A binary string is a finite word over the alphabet . There are strings of length . Finite binary strings have an effective bijection with natural numbers, so computable enumerability and immunity apply to sets of strings.
A binary string with zeroes and ones is determined by its one-positions, giving possibilities. Sum for and apply the hockey-stick identity. Equivalently, distribute at most ones among the gaps around the zeroes and add a slack count; stars and bars counts nonnegative counts summing to .
A self-delimiting code permits a decoder to identify the end of a code without an external length marker. Encoding a binary word of length by ones, a zero and the word uses bits. This yields an description of a natural number in an incompressibility proof.
For formal languages over the same alphabet, concatenate each word over an alphabet in with each word over an alphabet in . This operation is associative, has identity and absorbing element the empty language.
The formal language containing no words over an alphabet. It differs from the language containing only the empty word, which is a multiplicative identity for concatenation of formal languages.
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A formal language is a set of strings composed of symbols from a defined alphabet that follows specific syntactical rules or grammar. Unlike natural languages, which are used for everyday communication and can be ambiguous and variable, formal languages are precise and unambiguous. They are often used in mathematical logic, computer science, linguistics, and theoretical computer science. Key characteristics of formal languages include: 1. **Alphabet**: The basic set of symbols from which strings are formed.
