# What is computation?

Computation is adopted from computational science and is a broad subject that includes scientific, engineering, and financial analysis in which the computer plays an essential role. It involves the development of models, systems, algorithms, and simulations to solve problems. For any computational science, the principal emphasis is the solution to a problem and the creation of useful computational methods rather than the proof of a theorem. Computational theories are used in different domains like;

1. Computational optimization. This is an important subject in applied mathematics, operational research, and computer science. It theorizes the selection process whereby the users could select the best model from multiple models which are then applied for a particular job.
2. Numerical simulation. This is the application of mathematics to model different products and then testing it on a computer to examine its validity.
3. Data modeling. The data model is a description of data structure and organization. It is the process of defining, organizing, and structuring data for a particular need. Data modeling also defines restrictions and constraints on the data placed within the structure.

# Subdomains of computational science

There are several sub-domains that include algebraic techniques and semidefinite optimization, algorithms, automata, computability and complexity, the behavior of algorithms, combinatorial optimization, complexity theory, computation structures, computational commutative algebra and algebraic geometry, continuous and discrete signal processing, error-correcting codes, information and entropy, integer programming and combinatorial optimization, matching theory, numerical computation for mechanical engineers, paradox and infinity, parallel computing, primal-dual algorithm, quantum computation, quantum information science, etc.

# Theory of computation

The theory of computation is the branch that deals along with whether and how efficiently problems can be resolved on a model of computation, by using an algorithm. It is divided into 3 branches including;

1. Automata theory. This is the study of abstract machines and the computational problems that can be solved out by using these machines. These abstract machines are called automata. Automata come through the Greek word that means that something is doing something by itself. Automata theory is also nearly associated with 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.
2. Computability theory. This theory deals primarily with the question of the extent to which 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 the 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 with the branch of mathematical logic called recursion theory, which eliminates the limitation of studying only models of computation that are reducible to the Turing model. Various mathematicians and computational theorists who study recursion theory will refer to it as the computability theory.
3. 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 many several steps 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 for a particular number in a long list of numbers becomes harder as the list of numbers grows 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 requires to perform a number of steps that linearly grow in the size of the problem.

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