the two points if the switch is closed. When the switch is open, current cannot flow. Imagine also that this switch would be part of a circuit like in figure 1 (b). The light bulb in the drawing glows if, and only if, current can flow through the circuit and that only happens if, and only if, the switch is closed. (Figure 2) 7 Figure 2 (a) shows an electric circuit with two switches P and Q in series. The light bulb in this example will only turn on, if both switches are closed. The circuit in figure 2 is just slightly different with the switches in parallel alignment. It is obvious that the light bulb in this example turns only on if at least one of the switches is closed. The behaviour of the two figures is summarised like in the following tables (table 1: switches in series; table 2: switches in parallel): Switches Light bulb Switches Light bulb P Q State P Q State Closed Open Off Closed Open On Closed Closed On Closed Closed Off Open Open Off Open Open On Open Closed Off Open Closed Off 6 Drawing taken from (Epp, 2010) 7 Drawing taken from (Epp, 2010) Page 6 / 9
8 Now if we have a closer look onto the first table, we can determine, that if we are replacing the words Closed with True and Open with False the table becomes the truth table for the logical connective AND. If we are doing the same for the second table we are getting the truth table for OR. From this observation we can conclude that figure 2 (a) corresponds to the logical expression P Q and figure 2 (b) corresponds to P Q and Shannon, was indeed one of the first scientists who successfully applied Boolean algebra. Modern age In modern age, computers are based on the binary system of ones and zeros and are still using George Boole s logic concepts from the middle of the nineteenth century. In order to make a computer carry out arithmetically or logical operations, it uses computer circuits. These are built of logical gates and can be made from either electrical or mechanical switches. Both types work in a quite similar way, with the only difference that a mechanical switch uses lever or wheel positions and electrical switches voltage and current to represent binary digits. A bit sequence can express numbers and letters. A logical gate takes one or more inputs from a circuit and computes based on the underlying Boolean algebra exact one output value. It can also store a result into a so called flipflop (storage unit made from several gates). There are only three different basic types of logical gates AND, OR and NOT. All others such as NAND, NOR, EOR or for example ENOR gates can be made from the basic ones. Page 7 / 9
9 As time has gone by, symbolic logic and Boolean algebra have become the theoretical fundament for many domains in computer science such as artificial intelligence, automata theory, computability, digital logic circuit design and relational database theory. His work was theoretical; he was not an engineer and never actually built a computer or calculating machine. However, Boole s symbolic logic provided the perfect mathematical model for switching theory and for the design of digital circuits. 8 While reviewing the above mentioned works, it has been identified what an essential role deductive logic and Boolean algebra play in the history of computer science. At present time, it is frequent to find logical expressions based on Boolean algebra and deductive logic in form of conditions in the control structures of algorithms and computer programs as well as commands exempli gratia in querying databases. I would like to close this essay by stating that the more we look into a topic, the more questions arise from it and the more chances appear to challenge us in a way that will make us work further in order to improve technology. In line with that, what I have observed while researching, is that most of the time, challenges bring us to discoveries which we would have never even imagined that were possible if we hadn t been put under that pressure or extreme situation that brought us to question the status quo where we stood. I think that, ultimately, technology improvements are the result of challenging great minds, as history has already proven us. 8 (O'Regan, 2012) Page 8 / 9
10 Bibliography Epp, S. (2010). Discrete Mathematics with Applications (4 ed.). Boston: Wadsworth Publishing Co Inc. Gottlieb, P. (2007). Aristotle on Noncontradiction. Stanford Encyclopedia of Philosophy. Retrieved from Horn, L. R. (2006). Contradiction. Stanford Encyclopedia of Philosophy. Retrieved from Huntington, E. V. (1933). New sets of independent postulates for the algebra of logic, with special reference to Whitehead and Russell's Principia mathematica. Retrieved from O'Regan, G. (2012). A Brief History of Computing. Springer. Page 9 / 9
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