The Central Role of Discrete Mathematics in the Context of Information Technology and Communications Eugene Roventa Department of Computer Science and Engineering York University Toronto, Ontario roventa@yorku.ca Tiberiu Spircu Department of Medical Informatics C. Davila Univ. of Medicine and Pharmacy Bucharest, Romania tspircu@univermed-cdgm.ro «Aujourd'hui, on mesure le degré de la culture d'un pays par le niveau mathématique de ses habitants.» (A. Lichnerowicz) Abstract - The importance and the role of discrete mathematics topics in the domain of information technology and communications are discussed. Some connections and applications of these topics with the content of consecrated topics in an information technology program are explored. A special emphasis is given to the applications of classical and fuzzy logic in information technology. I. INTRODUCTION The objective of the Discrete Mathematics topics is to familiarize students with main mathematical concepts and techniques used in information technology. The goal is not to provide detailed mathematical proofs but rather to make students comfortable in an environment where those techniques are increasingly used. Therefore, the choice of presented topics is connected with the possibility of currently applying them in Information Technology and Communication. II. DESCRIPTION The computer has a finite structure. Therefore, its properties are well explained and interpreted in the context of finite mathematics. (Hence the focus will not be on such concepts as limit, derivative, integral, etc.) The fundamental notions that will be studied will use the mathematical properties of finite sets. This branch of mathematics that studies discrete objects, i.e. objects composed of distinct and dispersed elements, is called discrete mathematics [1], [2], [3]. The main topics include: classical and fuzzy logic, sets and combinatorics, functions, relations and matrices, induction, recursion, algorithms, graphs and trees, Boolean algebra, and some of their applications. A laboratory using the software Maple 7 or Mathematica will accompany the course. In each of these domains the studied concepts and their properties are chosen regarding their practical importance in Information Technology and Communication. In accordance with their importance we will aim to provide appropriate motivations and comments. In addition, numerous practical examples are inserted throughout the course to illustrate the current applications in other IT&C courses such as: knowledge management, databases, web search and web programming, modelling and simulations, data structures, computer architecture, etc. (Of course, when a finite structure is large, continuous and analytical concepts could offer better opportunities of study than discrete ones. We do not try to exclude the use of differential and integral calculus!) III. SOME APPLICATIONS IN IT&C The formal language that permeates information technology is the language of Predicate Logic (however, crisp predicate logic is a particular case of fuzzy predicate logic). The connections between logic and computers are a matter of engineering practice at every level of Computer Organization. A computer program is viewed as a Formula. A programming language, with its formal syntax and with the proof rules that define its semantics, is a formal system for which program execution provides only a model. The computers are conceived with the help of logical devices and are programmed in a logical manner. The importance of Discrete Mathematics topics in core areas of information technology is illustrated by: Knowledge Management (Intelligent Information Systems; Expert Systems; Knowledge Representation; Automated Reasoning; Natural Language Processing etc.) Database Systems (Data Models: relational, entityrelationship, etc.; Query-Processing Languages: model theory, completeness, consistency, deduction, etc.; Knowledge-Based Systems with incomplete, imperfect and tentative information requiring probabilistic, modal, possibilistic, belief and fuzzy reasoning; Natural Language Interfaces etc.) Software Engineering (Program Verification; Correctness; Formal Specifications; Formal Design; Requirements, etc.) Programming Languages/Data Structures (Language Design; Object-Oriented Approaches; Parallel Processing; Logic programming etc.) Hardware Topics (Circuit Design/Optimization; Hardware Design Languages; Processor verification; Correctness of OS kernel; Language Implementation of given processors etc.) 978-1-4244-2352-1/08/$25.00 2008 IEEE
A. Some applications of graphs and trees 1) Neural nets (NN) NN are interconnected assemblies of simple processing elements. Usually a NN is a special DAG (directed acyclic graph). The vertices are placed on several totally ordered layers, and arrows are allowed only between a vertex on layer l and a vertex on layer l1. It is supposed that vertices possess very small brains that process values outputted toward the immediate inferior layer. Each arrow has a weight (positive or negative) influencing the output of the source vertex in the processing done by the address vertex. The weights are modified in initial learning steps. 2) Semantic Nets (SN) SN are directed graphs where vertices represent objects and edges/arrows represent relationships between objects. 