Tag Archives: dfa

DFA state minimization

If you follow the generic instruction for converting a NFA to a DFA presented in my earlier post , the DFA you generate will work, but it will not be optimal. It will have too many states opposed to the bare minimum needed to do it’s job properly. This tends to happen more frequently if you give it an input of an expression that in some respect have redundancy. A prime example of this is the expression a+b+|ab . The |ab part is of course, totally useless. But the subset construction algorithm doesn’t know that.

To compare, here is the NFA generated by thompsons construction:

Here is the non-minimized DFA that the subset construction algorithm will generate:

And here is the minimum DFA required to do the job properly:

Quite a difference! So, how do we do this? There are several ways of accomplishing this, some more complex than others. I’m going to give you a working algorithm which admittedly isn’t the fastest. The faster versions are more complex, and would ruin the idea of explaining the algorithm.

For programmers

I’m willing to bet you’re like me, a programmer and not a mathematician. You can describe this algorithm in math terms to no end, and I’m willing to bet that most people are going to miss out on what is really not the most complex algorithm in the world. If you google this you’re going to get the algorithm described as some function which partitions states so that this and that is true. Perhaps this is jealousy on my part since I feel left out of the cool crowd with the greek symbols and all, but I think there are better ways of explaining this stuff.

The algorithm

It’s a bit backwards, but what we’re going to do is to assume that all states can be merged together and then rule out those that can’t. This is going to be represented by a table, where empty squares are those that we cannot tell apart. So, for now, this table looks like this:

+-----+
|s1|  |
+--+--+--+
|s2|  |  |
+--+--+--+--+
|s3|  |  |  |
+--+--+--+--+--+
|s4|  |  |  |  |
+--+--+--+--+--+
   |s0|s1|s2|s3|
   +--+--+--+--+

This table is going to be filled later on. There’s no reason for a full table, we only want the distinct pairs of states.

What we are interested in is to sort the states into groups that may be merged into a single state without altering the functionality of the DFA. So, what we really need to do is to find out how to find these state groups normally referred to as nondistinguishable states. The first rule of this is:

A state that is an accepting state may never be in the same group as a state which is not an accepting state.

So, in our graphs we may never merge a double-circle state with a single circle state. Let’s apply this to our table. We end up with this:

+-----+
|s1|  |
+--+--+--+
|s2|X |X |
+--+--+--+--+
|s3|  |  |X |
+--+--+--+--+--+
|s4|X |X |  |X |
+--+--+--+--+--+
   |s0|s1|s2|s3|
   +--+--+--+--+

We’re not out of the woods yet, still more work to do. There is another rule at work here. It’s easiest to express this in pseudocode. You will apply the algorithm repeatedly until the table stops changing. If you complete a full iteration without any entry getting filled in, you are done.

do {
for (int i = 0; i < numStates; ++i) 
{
  for (int j = i + 1; j < numStates; ++j)
  {
    if (table[i,j].IsEmpty) 
    {
      foreach (var inputsymbol in allValidInputs)
      {
        if (one of the states has a transition but the other one doesn't)
        {
          table[i,j] = X;
        }
        else if (both states have a transition) 
        {
          int iTarget = state at other end of transition from state i;
          int jTarget = state at other end of transition from state j;
          if (!table[iTarget, jTarget].IsEmpty)
          {
            table[i,j] = X;
          }
         }
        else
        {
           // For clarity, if none of the states have transitions
           // on this symbol, don't change the table
        }
      }
    }
  } 
}
} while (table.WasChanged)

What this does is to check if there is any significant difference by checking where the transitions go, and filling in the table as it goes along. Applying this changes the table to look like this:

+-----+
|s1|X |
+--+--+--+
|s2|X |X |
+--+--+--+--+
|s3|X |  |X |
+--+--+--+--+--+
|s4|X |X |  |X |
+--+--+--+--+--+
   |s0|s1|s2|s3|
   +--+--+--+--+

In this example, iterating the algorithm won’t change the table which means we are done. The states s1 and s3 can be merged into one state and s2 and s4 can be merged into another. To illustrate, here is the two nondistinct state groups marked out:

A final caveat is that you might get long chains of states. Say that s0 and s1 will be merged along with s1 and s2, which really means you need to merge s0, s1 and s2 into another state. This is pretty trivial, and I will not elaborate further on this, an example of this can be found in the class DFA of my parser library Piglet , which is available on GitHub.

