Nfa to dfa python

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Converting Epsilon-NFA to DFA using Python and Graphviz
Types of Finite Automata:
Deterministic Finite Automata (DFA)
Non-Deterministic Finite Automata (NFA)
Why the conversion?
Steps For Converting E-NFA to DFA :
For example,
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QuinnHerden/NFA-to-DFA
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Python Implementation of an algorithm which converts a Non-Deterministic Finite Automaton to a reduced Deterministic Finite Automaton as a part of ‘Automata Theory’ course assignment

narenakash/NFA-to-DFA-Converter

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README.md

CS1.301 Automata Theory: Course Assignment 01

Under Prof. Venkatesh Chopella, IIIT Hyderabad

An NFA which recognizes a language L is represented formally by a 5-tuple, (Q, Σ, Δ, q0, F). Then, DFA D = (Q’, Σ, Δ’, q0, F’) can be constructed for language L.

Step 01: Initially Q’ = ɸ. Add q0 to Q’.

Step 02: For each state in Q’, add the possible set of states for each input symbol using the transition function of NFA to Q’ if not present already.

Step 03: The final state of DFA will be all states containing F.

Note: The algorithm produces only the reduced transition function of the DFA.

About

Python Implementation of an algorithm which converts a Non-Deterministic Finite Automaton to a reduced Deterministic Finite Automaton as a part of ‘Automata Theory’ course assignment

Источник

Converting Epsilon-NFA to DFA using Python and Graphviz

Finite Automata (FA) is a simple machine used to match patterns in an input string. Finite Automata is a quintuple i.e., it has five elements. In this article we are going to see how to convert epsilon-NFA to DFA using Python and Graphviz.

The quintuple of FA is represented as :

$\langle Q, \Sigma, q_0, F, \delta \rangle$

The Five Elements are :

A finite set of States (Q)
A finite set of Input Alphabets ( $\Sigma$ )
Start State ( » width=»» height=»»/>)
A finite set of Final States (F)

Transition Function (

Types of Finite Automata:

Deterministic Finite Automata (DFA)

Deterministic Finite Automata (DFA) is a FA where there is only one fixed state the machine can go to for an input alphabet as defined in the transition function.

$<\epsilon data-lazy-src=$

DFA does not allow » width=»» height=»»/> (null) alphabet which means that the machine will not change state if no alphabet is detected.

Non-Deterministic Finite Automata (NFA)

Non-Deterministic Finite Automata (NFA) is a FA where the machine can go to more than one states for an input alphabet.

$\delta: Q X \Sigma\rightarrow 2^Q$

$\epsilon$

NFA which allows moves is called E-NFA or Epsilon-NFA.

$\epsilon$

NFA allowing (null) alphabet means that machine can change state even if no input alphabet is detected.

$\delta: Q X (\Sigma\cup\epsilon)\rightarrow 2^Q$

Example, NFA that accepts all the strings made up of f’s and g’s which contains “gfg” as a substring.

The is because, for every transition, there are 2 possibilities, to transit or not transit. And this is for every Q state. So will be the total possible configurations for each transition.

Why the conversion?

Computers can understand FA in its basic form that is in DFA. But because of the features of NFA, we humans can understand NFA and better E-NFA with ease. So we need to convert the E-NFA to DFA.

Steps For Converting E-NFA to DFA :

– closure : It is the set of states to which we can go without any input i.e., with moves.

$\epsilon$

Step 1: Find – closure of the start state of NFA and that will be the start state of DFA.

$\epsilon$

Step 2: Starting with this set, for each alphabet, evaluate – closure of the transition set for this alphabet.

Step 3: For each new set of closure set we come across, we will repeat Step 2 until no new set is left.

Step 4: The set in DFA which contains the final state of NFA will be the final set state.

For example,

$\langle Q, \Sigma, q_0, F, \delta \rangle$

NFA :

$\Sigma$

q_0

: A

$\delta$

NFA	a	b	c
A	A	–	–	BC
B	–	BD	–	–
C	–	–	CD	–
D	–	–	–	–

It is an NFA for language which accepts string of the type : » width=»» height=»»/> , where $<l\ge0,m\ge1,n\ge1 data-lazy-src=$ » width=»» height=»»/>>

Steps to convert :

States

$\epsilon$

Step 1: Find – closure of the start state of NFA and that will be the start state of DFA.

$\epsilon$

– closure of start state of NFA.

$\epsilon$

– closure (A) :

$\epsilon$

Steps 2,3: Starting with this set, for each alphabet, evaluate – closure of the transition set for this alphabet. and For each new set of closure set we come across, we will repeat Step 2 until no new set is left.

Current set of states of DFA:

-> a =
-> b = BD
-> c =
-> a =
-> b =
-> c = CD

$\delta$

Transition Function :

DFA	a	b	c
ABC	ABC	BD	CD
BD		BD
CD			CD

Step 4: The set in DFA which contains the final state of NFA will be the final set state.

D was the final state in NFA. So all the states having D in its set will be final state in DFA.

DFA obtained :

$\phi$

is also known as dead state because there is no outgoing edge from it. So after machine arrives in the dead state, it cannot reach the final state.

