Daniel M. Dubois

Computing anticipatory systems with incursion and hyperincursion

An anticipatory system is a system which contains a model of itself and/or of its environment in view of computing its present state as a function of the prediction of the model. With the concepts of incursion and hyperincursion, anticipatory discrete systems can be modelled, simulated and controlled. By definition an incursion, an inclusive or implicit recursion, can be written as: x(t+1)=F[…,x(t−1),x(t),x(t+1),…] where the value of a variable x(t+1) at time t+1 is a function of this variable at past, present and future times. This is an extension of recursion. Hyperincursion is an incursion with multiple solutions. For example, chaos in the Pearl-Verhulst map model: x(t+1)=a.x(t).[1−x(t)] is controlled by the following anticipatory incursive model: x(t+1)=a.x(t).[1−x(t+1)] which corresponds to the differential anticipatory equation: dx(t)/dt=a.x(t).[1−x(t+1)]−x(t). The main part of this paper deals with the discretisation of differential equation systems of linear and non-linear oscillators. The non-lin...

Mathematical Foundations of Discrete and Functional Systems with Strong and Weak Anticipations

This paper deals with some mathematical developments to model anticipatory capabilities in discrete and continuous systems. The paper defines weak anticipation and strong anticipation and introduces the concepts of incursive and hyperincursive discrete processes as an extension to recursion. Functional systems represented by differential difference equations with anticipation and/or delay seem to be a very useful tool for describing strong anticipation. Anticipation and delay play a complementary role and synchronization mechanisms seem to be a powerful way to anticipate the evolution of systems with delay. This paper shows finally that the modelling of anticipation in predictive control is the basic mechanism for enhancing the control of the trajectory of systems toward a target.

A model of patchiness for prey—predator plankton populations

Abstract A non-linear model is proposed to explain the horizontal structuration of prey—predator populations in a turbulent sea. The model is represented by a system of partial differential equations taking into account advection, due to residual currents, and eddy diffusivity. The ecological interactions are assumed to be of the Lotka-Volterra type. Starting from an initial small patch of prey—predator populations, numerical simulations show two distinct phases: 1. (i) an “explosive” phase corresponding to a bloom of phytoplankton and, 2. (ii) after that, a decrease of the quantity of plankton at the centre of the patch leading to a ring structure. The ring propagates in increasing its radius with constant intensity and velocity. The prey behaves like an “activator” and the predator like an “inhibitor”. The ring is thus similar to an active wave. When two waves meet each other, the simulation shows their “annihilation”. All these new properties of the prey-predator equations are in agreement with experimental data. Finally, the physical mechanism for initiation of patchiness is discussed.

Generation of fractals from incursive automata, digital diffusion and wave equation systems

This paper describes modelling tools for formal systems design in the fields of information and physical systems. The concept and method of incursion and hyperincursion are first applied to the fractal machine, an hyperincursive cellular automata with sequential computations with exclusive or where time plays a central role. Simulations show the generation of fractal patterns. The computation is incursive, for inclusive recursion, in the sense that an automaton is computed at future time t + 1 as a function of its neighbouring automata at the present and/or past time steps but also at future time t + 1. The hyperincursion is an incursion when several values can be generated for each time step. External incursive inputs cannot be transformed to recursion. This is really a practical example of the final cause of Aristotle. Internal incursive inputs defined at the future time can be transformed to recursive inputs by self-reference defining then a self-referential system. A particular case of self-reference with the fractal machine shows a non deterministic hyperincursive field. The concepts of incursion and hyperincursion can be related to the theory of hypersets where a set includes itself. Secondly, the incursion is applied to generate fractals with different scaling symmetries. This is used to generate the same fractal at different scales like the box counting method for computing a fractal dimension. The simulation of fractals with an initial condition given by pictures is shown to be a process similar to a hologram. Interference of the pictures with some symmetry gives rise to complex patterns. This method is also used to generate fractal interlacing. Thirdly, it is shown that fractals can also be generated from digital diffusion and wave equations, that is to say from the modulo N of their finite difference equations with integer coefficients.