As a final note, it can be proven that this algorithm generates the smallest possible DFA that accepts the given grammar. All in a few lines of code. Try the algorithm out yourself at the regex visualizer – make a redundant expression and uncheck the Minimize checkbox.

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Converting a DFA to a table, Regular expressions part 3

So, last time we explored the wonderful world of regular expressions we ended on a nice looking graph representing our deterministic machine that recognized the regular expression that generated it. That’s all fine, but in reality we’d like a table representation for our machine. Tables offer some significant benefits, most prominently they’re really fast with O(1) access times. They are, however, not very compact. That flaw can be remedied with table compression though, which I will not delve into in this post. Mostly because I do not completely understand it yet.

Anyway, here’s where we ended up last time:

First order of business is to assign state numbers to each state. It’s really arbitrary what numbers you give the states as long as they are unique, but keeping with tradition we’re going to assign 0 to the starting state. The graph now looks like this:

Now we can create a table. On the X-axis we will use the character input, on the Y axis we will use the state number. For each state, if there is a transition to another state using a given input character, we fill in the state number to go to in the corresponding column. If the state is an accept state, add a special ACC for accept. The table will look like this:

           INPUT 
------+---+---+---+---+
STATE | a | b | c | d |
------+---+---+---+---+
  0   | 1 | 2 |   |   |
------+---+---+---+---+
  1   | 1 | 2 |   |   |
------+---+---+---+---+
  2   | 1 | 3 |   |   |
------+---+---+---+---+
  3   | 2 | 3 | 4 |   |
------+---+---+---+---+
  4   |   |   |   |ACC|
------+---+---+---+---+

Apply the pseudocode algorithm outlined in the first post to this table, and you’ve got your very own regular expression engine!

Wrapping up, there’s of course more to do. This engine does not handle things such as capture groups, lookahead and all that advanced stuff. But it isn’t that hard to add, and nothing of what’s been said will not hold true for the more full-featured engines. Another option is to never do the table generation stuff and instead emit actual IL code that will execute the DFA straight up, which would be incredibly cool stuff. As I said earlier these tables end to be very sparse, since the input requited normally is at least 256 different characters and only a few are legal for each state. This means that there’s a lot to do on the compression side as well.

If there’s any questions or requests for elaborations on all of this, I’ll happily accept and answer. Otherwise, I think it’s time to continue with regular expressions big brother – parsers.

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Converting a NFA to a DFA by subset construction, Regular expressions part 2

Welcome again to the understanding regular expressions series, the previous part is located here if you missed it. Much like the dragon book you probably really can’t read these things out of order. Last time we ended with a really nice non deterministic finite automata, but in order to make it work in a deterministic fashion we will need to transform it. Again, a deterministic machine is the mirror image of a nondeterministic machine, it will achieve exactly the same thing but in a different fashion.

For reference, a nondeterministic finite automata:

  • One entry state
  • May contain epsilon transitions
  • One input may lead to several new states
  • One accepting state

It’s cousin the deterministic finite automata (DFA) has these properties:

  • One entry state
  • May not contain epsilon transition
  • One input may only lead to one state
  • May have several accepting states

The key thing about the DFA is that we can easily convert it to a table and run it through our very simple engine, giving us very nice run-times of O(n). So how to convert this machine? The algorithm used here is called closure and it is very cool indeed. It is also used in a different variation for parser generation, which is why it’s almost a must to understand regular expressions before starting to truly understand parser generators. Here it is

Given a list of states the closure set is all the sets that can be reached without any input, including the input states themselves

The function will have an input of one or more states, and return a set of states which will at least contain the input states. In pseudo C# code it looks like this. I myself much prefer pseudocode that is much closer to real code – it leaves much less to the imagination of the reader.

