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(Čapek,, 1920 . Published in 1920 and first performed in 1921, the play introduced the word “robot” into the English language. The robots were constructed from biochemical components and designed to resemble humans, but lacked “superfluous” capacities such as feelings or the capacity to reproduce. They were mass-produc... |
One of the scientists in the factory experiments in making robots with more human-like feelings such as pain and irritability, but this results in unintended and ultimately disastrous consequences when the robots come to despise their human masters and rise up against them. This eventually leads to a stand-off where th... |
The climax of the play thus revolves around a struggle for the ownership of the written instructions that would allow the robots to collectively produce more of themselves—a struggle for the ownership of the robot’s DNA, as it were. This idea of the collective reproduction of a society of robots reflects some of Butler... |
Early American Science Fiction (1920s-1950s) The appearance of American pulp science fiction magazines in the 1920s, and their growing popularity over the decades that followed, provided a medium in which many writers explored the idea of self-reproducing robots and evolving machines. Perhaps the first example in this ... |
(Wright,, 1929 . With echoes of Samuel Butler, the story extrapolates the observed accelerating pace of technological development of the time into the far future, to a point when machines no longer rely on humans to service them. The machines become not only self-reproducing, but also able to design their own offspring... |
“Even in the early days of the Twentieth Century man had stood in silent adoration around the machines that had self-produced a newspaper or a needle … And at that time they could no more have conceived what was to follow than the first ape that drew the sheltering branches together could foresee the dim magnificence o... |
Automata (Wright,, 1929 , p. 344) Three years later, in 1932, the influential American sci-fi writer and editor John W. Campbell published The Last Evolution |
(Campbell,, 1932 , which also anticipated the eventual replacement of the human race by self-reproducing and self-designing machines. However, Campbell’s story is more optimistic than Wright’s, foreseeing a period where humans live in peaceful and co-operative coexistence with intelligent machines, with human creativit... |
“I think … that this is the end …of man … But not the end of evolution. The children of men still live—the machines will go on. Not of man’s flesh, but of a better flesh, a flesh that knows no sickness, and no decay, a flesh that spends no thousands of years in advancing a step in its full evolution, but overnight leap... |
The Last Evolution (Campbell,, 1932 , p. 419) Campbell’s vision of a complementary coexistence of humans and intelligent machines is replaced by a less positive image in his 1935 story The Machine (written under the pseudonym Don A. Stuart) |
(Campbell,, 1935 . In the story a human-like race on a distant planet design a thinking machine that is set the task of making better versions of itself. The outcome is a machine that takes care of all of the race’s basic needs. However, this ultimately leads to the degeneration of the race’s intelligence, civility, an... |
Laurence Manning’s The Call of the Mech-Men (Manning,, 1933 also mirrors ideas first aired by Butler 60 years earlier. Two explorers discover a group of extraterrestrial robots who have been living in underground caverns on Earth since their spaceship was damaged many tens of thousands of years earlier. The robots are ... |
Recurring themes of machine evolution and self-reproduction are seen in stories over the following years. An example is Joseph E. Kelleam’s Rust , set on a post-apocalyptic Earth where human-designed robots have survived after humankind has been wiped out (Kelleam,, 1939 . The robots try to design and build more of the... |
(Williams,, 1938 , three robots from a faraway planet travel to Earth in search of information about their origins many thousands of years earlier. To their surprise, they discover that they were originally designed by humans, and had been sent into space to accompany their creators in escaping a dying Earth. The human... |
The most explicit exploration of machine self-reproduction and evolution in early science fiction is found in Philip K. Dick’s Second Variety |
