Automata Toolkit

Automata toolkit is a small library which contains tools to convert, minimize and visualize Regular Expressions, NFA and DFA.

Installation

Prerequisites

sudo apt install graphviz
pip install graphviz==0.16

Install

pip install automata-toolkit

Modules

regex_to_postfix

• This module converts any given regular expression to its equivalent postfix expression.
• The conversion from regular expression to postfix representation makes use of Shunting Yard Algorithm which arranges the expression from left to right based on the priority order of operations.
• Functions are as follows:
  • regex_to_postfix(reg_exp)
  • is_alphabet(c)

regex_to_nfa

• This module converts any given regular expression to its equivalent NFA.
• The conversion process is split into two parts: conversion of given expression to its postfix representation, and then using that postfix representation to create NFA.
• The conversion from regular expression to postfix representation makes use of Shunting Yard Algorithm which arranges the expression from left to right based on the priority order of operations.
• Then this postfix representation is converted to NFA using Thompson's construction algorithm, where characters from postfix representation are pushed into a queue, and operators are pushed into a stack. This stack is emptied and operations gets applied to the elements in the queue, once any opertor having lower or equal priority is about to get pushed into the stack.
• This is how it gives us an equivalent Non Deterministic Finite Automata.
• Functions are as follows:
  • regex_to_nfa(reg_exp)
  • get_alphabet_nfa(character, alphabets)
  • concat_nfa(nfa1, nfa2)
  • union_nfa(nfa1, nfa2)
  • cleene_star_nfa(nfa1)

nfa_to_dfa

• This module converts a given NFA to its equivalent DFA.
• Initially the epsilon enclosure of all the states is calculated and stored in a dictionary.
• Then, DFA[start_state] = Epsilon[NFA[start_state]]
• Then for calculating the transitions, the program first calculates the Epsilon enclosure of that state, and then checks for alphabet based state transitions.
• And this is how we obtain a deterministic finite state automata. Note that DFA obtained may not be the minimal one.
• Functions are as follows:
  • nfa_to_dfa(nfa)
  • get_epsilon_closure(nfa, dfa_states, state)

dfa_to_regex

• This module obtains a regular expression of the given DFA.
• Initially a GNFA is created by removing all the unreachable and dead states. Then we add a new start state with epsilon transition to original start stare, and connect all the original final states to a new final state using epsilon transitions.
• Now, all the states except the new initial, and final state, are removed one by one.
• While removing a particular state, the program first checks for any self loops. If multiple self loops exist, then program does a union of all these parallel transitions alphbets, and thus adds a cleene star over it.
• Then this cleen star value is concatenated to alphabets/strings of all the outgoing states.
• At the end, the incoming states'alphabets/strings are concatenated with the alphabets/strings of all the outgoing states, and then this state is removed.
• Upon iterating same procedure for all middle states, we arrive at a point when only initial and final states are left.
• The transition string between them is the required regular expression. Note that this regular expression might not be the simplified one.
• Functions are as follows:
  • dfa_to_regex(dfa)
  • union_regex(a, b)
  • concat_regex(a, b)
  • cleene_star_regex(a)
  • bracket(a)

dfa_to_efficient_dfa

• This module calculates the minimal equivalent DFA for a given DFA.
• It makes use of Myhill Nerode theorem or in simple words, table filling algorithm.
• Initially a states*states sized table is initialized with 0 value in all the cells.
• Then cell with final state, non-final state pairs are marked with 1.
• Then we check for unmarked state pairs, that whether any of their transition state pairs based on a particular alphabet value results in a marked cell.
• If yes, then mark that cell as 1, otherwise continue.
• This procedure is repeated multiple times until all the cell values achieve a stable state, i.e. they donot change.
• Now we merge all the unmarked state pairs. In order to merge this efficiently, this program uses union find data structure.
• After merging all the unmarked pairs, we obtain a minimal equivalent DFA.
• Functions are as follows:
  • dfa_to_efficient_dfa(dfa)

visual_utils

• This module contains functions to visualize the NFA or DFA using graphviz library.
• Functions are as follows:
  • draw_nfa(nfa, title="")
  • draw_dfa(dfa, title="")

Input Format

Regular Expression

• string
• Input regular expression should be syntactically correct

NFA

{
   "states": [
       <state_ids>,
       ...
   ],
   "initial_state": <initial_state_id>,
   "final_states": [
       <state_ids>,
       ...
   ],
   "alphabets": [
      "$",
       <alphabets>,
      ...
   ],
   "transition_function": {
       <state_id>: {
           <alphabet>: [
               <state_ids>,
           ],
           ... # transition for all alphabets shoud be present here
       },
       ...
   }
}

DFA

{
   "states": [
       "phi",
       <state_ids>,
       ...
    ],
    "initial_state": <state_id>,
    "final_states":[
       <state_ids>,
       ...
    ],
    "alphabets": [
       <alphabets>,
       ...
    ],
    "transition_function": {
        <state_id>: {
            <alphabet>: <state_id>,
            ... # transition for all alphabets shoud be present here. In case of no transition, alphabet must point to phi
        },
        ...
    },
    "reachable_states": [
        <state_ids>,
        ...
    ],
    "final_reachable_states": [
        <state_ids>,
        ...
    ],
}

Dependencies:

• Python 3
• Graphviz

Note: Visualizer can generate visualizations but will not get triggered in WSL (Windows Subsystem for Linux). This library has been tested in Ubuntu, Elementary OS.

Author

This project is developed by Shlok Pandey aka b30wulffz and is licensed under the MIT License.