In our previous blog, we saw that it is possible to provide multiple outputs, each one for a specific use case. So far, the two binary use cases, winning and winner, and the categorical use case, move, do not have any mutual dependencies between them: Their result are all derived from the original inputs.

Let’s now see, if stacking the layers may help to reduce the model’s complexity whilst still getting the same outputs with the same quality (accuracy = 1.0).

As we have seen in the blog posts before, determining whether a board has a winner, determining which player the winner is, and with which move the winner won the game can all be done using neural networks. These networks only require Dense layers. However, each case has an inherent complexity, so a “minimal” number of units in these layers (and the number of layers) are necessary:

This suggests a kind of “complexity ranking” for the three challenges: The task with the highest complexity is the “winning problem”, followed by the “move problem”. Finally, the “winner problem” is the easiest to solve, because it may only be answered accurately with a single small Dense layer.

Using the principle of multi-output, you may ask, whether it is possible to retrieve all three pieces of information within a single model. Well, let us try this:

After having implemented neural networks for determining whether a Tic-Tac-Toe board has a winner or not, and which player the winner is, it is now time to have a look at which is the winning move.

Looking at this case, you will notice that there are only three possible cases:

• The winner is clear after the third move (of either “X” or “O”),
• the winner is determined after the fourth move (of either “X” or “O”), or
• the winner is determined after the fifth move (only “X” may achieve that state, as there is an odd number of fields available on the board).

Additionally, there is the special case that no winner exists, which we deserved the special identifier of “nine moves” (being the last single digit value). So, in total there are four possible states. There can only be one single state valid at a time (“one-hot case”).

However, this also means that we need to adjust our data preparation:

After having determined if a board has a winner using TensorFlow in the previous blog post, let us tackle a very similar question: Who is the winner?

Again this is a binary decision: Either X (“0”) or O (“1”) may win a board. It is also possible that we run into a tie, and therefore no one is a winner. For the sake of simplicity, let’s then still say “0” as a result. If the board really has a winner, we have already found a high-accuracy neural network to decide that in first place.

Using the same imports and setups as before, we again prepare our data. This time, we are interested in the information about the winner for our labels:

As a first learning task for using AI with Tic-Tac-Toe, let us take the task of determining whether a board is a winning one or not (i.e. either X or O has won the game). We cannot directly tell the neural network what the rules are for an assignment to be winning. Instead, we need to train it by examples. For that we already have prepared some data in a previous post.

The idea is a to train a TensorFlow model with several Dense layers. By varying the model’s configuration, we want to determine how complex (e.g. how many parameters we need) such a model needs to be to fulfill this requirement. Also looking at the accuracy will be an interesting topic.

As an information in advance: The computation were done using TensorFlow 2.13.1 on a Windows WSL2 machine having an NVIDIA Geforce RTX 4060 (8GB) installed. Mixed Precision was not enabled.

As usual you may download the entire example using the following link:

  tictactoetf.zip (9.6 KiB, 1,121 hits)

Let’s get started…

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You might still remember the times when you played Tic-Tac-Toe (a.k.a. noughts and crosses) in your childhood:

In a series of blog posts I want to apply neural networks on this well-known game. But before we may do that, we need to do some preparations.

Eventually, we want to teach a neural network to determine:

• if a board has a winner,
• who the winner is, and
• after which move the winner has won the board.

To be able to properly describe such a board, we need to define an order on the board. I decided to take the order “top-left to bottom-right” like this:

You may also apply a different, more sophisticated scheme for defining the position, but let’s keep it simple.

With this nomenclature, we now can describe the board mentioned initially using a set of integers: 1,6,9,5,4,7,3,8,2. Even more interesting this become, if you consider this not being a set but a list (including order). That is important, because a board may have two “winners”, depending on which player came first. Note that

• by definition, let us assume that X starts the game,
• every other round, it is the other’s player to move (even positions are ‘O’ moves, odd positions are ‘X’ moves), and
• that there are always exactly nine positions in total until the board is fully populated (also called an assignment).

Moreover, the list must not have any duplicates: 1,1,1,1,1,1,1,1,2 may also be a list of numbers, but this does not describe a proper Tic-Tac-Toe game, because the position 1 appears more than once. Therefore, the generation of the list represents a system of choosing without repetition. That brings us to another aspect: We first need to determine, if a list of integers is a valid board after all.

[to be continued on the next page]