# Science.-Solutions for a fluid capable of mimicking any Turing machine

Madrid, 5 (Europe Press)

Four Spanish researchers have, for the first time, built solutions for a liquid capable of mimicking any Turingne machine.

As ICMAT (Institute of Mathematical Sciences) has shown, a Turing machine is an abstract construct capable of simulating any algorithm. As input data, it receives a series of zeros and ones and, after a number of steps, returns a result, also in the form of zeros and ones.

The fluid that the researchers studied can be considered as a hydroaccumulator; You take a point in space as input data, process it – following the path of the fluid through that point – and present as a result the next region the fluid has traveled to. The result is an incompressible, non-viscous liquid – the Navier – Stokes equations consider viscosity – in the third dimension. It is the first time that a water machine has been designed.

One of the main consequences of the result is that it makes it possible to demonstrate that some hydrodynamic phenomena are unreportable. For example, if a message is released inside a bottle, it cannot be guaranteed that it will reach the addressee. Something similar happened to 29,000 rubber ducks that fell from a cargo ship during a storm and got lost in the ocean in 1992: Nobody can predict where they will appear.

That is, there is no algorithm that allows us to guarantee whether a liquid particle will pass through a specific region of space at a specific time. “This unpredictability, which differs from that established by chaos theory, is a new manifestation of the disordered behavior of liquids,” the researchers say in a statement.

The team consists of Eva Miranda, a professor at the Polytechnic University of Catalonia (UPC), a member of the Institute of Mathematical Sciences (ICMAT), the Paris Observatory (France) and the Center for Mathematical Research (CRM); Daniel Peralta Salas (ICMAT-CSIC); Robert Cardona (BGSMath – UPC) and Francesco Brisas (ICMAT-CSIC) published the results Thursday in the journal “Proceedings of the National Academy of Sciences (PNAS)”.

“ In chaos theory, the unpredictability of a system’s extreme sensitivity correlates with initial conditions – the flapping of a butterfly can generate a tornado – and in this case it goes further: We have demonstrated that there can be no algorithm that can solve the problem, not a limitation of our knowledge, But to mathematical reasoning itself, “Miranda and Peralta Salas assert. This illustrates the complexity of fluid behavior, which manifests itself in various fields, from weather forecasting to flow and waterfall dynamics.

Regarding its relationship to the Navier-Stokes problem, which is on the Clay Foundation’s Millennium Problems List, researchers are cautious. They say “Tao’s proposal is, for now, hypothetical.” His idea is to use a water computer to force the fluid to accumulate more and more energy in smaller and smaller areas, until singularities are formed, i.e. a point where the energy becomes unlimited. The presence or absence of singularities in the equations is precisely the Navier-Stokes problem. However, scientists who have discussed their findings with Tao say: “At the moment we don’t know how to do that with Euler’s or Navier-Stokes equations.”

The Cardona, Miranda, and Peralta-Salas y Presas water machine – the first to exist – are guided by Euler’s equations but their solutions do not contain singularities. For its design, the tools of engineering, topology, and dynamic systems developed in the past 30 years were fundamental. Specifically, affective engineering, communication engineering, and fluid dynamics are combined with computer science theory and mathematical reasoning. The scientists conclude, “It took us more than a year to understand how the various cables were connected to the demonstration.”

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