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[EQUATION] [EQUATION] по формуле полной вероятности. Аналогичные рассуждения для (1+1) EA дают [EQUATION] где [MATH] [MATH] [MATH] , при [MATH] |
Условие доминирования означает, что для любых [MATH] , выполняется [MATH] . Пусть [MATH] для каждого [MATH] . Тогда [MATH] [EQUATION] |
Используя эти свойства, выражения ( 56 ) и 57 ), получаем: [EQUATION] [EQUATION] [EQUATION] [EQUATION] 2. Необходимость. Рассмотрим алгоритм EA, где [MATH] , а оператор селекции таков, что всякий раз выбираются [MATH] последних построенных EA генотипов из последовательности [MATH] Согласно условиям теоремы при любой [M... |
[EQUATION] [EQUATION] т. е. оператор [MATH] доминируется оператором [MATH] [MATH] было показано, что при [MATH] оператор мутации из КГА для задачи ONEMAX доминирует сам себя, т.е. является монотонным . Аналогичный результат получен для ЗВП на графах специальной структуры. Большинство известных операторов мутации не име... |
15.1 Монотонные операторы воспроизведения В некоторых случаях для оператора [MATH] существует достаточно простой способ построения оператора мутации, доминирующего его. Определим оператор мутации, соответствующий оператору [MATH] следующим образом |
[EQUATION] где [MATH] . Таким образом, в [MATH] сначала оператор воспроизведения [MATH] применяется к набору идентичных родительских генотипов, и после этого результат выбирается как генотип с наибольшей приспособленностью среди генотипов, имеющихся на входе и выходе [MATH] . В случае, если при этом обнаружится несколь... |
Определение 15.3 Оператор воспроизведения [MATH] называется монотонным, если для любых двух последовательностей из [MATH] генотипов [MATH] |
[MATH] , таких что [EQUATION] следующие условия выполняются для всех [MATH] [EQUATION] где [MATH] [MATH] Условие монотонности оператора воспроизведения означает, что замена родительских генотипов на генотипы большей или равной приспособленности не приводит к снижению шансов получения достаточно приспособленных потомков... |
Заметим, что если условия ( 58 ) выполнены как равенства для родительских генотипов [MATH] [MATH] , то ввиду ( 59 распределения вероятностей приспособленности лучшего потомка для |
[MATH] [MATH] должны совпадать. Если оператор [MATH] является монотонным, то [MATH] доминирует [MATH] по построению, в таком случае алгоритм EA с оператором воспроизведения [MATH] может сравниваться по теореме 15.1 |
(1+1) EA , где используется оператор мутации [MATH] . Следующее следствие непосредственно вытекает из теоремы 15.1 Следствие 15.1 |
Пусть в EA используется монотонный оператор воспроизведения [MATH] , и оператор [MATH] используется в (1+1) EA . Пусть, кроме того, |
(1+1) EA начинает работу с генотипа [MATH] такого что [MATH] . Тогда для всех [MATH] и всех [MATH] выполняется [EQUATION] Частным случаем монотонного оператора воспроизведения при |
[MATH] является монотонный оператор мутации. Непосредственно из определений следует, что монотонный оператор мутации доминирует сам себя, и по следствию 15.1 таком случае (1+1) EA является <<наилучшим>> методом в классе эволюционных алгоритмов с неограниченной памятью. |
Приведем простой пример монотонного оператора воспроизведения, где [MATH] . Пусть [MATH] и оператор [MATH] построен на основе равномерного кроссинговера, однако перед применением равномерного кроссинговера к генам одного из двух родительских генотипов применяется перестановка, выбранная равновероятно. В качестве резуль... |
Заметим, что без использования случайной перестановки такой оператор не был бы монотонным (например, для функции [MATH] условие ( 59 нарушается на родительских генотипах [MATH] |
[MATH] при [MATH] ). Все упомянутые выше примеры относятся к задачам регулярной структуры. На практике, однако, задачи комбинаторной оптимизации, как правило, не имеют такой структуры и свойство монотонности оператора воспроизведения выполняется редко. В частности, при наличии локальных оптимумов функции приспособленно... |
15.2 Среднее время достижения оптимума и средняя приспособленность на заданной итерации Рассмотрим среднее число обращений к оператору воспроизведения до достижения множества генотипов требуемого качества (при этом наибольший интерес представляет множество генотипов, кодирующих оптимальные решения). Вместе с тем, будем... |
Обозначим через [MATH] среднее число итераций (1+1) EA до получения генотипа с приспособленностью не ниже [MATH] . Среднее число итераций ЭА с неограниченной памятью до получения генотипа с приспособленностью не ниже [MATH] обозначим через [MATH] |
Следствие 15.2 Пусть оператор [MATH] доминируется оператором [MATH] [EQUATION] тогда: (i) всех [MATH] выполняется [EQUATION] (ii) если [MATH] конечно, то [MATH] |
Доказательство. (i) Пусть [MATH] [MATH] обозначают функции распределения случайных величин [MATH] [MATH] соответственно. Тогда из теоремы 15.1 следует неравенство [MATH] для любого [MATH] . Таким образом, по свойствам математического ожидания (см., например, |