3) Bayesian (Causal) Networks (BN) BN are directed acyclic graphs where each vertex represents a random variable and the arrows represent conditional dependence. 4) Minimum Spanning Trees MST are special sub-graphs of a weighted and connected graph; they are trees and contain all vertices (e.g. optimization of total cost of an electrical network in a rural area, etc.). 5) Binary Trees In a BT every internal vertex has at most 2 children (and at least one!). Representation of binary trees in the memory of a computer is done after imposing an ordering between the two children of a vertex (if they exist). One of them is the left child, the other is the right child. Mathematical and logical expressions (i.e. programs!) possibly involving parentheses could be represented mostly as binary trees. Internal vertices represent operators; leaves represent values (numbers and/or symbols) see figure 1 for simple examples. 6) Compression of (stored or transmitted) data using Huffman algorithm Let us remind that an extended ASCII character is stored on 8 bits. Suppose we have a long message composed of extended ASCII characters (for example the content of a text file is such a message). If our message has n characters, then to store it we need less than 8n bits! (Each character is stored on 8 bits.) In our message every character could have a different frequency. Suppose we know the (relative) frequencies of all characters f 0, f 1, f 2,, f 255 where f i denotes the frequency of character i. (For example, in English texts the character e appears with frequency approx 0.13). We create a binary tree in which the characters are the leaves. The content of each vertex is the frequency (for internal vertices it is the cumulative frequency of all descendent leaves). The Huffman code replaces each character with the code of the unique path from the root (ending in the leaf representing the character). In a path 00101110 0 means left and 1 means right (see the figure 2 below). If some rules are respected, then a message codified using Huffman code occupies less space and the de-codifying procedure is possible without misinterpretation. 7) Creating a totally ordered list and searching items Suppose we have items identified by a key, and the keys are totally ordered. Suppose we have items identified by a key, and the keys are totally ordered. Examples: (1) items are personal data of Toronto citizens, keys = SIN; (2) items are explanations of words in an encyclopedia, keys = words. It is a bad idea to search for an item by controlling key after key; it is a better idea to check in the middle. To search in an efficient way, the inputted data should be indexed in a binary tree form. When inserting a new item, its key is compared with the key of the root. If it is smaller the left subtree will be considered, else the right sub-tree will be considered. If the left (res. right) sub-tree does not exist, then our new item will be placed as a leaf in this position. 8) The reverse Polish notation The reverse Polish notation (obtained by using post-order traversal) is used, together with an execution stack, to obtain the result of a composed arithmetical expression. For example, (x y) z, written as x y z, is calculated as in figure 3. 9) Decision trees (DT) DT are rooted trees in which, alternatively, internal vertices correspond to a decision res. to a possible result. Leaves represent final states. a 3 b (a 3) b a b 00 01 11 3 b a 3 Fig. 1 Representation of mathematical expressions as trees 100 101 Fig. 2 Creating Huffman codes
y y z z x x x adder sum sum sum multi -plier result Fig. 3 Computing arithmetical expressions B. Boolean Algebra Computers are built up from integrated circuits (CPU, memory, etc.). The integrated circuits are made up of transistors, resistors, capacitors, and other electronic components that are combined into circuits. Transistors can act as amplifiers or switches. In the computers, these transistors switches are combined to form logic gates, which model operations in Boolean algebra. Boolean algebra is the theoretical basis for computer logic design. Transistors are the bricks for implementation. Digital circuits are used to perform arithmetic, to control the movement of data within the computer, to compare values for decision-making, etc. Combinatorial circuits are circuits in which the results of an operation depend only of the present inputs to the operation (e.g. arithmetic circuit). A sequential circuit is dependent on the previous state of an operation as well as the current sets of inputs (e.g. counter circuit, memory circuit, etc.) C. Logic Formal logic is the science that studies by means of mathematical tools the notions, judgments and reasoning as well as the laws of correct reasoning. It investigates how sentences are combined and connected, how theorems can be deduced formally from certain axioms, and what kind of object constitutes a proof. Formal logic presents two parts: the formal language and its intended meaning, which refers to truth-values and is therefore called the truth structure. Specification of a logic is in its syntax and semantic. For all kinds of logics one defines mathematically: Formulas (certain strings of symbols), Inferences (derivations of new formulas from old ones). D. Objectives of studying logic To familiarize students with formal logic which is a systematization of much of what we do daily when we communicate in natural language and when we draw