*There are several models in use, but the most commonly examined is the Turing machine.Computer scientists study the Turing machine because it is simple to formulate, can be analyzed and used to prove results, and because it represents what many consider the most powerful possible "reasonable" model of computation (see Church–Turing thesis).*

*There are several models in use, but the most commonly examined is the Turing machine.*Computer scientists study the Turing machine because it is simple to formulate, can be analyzed and used to prove results, and because it represents what many consider the most powerful possible "reasonable" model of computation (see Church–Turing thesis).

Much of computability theory builds on the halting problem result.

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

Because automata are used as models for computation, formal languages are the preferred mode of specification for any problem that must be computed.

is one of the most important results in computability theory, as it is an example of a concrete problem that is both easy to formulate and impossible to solve using a Turing machine.

A computation consists of an initial lambda expression (or two if you want to separate the function and its input) plus a finite sequence of lambda terms, each deduced from the preceding term by one application of Beta reduction.-calculus).

Combinatory logic was developed with great ambitions: understanding the nature of paradoxes, making foundations of mathematics more economic (conceptually), eliminating the notion of variables (thus clarifying their role in mathematics).a computation consists of a mu-recursive function, i.e.its defining sequence, any input value(s) and a sequence of recursive functions appearing in the defining sequence with inputs and outputs.Thus, if in the defining sequence of a recursive function appear, then terms of the form 'g(5)=7' or 'h(3,2)=10' might appear.Language theory is a branch of mathematics concerned with describing languages as a set of operations over an alphabet.It is closely linked with automata theory, as automata are used to generate and recognize formal languages.Computability theory is closely related to the branch of mathematical logic called recursion theory, which removes the restriction of studying only models of computation which are reducible to the Turing model.Complexity theory considers not only whether a problem can be solved at all on a computer, but also how efficiently the problem can be solved.We thus say that in order to solve this problem, the computer needs to perform a number of steps that grows linearly in the size of the problem.To simplify this problem, computer scientists have adopted Big O notation, which allows functions to be compared in a way that ensures that particular aspects of a machine's construction do not need to be considered, but rather only the asymptotic behavior as problems become large.In theoretical computer science and mathematics, the theory of computation is the branch 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 languages, computability theory, and computational complexity theory, which are linked by the question: "What are the fundamental capabilities and limitations of computers? In order to perform a rigorous study of computation, computer scientists work with a mathematical abstraction of computers called a model of computation.

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