(Dick,, 1953 . The story is set on Earth at the end of a long-running war between East and West, in which Western forces are driven to design killer robots to turn the tide on the battlefield. The robots are highly autonomous, with each generation of design becoming more sophisticated, including powers of self-repair a... |
and Campbell’s The Last Evolution , the robots in Second Variety eventually develop the ability to design their own offspring, and increasingly sophisticated and human-like species of killer robots begin to emerge. Echoes of these earlier stories are also seen when one of the human characters remarks “It makes me wonde... |
new species. Evolution. The race to come after man” (Dick,, 1953 Themes of machine self-repair, self-reproduction and evolution were central to various subsequent works by Dick. Another notable example is Autofac |
(Dick,, 1955 , which ends with a vision of the seeds of self-reproducing manufacturing plants being launched into space. J. D. Bernal: The World, The Flesh and the Devil (1929) |
In addition to the fictional explorations of machine self-reproduction and evolution described above, we also see continued interest in these topics from scientists in the early 1900s. John Desmond Bernal (1901–1971) was an influential researcher who conducted pioneering work on structural crystallography. Later in his... |
(Bernal,, 1951 . In addition to his experimental work, he wrote many works on science and society; his first monograph, and yet perhaps his most futuristic writing, was entitled “ The World, the Flesh and the Devil: An Enquiry into the Future of the Three Enemies of the Rational Soul (Bernal,, 1929 |
In this work, Bernal discusses how one might examine the future of humanity in a scientifically defensible way. After sign-posting the methodological and intellectual dangers to be avoided, and discussing the unavoidable limitations, he proceeds to explore what might be said of the three major kinds of struggle facing ... |
Writing before the advent of space travel, atomic energy or computers, Bernal first tackles how humankind might overcome the challenges that arise from the material world. He argues that limitations of land and energy in the world will eventually compel us to colonise space: “On earth, even if we should use all the sol... |
Bernal proposes a “spherical shell ten miles or so in diameter” (Bernal,, 1929 , p. 23) which could provide a habitable environment for twenty or thirty thousand inhabitants. After discussing how the construction of a sphere might be bootstrapped from a basic design built largely of materials mined from an asteroid, Be... |
(Bernal,, 1929 , p. 23) with a protective “epidermis”, complete with regenerative mechanisms to protect against meteorites, mechanisms for the capture of meteoric matter to be used as raw material for the growth and propulsion of the sphere, systems for energy production from solar energy, stores for basic goods such a... |
The inhabitants of these globes in space would not be isolated, but would be in wireless communication with other globes and with the earth. In addition, there would be a constant interchange of people between the globes and the earth via interplanetary transport vessels. Having set out how the globes might function to... |
“However, the essential positive activity of the globe or colony would be in the development, growth and reproduction of the globe. A globe which was merely a satisfactory way of continuing life indefinitely would barely be more than a reproduction of terrestrial conditions in a more restricted sphere.” |
The World, The Flesh and the Devil (Bernal,, 1929 , p. 27) Hence, the globe is conceived of as a fully self-maintaining and self-reproducing unit—what might now be described as an |
autopoietic organisation (Maturana and Varela,, 1972 . Bernal discusses methods by which a globe might construct another globe, and then envisages how an evolutionary pressure to explore might arise among a population of globes: |
“As the globes multiplied they would undoubtedly develop very differently according to their construction and to the tendencies of their colonists, and at the same time they would compete increasingly both for the sunlight which kept them alive and for the asteroidal and meteoric matter which enabled them to grow. Soon... |
The World, The Flesh and the Devil (Bernal,, 1929 , p. 29) The enormous challenges of travelling interstellar distances are addressed, but Bernal argues that such a vision is nevertheless reasonable to consider: “once acclimatized to space living, it is unlikely that man will stop until he has roamed over and colonized... |