[EQUATION] (ii) Аналогично неравенству ( 62 ) при конечном [MATH] имеем [EQUATION] [EQUATION] Применение теоремы 15.1 завершает доказательство. |
[MATH] Теорема 15.1 и следствие 15.2 показывают, что если требуется сделать выбор между ЭА и эволюционной стратегией (1+1) EA , то в условиях применимости этих результатов предпочтение следует отдать (1+1) EA |
Приложение 1. Список задач Упражнение 15.1 Показать, что при известных значениях приспособленности особей текущей популяции [MATH] селекция всех родительских особей для построения новой популяции [MATH] может быть выполнена в КГА за время [MATH] |
Упражнение 15.2 Предложить взаимно-однозначное представление решений задачи о максимальном разрезе в графе при [MATH] Упражнение 15.3 |
Показать, что для задачи о наименьшем вершинном покрытии при [MATH] задача оптимальной рекомбинации эффективно разрешима. Упражнение 15.4 Описать алгоритм, осуществляющий мутацию <<2-замена>> в кодировке решений задачи коммивояжера с помощью перестановок. |
# Source: arxiv 1512.02100 # Title: Digital Genesis: Computers, Evolution and Artificial Life # Sections: all # Downloaded: 2026-03-03T01:55:26.341250+00:00 |
Digital Genesis: Computers, Evolution and Artificial Life Abstract The application of evolution in the digital realm, with the goal of creating artificial intelligence and artificial life, has a history as long as that of the digital computer itself. We illustrate the intertwined history of these ideas, starting with t... |
Introduction In The Origin of Species , Darwin introduced his theory of natural selection as an explanation of the complexity of the biological world (Darwin,, 1859 . Simply put, in a population where heritable variation exists in the characteristics of individual organisms, if one variety of a particular characteristi... |
The logic of Darwin’s argument seems to apply to any system of entities which possesses the three fundamental features of variation differential reproduction , and |
inheritance . The beautiful simplicity of this picture raises the alluring question of whether it would be possible to create virtual worlds instilled with these features, that might give rise to the evolution of complex digital life. |
Digital Origins The idea of applying an evolutionary process in a digital world dates back to the origins of the digital computer itself. Over the 1940s and 1950s the idea appears to have arisen, independently, as many as ten times Fogel, 1998b, , p.4) |
The earliest substantial theoretical work in this area was developed by John von Neumann. In the late 1940s, he became interested in the question of how complicated machines could evolve from simpler ones |
(von Neumann,, 1966 He was interested in self-reproducing machines that were robust in the sense that they could withstand some types of mutation and pass these mutations on to their offspring; such machines could therefore participate in a process of evolution. Looking for a suitable formalism that was both simple and... |
Although the design was not implemented on a computer before his death in 1957, von Neumann’s work can be regarded as the first attempt to instantiate an evolutionary process in the context of a modern, digital computational framework. |
At around the same time, Alan Turing also considered the application of evolution to computers. In his seminal paper Computing Machinery and Intelligence he described a method of machine learning involving mutations (random or otherwise) to a computer program and feedback from a human experimenter (Turing,, 1950 . Turi... |
However, it was not long until more substantial experiments with evolution on computers commenced. The first were conducted by Nils Aall Barricelli while working in von Neumann’s group at the Institute of Advanced Studies (IAS) in Princeton over the period 1953–1956 (Barricelli,, 1954 1962 1963 Barricelli employed a on... |
(Barricelli,, 1962 In later work, Barricelli experimented with giving his symbioorganisms greater opportunities for evolving complex phenotypes. In particular, if two symbioorganisms attempted to reproduce into the same space, their genotype was decoded into a strategy for playing a simple game (called “Tac Tix”), and ... |
(Barricelli,, 1963 Barricelli’s pioneering work was therefore very much focussed on replicating the dynamics of biological evolution in a digital medium, and in creating an “unlimited evolution” process in which complex digital lifeforms (“numerical symbioorganisms”) would emerge. |
Following Barricelli’s work at IAS, research on the application of evolution on computers has flourished. From the mid-1950s to the mid-1980s, the majority of this research effort focussed on using evolution as a practical tool for optimisation rather than the more lofty goals of Barricelli and von Neumann. |
Fogel, 1998a provides a good review of pioneering work from this period. In the mid-1980s the field of Artificial Life was reborn, stimulated by a workshop in 1987 (Langton,, 1989 This has led to a renewed interest in the kinds of ideas first explored by Barricelli, including attempts to create an open-ended evolutiona... |
