01 Sep 2021

Abstract

To prove that any "artificial neural network" is just a statistically programmed (virtual) machine whose model software is a derivative work of the source dataset used during its "training", we provide a small suite of tools to assemble and program such machines and a decompiler that reconstruct the source dataset from the cryptic matrices that constitute the software executed by them.
Finally we test the suite on the classic MNIST dataset and compare the decompiled dataset with the original one.

Keywords

Artificial Intelligence, Machine Learning, Artificial Neural Networks, Microsoft, GitHub Copilot, Python, Statistical Programming, Vector Mapping Machine

Introduction

Despite the hazy anthropomorphic language and the unprofessional narrative, in the last few years the field of Informatics that goes under the names of "Artificial Intelligence" and "Machine Learning" established a long record of self-appointed successes. Behind the hype and the sci-fi depictions of marvelous futures where virtual humans will explore the Universe on behalf of a mostly extincted humanity spread to slow down any effective solution that could save the lifes of billions of real people, there are few misunderstood (and often misused) techniques that actually bring some innovation to the table.

The most iconic of such innovations is improperly named "artificial neural network", after a wrong model of the human brain that inspired their creation back in the sixties. On top of this class of tools, composed and programmed in a multitude of ways, several useful services are being commoditized by large corporations with access to huge dataset, money and computational power.

Among them, Microsoft has recently released GitHub Copilot, a Service as a Software Substitute that suggests to hasty programmers code snippets extracted and automatically adapted from the vast codebase uploaded on GitHub. From the very beginning Microsoft recognised that Copilot can recite verbatim code from the GitHub repositories in 0.1% of the cases, but it didn't take long before a developer have been suggested a particularly valuable piece of free software with a permissive license instead of the strong copyleft decided by the authors.

After the obvious backslash from the Free Software community, several lobbists, lawyers and even some politicians run to justify the poor company, basically arguing that using copyrighted work for "machine learning" is actually fair, so developers should not annoy the corporation.

As always, the problem is that people who do not really understand how "machine learning" works, will trust and spread the words of such misinformed "experts" and this will inevitably hurt the exploited "humans in the loop".

In fact, Microsoft did not use the huge amount of high quality source code of its own products to train Copilot, but the free software and the open source software uploaded on GitHub. This despite the fact that only a tiny minority of projects on GitHub go through the long reviews and the quality assurances that a company like Microsoft can afford to set up, and since Heartbleed we all know how the " Linus's law" is just a bedtime story for naive kids.

There is no "learning" in "artificial neural network"

Most of the arguments in support of Microsoft's new service comes from people who really do not understand the inner working of a "artificial neural network". An ANN is just a virtual machine designed to be programmed statistically, by examples.

Such kind of virtual machines are composed of tiny devices that are improperly named "artificial neurons" or "perceptrons", but they are simply vector reducers that, given a non-linear differentiable function, fold a vector from a certain N-dimensional space into a 1-dimensional one.

A vector, as you might know, is just a list of numbers that can be interpreted by a human as a descriptions of something. For example I might be described by (40, 20, 16), respectively my age, the number of years I worked as a programmer, and the number of programming languages I know.

The inner working of a vector reducer is trivial: you just compute the dot product between the input vector and a parametric one and then apply the differentiable function to get the magnitude of the output vector, but the interesting bit is how you can statistically program the parametric vector by distributing the observed errors on a known dataset so that, if you are lucky enough, after some iterations over the dataset the vector reducer might approximate the desired output for a given vector.

Such vector reducers can be easily composed in a variety of topologies and programmed in a variety of ways, so that a whole "artificial neural network" can be statistically programmed ("trained", in AI/ML parlance) to approximate one of the possible translations from a vector space to a different one.

Yet, there is no "learning" ongoing: just the iterative tuning of parametric vectors to approximate a certain output. Since the various entries of the parametric vectors are usually called "weights", people who try to avoid the anthropomorphic jargon of AI usually call the process "calibration", but as we will see, it's important to use "statistical programming" instead to clarify to untrained people the components, the processes, the roles and the responsibilities that characterize such technology.

Indeed the only "artificial intelligence" in a vector mapping machine is the one that people are trained to imagine through powerful marketing, confusing propaganda and a poor technical language.

Vector Mapping Machines: engineering and programming

Vector mapping machines (improperly known as "artificial neural network") have been usually implemented as software, so they are basically specialized (and opaque) virtual machines.

In the attempt to preserve the "machine intelligence" pipedream, the design of such machines has been so much conflated with their programming, that we needed to invent hyperbolic definitions such as "hyper-parameters" to let "data scientist" distinguish and comunicate about the design of a specific machine, its programming (imprerly named "training") and its inner working at runtime.