(Bernal,, 1929 , p. 30) Moving next to the possibilities of how our own bodies might develop in the distant future, Bernal imagines that we will increasingly replace and augment body parts with synthetic alternatives. Turning to the activities such advanced beings might pursue, Bernal suggests that, among other importa... |
Bernal’s vision of the relationship between the future evolution of humans and machines is more symbiotic than the futures imagined by Forster and Čapek: “Normal man is an evolutionary dead end; mechanical man, apparently a break in organic evolution, is actually more in the true tradition of a further evolution” |
(Bernal,, 1929 , p. 42) This perspective is more in line with the ideas expressed by Butler in Lucubratio Ebria , and with those of sci-fi authors such as John W. Campbell. Bernal sees the main barriers towards progress in these areas arising from human psychology—in addition to having the desire for progress, we must ... |
He also considers a third possibility, “the most unexpected, but not necessarily the most improbable” (Bernal,, 1929 , p. 56) , that human evolution might diverge, with one race following the natural path and another race following the intellectual and technological path. |
More Recent Work (1950s–present) The 1940s and, in particular, the 1950s saw the emergence both of the first rigorous theoretical work on the design of self-reproducing machines, and of the first implementations of artificial self-reproducing systems in software and in hardware (Taylor and Dorin,, 2018 . This has been ... |
Discussion As demonstrated in the preceding sections, the early history of thought about self-reproducing and evolving machines unveils a diverse array of hopes and fears. These contributions demonstrate that current debates about the implications of AI and ALife for the future development of humankind are actually a c... |
Takeover by intelligent machines The most prominent theme apparent in this work is the fear that machines might evolve to a level where they displace humankind as the dominant intelligent species. While some writers proposed more positive, co-operative alliances between humans and machines (e.g. Butler, Marshall, Wrigh... |
The idea that we ourselves are creating our own successors can be seen in the work of Butler, Eliot, Čapek, Wright and Campbell. Some saw this not as a development to be feared, but rather as a way in which the reach of humankind might be extended beyond the extinction of our species (e.g. Čapek, Campbell [ The Last Ev... |
Most saw the evolution of increasingly intelligent machines as an inevitable process. In the work reviewed, only Čapek engages significantly with the idea that humans might exert some control over the robots’ reproduction. Butler and Bernal thought this could likely only be achieved by humans forsaking the development ... |
The idea of self-repairing machines is present in the work of Eliot, Forster, Campbell [ The Machine ] and Bernal, and this is indeed a theme in current evolutionary robotics research, e.g. (Bongard et al.,, 2006 (Cully et al.,, 2015 . In contrast, we are unaware of any serious scientific investigation of the idea of |
self-designing machines, which appears in the work of Wright, Campbell and Dick. These authors portray self-design as a route by which the pace of machine evolution can accelerate—these works, and Butler’s before, strongly foreshadow current interest in the idea of the technological singularity . The concept has attrac... |
Kurzweil, ( 2005 and Bostrom, ( 2014 . However, such speculations date back at least two hundred years; for a good discussion of the history of these ideas, see (Eden et al.,, 2013 |
Implications for human evolution Beyond the idea that machines might become the dominant intelligent species, the reviewed works have explored a number of potential implications of self-reproducing machines for the future direction of human evolution. |
In Erewhon Butler envisaged that humans might become weaker and physically degenerate due to reduced evolutionary selection pressure brought about by all-caring machines. Eliot and Forster foresaw a similar outcome. In contrast, an alternative outcome explored by Butler [ Lucubratio Ebria ] and Bernal is that human abi... |
enhanced by the incorporation of increasingly sophisticated cyborg technology. Several authors emphasised that humans and machines are engaged in a |
