Computable Function - Formal Languages

Formal Languages

In computability theory in computer science, it is common to consider formal languages. An alphabet is an arbitrary set. A word on an alphabet is a finite sequence of symbols from the alphabet; the same symbol may be used more than once. For example, binary strings are exactly the words on the alphabet {0, 1} . A language is a subset of the collection of all words on a fixed alphabet. For example, the collection of all binary strings that contain exactly 3 ones is a language over the binary alphabet.

A key property of a formal language is the level of difficulty required to decide whether a given word is in the language. Some coding system must be developed to allow a computable function to take an arbitrary word in the language as input; this is usually considered routine. A language is called computable (synonyms: recursive, decidable) if there is a computable function f such that for each word w over the alphabet, f(w) = 1 if the word is in the language and f(w) = 0 if the word is not in the language. Thus a language is computable just in case there is a procedure that is able to correctly tell whether arbitrary words are in the language.

A language is computably enumerable (synonyms: recursively enumerable, semidecidable) if there is a computable function f such that f(w) is defined if and only if the word w is in the language. The term enumerable has the same etymology as in computably enumerable sets of natural numbers.

Read more about this topic:  Computable Function

Other articles related to "words">formal languages, languages, language, formal":

Semantic Gap - Formal Languages
... Real world tasks are formalized by programming languages, which are executed on computers based on the von Neumann architecture ... Since programming languages are only comfortable representations of the Turing machine any program on a von Neumann computer has the same properties and limitations as the Turing machine or ... Consequently every programming language such as CPU level machine code, assembler, or any high level programming language has the same expressional power ...
Important Distinctions in Metalogic - Syntax–semantics
... In metalogic, 'syntax' has to do with formal languages or formal systems without regard to any interpretation of them, whereas, 'semantics' has to do with interpretations of formal languages ... scope than 'proof-theoretic', since it may be applied to properties of formal languages without any deductive systems, as well as to formal systems ...
Logical Connectives - In Language - Formal Languages
... In formal languages, truth functions are represented by unambiguous symbols ... These symbols are called "logical connectives", "logical operators", "propositional operators", or, in classical logic, "truth-functional connectives" ...

Famous quotes containing the words languages and/or formal:

    Wealth is so much the greatest good that Fortune has to bestow that in the Latin and English languages it has usurped her name.
    William Lamb Melbourne, 2nd Viscount (1779–1848)

    Two clergymen disputing whether ordination would be valid without the imposition of both hands, the more formal one said, “Do you think the Holy Dove could fly down with only one wing?”
    Horace Walpole (1717–1797)