However, if we get rid of the sci-fi narrative and talk about opaque virtual machine specialized in approximated mappings between vector spaces, we can build on the usual language of Informatics to describe and understand these tools and their impact.

Infact, designing a vector mapping machine is an engineering act composed of several steps such as deciding the number and kinds of filters, their connections, the differentiable function to use, the initialization process of the parametric vectors of each vector reducer and so on.

The output of such engineering process is a machine that can be built on hardware or emulated through software, but it's just as useful as a computer without any software on it.

It's important to note tha when such virtual machine is emulated throught software, the programming of such emulator obviously grant to the authors a copyright on it, just like with the programming of any other virtual machine. But like any other virtual machine, the copyright holder of the software emulating the machine cannot claim rights on the software run by such machine.

In fact, despite conflated by primitive jergon and practices, the engineering of vector mapping machines is a totally distinct and different process from their programming.

The vector mapping machine is a different intellectual artifact from the program it runs and constitute a creative work on its own. Indeed the same machine can be programmed to produce different vector mapping task by providing different source dataset (with the exact same dimensionality on the input and output vectors).

In other words, an "artificial neural network" does not learn anything and it's not a network of neurons or anything like that. It does not understand anything about the data and the output vectors has no inherent meaning: its semantic is always attributed by humans according to their insights about the statistical program they uploaded into the vector mapping machine.

As we will see, to program of a vector mapping machine (aka ANN), you run it to compute an output and then automatically adjusts the parametric vectors of its reducers backwards, from the last filter to the first one, so that after each iteration, its output is a bit more similar to the intended one. It's an iterative process based on examples whose recurrent relations are "absorbed" by the vector reducers' parametric vectors and then reproduced at runtime.

That's why it's statistical programming: you start with a source dataset (improperly named "training set" in the AI/ML parlance) and, after a compilation process that is specifically designed for that specific virtual machine, you get a binary that such machine can run.

While such binary is not constituted by a sequence of instructions, it's still just an algorithmic transformation of the original source that, despite being expressed as cryptic matrices, still contains all the relevant features of the sources.

The software run by a vector mapping machine is not a "model", since it does not give any insight about the relations it absorbed during its statistical programming. Instead it is just a software like any other that describes a rigorous (if unknown) process that a specific machine has to follow to automatically and deterministically compute (or rather approximate) a desired output from a given input.

This means for example, that Copilot's "model" is an algorithmic derivative work of all the public repositories uploaded to GitHub and as such, Microsoft should comply with all network-copyleft that requires such derivative work to be distributed to the users.

A Vector Mapping Machine in Python

To show how the "model" of an "artificial neural network" contains substantial and valuable parts of the sources used to program it, we assemble and program a simple emulator of a VMM engineered to compute a classic classification task: the handwritten digits "recognition" from the MNIST database.

The code is overly simple, avoiding any external dependency except for the Python standard libary and avoiding fancy mathematical optimizations that would make the process more difficult to follow, but it shows how to compile the source dataset into a working binary, how to check its accuracy and how to decompile it.

While this is a pretty simple "deep learning" proof of concept, it's always possible (if expensive) to build a compiler/decompiler suite for any "machine learning" system, including OpenAI and the specific vector mapping machines operating GitHub's Copilot.

Assembling a Vector Mapping Machine

The vmm.py module describes the building blocks of a vector mapping machine.

We define VectorMapper any device that turns a vector from a certain vector space (of inputSize dimension) into another vector from another space (of outputSize dimension).

An example is the VectorReducer, a simple device that constitutes the basic building block of vector mapping machines: it's the mythical "perceptron" also known as "artificial neuron" in AI/ML parlance.
For the sake of simplicity, it embeds the classic sigmoid "transfer" function.

The ParallelVectorMapper is a vector mapper that compose several VectorReducer and run them on the same input vector (potentially, in parallel), returning a multi-dimensional vector composed of their output.

The VectorMappingMachine is just a pipeline of ParallelVectorMapper that can map a vector from the input space in a vector from the output space by feeding to each filter (aka "layer" in AI/ML parlance) in the pipeline the output vector returned by the preceding one.

Once we have designed our VMM of choice for the task, we can assemble it through assemble.py:

$ ./assemble.py classifier-784-32-20-10.vm 784 32 20 10

will create a "deep learning" vector mapping machine named classifier-784-32-20-10.vm in your current directory.

Once the virtual machine is ready, we need a source of samples to program it.

Programming a VMM

To program a Vector Mapping Machine you need a source dataset.

Google, for example, uses the personal data extracted from unaware people to program its marketing VRMs just like Microsoft used the creative work of tons of unaware developers to create GitHub Copilot.

In our case we will simply use the MNIST dataset.

Once you have a source dataset, you often need to turn it into a set of vectors that you can use to actually program the machine: that's where several processes such as dimensionality reduction, word embeddings and so on take place.