Digital Future There has been renewed interest in the open-ended evolution of digital life but a convincing argument about whether or not such a system has been, or even can be created digitally, hinges on identifying a satisfactory set of criteria for judging its success. To date this has been elusive. |
Many digital evolutionary systems generate an initial burst of interesting activity, but then seem to reach a quasi-stable state beyond which no further qualitative changes are observed. Intuitively, these systems don’t seem to be open ended. This suggests that more features of biological evolution must be incorporated... |
We argue that a more principled, ecologically-inspired approach to modelling energy and matter is important, along with a more careful consideration of the “physical” dynamics of the environment and of the modelling relationship between organisms and environment (Dorin et al.,, 2008 ; Korb and Dorin,, 2011 ; Taylor,, 2... |
Looking forward, with the increasing importance in many application areas of systems that can autonomously learn and adapt, we see the close relationship between computers, evolution and artificial life only growing stronger. |
# Source: arxiv 1512.04478 # Title: Population Dynamics of Self-Replicating Cell-like Structures Emerging from Chaos # Sections: all # Downloaded: 2026-03-03T01:58:58.808369+00:00 |
Population Dynamics of Self-Replicating Cell-like Structures Emerging from Chaos submitted [MATH] December 2015 Abstract We present here a system of self-propelled particles that follow a very simple motion law in continuous space in a deterministic and asynchronous way. This system of particles is capable of producing... |
Keywords: Self-organization, Emergent pattern formation, Self-replication, Protocell model, Artificial Life, Emergence of Life, Morphogenesis |
Introduction At the very beginning of life we assume a biological singularity: Based on simple local interaction rules initially randomly distributed particles (molecules) spontaneously self-organized into aggregates of higher order citation:1 . These compounds first managed to withstand the thermodynamic path towards ... |
In this study we present a simple model that demonstrates that such self-reproducing protocells can emerge spontaneously from a population of homogenous, purely reactive, deterministically moving (self-propelled) particles citation:6 citation:7 following one single very simple rule. Following this approach we looked fo... |
II The mathematical model A PPS models a population of particles moving deterministically and asynchronously in a continuous toroidal wrapped space. Each particle is defined by its position [MATH] , by its heading [MATH] and by its movement with constant velocity [MATH] , thus [MATH] , assuming an open system allowing ... |
[EQUATION] where [MATH] represents a fixed rotation and [MATH] models a rotation proportional to local neighborhood size. Neighborhood configuration affects [MATH] in each timestep, in turn changing a particle's position, ultimately yielding new local configurations. This feedback loop governs the self-organization of ... |
With [MATH] [MATH] , isolated particles hold position within 2 timesteps. Only when other particles enter their neighborhood they start to move away: PPS with [MATH] |
[MATH] “mirror” the behaviour of particles in PPS with [MATH] [MATH] on the microscopic level. Spatial implementation of the model: In our PPS each particle holds position [MATH] and heading [MATH] at every time step t. The change of this heading ( [MATH] ) is modeled in our model equation . Every positive heading chan... |
Listing 1: Explanatory implementation of a PPS as pseudo-code loop foreach timestep loop foreach particle count left neighbors in |
radius count right neighbors in radius delta_phi alpha beta sign turn right delta_phi move forward }\ end lstlisting We discovered |
an exceptionally interesting system with the parameter set $PPS =\ langle =5,\ alpha =180^{\ circ }, beta =17^{\ circ }, =0.67 rangle |
used as default PPS parameters here For visualization we color code particles by their local neighborhood size $color_t if 16< N_ =5}\ leq35 |
blue else if N_ =5}>35 yellow else if N_ =1.3}>15 magenta else if \\ 13 leq N_ =5}\ leq15 brown else green$ These colors indicate |
an exhaustive classification of each particles rotational power We used density dependent population model to interpret observed |
population dynamics of emerging structures cite citation :8}: Delta /\ Delta cdot (1- )\ cdot X$ where the variable $X$ models the |
population size the constant $a$ resembles the maximum reproduction rate per step and the constant $K$ is the carrying capacity of |
the habitat Fig ). Best fit was found by applying the minimum residuals method using least squares %\ section Results %\ end multicols |
includegraphics scale =0.08]{ lifecycle35m %\ begin multicols }{2} %\ section Discussion %\ subsection Subsection One %\ subsection Subsection |
