finding good hash function for languages accepted by finite state automata

Question

I'm working on project in Java (but I think it doesn't depend on the language) where I'm generating small (4 states max) nondeterministic finite state automata on binary alphabet and I have to check fast the generated automaton for equivalence with the previous ones. Therefore, I have to use some good hash function, to avoid compairing with too many automatas.

My first thought was doing a DFS on the transitions and finding all the accepted words until length max. 5 and then I map the set of accepted words to a 64-bit long (the amount of binary words of length max. 5). But it seems to produce too many collisions on NFAs with 4 states. Increasing the length results in making the computing of the hash code too slow for practical use.

Another approach was having a set of words and testing which of them the automaton accepts but finding the right ones, I think, isn't that trivial.

Do you have any idea how to improve the hash function to avoid too many collisions without a significant loss of speed?

Thanks in advance

Answer 1

I was thinking further (thanks @justhalf and @templatetypedef) and I have an idea - an injective function of any NFA (or more precisely, language accepted by it) to integers - Let's have an NFA A. Let's construct minimal DFA A_min accepting the same language with complete delta-function. As a consequence of Myhill-Nerode theorem, this automaton should be unambiguous except isomorphism. Do a BFS from the initial state giving priority to the edges(transitions) based on some fixed order of characters in the alphabet (for example first 0, second 1). And renumber the states based on the order of visiting. Now we have a canonical minimal DFA and we can map the incidence matrix of states to an integer and append enumeration of final states (or better make a tuple, to avoid collision). This integer could be then used for deciding equivalence of two NFAs. Do you think, it is ok or have any other idea?