SE 2FA3: Discrete Mathematics and Logic II. Teaching Assistants: Yasmine Sharoda,

Transcription

1 SE 2FA3: Discrete Mathematics and Logic II Instructor: Dr. Ryszard Janicki, ITB 217, tel: ext: 23919, Teaching Assistants: Yasmine Sharoda, Course website: Lectures: Monday: 12:30-13:20, Tuesday: 13:30-14:20, Thursday: 12:30-13:20, in T13/125 Tutorials: Wednesday: 16:30-17:20 in BSB/119, starts Wednesday January 14. Office hours: to be arranged, see the website.

2 Course Outline (Tentative): 1. Formal Logic: Formal logical systems, constructive logics 2. Predicate Calculus: Types, syntax and interpretation of quantification, rules about quantification, manipulating ranges, universal quantification, existential quantification, English to predicate logic 3. Predicates and programming: Specification of programs, reasoning about the assignment statement, calculating parts of assignments, conditional statements and expressions 4. Mathematical induction: The principle of induction, course-of-values induction, induction and well-founded sets, the correctness of loops, proof calculus for partial correctness, proof calculus for total correctness 5. Recursion: Recursive definitions, using recursion to solve problems, recursion as a programming technique 6. Automata and regular sets: Finite automata, Regular sets, non-deterministic finite automata, pattern matching, regular expressions and finite automata, limitations of finite automata, pumping lemma, DFA state minimization, Myhill-Nerode theorem 7. Pushdown automata and context-free languages: Context-free grammars and languages, normal forms, pushdown automata, deterministic pushdown automata, parsing 8. Turing machines and effective computability: Turing machine, Church s thesis, equivalent models to Turing machine, decidable and undecidable problems

3 Prerequisite: See Software Engineering Program. Texts: 1. David Gries and Fred B. Schneider, A Logical Approach to Discrete Mathematics, Springer 1993 (it will be used for topics 1-5). 2. Dexter C. Kozen. Automata and Computability, Springer 1997 (it will be used for topics 6-9). The course may not always follow text-books closely. Often only blackboard will be used. Lecture Notes will be on the website a few days after a class. Evaluation: There will be a three hours (one double sided cheat sheet will be allowed) final examination (54%), 1 hour midterm test (18%, closed book) and four assignments (4H7=28%). Detailed grading scheme: Grade = 0.54Hexam Hmidterm H(assg1+assg2+assg3+assg4) Late assignments will not be accepted because solutions will be posted on the website a day after the due day. Although you may discuss the general concept of the course material with your classmates, your assignment must be your individual effort. The exact date of the midterm will be announced soon

4 Discrete Mathematics and Logic II. Introduction SFWR ENG 2FA3 Ryszard Janicki Winter 2014 Ryszard Janicki Discrete Mathematics and Logic II. Introduction 1 / 16

5 General Introduction Why software Engineers need mathematics? Formal specication of systems Formal verication of systems Understand the various models of computation and their limitations... Ryszard Janicki Discrete Mathematics and Logic II. Introduction 2 / 16

6 General Introduction Example (Shut o the pumps if the water level is above 100 meters for 4 seconds.) There are several reasonable interpretations for this sentence. 1 Shut o the pumps if the mean water level over the past 4 seconds was above 100 meters. [ ( T T 4 WL(t)dt ) ] 4 > Shut o the pumps if the median water level over the past 4 seconds was above 100 meters. (Max [t 4,t](WL(t)) + Min [t 4,t](WL(t))) 2 > Shut o the pumps if the minimum water level over the past 4 seconds was above 100 meters. Min [t 4,t](WL(t)) > 100 Ryszard Janicki Discrete Mathematics and Logic II. Introduction 3 / 16

7 General Introduction Example (Assignments of TV Channels) Television channels are assigned to broadcasting stations by a governmental agency. Obviously, two stations in geographic proximity must get dierent channels, to avoid reception interference. Suppose that the rule has been adopted that stations within 140 miles of each other must have dierent channels. The grid shows the locations of 15 hypothetical stations. Each square is 50 miles on a side. How many channels are required, and how can they be assigned to comply with the rule? B C E N D F G H J L A I K M O Ryszard Janicki Discrete Mathematics and Logic II. Introduction 4 / 16

8 General Introduction Example (A Partial Solution) We build a graph with locations as vertexes and edges connecting locations that are closer than 140 miles. We want a coloured graph such that no edge connects vertexes with the same colour and the number of colours is minimal. C E N B G J D F H L A I M K O Ryszard Janicki Discrete Mathematics and Logic II. Introduction 5 / 16

