How useful is Turing completeness? are neural nets turing complete?

While reading some papers about the Turing completeness of recurrent neural nets (for example: Turing computability with neural nets, Hava T. Siegelmann and Eduardo D. Sontag, 1991), I got the feeling that the proof which was given there was not really that practical. For example the referenced paper needs a neural network which neuron activity must be of infinity exactness (to reliable represent any rational number). Other proofs need a neural network of infinite size. Clearly, that is not really that practical.

But I started to wonder now if it does make sense at all to ask for Turing completeness. By the strict definition, no computer system nowadays is Turing complete because none of them will be able to simulate the infinite tape.

Interestingly, programming language specification leaves it most often open if they are turing complete or not. It all boils down to the question if they will always be able to allocate more memory and if the function call stack size is infinite. Most specification don't really specify this. Of course all available implementations are limited here, so all practical implementations of programming languages are not Turing complete.

So, what you can say is that all computer systems are just equally powerful as finite state machines and not more.

And that brings me to the question: How useful is the term Turing complete at all?

And back to neural nets: For any practical implementation of a neural net (including our own brain), they will not be able to represent an infinite number of states, i.e. by the strict definition of Turing completeness, they are not Turing complete. So does the question if neural nets are Turing complete make sense at all?

The question if they are as powerful as finite state machines was answered already much earlier (1954 by Minsky, the answer of course: yes) and also seems easier to answer. I.e., at least in theory, that was already the proof that they are as powerful as any computer.

Some other questions which are more about what I really want to know:

• Is there any theoretical term which can say something more specific about the computational power of a computer? (given its limited memory space)

• How can you compare the computational power of practical implementations of neural nets with computers? (Turing-completeness is not useful as argumented above.)

Solution 1:

The point of stating that a mathematical model is Turing Complete is to reveal the capability of the model to perform any calculation, given a sufficient amount of resources (i.e. infinite), not to show whether a specific implementation of a model does have those resources. Non-Turing complete models would not be able to handle a specific set of calculations, even with enough resources, something that reveals a difference in the way the two models operate, even when they have limited resources. Of course, to prove this property, you have to do have to assume that the models are able to use an infinite amount of resources, but this property of a model is relevant even when resources are limited.

Solution 2:

When modern computers are said to be Turing Complete there is an unspoken exception for the infinite storage device Turing described, which is obviously an impossibilty on a finite physical computation device. If a computation device can do everything a Turing machine can do (infinite storage not withstanding) it is Turing complete for all practical intents and purposes. By this less strict definition of Turing completeness, yes, its possible that many neural networks are Turing complete.

It is of course possible to create one that is not Turing complete.

Solution 3:

Turing-completeness of recurrent neural networks could mean: the (finite) transition tables of each and every Turing machine (with a finite-state head and an infinite tape) can be modelled by a finite recurrent neural network (finitely many neurons with finitely many states, especially only two states). The transition tables define three functions:

• next-state(current-state,current-symbol)

• next-symbol(current-state, current-symbol)

• direction(current-state,current-symbol)

This is how a recurrent neural network may perform this task (just a very raw sketch):

The green neurons read the symbol in the current cell (in binary representation), the gray neurons (initally mute) determine the current state, the red neurons write the new symbol to the current cell, the yellow neurons determine whether to go left or right. The blue neurons are the inner neurons (initially mute).

The claim is, that for each and every Turing machine there is such a recurrent neural network.

I wonder if there is a systematic way to construct such a network from given transition tables.

Solution 4:

Regular feedforward neural networks are not turing complete. They are, in effect, equivalent to a single complicated mathematical function that may do quite a lot of calculations but doesn't have any ability perform looping or other control flow operations.

However, if you wire up a neural network with some way to access a stateful environment then it can be be made into a turing complete machine.

As a most trivial example, you could recreate a classic-style Turing machine where:

• the input to the neural network is the value on the tape and the previous state
• the output of the neural network is the next state and the action

You could then train the neural network to emulate the actions of any desired turing machine state table / configuration (perhaps by supervised learning on the actions of another turing machine?)

Note: The idea of running a feedforward net repeatedly with some form of state feedback is essentially equivalent to a recurrent neural network. So you can think of a recurrent neural network plus the logic that runs it repeatedly as being Turing complete. You need the extra logic (over and above the network itself) to ensure Turing completeness because it is necessary to handle things like termination, repetition and IO.

Solution 5:

To partially address your second question:

Neural Networks have the property of being universal approximators - that is, they can approximate any function to an arbitrary degree of precision. It's the "degree of precision" part that keeps the Neural Network from needing to be infinite.