co -evolutionary process. In Lucubratio Ebria Butler suggests that this closely coupled evolution of humans and machines might increase our physical and mental capabilities. In particular, he suggests that intelligent machines change the environment in which humans develop and evolve—foreshadowing the modern idea of bi... |
(Odling-Smee et al.,, 2003 In The Last Evolution Campbell envisaged a positive outcome of this co-evolution, with human creativity working in harmony with machine logic and infallibility. Butler in Erewhon , however, was more dubious of the process, conjuring an image of machines as parasites benefiting from the unwitt... |
The significance of self-reproducing machines as a technology to allow humankind to explore and colonise other planets is a theme covered in various works. The properties of self-repair and multiplication by self-reproduction are seen as essential for attempts to traverse the immense distances of inter-stellar—or even ... |
Implications for human society In addition to imagining consequences for human evolution, these authors also envisaged how human society and the lives of individuals might be affected by the existence of super-intelligent machines. |
The prospect of humans becoming mere servants to machines was raised by Butler [ Darwin Among The Machines ], Wright and Manning. However, Butler suggests that this might not necessarily be a detrimental development—the machines would likely take good care of us, at least for as long as they still rely upon humans for ... |
Many of the works explore how humans might spend their time in a world where all of their basic needs are taken care of by beneficent machines. In Forster’s work, humans engage in the exchange of ideas and academic learning (mostly about the history of the world before the Machine existed). Similarly, Bernal suggests t... |
Likewise, Butler [ Erewhon ] and Bernal discuss the possibility that humans might separate from machines at some point in the future, although in their works, in contrast to Campbell’s, this is a decision made by the humans rather than the machines. Bernal also considers the possibility that the human species might ult... |
Conclusion Concern about the impact of self-reproducing and evolving machines on human society and our future evolution has a surprisingly long history. As we have shown, this dates back at least as early as the Industrial Revolution in Britain, and gains momentum with the publication of The Origin of Species . Modern ... |
There is a possible dystopian bias in the works reviewed, which were predominantly written by young, white men (Roberts,, 2018 . It is indeed true that the large-scale mechanical self-reproducing machines envisaged by these authors have not yet been realised. Nevertheless, technological advances in recent years have ma... |
# Source: arxiv 1806.01398 # Title: Pseudofinite H-structures and groups definable in supersimple H-structures # Sections: all # Downloaded: 2026-03-03T02:36:28.136852+00:00 |
Pseudofinite [MATH] -structures and groups definable in supersimple [MATH] -structures Abstract In this paper we explore some properties of [MATH] -structures which are introduced in |
We describe a construction of [MATH] -structures based on one-dimensional asymptotic classes which preserves pseudo-finiteness. That is, the [MATH] -structures we construct are ultraproducts of finite structures. |
We also prove that under the assumption that the base theory is supersimple of [MATH] -rank one, there are no new definable groups in [MATH] -structures. This improves the corresponding result in |
Acknowledgements The author wants to thank her supervisor Frank Wagner for proposing the interesting question that initiated this project, for contributing at least half of the ideas of this paper. She also wants to thank the referee for the valuable comments and corrections. She is thankful to Gareth Boxall for sugges... |
Introduction [MATH] -structures are introduced in . They are based on a geometric theory, where algebraic closure satisfies the exchange property and [MATH] is eliminated. When a dense and co-dense independent subset is added to a model of this theory, the resulting structure is an [MATH] -structure. Strongly minimal t... |
In the following, we will recall the definition of [MATH] -structures and some of their main properties. Let [MATH] be a complete geometric theory in a language [MATH] . Let [MATH] be a unary predicate and put [MATH] . Let [MATH] ; we say that [MATH] is finite dimensional if [MATH] for some [MATH] . For a tuple [MATH] ... |
Definition 1 We say that [MATH] is an H-expansion of [MATH] if: 1. [MATH] 2. [MATH] is an [MATH] -independent subset of [MATH] 3. |