In the previous example we created a "classifier" that can map a vector space with 784 dimensions to a vector space with 10 dimensions: it could be programmed with any source dataset that match such definition (784 numeric input features and 10 output categories, with one-hot encoding for simplicity) but we are going to use the classic MNIST dataset consisting of 60000 grey-scaled 28x28 images of handwritten digits.

Since the MNIST database is distributed in a custom binary form and we want to see how to get back readable sources from the "model" of an "artificial neural network", the mnist/ folder contains a mnist2csv.py script that converts the included dataset into a human readable CSV: the first 784 integers of each row correspond to the pixels of a 28x28 raster image while the digit in the last column corresponds to the intended digit.

The command

$ ( cd mnist && ./mnist2csv.py ) 

will create source.csv and test.csv in the mnist/ directory.

A very simple plot.py script (depending on matplotlib) is provided to let you inspect the digit at each sample if you feel so inclined.

Compiling the sources

Now that we have a source dataset that we can legally use to program our "digits classifier", we just need to compile it into a binary.

The compile.py script produce a binary that can be run by a target Vector Mapping Machine from a source dataset in CSV format. The executable format is described by the classes in ef.py.

The loadSamples procedure loads the rows of the CSV file into a list of Sample, normalizing each column and collecting useful SourceStats such as the maximum and minimum value of each input column and the ordered list of output categories. Such statistics will be used at runtime by the VMM to convert the input CSV rows provided by the users to input vectors composed of float in the 0 - 1 range.

The compile procedure is where the parametric vectors of the every VectorReducer that constitute the machine are iteratively changed to better approximate the desired output.

Technically speaking, it executes a classic stochastic gradient descent with back propagation of errors. Simply put, you iterate a few times (aka "epochs") over your source dataset, try to see what output your VMM produce for each sample, compute the difference with the expected output and then go backwards filter by filter, modifying a little the parametric vector of each reducer so that it would produce a slightly better output given such input.

There is no magic here, no one is "learning" anything: we are just iteratively changing the numbers that each vector reducer multiply for the elements of the input vector so that it will produce an output more similar to the one we want to obtain.

As we will see later, it's worth noticing that the compile procedure does not throw away all informations about how each sample impacts the final executable, but record a few logs during the last "epoch".

So to compile the source dataset you just need to run

$ ./compile.py classifier-784-32-20-10.vm mnist/source.csv mnist/digit-reader.bin

After a while (it may take a few of hours on slow machines), you'll get your program in mnist/digit-reader.bin.

Then you can check the program accuracy by running

$ ./test.py classifier-784-32-20-10.vm mnist/digit-reader.bin mnist/test.csv
Correct "predictions": 9530/10000 (95.3%)

Did our "deep learning" "neural network" "learnt" how to "read" handwritten digits?

No.

We just statistically programmed a Vector Mapping Machine to fold a vector space with 784 dimensions into a vector space with 10 dimensions. We did so by leveraging the same kind of regularieties among the source dataset that the human mind exploits to map a symbol to a digit when such dataset is plotted into a raster screen, since such regularities are obviously prominent into a database designed to contain "handwritten digits" and they characterize our samples.

Indeed such characteristic regularities are statistically relevant into the source dataset so they get "absorbed" into the weights of the reducers that compose the machine during the compilation.

These are fundamental aspects to understand and remember when thinking about these kind of technologies.

First, the VMM doesn't learn anything: it always reproduces a vector reduction that is statistically relevant into the source dataset, just because we statistically programmed it to do so. In this case, such reduction is possible because humans designed a regular set of 10 signs to comunicate digits and the MNIST source dataset constains 60 thousands of such signs drawn by humans for humans. To comunicate with other humans, such humans had to reproduce signs similar to the ones that they would understand and this produce the regularities that can be "absorbed" and reproduced by a Vector Mapping Machine.
In other words, there is no intelligence at work here except for the human one.

Second, the regularities "absorbed" by the weights of the vector reducers are those that characterize the creative work done by humans: the creative work that produced the symbols we use as numerical digits and the creative work of the people who actually draw those digits and the work of people who collected them into the source dataset. We can legally use these works for different reasons, but at the end of the day, all of such reasons are based on the consent of the people who contributed to them.

This is relevant because, whenever the source dataset used to program a VMM is not composed of data emitted by observed objects or subject (such as personal data or data about a physical phenomenon), the executable produced during the compilation is always a derivative work of the source dataset, just like a binary is a derivative work of the source code used to compile it.

Moreover, in this second case, the program executed by a VMM/ANN incorporates exactly those characteristics of the source dataset that constitutes the creative value of the work.

