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String homomorphisms are monoid morphisms on the free monoid, preserving the empty string and the binary operation of string concatenation. Given a language , the set () is called the homomorphic image of . The inverse homomorphic image of a string is defined as
Also apophthegm. A terse, pithy saying, akin to a proverb, maxim, or aphorism. aposiopesis A rhetorical device in which speech is broken off abruptly and the sentence is left unfinished. apostrophe A figure of speech in which a speaker breaks off from addressing the audience (e.g., in a play) and directs speech to a third party such as an opposing litigant or some other individual, sometimes ...
The occurrences of a given pattern in a given string can be found with a string searching algorithm. Finding the longest string which is equal to a substring of two or more strings is known as the longest common substring problem. In the mathematical literature, substrings are also called subwords (in America) or factors (in Europe). [citation ...
Certain operations on languages are common. This includes the standard set operations, such as union, intersection, and complement. Another class of operation is the element-wise application of string operations. Examples: suppose and are languages over some common alphabet .
Edit distance with non-negative cost satisfies the axioms of a metric, giving rise to a metric space of strings, when the following conditions are met: [1]: 37 Every edit operation has positive cost; for every operation, there is an inverse operation with equal cost. With these properties, the metric axioms are satisfied as follows:
For example, the alphabet of lowercase letters "a" through "z" can be used to form English words like "iceberg" while the alphabet of both upper and lower case letters can also be used to form proper names like "Wikipedia". A common alphabet is {0,1}, the binary alphabet, and a "00101111" is an example of a binary string.
The suffix tree for the string of length is defined as a tree such that: [7]. The tree has exactly n leaves numbered from to .; Except for the root, every internal node has at least two children.
The similarity of two strings and is determined by this formula: twice the number of matching characters divided by the total number of characters of both strings. The matching characters are defined as some longest common substring [3] plus recursively the number of matching characters in the non-matching regions on both sides of the longest common substring: [2] [4]