9 General Introduction Example (A Possible Solution) Colours = {Black, Red, Yellow, Green, Blue, Brown, Magenta} Assignments: 1 Black = {A, G, N, O} 2 Red = {M, F } 3 Yellow = {L, D} 4 Green = {J, C} 5 Brown = {I } 6 Magenta = {H, B} Ryszard Janicki Discrete Mathematics and Logic II. Introduction 6 / 16

10 Distinction between discrete and continuous mathematics Discrete mathematics is the study of mathematical structures that are fundamentally discrete The objects studied are countable sets, such as integers, nite graphs, and formal languages Discrete mathematics has become popular in recent decades because of its applications to computer science Concepts and notations from discrete mathematics are useful to study or describe objects or problems in computer algorithms and programming languages Ryszard Janicki Discrete Mathematics and Logic II. Introduction 7 / 16

11 Distinction between discrete and continuous mathematics Discrete mathematics includes the following topics: Logic - a study of reasoning; Set theory - a study of collections of elements; Number theory - a study of the properties of numbers in general, and integers in particular; Combinatorics - a study of counting; Graph theory - a study of graphs and the algorithms on graphs; Algorithmics - a study of methods of calculation; Computability and complexity theories - dealing with theoretical and practical limitations of algorithms; Algebras etc. Ryszard Janicki Discrete Mathematics and Logic II. Introduction 8 / 16

12 Distinction between discrete and continuous mathematics Continuous mathematics is the study of mathematical structures that are fundamentally continuous The objects studied are uncountable sets, such as reals Continuous mathematics includes the following topics: Analysis - Limits, continuity and dierentiability, Taylor series, Complex numbers, etc. Fourier series Representation of signals - Fourier transforms, Laplace transforms, etc. Ryszard Janicki Discrete Mathematics and Logic II. Introduction 9 / 16

13 Recommendations No mathematics can be made easy, and discrete mathematics is no exception read with pencil in hand and a pad of paper beside you Try to solve the problem before reading its solution you may be able to solve it = you understand the concepts After working on it, you are not able to solve it = you put your nger on what you do not understand You need to pay careful attention to being precise You should make up your own dictionary as you study Justify your answers DO THE EXERCISES Ryszard Janicki Discrete Mathematics and Logic II. Introduction 10 / 16

14 Conventions and notation Conventional Proof of A (B C) = (A B) (A C) We rst show that A (B C) (A B) (A C). If x A (B C), then either x A or x B C. If x A, then certainly x A B and x A C, so x (A B) (A C). On the other hand, if x B C, then x B and x C, so x A B and x A C, so x (A B) (A C). Hence, A (B C) (A B) (A C). Conversely, if y (A B) (A C), then y A B and y A C. We consider two cases: y A and y A. If y A, then y A (B C), and this part is done. If y A, then, since y A B we must have y B. Similarly, since y A C and y A, we have y C. Thus, y B C, and this implies y A (B C). Hence(A B) (A C) A (B C). Ryszard Janicki Discrete Mathematics and Logic II. Introduction 11 / 16

15 Conventions and notation Calculational Proof of A (B C) = (A B) (A C) x A (B C) Denition of x A x B C Denition of x A x B x C Distributivity of over (x A x B) (x A x C) Denition of, twice x (A B) x (A C) Denition of x (A B) (A C) Ryszard Janicki Discrete Mathematics and Logic II. Introduction 12 / 16

16 Conventions and notation The last presentation of the proof is obvious and straightforward Anyone with a little experience in such calculational proofs will have no diculty reproducing them These proofs are rigorous and could be checked by a mechanical proof checker Ryszard Janicki Discrete Mathematics and Logic II. Introduction 13 / 16

17 Conventions and notation expression 0 op 0 hint 0 expression 1 op 1 hint 1 expression 2 expression n where op i, i = 0,, n 1, is a relational operator =, <, >,,,,, like it could be a logical operator = and. Ryszard Janicki Discrete Mathematics and Logic II. Introduction 14 / 16

18 Conventions and notation Example Prove that 2 (5 x 2 2 x + 6 x + X 2 4) = 12 X 2 + 8(X 1) 2 (5 x 2 2 x + 6 x + X 2 4) = Distributivity of * over + & calculus 10 x 2 4 x + 12 x + 2 X 2 8 = Commutativity of + & calculus 12 X x 8 = Distributivity of * over + 12 X (x 1) Ryszard Janicki Discrete Mathematics and Logic II. Introduction 15 / 16

19 Conventions and notation more examples Example 3x + 3 = 0 3x = 3 Add 3 in the two sides of the equation & 3 3 = 0 & 0 is the neutral element for & divide by 3 the two sides of the equation & 3 3 = 1 x = 1 When appropriate, we adopt this calculational way to present proofs. Ryszard Janicki Discrete Mathematics and Logic II. Introduction 16 / 16