conclusions using informal reasoning; To teach students to use the English language correctly, to understand the notion of formal language, to reason correctly, and to understand a mathematical proof; To provide applications of formal logic in information technology: Switching and logic networks; Introduction to PROLOG a programming language based on logic; Verification of correctness of computer programs. Predicate logic means a system of formal logic which, in order to represent more complex sentences and reasoning, models the properties of connectives and, or, not, implies, equivalent and of quantifiers (both existential and universal). It is characterized by a language (which includes variables, terms, functions, predicates, connectives and quantifiers) and it associates a truth structure. Predicates are statements about objects, their properties and their relationship with others objects. Predicates can have variables as arguments (they can make statements about sets of objects) and return a value of true or false. Remark. In predicate logic or first order predicate calculus the only things that can be quantified are individual variables, and the only things that can be arguments for predicates are terms (i.e., constants, variables or functors with terms as arguments). If one allows formulas to quantify predicates or functors, or if one allows the predicates to take arguments that are predicates or functors, then one has higherorder logic. E.g. the following sentence: There is a function f that is larger than the function g is formalized as follows f x (f(x) g(x)) or f x P(f(x), g(x)). However, even predicate logic is not powerful enough to be able to express the imperfection of information (partial truths). E.g. from Most young men are healthy and It is likely that Alain is a young man in predicate logic we cannot infer that It is likely that Alain is healthy. Fuzzy logic [4] is a logical system that generalizes the classical (two-valued) logic and includes many-valued logic, i.e. a body of concepts, constructs and techniques which relates to modes of reasoning which are approximate rather than exact in order to deal with the imprecision of information. A proposition is viewed as a fuzzy constraint and membership in a set is a matter of degree. Fuzzy logic plays a pivotal role in AI to knowledge representation and to make inferences from information that is imprecise, incomplete, uncertain or partially true.
In order to study imprecise knowledge (not inexact!), we can introduce certain functions of membership. The values of these functions, contained within the interval [0, 1] will indicate the degree of membership of an element to a fuzzy sub-set. To better understand the notion of fuzzy subsets, we consider space X of generic element x. We know that the classical subset family of X, (X) = { }, with its reunion, intersection and complementation operations {P(X),, }, forms a Boolean algebra. We can easily establish the one-to-one correspondence that A (x) = 1, if x A and 0 if x A. ({0,1} X,,, ), where = max, = min, A = 1 A is also a Boolean algebra. Since this oneto-one correspondence commutes with the operations, the two sets are isomorphic, like Boolean algebras. We can therefore identify a usual set A with its characteristic function A. The fuzzy subsets of space X are in one-to-one correspondence with the richer category of functions { : X [0,1]} = [0,1] X, which are called membership functions. Definition. A fuzzy subset of X is a subset F X [0, 1], such as: (i) pr ox (F) = X; (ii) (x,y 1 ) F and (x,y 2 ) F y 1 = y 2 where pr ox is the projection along the axis 0x. Therefore, F [0, 1] X, x F(x) [0,1]. We also know that F = (x, F(x)), where F is called the membership function. Remarks. (1) The theory of fuzzy subsets is a generalization of the usual sets theory (conventional or classical). (2) The notion of membership function is crucial to this whole theory. The construction of membership functions is subjective (for instance, through taking a survey of expert opinions) and reflects the context within which the concrete problem is being studied. E.g. the set of real numbers much larger than 1 can be given by A(x) = {x R x 1} = 0, x 1 and = (x 1)/x, otherwise. By discretizing on N, we get: A = {(1, 0), (2, 0.5), (3, 0.6), (4, 0.7), }. The precise proposition A = John is between 20 and 25 years old is A(x) = 1, x [20, 25] and A(x) = 0, x [20, 25]. The imprecise proposition F = the young men category can be given by F(x) = 1, x 25 and (1 (x 25) 2 ) 1, otherwise. Remark. It is important to note the net difference between the theory of probability and the mathematical techniques that operate with fuzzy concepts. The problematic random element, for example, results from incertitude as to its belonging or non-belonging to a classical set, whereas a fuzzy phenomenon typically shows the existence of various degrees of belonging. The notion of belonging then does not play the role here that it has traditionally played in the theories based on classical sets. It makes no sense, where a fuzzy set F is concerned, to state whether or not x belongs to F. E. Fuzzy Logic (FL) versus Probability Theory (PT) Fuzzy Logic and Probability Theory are distinct (they describe different kinds of uncertainty). They are complementary rather than competitive [5]. PT is the theory of random events (deals with the expectation of a future event, based on something known now). FL deals with the uncertainty resulting from the imprecision of meaning of a concept expressed by a linguistic term in a natural language, such as tall, warm, very warm, rapidly increasing, etc. 1) PT does not support the concept of fuzzy event (e.g. the prices will stabilize in the long run ). 