Decompiling (the software run by) an "artificial neural network"

To prove that an "artificial neural network" contains the source dataset used to "train" it in a cryptic (but not encrypted) form, we will decompile the executable produced and get back the source dataset with a simple command:

$ ./decompile.py mnist/digit-reader.bin mnist/digit-reader.sources.csv

This will create a new CSV file named digit-reader.sources.csv that can be compared to the MNIST dataset in mnist/sources.csv.

In our example, 59993 decompiled samples perfectly match the one from the original MNIST dataset (~99.99%), and only seven images does not match perfectly the original ones, but the differences are imperceptible to the eye. Infact, in those images, the slight variations on the gray scale of some pixels are due to rounding errors during floating point operations. Such errors grow with the raise of the "prediction accuracy" of the executable, but they are just implementations issues that could be fixed by using arbitrary precision floating-point arithmetic libraries.

So we can see that the program executed by the "artificial neural network" actually contains the whole "training" dataset despite arranged in a pretty cryptic set of numerical matrices.

Most of the decompilation process occurs in the decompile procedure that traverse the logs recorded during the last iteration over the source dataset (the last "epoch" in AI/ML parlance) and revert the compilation process.

For each log line, the VMM is traversed backwards, from the last filter to the first one.

For each filter we use the recorded output, the error and the weights' variations of a sentinel node to compute the input vector received from the preceeding filter. Then we compute the weight variations for every other reducer of the filter and apply them.

Now, if you are too fond in the AI narrative, if you worship Futurology or if you are a transumanist scared about imaginary "existential risks", you should be seriously outraged here: the "neural network" is "unlearning"!
We are literally "brain washing" the machine!
It could happen to your imaginary simulated-heirs too!
And there's nothing you can do about it! :-PP

More seriously, when all the reducers of the filters have been restored to the previous parameters, we compute the errors for the previous filter and then repeat the process with it.

Finally, when all the filters have been processed, the input of the first filter is rescaled and saved in the output csv and we move to the next log.

Where is the trick?

Usually, during the compilation process (aka "training"), each sample from the source dataset is turned into a miriad of numeric vectors that are "absorbed" by the parametric vectors of the reducers in the machine... and then discarded.

By recording very few of such vectors (one for each filter, to be precise) and reverting the compilation process, we can compute all the other vectors and the original input.

In particular, for each sample compiled during the last iteration over the dataset, the executable will retain:

• the output vector produced by the VMM
• the error vector (the difference between the computed output and the one provided by the sample sources)
• the variation of weights applied to one single VectorReducer for each ParallelVectorMapper of the program.

It does not matter how complex or large is a "neural network", the changes to the parametric vector of a single "neuron" of each "layer" is enough.

Note that such vectors are not enough, by themselves, to get back the original dataset: the final weights used by the reducers that compose the machine are also needed during the decompilation process because they hold fundamental projections of the dataset.

In other words, any "model" of an "AI" is a substantial portion of a larger derivative work based on the dataset used during the "training".

Discarding the rest of the work (the logs that compiler.py preserved) is just a way to hide the proofs of such derivation.

That's why nobody should be surprised when GitHub Copilot distributes suggests to include copylefted code into proprietary software: it has literally been "trained" to violate hackers' copyright from the very beginning!

In fact, it still contains substantial portions of the copylefted code uploaded on GitHub, but it does not comply with the licenses.

Conclusions

"GitHub Copilot" should be probably rebranded as "Microsoft Copy(a)lot". ;-)

Its software, like any other "model" produced though those techniques that currently go under the umbrella of "artificial intelligence", is a derivative work of the source dataset, just like any binary produced by a compiler is a derivative work of the sources.

Thus Microsoft should either get a license from the authors of the sources they compiled into such software, or comply with all their licenses.

On the other hand, if Microsoft is right and programming a Vector Mapping Machine is enough to remove the copyrights from a work, we could use the technique shown here to compile texts, songs, movies and even proprietary software into a program for a standard VMM and then decompile it when required, to play or execute it: the "artificial intelligence" would remove any legal risk for the people distributing the compiled binary.

In fact, if what Microsoft did was actually "fair use", every work under copyright can be used in the same way. In other words, if Microsoft is right, they made obsolete most of the oscurantist "intellectual property" laws that plague our cybernetic society: a huge step forward that we should all welcome with gratitude.

Further research

The technique shown here is pretty simple but it can be adapted to any other "machine learning" technique. You just need to not throw away relevant data during the compilation of the source dataset.

Furthermore, such technique could be extended to verify, under appropriate conditions, if and how a certain set of samples has been used during the compilation of a given "artificial intelligence" during forensic analysis.

Finally, if the required data has been dutyfully preserved during the realization of a "machine learning" system, such technique could be adapted to remove problematic samples from the program executed from the system.

Licensing

This essay and the all the Python code distribuited with it is licensed to you under the terms of the Hacking License (see HACK.txt).

The MNIST dataset has been copied in the repository as requested by its maintainers.