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**Theory of Computation/Automata:**

In theoretical computer science & mathematics, the theory of computation is the branch which deals along with whether and how efficiently problems can be resolved on a model of computation, by using an algorithm. The field is divided into three main branches:, computability theory, automata theory and computational complexity theory.

To perform a rigorous study of computation, computer scientist’s work along with a mathematical abstraction of computers called as model of computation. There are various models in use, but the most commonly analysis is the Turing machine. Computer scientists examine the Turing machine because it is easy to formulate, can be analyzed and utilized to prove results, and because this represents what various consider the most powerful possible "reasonable" model of computation (see Church–Turing thesis). It might appear that the potentially infinite memory capacity is an unrealizable attribute; however any decidable problem solved by a Turing machine will always require only a finite amount of memory. So in principle, any difficulty that can be resolved (decided) by a Turing machine can be solved out by a computer that contains a bounded amount of memory.

**Automata theory:**

This is the study of abstract machines (or more suitably, abstract 'mathematical' machines or systems) and the computational problems that can be solved out by using these machines. These abstract machines are called as automata. Automata come through the Greek word that means that something is doing something by itself. Automata theory is also nearly associated to formal language theory, as the automata are frequently classified by the class of formal languages they are able to recognize. An automaton may be a finite representation of a formal language that can be an infinite set.

**Computability theory:**

This deals primarily with the question of the extent to which a difficulty is solvable on a computer. The statement that the halting problem can’t be solved out by a Turing machine is one of the most significant results in computability theory, as it is an instance of a concrete problem that is both simple to formulate and impossible to solve by using a Turing machine. Much of computability theory prepares on the halting problem result.

Another significant step in computability theory was Rice's theorem, which states that for all of the non-trivial properties of partial functions; it is undecidable whether a Turing machine computes a partial function with that property.

It is closely associated to the branch of mathematical logic called recursion theory, which eliminates the limitation of studying only models of computation which are reducible to the Turing model. Various mathematicians and computational theorists who study recursion theory will refer to it like computability theory.**Computational complexity theory:**

Complexity theory assumed not only whether a problem can be solved out at all on a computer, but also how efficiently the problem can be solved out. Two main aspects are considered: time complexity and space complexity, which are respectively how several steps does it take to perform a computation, and how much memory is needed to perform that computation.

To analyze how much time and space a given algorithm needed, computer scientists express the time or space needed to solve the problem as a function of the size of the input problem. For instance, searching a particular number in a long list of numbers becomes harder as the list of numbers grow up larger. If we say there are n numbers in the list, then if the list is not indexed or sorted in any way we might have to look at each number in order to discover the number we're seeking. Thus we say that to solve this problem, the computer require to perform a number of steps that linearly grows in the size of the problem.

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