A computation is any type of arithmetic or non-arithmetic calculation that is well-defined.
Mechanical or electronic devices (or, historically, people) that perform computations are known as . Computer science is an academic field that involves the study of computation.
Introduction
The notion that mathematical statements should be 'well-defined' had been argued by mathematicians since at least the 1600s,
but agreement on a suitable definition proved elusive.
A candidate definition was proposed independently by several mathematicians in the 1930s.
The best-known variant was formalised by the mathematician
Alan Turing, who defined a well-defined statement or calculation as any statement that could be expressed in terms of the initialisation parameters of a
Turing machine.
Other (mathematically equivalent) definitions include
Alonzo Church's
Lambda calculus,
Herbrand-Gödel-
Kleene's
general recursiveness and
Emil Post's
1-definability.
Today, any formal statement or calculation that exhibits this quality of well-definedness is termed computable, while the statement or calculation itself is referred to as a computation.
Turing's definition apportioned "well-definedness" to a very large class of mathematical statements, including all well-formed equations, and all statements written in modern computer programming languages.
Despite the widespread uptake of this definition, there are some mathematical concepts that have no well-defined characterisation under this definition. This includes the halting problem and busy beaver. It remains an open question as to whether there exists a more powerful definition of 'well-defined' that is able to capture both computable and 'non-computable' statements.[The study of non-computable statements is the field of hypercomputation.]
Some examples of mathematical statements that are computable include:
-
All statements characterised in modern programming languages, including C++, Python, and Java.
-
All calculations carried by an electronic computer, calculator or abacus.
-
All calculations carried out on an analytical engine.
-
All calculations carried out on a Turing Machine.
-
The majority of mathematical statements and calculations given in maths textbooks.
Some examples of mathematical statements that are not computable include:
-
Calculations or statements which are ill-defined, such that they cannot be unambiguously encoded into a Turing machine: ("Paul loves me twice as much as Joe").
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Problem statements which do appear to be well-defined, but for which it can be proved that no Turing machine exists to solve them (such as the halting problem).
Computation can be seen as a purely physical process occurring inside a closed physical system called a computer. Turing's 1937 proof, On Computable Numbers, with an Application to the Entscheidungsproblem, demonstrated that there is a formal equivalence between computable statements and particular physical systems, commonly called computers. Examples of such physical systems are: Turing machines, human mathematicians following strict rules, , mechanical computers, and others.
Alternative accounts of computation
The mapping account
An alternative account of computation is found throughout the works of
Hilary Putnam and others. Peter Godfrey-Smith has dubbed this the "simple mapping account."
Gualtiero Piccinini's summary of this account states that a physical system can be said to perform a specific computation when there is a mapping between the state of that system and the computation such that the "microphysical states of mirror the state transitions between the computational states."
The semantic account
Philosophers such as
Jerry Fodor have suggested various accounts of computation with the restriction that
semantic content be a necessary condition for computation (that is, what differentiates an arbitrary physical system from a computing system is that the operands of the computation represent something). This notion attempts to prevent the logical abstraction of the mapping account of pancomputationalism, the idea that everything can be said to be computing everything.
The mechanistic account
Gualtiero Piccinini proposes an account of computation based on mechanical philosophy. It states that physical computing systems are types of mechanisms that, by design, perform physical computation, or the manipulation (by a functional mechanism) of a "medium-independent" vehicle according to a rule. "Medium-independence" requires that the property can be instantiated by multiple realizers and multiple mechanisms, and that the inputs and outputs of the mechanism also be multiply realizable. In short, medium-independence allows for the use of physical variables with properties other than voltage (as in typical digital computers); this is imperative in considering other types of computation, such as that which occurs in the
brain or in a
quantum computer. A rule, in this sense, provides a mapping among inputs, outputs, and internal states of the physical computing system.
Mathematical models
In the theory of computation, a diversity of mathematical models of computation has been developed.
Typical mathematical models of computers are the following:
Giunti calls the models studied by computation theory
computational systems, and he argues that all of them are mathematical
with discrete time and discrete state space.
He maintains that a computational system is a complex object which consists of three parts. First, a mathematical dynamical system
with discrete time and discrete state space; second, a computational setup
, which is made up of a theoretical part
, and a real part
; third, an interpretation
, which links the dynamical system
with the setup
.
See also
Notes
