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Lecture 1 Theory of Computation

Lecture 1 Theory of Computation. Yasir Imtiaz Khan. Goals of Theory of Computation. 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.

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Lecture 1 Theory of Computation

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  1. Lecture 1Theory of Computation Yasir Imtiaz Khan

  2. Goals of Theory of Computation • 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.

  3. Theory of Computation • 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.

  4. Central Areas of The Theory of Computation • Automata Theory • Computability Theory • Complexity Theory

  5. Automata Theory • 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

  6. Computability Theory • Study of computable functions and Turing degrees. • Classification of problems is by those that are solvable and those that are not.

  7. Complexity Theory • Classify the easy problems and hard ones. • Some problems are hard even we are unable to prove • Cryptography is application area of complex computation

  8. Sets • 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}

  9. Sets Continued… • The membership is denoted by ϵ symbol. For example, 21 ϵ 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 2A

  10. Examples of Sets • 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

  11. Set Operations

  12. Venn Diagrams

  13. Sequence and Tuples • 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.)

  14. Functions

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