2) PT offers no techniques for dealing with fuzzy quantifiers like many, most, several, few. 3) PT does not provide a system for computing with fuzzy probabilities expressed as likely, unlikely, not very likely. 4) PT is not sufficient expressive as a meaningrepresentation language (where we deal with sentences as it is not likely that there will be a sharp increase in the price of the oil in the near future ). 5) PT cannot handle some kind of approximate reasoning. E.g. Usually (X is not very small) Usually (X is not very large) ---------------------------------- X is? The assimilation of classical (bivalent) logic is a necessary premise for studying other logics, the non-standard logics such as intuitionistic logic, modal logic, temporal logic, multi-valued logic, fuzzy logic, etc. The idea behind fuzzy logic is to associate a fuzzy set (semantic) to any multi-valued (fuzzy) predicate. The value of the membership function of a fuzzy set in an element x of the domain shows the truth of the statement x verifies the predicate. The differences between classical and fuzzy logic include: The fuzzy logic inference is based not only on the structure of the statements, but also on the semantic of every fuzzy predicate. The fuzzy logic inference is approximate, i.e., from imprecise premises it derives imprecise conclusions. Fuzzy logic can be analysed from logical, practical and philosophical point of view. From a logical point of view: 1. Fuzzy logic is not a logic but an entire family, where the connectives ( and, or ) can be generated by t-norms and t-conorms, the implication can be generated by a function, the negation can be generated by another function and a Generalized Modus Ponens is used. Obviously, some of the classical logical laws are lost (e.g. the law of excluded middle, De Morgan laws, etc.). 2. Fuzzy logic is not complete. 3. In fuzzy logic many partially inconsistent propositions can be derived. From a practical point of view: 1. Fuzzy logic deals with fuzzy predicates and thus it can be used to model linguistic human reasoning.
2. Fuzzy logic allows implementing approximate reasoning. Fuzzy logic is a very general knowledge representation model (fuzzy rules are successfully used in expert systems). 3. Most important is the fact that fuzzy logic methods are often computationally simpler and faster than other, including PT, methods. From philosophical point of view: 1. Fuzzy logic depends on the semantic interpretation of every fuzzy predicate. 2. The fuzzy inferences are not fuzzy predicate-based statements but fuzzy set interpreted predicate-based statements. However, Fuzzy Logic is a rigorously mathematical system. Its main capacity is to model linguistic knowledge. Its main limit is that the number of rules grows exponentially with the accuracy level. In a more precise manner, a specialist in information technology needs logical competence for: Showing that a problem can be solved by a computer program; Translating a problem description in a programming language; Arguing that a computer program is correct and efficient; Applying the new techniques of programming which require the mastery of different aspects of logic (e.g. in PROLOG, in knowledge representation within AI, in the database query languages, etc.) The impressive advances in formal logic and the spectacular progresses in the chip technology made possible the development of today's computers. Computers themselves are designed and built with the help of logical (crisp and fuzzy) devices and they are programmed in a logical (crisp and fuzzy) fashion. IV. FINAL REMARKS The main topics covered by a Discrete Mathematics course include: classical and fuzzy logic, sets and combinatorics, functions, relations and matrices, induction, recursion, algorithms, graphs and trees and Boolean algebra. It is important to notice that in the context of information technology, the modelling computation topics (languages and grammars, finite-state machines and Turing machines) are not covered being mainly theoretical developments. A special emphasis is given to fuzzy logic techniques, which generalize crisp logic, model human perceptions (computing with words, precisiated natural language, generalized theory of uncertainty, computational theory of perceptions, etc. [6]) and plays a major role in Soft Computing with numerous practical implementations. The differences between Fuzzy Logic and Probability Theory are stressed. REFERENCES [1] Gersting, J., Mathematical Structures for Computer Science, Freeman & Cie, 2003. [2] Hein, J. L., Discrete Mathematics, Jones and Bartlett Publishers, 1996. [3] Rosen, H. K., Discrete Mathematics, McGraw-Hill, 2003. [4] Reghis, M. Roventa, E., Classical and Fuzzy Concepts in Mathematical Logic and Applications, CRC Press 1998. [5] Zadeh, L. A., Probability Theory and Fuzzy Logic are Complementary rather than Competitive, Technometrics, August 1995, Vol. 37, No. 3. [6] Zadeh, L. A., Toward a Generalized Theory of Uncertainty (GTU) An Outline, Information Sciences, Elsevier, Vol. 172, pp. 1-40, 2005.