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      In formal language theory, the empty string, or empty word, is the unique string of length zero.


      Formal theory


      Formally, a string is a finite, ordered sequence of characters such as letters, digits or spaces. The empty string is the special case where the sequence has length zero, so there are no symbols in the string.
      There is only one empty string, because two strings are only different if they have different lengths or a different sequence of symbols.
      In formal treatments, the empty string is denoted with ε or sometimes Λ or λ.
      The empty string should not be confused with the empty language ∅, which is a formal language (i.e. a set of strings) that contains no strings, not even the empty string.
      The empty string has several properties:

      |ε| = 0. Its string length is zero.
      ε ⋅ s = s ⋅ ε = s. The empty string is the identity element of the concatenation operation. The set of all strings forms a free monoid with respect to ⋅ and ε.
      εR = ε. Reversal of the empty string produces the empty string, so the empty string is a palindrome.




      ∀
      c
      ∈
      s
      :
      P
      (
      c
      )


      {\displaystyle \forall c\in s:P(c)}

      . Statements that are about all characters in a string are vacuously true.
      The empty string precedes any other string under lexicographical order, because it is the shortest of all strings.
      In context-free grammars, a production rule that allows a symbol to produce the empty string is known as an ε-production, and the symbol is said to be "nullable".


      Use in programming languages


      In most programming languages, strings are a data type. Strings are typically stored at distinct memory addresses (locations). Thus, the same string (e.g., the empty string) may be stored in two or more places in memory.
      In this way, there could be multiple empty strings in memory, in contrast with the formal theory definition, for which there is only one possible empty string. However, a string comparison function would indicate that all of these empty strings are equal to each other.
      Even a string of length zero can require memory to store it, depending on the format being used. In most programming languages, the empty string is distinct from a null reference (or null pointer) because a null reference points to no string at all, not even the empty string.
      The empty string is a legitimate string, upon which most string operations should work. Some languages treat some or all of the following in similar ways: empty strings, null references, the integer 0, the floating point number 0, the Boolean value false, the ASCII character NUL, or other such values.
      The empty string is usually represented similarly to other strings. In implementations with string terminating character (null-terminated strings or plain text lines), the empty string is indicated by the immediate use of this terminating character.
      Different functions, methods, macros, or idioms exist for checking if a string is empty in different languages.


      = Representations of the empty string

      =

      The empty string is a syntactically valid representation of zero in positional notation (in any base), which does not contain leading zeros. Since the empty string does not have a standard visual representation outside of formal language theory, the number zero is traditionally represented by a single decimal digit 0 instead.
      Zero-filled memory area, interpreted as a null-terminated string, is an empty string.
      Empty lines of text show the empty string. This can occur from two consecutive EOLs, as often occur in text files. This is sometimes used in text processing to separate paragraphs, e.g. in MediaWiki.


      See also


      Empty set
      Null-terminated string
      Concatenation theory
      String literal


      References

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    formal languages - Importance of the empty string - Computer …

    $\begingroup$ If the empty string is not allowed, my definition of proper prefix is the usual definition of prefix (when empty string is allowed), and my definition of prefix is that u is a prefix of v iff u is a proper prefix of v or equal to v. The point is to remain semantically consistent with the theory that allows for the empty string ...

    What is the difference between the set containing the empty …

    Jan 11, 2023 · Think of the set containing the empty string as one, and the empty set as zero. A Kleene algebra is essentially an idempotent semi-ring, plus the Kleene closure operator. Ignoring the Kleene closure (a.k.a. Kleene star) operator, the ordered "multiplication" corresponds to string concatenation and "addition" corresponds to set union.

    regular languages - Empty string in an ambiguous grammar?

    Nov 15, 2021 · In a grammar when a rule transitions to the empty string, that means that the symbol just disappears. For example, here is a possible derivation: A' A if A B if null B if null Notice that on the last line, B is replaced by the empty string -- nothing. It goes away.

    regular languages - What does empty string ε actually mean?

    Feb 9, 2024 · $\begingroup$ There is a bit of an abuse of notations between the empty string, the regular expression that matches just the empty string, and the singleton set that contains only the empty string. $\endgroup$ –

    formal languages - What is the origin of λ for empty string?

    Oct 19, 2016 · $\begingroup$ 1 is particularly nice when you're defining Regular Expressions algebraically. 1 is the empty string, 0 is the empty language, concatenation is $\cdot$ and union is $+$, and you get a Semi-ring. $*$ makes things a bit more complicated though. $\endgroup$ –

    Empty regular expression - Computer Science Stack Exchange

    May 26, 2023 · In many programming languages with "regular expression" (regex) support, the default when you match a regex against a string is to consider it a match if any substring matches. Since every string has the empty string as a substring, the empty regular expression will match any string. But that convention isn't standard in computer science.

    normal forms - Is the empty string a terminal symbol? - Computer ...

    And A,B and C are non-terminal symbols, B and C are different from S and a is a terminal symbol. The definition also states that the last rule can only be present if the language of the grammar accepts the empty string . Now if $\epsilon$ is a terminal symbol, than this is a contradiction as the second rule can be transformed to S $\rightarrow ...

    Why does this automation accept the empty string?

    Jan 31, 2014 · An automaton accepts $\lambda$, the empty string, if and only if the starting state is a final state, or there is some final state in the empty-string-closure of the starting state (in the non-deterministic case). You don't need to walk from the start state, since it is already final.

    Difference between a TM with an empty language and the one …

    A TM accepting the empty language may move directly into the halt reject state, regardless of what may or may not be on the input tape. A TM accepting the language consisting of only the empty string may examine the first tape symbol. If the symbol is a blank, it could move to the halt accept state. Otherwise, it would move to the halt reject ...

    Is the empty string ε in regular languages implicitly between any …

    Jan 21, 2017 · The empty string is not there between every two characters implicitly but quite explicitly. In fact, there are two empty strings between any two characters, and between those two empty strings there is another, third, empty string. However, since empty strings are empty, them being there is like them not being there at all.