1. Donghyun (David) Kim Department of Mathematics and Physics North Carolina Central University 1 Chapter 0 Introduction Some slides are in courtesy of Prof. Xinhua Zhuang at Dept. of CECS at University of Missouri-Columbia Spring 2016 CISG 5115 Theory of Computation

2. Spring 2016 CISG 5115 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Overview Fundamental mathematical models of computation what can and cannot be computed? what computational models? how quickly? how much memory? Examples a compiler detecting the presence of infinite loops? a program checking whether or not two programs compute the same function? a program determining the shortest program to compute a given function? 2

3. Spring 2016 CISG 5115 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Motivation and Overview Assess the difficulty of a given problem Measure the quality of a design Two major tasks for the theory of computation devise a universal model of computation characterize problems as solvable, efficiently solvable, simply solvable, etc. Tools for practice grammars: programming languages finite automata: string searching, pattern matching 3

4. Spring 2016 CISG 5115 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Automata, Computability, and Complexity What are the fundamental capabilities and limitations of computers? Automata theory definitions and properties of mathematical models of computation finite automaton, push-down automaton, Turing machine Computability theory distinguish solvable problems from unsolvable ones Complexity theory what makes problems computationally hard? examples: sorting problems vs. scheduling problems games: tic-tac-toe, checker, chess, go 4

5. Spring 2016 CISG 5115 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University History Late 19th to early 20th century questions about the nature of computing in the context of pure mathematics rigor and formalism in mathematical arguments some great mathematicians: Hilbert, Cauchy, Gödel 1930s and 1940s several universal models of computation (Church-Turing thesis) Gödel and Kleen: the theory of partial recursive functions Alonzo Church: lambda calculus Alan Turing: Turing machine 5

6. Spring 2016 CISG 5115 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Recent Development 1950s and 1960s from computability to complexity computing Pioneers: Alan Turing, John von Neumann Stephen Cook and Richard Karp: NP-complete problems Recent years alternative models of computation parallel computing quantum computing DNA computing proof theory randomized computing 6

7. Spring 2016 CISG 5115 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Terminologies 7

8. Spring 2016 CISG 5115 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Sequences A sequence: a list of objects in some order Tuples : a sequence of k elements is a k -tuple. Repetition is allowed. 8

9. Spring 2016 CISG 5115 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Relations Let A and B be sets. A relation R from A to B is a subset. Given an ordered pair, x is related to y by R, written if and only if. The set A is called the domain of R and the set B is called its co- domain. Ex. 9

10. Spring 2016 CISG 5115 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University n -ary Relation A more formal way to refer to the kind of relation defined in the previous slide is to call it a binary relation because it is a subset of a Cartesian product of two sets. An n -ary relation to be a subset of a Cartesian product of n sets, where n is any integer greater than or equal to two. Roughly, an n -ary relation to be a subset of a Cartesian product of n sets, where n is any integer greater than or equal to two. 10

11. Spring 2016 CISG 5115 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Reflexive Relation Informally, a relation R is Reflexive: Each element is related to itself. 11

12. Spring 2016 CISG 5115 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Symmetric Relation Informally, a relation R is Symmetric: If any one element is related to any other element, then the second element is related to the first. 12

13. Spring 2016 CISG 5115 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Transitive Relation Informally, a relation R is Transitive: If any one element is related to a second and that second element is related to a third, then the first element is related to the third. 13

14. Spring 2016 CISG 5115 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Equivalence Relation Let A be a set and R a relation on A. R is an equivalence relation if, and only if, R is reflexive, symmetric, and transitive. Example Q) Let X be the set of all nonempty subset of {1, 2, 3}. Define a relation R on X as follows: For all A and B in X, Prove that R is an equivalence relation on X. 14 Show R is reflexive, symmetric, and transitive!

15. Spring 2016 CISG 5115 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Function A function f from a set X to a set Y, denoted is a relation from X, the domain, to Y, the co-domain, that satisfies following two properties: 1) every element in X is related to some element in Y 2) no element in X is related to more than one element in Y. 15 (Arrow Diagram)

16. Spring 2016 CISG 5115 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Function – cont’ 16 The set of values of f taken together is called the range of f or the image of x under f. Followings are same range of f Image of x under f Range: { a,c }

17. Spring 2016 CISG 5115 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Function – cont’ Predicate (property) : a function whose range is {True, False}. k -ary function: a function with k arguments. Input:, a k -tuple unary function if k = 1 binary function if k = 2 17

18. Spring 2016 CISG 5115 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Graphs A graph consists of a finite set of vertices with lines connecting some of them (edges). Undirected graph degree of a node: the number of edges path: a sequence of nodes connected by edges cycle: a path starts and ends in the same node tree: no cycle Directed graph: in-degree and out-degree directed path directed acyclic graph (DAG) 18

19. Spring 2016 CISG 5115 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Strings Strings of characters Alphabet : any finite set, and A string over an alphabet: a finite sequence of symbols from the alphabet Length of a string Empty string: 19

20. Spring 2016 CISG 5115 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Strings and Languages Reverse of Substring Concatenation Lexicographic ordering of strings Language : a set of strings 20

21. Spring 2016 CISG 5115 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Languages Language : a set of strings Finite / infinite Kleene closure of alphabet, the set of all strings of all lengths over A (arbitrary length) A language L over the alphabet A is a subset of 21

22. Spring 2016 CISG 5115 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Proofs Proof: a convincing logical argument that a statement is true. convincing in an absolute sense Methods of proof The pigeonhole principle Proof by construction Proof by contradiction Proof by induction 22

23. Spring 2016 CISG 5115 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University The Pigeonhole Principle If there are n pigeonholes, n + 1 or more pigeons, and every pigeon occupies a hole, then some hole must have at least two pigeons. Example To show a 9-color pie chart on a 8-color monitor, two parts must have the same color. 23

24. Spring 2016 CISG 5115 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Proof by Construction Prove a particular type of objects exists by constructing the object. Example For each even number n > 2, there exists a 3-regular graph (no loop, no multiple edges) with n nodes. Proof: construct a graph G = ( V, E ) with n nodes as follow. 24

25. Spring 2016 CISG 5115 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Proof by Contradiction Assume a theorem is false and then show that this assumption leads to a false consequence. Example The language contains strings with an even number of 1's. Proof: Assume L contains only strings with an odd number of 1's. However, L contains the string 11, which contradicts with the assumption. 25

26. Spring 2016 CISG 5115 Department of Mathematics and Physics Donghyun (David) Kim North Carolina Central University Proof by Induction A proof by induction has a predicate P, a basis - is true an induction hypothesis - for some are true. an inductive step - is true given the induction hypothesis. Example Proof: Basis - Induction hypothesis - for Inductive step - 26