Regular Languages – Deterministic Finite Automata – Nondeterministic Finite Automata • Today: – Equivalence of NFA and DFA – Regular Expressions
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The NFA definitely provides an accepting state for the symbol It is proven (Kleene's Theorem) that RE and FA are equivalent language definition methods In following this progression an NFA is constructed first from a regular expression, then the NFA is reconstructed to a DFA, and finally a Transition Table is built
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NFA to DFA conversion and regular expressions CSCI 3130 Formal Languages and Automata Theory Siu On CHAN Fall 2020 Chinese University of Hong
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Convert the following NFA into an equivalent DFA? Nondeterministic Finite Automata with empty moves (?-NFA) Definition A nondeterministic finite automaton
Compact L
Abstract There are two principal methods for turning regular expressions into NFA's - one due to testing, and NFA-to-DFA conversion by subset construction
Deterministic Finite Automata (DFAs) – Non-deterministic Finite Automata (NFAs ) • Implementation of regular expressions RegExp => NFA => DFA => Tables
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Since DFAs are special cases of NFAs, NFAs recognize at least the DFA- recognizable (regular) languages • Nothing else • Theorem: If M is an NFA then L(M) is
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Given DFA M, will construct regular expression R such that L(M) = L(R) 2 Regular Expressions to NFA Regular Expressions to Finite Automata
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Regular Expressions • DFA Find the longest match Deterministic, fast, state NFA Leftmost, longest (in order) ordering of RE changes pattern matched
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Non-deterministic Finite Automata (NFA) is exactly one edge out of s labeled a Regular Expression NFA DFA For real languages NFA and DFA have
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Nondeterministic Finite Automata. • Today: – Equivalence of NFA and DFA. – Regular Expressions. – Equivalence to Regular Languages. Equivalence of Machines.
every DFA is also an NFA this then shows that there is an NFA
By induction on the number of operators in the regex. Base case: L has a regular expression with 0 operators. Then the regex should be one of Ø ?
Label on remaining edge between start and accept states is a regular expression for language of original DFA. Remark: Method also can convert NFA into a regular
22 thg 7 2002 The technique described in this thesis converts a Finite Automata (as either a DFA or an. NFA)
Given DFA M will construct regular expression R such that L(M) = L(R). 2 Regular Expressions to NFA. Regular Expressions to Finite Automata.
Convert Regular Expressions to Finite Automata. • High-level sketch. Regular expressions. NFA. DFA. Lexical. Specification. Table-driven.
In general how do we convert a regular expression to an NFA? A regular expression over ? is an expression formed by the following rules. ? The symbols ? and
First the regular expressions are studied then the Thomp- son [1] method is introduced to produce NFA from DFA. Similarly
regular expression – Build NFA – Convert NFA to DFA using subset construction – Minimize resulting DFA Theorem: A language is recognized by a DFA (or NFA) if and only if it has a regular expression You need to know this fact but we won’t ask you anything about the “only if” direction from DFA/NFA to regular expression For fun we
any regular expression is regular: Theorem: For any regular expression R there is an NFA N such that ?(R) = (? N) N has exactly one accepting state N has no transitions into its start state N has no transitions out of its accepting state
We will convert a DFA to a regular expression as follows: (A) Convert DFA to a NFA adding new initial and ?nal states (B) Remove all states one-by-one until we have only the initial and ?nal states
Lecture 24: NFAs Regular expressions and NFA ?DFA Nondeterministic Finite Automata (NFA) described by a regular expression there is an NFA that recognizes
Lecture 24: NFAs Regular expressions and NFA ?DFA Nondeterministic Finite Automata (NFA) • Graph with start state final states edges labeled by symbols (like DFA) but – Not required to have exactly 1 edge out of each state labeled by each symbol---can have 0 or >1 – Also can have edges labeled by empty string ?
How to convert a regular expression to a DFA?
Humans can convert a regular expression to a DFA fairly easily using just a few heuristic rules. But a computer needs an algorithm. There are two steps: 1. Convert the regular expression to an NFA. 2. Convert the NFA to a DFA. This video covers step 1, converting a regular expression to an NFA. Related videos cover the second step.
What is the difference between NFA and DFA?
The finite automata are called NFA when there exist many paths for specific input from the current state to the next state. 3. Every NFA is not DFA, but each NFA can be translated into DFA. 4. NFA is defined in the same way as DFA but with the following two exceptions, it contains multiple next states, and it contains transition.
When is a string accepted by DFA D?
So a string is accepted by DFA D if, and only if, it is accepted by NFA N. Steps for Converting NFA to DFA: Step 1: Initially Q' = ?
Is there a DFA that also accepts L?
There exists a DFA D= (?, Q’, q’ 0 , F’, ’) that also accepts L. (L(N) = L(D)). Proof: By allowing each state in the DFA D to represent a set of states in the NFA N, we are able to prove through induction that D is equivalent to N.
CS 301 - Lecture 3 NFA DFA Equivalence Regular Expressions