Two section The study After we discovered the PPS in general and the especially interesting parameter set of PPS =\ langle =5,\ alpha =180^{\ circ },\ beta =17^{\ circ }, =0.67\ rangle |
we investigated set of research questions to understand the properties of the emerging phenomena Our research questions were \\ setlength {\ parindent }{0 pt |
Q1 Do ordered structures emerge from random initial particle configurations \\ Q2 Do such populations of structures grow over time |
\\ Q3 Is the system converging to certain ratio between different structure types \\ Q4 Do the observed populations of structures |
follow dynamics known from nature \\ To investigate those questions we started such PPS with particle density of 0.08 particles per |
space unit this means 8000 particles in space of 250 units by 250 units distributed randomly uniform random distribution and with |
randomized initial heading The PPS does not contain any stochastic component thus it is fully deterministic motion law All particles |
move asynchronously in randomized order We performed such simulation runs and recorded all color transitions of particles which express |
significant changes in local density We recorded the spatial evolution of the system picture sequences and used previously determined |
average number of blue and yellow particles in cell structures (48 such particles per cell and previously determined average number |
of pink particles in spore structures (18 particles per spore to record population dynamics of those structures To compare populations |
of different type and times we used Mann Whitney tests after applying Bonferroni correction as we made ad hoc tests on our collected |
data to prevent false positives type -1 errors due to multiple testing section Results Figure shows the resulting distributions of |
particles after 100,000 timesteps in runs each started from random initial distribution It is clearly visible that in all runs both |
typical structures very dense magenta spores and extended cell like structures in blue yellow emerged and stayed alive for this extended |
time span In all runs those structures started by initially building single spore somewhere at random place in the environment Afterwards |
this structure grew to cell which then replicated into more cells or spores After some runtime population of such structures inhabits |
the whole habitat These structures are areas of denser particle populations thus the density of particles outside of these structures |
goes down in consequence as the system is closed system material wise However it is an open system energy wise as particles are self propelled |
The lowered density of the free particles allows them to arrange in rather regular hexagonal matrix like configuration \\ Figure |
shows that the estimated populations of cells and spores show typical logistic growth dynamics comparison of populations of cells |
and spores in the initial phase shows that the system produces significantly more cells than spores fact that we observe also for |
later points in time Mann Whitney test $N_1 N_2 =9 runs $p <0.001 for all those comparisons ). Comparing the populations of those |
structures in an initial phase step 2000) to later periods step 5000 and step 100,000) shows that there is an initial growth phase |
Mann Whitney test $N_1 N_2 =9 runs $p <0.001 comparing consecutive periods ). Later periods in longer runs show no significant differences |
in population size anymore data not shown ). Figure shows that the observed populations of spores and cells can be fitted to the |
classical top down model of sigmoid density dependent growth of natural populations see for example cite citation :9}) yielding growth |
rate of $a =7.1 cdot 10^{-4} per step for cells and $a =4 cdot 10^{-4} per step for spores and carrying capacities of $K =18.21 spores |
and $K =50.78 cells Figure shows microscopic observation of state transitions of particles as the different colors indicate different |
states of local densities In the growth phase of the population the majority of state changes happen from green very low density |
to brown slightly higher density to blue medium density ). After the growth phase of the population there is an almost balanced flow |
through brown rightarrow$ blue rightarrow$ magenta rightarrow$ brown and green rightarrow$ brown rightarrow$ blue rightarrow$ green |
whereby blue indicates high particle density and magenta indicates very high particle density section Discussion and Conclusion Our |
general and overall results are the novel discovery that this simple motion law which we call PPS is capable to self generate ordered |
structures cells spores from initial randomized spatial configuration of particles This is an emergent phenomenon of the system Figure |
shows that research question Q1 was affirmed in out of tested runs We further demonstrate that the arising populations differ between |
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