Recursive language
In mathematics, logic and computer science, a formal language (a set of finite sequences of symbols taken from a fixed alphabet) is called recursive if it is a recursive subset of the set of all possible finite sequences over the alphabet of the language. Equivalently, a formal language is recursive if there exists a total Turing machine (a Turing machine that halts for every given input) that, when given a finite sequence of symbols as input, accepts it if it belongs to the language and rejects it otherwise. Recursive languages are also called decidable.
The concept of decidability may be extended to other models of computation. For example one may speak of languages decidable on a non-deterministic Turing machine. Therefore, whenever an ambiguity is possible, the synonym for "recursive language" used is Turing-decidable language, rather than simply decidable.
The class of all recursive languages is often called R, although this name is also used for the class RP.
This type of language was not defined in the Chomsky hierarchy of (Chomsky 1959). All recursive languages are also recursively enumerable. All regular, context-free and context-sensitive languages are recursive.
Contents
1 Definitions
2 Examples
3 Closure properties
4 See also
5 References
Definitions
There are two equivalent major definitions for the concept of a recursive language:
- A recursive formal language is a recursive subset in the set of all possible words over the alphabet of the language.
- A recursive language is a formal language for which there exists a Turing machine that, when presented with any finite input string, halts and accepts if the string is in the language, and halts and rejects otherwise. The Turing machine always halts: it is known as a decider and is said to decide the recursive language.
By the second definition, any decision problem can be shown to be decidable by exhibiting an algorithm for it that terminates on all inputs. An undecidable problem is a problem that is not decidable.
Examples
As noted above, every context-sensitive language is recursive. Thus, a simple example of a recursive language is the set L={abc, aabbcc, aaabbbccc, ...};
more formally, the set
- L={w∈{a,b,c}∗∣w=anbncn for some n≥1}{displaystyle L={,win {a,b,c}^{*}mid w=a^{n}b^{n}c^{n}{mbox{ for some }}ngeq 1,}}
is context-sensitive and therefore recursive.
Examples of decidable languages that are not context-sensitive are more difficult to describe. For one such example, some familiarity with mathematical logic is required: Presburger arithmetic is the first-order theory of the natural numbers with addition (but without multiplication). While the set of well-formed formulas in Presburger arithmetic is context-free, every deterministic Turing machine accepting the set of true statements in Presburger arithmetic has a worst-case runtime of at least 22cn{displaystyle 2^{2^{cn}}}
Closure properties
Recursive languages are closed under the following operations. That is, if L and P are two recursive languages, then the following languages are recursive as well:
- The Kleene star L∗{displaystyle L^{*}}
- The image φ(L) under an e-free homomorphism φ
- The concatenation L∘P{displaystyle Lcirc P}
- The union L∪P{displaystyle Lcup P}
- The intersection L∩P{displaystyle Lcap P}
- The complement of L{displaystyle L}
- The set difference L−P{displaystyle L-P}
The last property follows from the fact that the set difference can be expressed in terms of intersection and complement.
See also
- Recursively enumerable language
- Recursion
References
Michael Sipser (1997). "Decidability". Introduction to the Theory of Computation. PWS Publishing. pp. 151–170. ISBN 0-534-94728-X..mw-parser-output cite.citation{font-style:inherit}.mw-parser-output q{quotes:"""""""'""'"}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-limited a,.mw-parser-output .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}
Chomsky, Noam (1959). "On certain formal properties of grammars". Information and Control. 2 (2): 137–167. doi:10.1016/S0019-9958(59)90362-6.
Fischer, Michael J.; Rabin, Michael O. (1974). "Super-Exponential Complexity of Presburger Arithmetic". Proceedings of the SIAM-AMS Symposium in Applied Mathematics. 7: 27–41.
Oppen, Derek C. (1978). "A 222pn Upper Bound on the Complexity of Presburger Arithmetic" (PDF). J. Comput. Syst. Sci. 16 (3): 323–332. doi:10.1016/0022-0000(78)90021-1.
