Introduction to theory of computation
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The Theory of Computation deals with how efficiently problems can be solved using algorithms on computational models. It is divided into three branches: automata theory, computability theory, and complexity theory. Complexity theory analyzes problem difficulty and classifies problems as easy or hard to solve efficiently. Computability theory determines if problems are solvable or unsolvable. Automata theory studies the properties of computation models like finite automata and Turing machines. The overall purpose is to develop mathematical models of computation that reflect real-world computers and determine computational limitations.
Introduction to theory of computation
- 1. Theory of Computation The Theory of Computation is the branch of computer science that deals with how efficiently problems can be solved on a model of computation, using an algorithm. The field is divided into three major branches: Automata theory and language Computability theory Complexity theory
- 2. Complexity theory The main question asked in this area is “What makes some problems computationally hard and other problems easy?” A problem is called “easy”, if it is efficiently solvable. Examples of “easy” problems are (i) sorting a sequence of, say, 1,000,000 numbers, (ii) searching for a name in a telephone directory. A problem is called “hard”, if it cannot be solved efficiently, or if we don’t know whether it can be solved efficiently. Examples of “hard” problems are (i) factoring a 300-digit integer into its prime factors. Central Question in Complexity Theory: Classify problems according to their degree of “difficulty”. Give a proof that problems that seem to be “hard” are really “hard”.
- 3. Computability Theory Computability theory In the 1930’s, G¨odel, Turing, and Church discovered that some of the fundamental mathematical problems cannot be solved by a “computer”. To attack such a problem, we need formal definitions of the notions of computer, algorithm, and computation. The theoretical models that were proposed in order to understand solvable and unsolvable problems led to the development of real computers. Central Question in Co m putability The o ry: Classify problems as being solvable or unsolvable.
- 4. Automata theory Automata Theory deals with definitions and properties of different types of “computation models”. Examples of such models : Finite Automata :These are used in text processing, compilers, and hardware design. Context-Free Grammars: These are used to define programming languages and in Artificial Intelligence. Turing Machines: These form a simple abstract model of a “real” computer, such as your PC at home. Central Question in Automata Theory: Do these models have the same power, or can one model solve more problems than the other?.
- 5. Theory of Computation Purpose and motivation : oWhat are the mathematical properties of computer hardware and software ? oWhat is a computation and what is an algorithm? Can we give mathematical definitions of these notions? oWhat are the limitations of computers? Can “everything” be computed? Purpose of the TOC: Develop formal mathematical models of computation that reflect real-world computers.
- 6. Theory of Computation Purpose and motivation : oWhat are the mathematical properties of computer hardware and software ? oWhat is a computation and what is an algorithm? Can we give mathematical definitions of these notions? oWhat are the limitations of computers? Can “everything” be computed? Purpose of the TOC: Develop formal mathematical models of computation that reflect real-world computers.