Introduction of the Aristotle's Final Causation in CAST: Concept and Method of Incursion and Hyperincursion

This paper will analyse the concept and method of incursion and hyperincursion firstly applied to the Fractal Machine, an hyperincursive cellular automata with sequential computations where time plays a central role. This computation is incursive, for inclusive recursion, in the sense that an automaton is computed at the future time t+1 in function of its neighbour automata at the present and/or past time steps but also at the future time t+1. The hyperincursion is an incursion when several values can be generated at each time step. The incursive systems may be transformed to recursive ones. But the incursive inputs, defined at the future time step, cannot always be transformed to recursive inputs. This is possible by self-reference. A self-reference Fractal Machine gives rise to A non deterministic hyperincursive field rises in a self-reference Fractal Machine. This can be related to the Final Cause of Aristotle. Simulations will show the generation of fractal patterns from incursive equations with interference effects like holography. The incursion is also a tool to control systems. The Pearl-Verhulst chaotic map will be considered. Incursive stabilisation of the numerical instabilities of discrete linear and non-linear oscillators based on Lotka-Volterra equation systems will be simulated. Finally the incursive discrete diffusion equation is considered.

A Semantic Logic for CAST Related to Zuse, Deutsch and McCulloch and Pitts Computing Principles

The goal of CAST research and development is to provide modelling tools for formal systems design in the field of information and systems engineering. This paper deals with such modelling tools for formal systems related to Zuse, Deutsch and McCulloch and Pitts computing principles. The semantic logic of such systems can be exhibited in replacing the differential equations by digital cellular automata. K. Zuse proposed such a method for representing physical systems by a computing space. I show that the digital wave equation exhibits waves by digital particles with interference effects. The logical table of the wave equation shows the conservation of the parity related to exclusive OR. The Fractal Machine proposed by the author deals with a cellular automata based on incursion, an inclusive recursion, with exclusive OR. In this machine, the superimposition of states is related to the Deutsch quantum computer. Finally, it is shown that the exclusive OR can be modelled by a fractal non-linear equation and a new method to design digital equations is proposed to create McCulloch and Pitts formal neurons.

Hyperincursive McCulloch and Pitts neurons for designing a computing flip-flop memory

This paper will firstly review a new theoretical basis for modelling neural Boolean networks by non-linear digital equations. With integer numbers, these digital equations are Heaviside Fixed Functions in the framework of the Threshold Logic. These can represent non-linear neurons which can be split very easily into a set of McCulloch and Pitts formal neurons with hidden neurons. It is demonstrated that any Boolean tables can be very easily represented by such neural networks where the weights are always either an activation weight +1 or an inhibition weight −1, with integer threshold. A fundamental problem in neural systems is the design of memory. This paper will present new memory neural systems based on hyperincursive neurons, that is neurons with multiple output states for the same input, instead of synaptic weights. Finally, a differential equation of membrane neural potential is used as a model of a brain, the incursive, that is the implicit recursive, computation of which gives rise to non-localit...

Incursive anticipatory control of a chaotic robot arm

This paper deals with an innovative mathematical tool for modelling, simulating and controlling systems in automation engineering. Classically, feedback processes are based on recursive loops where the future state of a system is computed from the present and past states. With the new concept of incursion, an inclusive recursion, the future state of a system is taken into account for computing this future state in a self-referential way. The future state is computed from the mathematical model of the system. With incursion, numerical instabilities in the simulation of finite difference equations can be stabilised. The incursive control of systems can also stabilise feedback loops by anticipating the effect of the control what I call a feed-in-time control. In this short paper, the particular case of the modelling, simulation and control of a robot arm in a working space is studied. The highly non-linear model is based on recursive finite difference equations which give rise to instabilities, bifurcations ...

Anticipation, Orbital Stability, and Energy Conservation in Discrete Harmonic Oscillators

We make a systematic analysis of the dual incursive model of the discrete harmonic oscillator. We derive its closed form solution, and identify its natural frequency of oscillation. We study its orbital stability, and the conservation of its total energy. We finally propose a superposed model that conserves energy with absolute precision, and exhibits a high degree of orbital stability. Within the conjecture that spacetime is discrete, the above results lead to the conclusion that discretization must be accompanied by anticipation, in order to guarantee orbital stability and energy conservation.

Emergence of Chaos in Evolving Volterra Ecosystems

This paper is an attempt to give new avenues of research and development in the mathematical modelling of evolving systems. The best example of a natural evolution deals with the successive appearance, transformation and extinction of biological species on earth. The fundamental causes of such changes are not yet well established. Many theories were proposed to explain evolution. The first two were Lamarckism and Darwinism. Lamarck thought that the function creates the organ, but this cannot explain the memorization of the new characters in the genetic code. Darwin considered two complementary processes: on the one hand, an evolution of species by some random mutations and, on the other, a natural selection of the species which fit best their environment.