The area of theoretical computer science concerning what problems can be solved by any computer.
A function is computable if an algorithm can be implemented which will give the correct output for any valid input.
Since computer programs are countable but real numbers are not, it follows that there must exist real numbers that cannot be calculated by any program. Unfortunately, by definition, there isn’t an easy way of describing any of them!
In fact, there are many tasks (not just calculating real numbers) that computers cannot perform. The most well-known is the halting problem, the busy beaver problem is less famous but just as fascinating.
[“Computability”, N.J. Cutland. (A well written undergraduate-level introduction to the subject)].
[“The Turing Omnibus”, A.K. Dewdeney].
[kuh m-pyoot] /kəmˈpyut/ verb (used with object), computed, computing. 1. to determine by calculation; reckon; calculate: to compute the period of Jupiter’s revolution. 2. to determine by using a or calculator. verb (used without object), computed, computing. 3. to reckon; calculate. 4. to use a or calculator. 5. Informal. to make sense; add up: His […]
[kom-pyoo-tey-shuh n] /ˌkɒm pyʊˈteɪ ʃən/ noun 1. an act, process, or method of ; calculation. 2. a result of . 3. the amount . /ˌkɒmpjʊˈteɪʃən/ noun 1. a calculation involving numbers or quantities n. early 15c., from Middle French computation, from Latin computationem (nominative computatio), noun of action from past participle stem of computare “to […]
[kom-pyoo-tey-shuh n] /ˌkɒm pyʊˈteɪ ʃən/ noun 1. an act, process, or method of ; calculation. 2. a result of . 3. the amount . /ˌkɒmpjʊˈteɪʃən/ noun 1. a calculation involving numbers or quantities adj. 1857, from computation + -al (1). Related: Computationally. n. early 15c., from Middle French computation, from Latin computationem (nominative computatio), noun […]
- Computational adequacy theorem
This states that for any program (a non-function typed term in the typed lambda-calculus with constants) normal order reduction (outermost first) fails to terminate if and only if the standard semantics of the term is bottom. Moreover, if the reduction of program e1 terminates with some head normal form e2 then the standard semantics of […]