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To decide whether two given regular expressions describe the same language, each can be converted into an equivalent minimal deterministic finite automaton via Thompson's construction, powerset construction, and DFA minimization. If, and only if, the resulting automata agree up to renaming of states, the regular expressions' languages agree.
Regular languages are a category of languages (sometimes termed Chomsky Type 3) which can be matched by a state machine (more specifically, by a deterministic finite automaton or a nondeterministic finite automaton) constructed from a regular expression. In particular, a regular language can match constructs like "A follows B", "Either A or B ...
For certain regular expression operators like | (the operator for alternation or logical disjunction) it is superior to PCRE. Unlike PCRE, which supports features such as lookarounds, backreferences and recursion , RE2 is only able to recognize regular languages due to its construction using the Thompson DFA [ 4 ] algorithm.
The system uses a DFA for lexical analysis and the LALR algorithm for parsing. Both of these algorithms are state machines that use tables to determine actions. GOLD is designed around the principle of logically separating the process of generating the LALR and DFA parse tables from the actual implementation of the parsing algorithms themselves.
The DFA can be constructed explicitly and then run on the resulting input string one symbol at a time. Constructing the DFA for a regular expression of size m has the time and memory cost of O(2 m), but it can be run on a string of size n in time O(n). Note that the size of the expression is the size after abbreviations, such as numeric ...
Therefore, the length of the regular expression representing the language accepted by M is at most 1 / 3 (4 n+1 (6s+7)f - f - 3) symbols, where f denotes the number of final states. This exponential blowup is inevitable, because there exist families of DFAs for which any equivalent regular expression must be of exponential size.
In particular, every DFA is also an NFA. Sometimes the term NFA is used in a narrower sense, referring to an NFA that is not a DFA, but not in this article. Using the subset construction algorithm, each NFA can be translated to an equivalent DFA; i.e., a DFA recognizing the same formal language. [1] Like DFAs, NFAs only recognize regular languages.
For each regular language, there also exists a minimal automaton that accepts it, that is, a DFA with a minimum number of states and this DFA is unique (except that states can be given different names). [2] [3] The minimal DFA ensures minimal computational cost for tasks such as pattern matching.