Set<State> Closure(List<State> inputStates)
{
    var output = new Set<State>();
    output.AddRange(inputStates);

    // Keeps states we are going to add later
    while (true) 
    {
        var statesToAdd = new Set<State>();
        foreach (var state in output)
        {
            foreach (var edge in output.EdgesGoingOut)
            {
                if (edge.IsEpsilonEdge)
                {
                    if (!output.Contains(statesToAdd)) 
                    {
                        statesToAdd.Add(edge.DestinationState);
                    }
                }
            }
        }
        if (!statesToAdd.Any())
            break; // Exit loop if there are no states to add
        output.UnionWith(statesToAdd); // Add all states to output
    }
    return output;
}

It’s helpful to visualize this using graphs. If we return to the regular expression (a|b)+bcd , the corresponding NFA looks like this

If we are going to compute closure for state 0 we will traverse the edges to 1 and 3, but not to 5, because that edge goes the wrong way. Then there are nowhere else to go, since the remaining transitions require inputs “a” or “b” respectively. So the closure of state 0 is {0, 1, 3} . Note that the input state is always included in the closure list. I like to think of this as a flood fill, because it looks like that if you have many epsilon transitions. This new set is actually the start of our deterministic machine! Yay. Lets begin to draw the machine to watch it grow.


Our first state

We now need to introduce a new operation called goto . I know, goto is a bad word these days. The algorithm apologizes profoundly for the association. The goto algorithm is real easy.

Given a set of states and an input symbol find all the states reachable by traversing the edges that require the given input symbol. Then apply the closure algorithm on the output

Think of this algorithm as move one step then flood fill again. In C# pseudocode it looks like this

List<State> Goto(List<State> inputState, char inputCharacter)
{
     var output = new Set<State>();
    foreach (var state in inputState)
    {
        foreach (var edge in state.EdgesGoingOut)
        {
            if (edge.Label == inputCharacter)
            {
                output.Add(edge.DestinationState);
            }
        }
    }
    return Closure(output);
}

So, given our first closure set {0, 1, 3} lets apply Goto on it for every input character. Nothing will happen except for characters “a” and “b”. Lets explore character “a”. The only transition we get to cross is the one going from state 3 to 4. So 4 is the only result of the Goto. We now apply closure to it, which gives us the list {4, 5, 6, 0, 1, 3} when we follow every arrow which is an epsilon. The input for “b” is very similar and yields state set {2, 5, 6, 0, 1, 3}. The next step is adding our new states to the new diagram with the input we gave to the Goto as the edge label. Now the graph looks like this:

You then repeat this again. Applying Goto on the state set {4, 5, 6, 0, 1, 3} and input “a” gives us exacly the same result as previously. This means that this state has an edge going to itself. Applying Goto on the same states with input “b” gives a new state set that we haven’t seen before {2, 5, 6, 7, 0, 1, 3}. Lets add this to our graph:

If we apply the same algorithm to the state list {2, 5, 6, 0, 1, 3} we find that it will only yield sets that we have already added to the graph, but it will give us new transitions. The graph now looks like this:

There is one unexplored set to apply goto on {2, 5, 6, 7, 0, 1, 3}. Doing this with “a” gets a transition back to a previously known state. “b” gives a transition back to the same state list. “c” gives a transition to a brand new set containing only state 8. Our graph now looks like this:

Still one new set to explore! Only one input gives a new set “d” gives a set containing only state 9. State 9 is an accepting state, so we add this to the graph.

Applying goto on this new set gives us nothing at all. We are done. Almost by magic we have transformed our non-deteministic machine into a deterministic machine. The reader is invited to try the same input as on the nondeterministic variant. It will respond the same. There is really only one bit left to do, we need to make this into a table that will run with our regular expression engine.

Next part can be found here

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