Tools for implementation with code:

Graphviz: It is a python library for visualizing graph diagrams.

To install graphviz in python, run this command in the terminal:

Prerequisite:

Number of States : no_state
Array of States : states
Number of Alphabets : no_alphabet
Array of Alphabets : alphabets
Start State : start
Number of Final States : no_final
Array of Final States : finals
Number of Transitions : no_transition
Array of Transitions : transitions
Transitions are of type : [From State, Alphabet, To State]

Dictionary/Map to get Index from state : states_dict
Dictionary/Map to get Index from alphabet : alphabets_dict
Transition Table Dictionary to get array of ‘to’ states from ‘state’ and ‘alphabet’ pair : transition_table
Digraph object to store graph of the NFA : graph

Methods/Functions :

Constructor of the class NFA : __init__()
It initializes all the input variables and evaluates the utility variables values from input.
Get input from user : fromUser()
Class Method to get input from user.
Representation of the quintuple of NFA : __repr__()
Find Epsilon Closure of a state : getEpsilonClosure(state)
To find epsilon closure of a state, we maintain a stack to get what state to evaluate next and a dictionary to track which states have been evaluated.
We start a while loop from the start state, and find its epsilon transition states.
We push all this states to the stack (stack.push(stata)) and mark them in the dictionary (dict[state]=0).
We also mark this state complete in the dictionary (dict[state]=1).
For each next iteration, we pop the top of the stack each time and evaluate it if it is not evaluated by checking from the dictionary.
Find appropriate name of state for diagram of converted DFA from array of states : getStateName(state_list)
As we will get a list of states from the evaluation, to display in the DFA diagram, we need a proper name which will be concatenation of all the names in the set.
For ex : To state set is = , then this function will return a string = “ABD”.
Check if the array contains a final state of NFA to find if the array will be final state in DFA : isFinalDFA(state_list)
This function checks if a list of states contains a state which is a final state in NFA, which in turn will tell if the set is final or not.
For ex : The set we are checking is = , and D is a final state in NFA. So this function will return True for this input set. So this set will be final state in DFA.

$\phi$

Make an object of NFA class: nfa, and initialize it with predefined values using a constructor or using user input.
Initialize the nfa.graph with nodes and edges according to the input values.
Display/Render the NFA graph.
Make another Digraph object to store values of the obtained DFA and to render the diagram: dfa.
Evaluate epsilon closure of all the states of NFA to not recalculate each time and store it in the dictionary with key-value pair as [state -> list of states of closure]: epsilon_closure<>
Make a stack to track which DFA state to evaluate next: dfa_stack[]
Add the epsilon closure of the start state of NFA as the start state of DFA.
Make a list to maintain all the states present in the DFA : dfa_states[]
Start a while loop which we continue till there are no new states in dfa_stack.
We pop the top of dfa_stack to evaluate the current set of states: cur_state.
We traverse through all the alphabets for the current set of states.
We make a set to maintain the epsilon closure of all the states in the current set : from_closure<>
If this set is not empty,
- We make another set to maintain the to_state set.
- If this set is not present in dfa_states, then we append it in dfa_stack and dfa_states.
- Then we add this node in dfa graph and add an edge between cur_state and to_state.
Else this set is empty, then
- This case is for dead state.
- If dead state not present in dfa, then we add the new state . And we make all transitions for all alphabets to itself so that the machine can never leave the dead state.
- We make transition of cur_state to this dead_state.
At last, all the states have been evaluated. So we will render/view this dfa graph.

Below is the full implementation:

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A tool to convert Non-deterministic Finite Automata to Deterministic Finite Automata

QuinnHerden/NFA-to-DFA

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README.md

This tool converts a Nondeterministic Finite Automata (NFA) into a Deterministic Finite Automata (DFA)

You may provide an input NFA in one of two ways.

Define the NFA via a CLI
- When invoking the program, provide a name for the output file
  - > python3 convert.py
- After walking you through the NFA definition, it will create a DFA in the form of .fa
Define the NFA via an input file
- When invoking the program, provide an input file of type .nfa
  - > python3 convert.py .nfa
- This will create a DFA in the form of .fa

The input .nfa should be formatted as follows:

The first line holds a set of one or more states the NFA could initially be on, delimted by spaces.
The second line holds a set of zero or more states that the NFA will accept, delimited by spaces.
The following lines will define the state transitions, as a tuple, for each state you wish to define.
- The first symbol is the current state.
- The second symbol will be the transition symbol.
- The following symbol (or symbols, delimted by spaces) will define the set of all states the automata will transition into.

1 2 3 1 a 1 1 b 2 2 a 2 3 2 b 2 3 a 3 3 b 2 3

About

A tool to convert Non-deterministic Finite Automata to Deterministic Finite Automata

Источник

Nfa to dfa python

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Converting Epsilon-NFA to DFA using Python and Graphviz

Deterministic Finite Automata (DFA)

Why the conversion?

Steps For Converting E-NFA to DFA :

For example,

Steps to convert :

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QuinnHerden/NFA-to-DFA

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