Regular Language Operations
Regular operations in formal languages and automata theory are fundamental tools for manipulating and describing sets of strings.
The main regular operations include:
Union
The union of two languages \( L_1 \) and \( L_2 \), denoted \( L_1 \cup L_2 \), is the set of all strings that belong to \( L_1 \), \( L_2 \), or both.
Example. Practically speaking, if \( L_1 = \{a, b\} \) and \( L_2 = \{b, c\} \), then \( L_1 \cup L_2 = \{a, b, c\} \).
Concatenation
The concatenation of two languages \( L_1 \) and \( L_2 \), denoted \( L_1 \cdot L_2 \) or simply \( L_1L_2 \), is the set of strings formed by joining every string of \( L_1 \) with every string of \( L_2 \).
Example. If \( L_1 = \{a, b\} \) and \( L_2 = \{c, d\} \), then \( L_1 \cdot L_2 = \{ac, ad, bc, bd\} \).
Kleene Star
The Kleene star of a language \( L \), denoted \( L^* \), is the set of all possible concatenations of zero or more strings from \( L \).
Example. For instance, if \( L = \{a\} \), then \( L^* = \{\epsilon, a, aa, aaa, \ldots\} \), where \( \epsilon \) represents the empty string.
The "star" operation is unique because it is unary, whereas union, intersection, concatenation, and complement are binary operations.
Intersection
The intersection of two languages \( L_1 \) and \( L_2 \), denoted \( L_1 \cap L_2 \), is the set of strings that belong to both \( L_1 \) and \( L_2 \).
Example. If \( L_1 = \{a, b, c\} \) and \( L_2 = \{b, c, d\} \), then \( L_1 \cap L_2 = \{b, c\} \).
Complement
The complement of a language \( L \) with respect to an alphabet \( \Sigma \), denoted \( \overline{L} \), is the set of all strings over the alphabet \( \Sigma \) that do not belong to \( L \).
Example. If \( \Sigma = \{a, b\} \) and \( L = \{a\} \), then \( \overline{L} = \{b, aa, ab, ba, bb, \ldots\} \).
These operations are essential for constructing and analyzing formal languages, automata, and grammars.
They are fundamental in many areas of computer science, such as compilation and programming language design.
Closure of Regular Operations
Regular operations on formal languages are closed, meaning that applying these operations to regular languages always produces a regular language.
For example, the union of two regular languages \( L_1 \) and \( L_2 \) is always a regular language. If \( L_1 \) and \( L_2 \) are recognized by automata \( A_1 \) and \( A_2 \) respectively, we can construct an automaton that recognizes \( L_1 \cup L_2 \) using a non-deterministic finite automaton (NFA) or by creating a new automaton that combines the states of \( A_1 \) and \( A_2 \). For instance, if we have two regular languages \( L_1 = \{a, b\} \) and \( L_2 = \{b, c\} \), the union \( L_1 \cup L_2 = \{a, b, c\} \). The same applies to other operations.
However, not all finite state automata can process the result of a regular operation.
A deterministic finite automaton cannot handle the concatenation of two languages because it wouldn't know where the string for the first automaton \( A_1 \) ends and where the string for the second automaton \( A_2 \) begins.
For example, if we have two regular languages \( L_1 = \{a, b\} \) and \( L_2 = \{b, c\} \), the concatenation of the string w1="ab" from \( L_1 \) with the string w2="b" from \( L_2 \) creates a new string w1w2="abb" in a new regular language. When does the first string end for the first automaton to process? The string "abb" could be the concatenation of "ab"·"b" or "a"·"bb". The deterministic automaton wouldn't know which part of the string to process with automaton \( A_1 \) and which with automaton \( A_2 \).
This requires a specific construction of the resulting automaton, not just a simple combination of the two original automata.
The idea is to build a new DFA that "simulates" the concatenation process. For example, one could create a data structure based on the pair of strings (w1, w2) rather than a concatenation w1w2.
