Complexity in Information Theory

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Loads of fun but little content, this is a short tutorial, much better than the one given by the Asian instructor in Coursera or Udacity but not as good as other courses online, even on YouTube. For that I think it does not deserve a very high mark, but because it was fun I am giving it 2 stars.

Information Theory Tutorial: Mutual Information

Great course! Very clear explanations, fun anegdotes and nice quizes. Professor Seth Lloyd is a great and engaging teacher. I found the knowledge gained in this course very useful. If you're interested in information theory I full heartedly recommend this course! Great course which gives you an overal introduction to information theory, which is pretty useful if you are starting to inmerse yourself in computer science or programming.

Professor Loyd gave a nice overview of the topic, his explanations where amazing!

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I took it in order to establish a research area in my university and we have done nice project. Very good course. Lloyd has an engaging and approachable style. His examples are clear and intuitive. The discussion is amenable. He clearly took care of preparing his content. Sign up for free. My Classes. On the contrary, this is not necessary in a conventional camera since the radiation emitted by the scene is much higher than the thermal noise level of the image sensor. This proofs that information emerges from physical reality. But we can go further, as information is the basis for describing natural processes.

Therefore, something that cannot be observed cannot be described. In short, every observable is based on information , something that is clearly evident in the mechanisms of perception. From the emerging information it is possible to establish mathematical models that hide the underlying reality , suggesting a functional structure in irreducible layers.

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Something that is generally extendable to all physical models. Another indication that information is a fundamental entity of what we call reality is the impossibility of transferring information faster than light. This would make reality a non-causal and inconsistent system. Therefore, from this point of view information is subject to the same physical laws as energy. And considering a behavior such as particle entanglement, we can ask: How does information flow at the quantum level? Based on these clues, we could hypothesize that information is the essence of reality in each of the functional layers in which it is manifested.

Thus, for example, if we think of space-time, its observation is always indirect through the properties of matter-energy, so we could consider it to be nothing more than the emergent information of a more complex underlying reality. This kind of argument leads us to ask: What is it and what do we mean by reality?

All this without losing sight of the knowledge provided by the different areas that study reality, especially physics. This is a very general review of the development of the theoretical and practical aspects that occurred throughout the twentieth century to the present day and which have led to the current vision of what information is. This is made from a more theoretical perspective based on the computation theory, information theory IT and algorithmic information theory AIT. But in this post, we will leave aside the mathematical formalism and expose some examples that will give us a more intuitive view of what information is and its relation to reality.

And above all try to expose what the axiomatic process of information means. This should help to understand the concept of information beyond what is generally understood as a set of bits. And this is what I consider one of the obstacles to establishing a strong link between information and reality. Nowadays, information and computer technology offers countless examples of how what we observe as reality can be represented by a set of bits.

Thus, videos, images, audio and written information can be encoded, compressed, stored and reproduced as a set of bits. This is possible since they are all mathematical objects, which can be represented by numbers subject to axiomatic rules and can, therefore, be represented by a set of bits.

Information theory and computation

However, the number of bits needed to encode the object depends on the coding procedure axiomatic rules , so that the AIT determines its minimum value defined as the entropy of the object. However, the AIT does not provide any criteria for the implementation of the compression process, so in general they are based on practical criteria, for example statistical criteria, psychophysical, etc. The AIT establishes a formal definition of the complexity of mathematical objects , called the Kolmogorov complexity K x.

For a finite object x , K x is defined as the length of the shortest effective binary description of x , and is an intrinsic property of the object and not a property of the evaluation process. The compression and decompression of video, images, audio, etc.

Bibliographic Information

In this context, both C and D are axiomatic processes, understanding as axiom a proposition assumed within a theoretical body. This may seem shocking to the idea that an axiom is an obvious and accepted proposition without requiring demonstration. To clarify this point I will develop this idea in another post, for which I will use as an example the structure of natural languages. In this context, the term axiomatic is totally justified theoretically, since the AIT does not establish any criteria for the implementation of the compression process.

And, as already indicated, most mathematical objects are not compressible. In such a way that a bit string has no meaning unless a process is applied that interprets the information and transforms it into knowledge. Thus, when we say that x is a video content we are assuming that it responds to a video coding system, according to the visual perception capabilities of humans.

And here we come to a transcendental conclusion regarding the nexus between information and reality. Historically, the development of IT has created the tendency to establish this nexus by considering the information as a sequence of bits exclusively. But AIT shows us that we must understand information as a broader concept, made up of axiomatic processes and bit strings. But for this, we must define it in a formal way. Thus, both C and D are mathematical objects that in practice are embodied in a set consisting of a processor and programs that encode the functions of compression and decompression.

If we define a processor as T and c and d the bit strings that encode the compression and decompression algorithms, we can express:. Therefore, the axiomatic processing would be determined by the processor T. And if we use any of the implementations of the universal Turing machine we will see that the number of axiomatic rules is very small.

Thus, any mathematical model that describes an element of reality can be formalized by means of a Turing machine. The result of the model can be enumerable or Turing computable, in which case the Halt state will be reached, concluding the process. On the contrary, the problem can be undecidable or non-computable, so that the Halt state is never reached, continuing the execution of the process forever. For example, let us weigh in the Newtonian mechanics determined by the laws of the dynamics and the attraction exerted by the masses.

Where x is the bit string encoding the laws of calculus, y the bit sequence encoding the laws of Newtonian mechanics and z the initial conditions of the masses constituting the system. It is frequent, as a consequence of the numerical calculus, to think that the processes are nothing more than numerical simulations of the models.

However, if z specifies more than two massive bodies, in general, the process will not be able to produce any result, not reaching the Halt state. This is because the Newtonian model has no analytical solution for three or more orbiting bodies, except for very particular cases, and is known as the three-body problem. But we can make x and y encode the functions of numerical calculus, corresponding respectively to the mathematical calculus and to the computational functions of the Newtonian model.

However, the process will not reach the Halt state, except in very particular cases in which the process may decide that the ephemeris is a closed trajectory. This behaviour shows that the Newtonian model is not computable or undecidable. This is extendable to all models of nature established by physics since they are all non-linear models. If we consider the complexity of the y sequence corresponding to the Newtonian model, both in the analytical or in the numerical version, it is evident that the complexity K x is small. If this were possible it would mean that w is an enumerable expression, which is in contradiction with the fact that it is a non-computable expression.

But this will be addressed another post. Georg Cantor.

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Co-creator of Set Theory Without going into greater detail, what should catch our attention is that this classification of numbers is based on positional rules, in which each figure has a hierarchical value. What does the AIT tell us?

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  • Consequently, the sequence p , which runs the Turing machine i to get x , will be composed of the concatenation of: The sequence of bits that encode the rules of calculus in the Turing machine i. The bitstream that encodes the compressed expression of x , for example a given numerical series of x. The length of the sequence x that is to be decoded and that determines when the Turing machine should reach the halt state, for example a googol 10 The random behavior of observable information emerging from quantum reality.

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