[PDF] equivalence of finite automata and regular expressions

Problem Description

Let be a finite automaton and the transition label of the branch going from to , where they denote any two states of . The conversion algorithm should return a regular expression defining the regular language that accepts. For example, let’s say we have the following automaton: The regular expression it’s equivalent to is . A use case for this is n...

Example

Let’s work out an example from a paper on state elimination. Our input finite-state machine is: First, we make it uniform: Now, we randomly choose a state. Let it be . The next step is to compute the regular expressions for paths between the states connected directly via . The new regular expression for the transition from to is: Substituting the l...

Complexity

In the worst-case scenario, all the states are directly connected in both directions ( and ), there’s a transition from each state to itself, and each state connects all the pairs of the automaton’s states ( and for any and ). Let be the number of states (excluding the initial and the final states). Since all the states are connected in both direct...

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How to construct finite automata from a regular expression?

As the regular expressions can be constructed from Finite Automata using the State Elimination Method, the reverse method, state decomposition method can be used to construct Finite Automata from the given regular expressions.

Which language is defined by a finite automata?

Theorem: Every language defined by a regular expression is also defined by a Finite Automata. Proof: Let’s assume L = L (R) for a regular expression R. We prove that L = L (M) for some ?-NFA M with: 1) Exactly one accepting state. 2) No incoming edges at the initial state. 3) No outgoing edges at the accepting state.

How do you prove a finite state automata?

Sec. 10.8 of the text proves that there is a finite state automata that recognizes the language generated by any given regular expression. The proof is by induction on the number of operators in the regular expression and uses a finite state automata with ? transitions.

What is a regular expression?

The regular expression it’s equivalent to is . A use case for this is natural language processing (NLP). NLP engineers often design string-matching patterns, and for them, it’s much easier to read a regular expression than to look at a complex finite automaton. 3. The State Elimination Method We first need to make uniform.

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