L ECTURE 1 T HEORY OF C OMPUTATION Yasir Imtiaz Khan

G OALS OF T HEORY OF C OMPUTATION What is computable? What can be computed efficiently within a certain and time constraints? The ultimate answer from the Turing machine test is that anything can be computed by ignoring time and space.

T HEORY OF C OMPUTATION The theory of computation or computer theory is the branch of computer science and mathematics that deals with whether and how efficiently problems can be solved on a model of computation, using an algorithm.

C ENTRAL A REAS OF T HE T HEORY OF C OMPUTATION Automata Theory Computability Theory Complexity Theory

A UTOMATA T HEORY Deals with the definitions and properties of mathematical model of computation. Examples : Finite automata, Context free grammars. Finite Automaton : Text Processing, Compilers Context Free grammars : Programming languages, AI

C OMPUTABILITY T HEORY Study of computable functions and Turing degrees. Classification of problems is by those that are solvable and those that are not.

C OMPLEXITY T HEORY Classify the easy problems and hard ones. Some problems are hard even we are unable to prove Cryptography is application area of complex computation

S ETS A set is a group of objects, called elements (or members) of this set. For example, the students in this room form a set. A set can be defined by listing all its elements inside braces, e.g.: S ={ 7,21,57} The order and repetitions of elements in sets do not matter – in particular, {7,21,57} = {21,57,7} = {21, 7, 57, 7, 21}

S ETS C ONTINUED … The membership is denoted by symbol. For example, 21 S but 10 not belong to S. For two sets A and B, we say A is a subset of B and write A subset B if every member of A is also a member of B. We say that A is a proper subset of B and write A proper B if A is a subset of B and not equal to B. The set of all subsets of a set A is called the power set of A and denoted 2 A

E XAMPLES OF S ETS The set with no elements is called the empty set and denoted The empty set is a subset of any other set. The set of natural numbers N (or N): N = {1, 2, 3,...} The set of integers Z (or Z): Z = {..., -2,-1, 0, 1, 2,…} It is clear that N subset of Z

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V ENN D IAGRAMS

S EQUENCE AND T UPLES A sequence is a list of objects in some order. For example, sequences of the students' names in alphabetic order such as (Alice,Bob). In contrast to sets, repetitions and order matter in sequences. The sequences (7, 21, 57) and (7, 7, 21, 57) are not equal. Finite sequences are called tuples. In particular, a sequence with k elements is called k-tuple (as well as pair, triple, quadriple, etc.)

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