Regular patterns in Coxeter groups

# Regular expressions

A language in an alphabet is any collection of strings in its symbols.
• A word is recognized by an automaton if there is a path from a start state to an accepting state labeled in order by its symbols.
• A language is associated to an automaton if its words are precisely those recognized by it.
• Any language is recognized by its tautologous automaton, that whose nodes are the prefixes of words in that language. The starting state is the empty string, and the accepting states are those which are actually words in the language. The ShortLex tree.
A language is regular if recognized by a finite automaton. The basic operations:
• A cat B (A . B): a word of A followed by a word of B
• A star (A+): 0 or more words of A
• A plus (A*): 1 or more words of A
• A alt B (A | B): a word of A or a word of B
• not A (!A): words with the same symbols but not in A
• A and B (A & B): words in both A and B
These generate regular expressions. The regular expression

( |+|-)(1|2|3|4|5|6|7|8|9)(0|1|2|3|4|5|6|7|8|9)*

recognizes decimal integers expressed conventionally.

• Regular expressions require only a finite amount of memory to recognize.
• The language of balanced parentheses ( ( ( ) ( ) ( ( ) ) ) ( ) ) is not regular, roughly because recognizing all such expressions requires a memory of arbitrary size.