DFA Synchronizing Word Finder

Description

This program checks if a given deterministic finite automaton (DFA) has a synchronizing word, and if so, it outputs the shortest such word. A synchronizing word is a sequence of input symbols that, when applied to any state of the DFA, leads to a specific state. If the DFA is not synchronizing, the program outputs the string "automaton isn't synchronizing". The program uses the fact that shortest synchronizing word of a DFA is determined by the shortest path in that DFA's power automaton.

Input File Format

The input file must describe a valid DFA and follow this format:

1. First Line: An integer n representing the number of states in the DFA.
2. Second Line: A string of k symbols representing the alphabet, with all symbols listed consecutively without spaces.
3. Third Line: A description of the transition function in the following format:
   • [list0, list1, ... listn-1], where each listi is a k-element list of tuples (c, m).
   • c is a Char representing a symbol from the alphabet, and m is an integer representing the target state (0 to n-1).
   • This means that the transition function sends state i to state m using symbol c (i.e., transitionFunction(i, c) = m).

Example Input

3 ab [[('a', 1), ('b',1)], [('a', 1), ('b', 2)], [('a', 0), ('b', 1)]]

Example Explanation:

• The DFA has 3 states (0, 1, 2).
• The alphabet consists of two symbols: a and b.
• The transition function is defined as:
  • From state 0: a leads to 1, b leads to 1
  • From state 1: a leads to 1, b leads to 2
  • From state 2: a leads to 0, b leads to 1

Output

• If the DFA has a synchronizing word, the program prints the shortest synchronizing word.
• If the DFA does not have a synchronizing word, the program prints "automaton isn't synchronizing".

Notes

• The main function and file handling assume that states are numbered from 0 to n-1. However, the functions implemented to find the shortest synchronizing word only require that the type representing states is comparable.