(Density/coheir property) If [MATH] is finite dimensional and [MATH] is non-algebraic, there is [MATH] such that [MATH] 4. (Extension property) If [MATH] is finite dimensional and [MATH] is non-algebraic, then there is [MATH] [MATH] and [MATH] |
Equivalently, we can replace density and extension properties with the following more general ones: (Generalised density/coheir property) If [MATH] is finite dimensional and [MATH] has dimension [MATH] , then there is [MATH] such that [MATH] |
(Generalised extension property) If [MATH] is finite dimensional and [MATH] is non-algebraic, then there is [MATH] [MATH] and [EQUATION] |
A structure [MATH] is called an [MATH] -structure if it is an [MATH] -expansion of some model of a geometric theory. [MATH] -structures are closely related to lovely pairs, where, instead of an independent subset, a dense and co-dense elementary substructure is added. We recall the definition of lovely pairs in the spe... |
Definition 2 Let [MATH] be a geometric theory in a language [MATH] and let [MATH] be the expansion of [MATH] by a unary predicate [MATH] . An [MATH] -structure [MATH] is a lovely pair of models of [MATH] , if |
1. [MATH] 2. [MATH] is an [MATH] -elementary submodel of [MATH] 3. (Density/coheir property) If [MATH] is finite dimensional and [MATH] is non-algebraic, there is [MATH] such that [MATH] |
4. (Extension property) If [MATH] is finite dimensional and [MATH] is non-algebraic, then there is [MATH] [MATH] and [MATH] Fact 3 |
Properties of [MATH] -structures and lovely pairs. Let [MATH] be a complete geometric theory in a language [MATH] [MATH] -expansions of models of [MATH] exist and all of them are [MATH] -elementary equivalent. Let [MATH] be the corresponding theory. Similarly, lovely pairs of models of [MATH] exist, and all of them are... |
If the geometry of [MATH] is nontrivial and [MATH] is strongly minimal/supersimple/superrosy of rank 1, then [MATH] is [MATH] -stable/supersimple/superrosy of rank [MATH] |
Let [MATH] be an [MATH] -structure. Then [MATH] is a lovely pair. Consider the theory of pseudofinite fields. It is supersimple of [MATH] -rank one. By the fact above, [MATH] -expansions and lovely pairs of pseudofinite fields exist. However, the proof of existence uses general model theoretic techniques such as satura... |
The answer turns out to be negative for lovely pairs. Lemma 4 If [MATH] is a lovely pair of pseudofinite fields, then it is not pseudofinite. |
Proof. Let [MATH] be a pair of pseudofinite fields with [MATH] such that [MATH] are finite fields for any [MATH] Suppose [MATH] . We will show that there are [MATH] and [MATH] in the language of rings such that [MATH] is non-algebraic, but there is no [MATH] such that [MATH] holds. However, as [MATH] is a lovely pair, ... |
[EQUATION] Therefore, [MATH] is not elementary equivalent to [MATH] As [MATH] , we may assume that [MATH] for all [MATH] . For any [MATH] take [MATH] with [MATH] . Let [MATH] be such that [MATH] and [MATH] . Let [MATH] [MATH] and [MATH] . Then [MATH] [MATH] and [MATH] . Define |
[EQUATION] We claim that [MATH] is non-algebraic in [MATH] . Since [MATH] for any [MATH] , we have [MATH] . Let [MATH] be such that there is no [MATH] with [MATH] . Then by , Proposition 4.3] , the ideal [MATH] is absolutely prime and does not contain [MATH] or [MATH] . Let [MATH] be the corresponding irreducible varie... |
[EQUATION] by [MATH] . As [MATH] , it is easy to see that [MATH] is a four-to-one function. By that [MATH] , we conclude that [EQUATION] |
Thus, [MATH] is non-algebraic. On the other hand, for any [MATH] we have [EQUATION] Therefore, there is no [MATH] such that [MATH] holds. |
The case of [MATH] is similar, using cubes instead of squares (and possibly going to some finite extension of [MATH] ). In view of the close connection between [MATH] -structures and lovely pairs, we might expect [MATH] -expansions of pseudofinite fields never to be pseudofinite. Luckily, this is not so. In fact, we ca... |
Definition 5 Let [MATH] be a geometric theory in a language [MATH] Let [MATH] be an infinite ultraproduct of finite structures. We call an [MATH] -expansion [MATH] an exact pseudofinite [MATH] -expansion of [MATH] if [MATH] with [MATH] for all [MATH] |
Remark: Let [MATH] be pseudofinite. Then an arbitrary pseudofinite [MATH] -expansion need not to be exact, since it need not be this particular ultraproduct. For example, let [MATH] be a nonprincipal ultrafilter on [MATH] and [MATH] an ultraproduct of finite vector spaces over [MATH] such that [MATH] . It is easy to bu... |
Let [MATH] be a one-dimensional asymptotic class and [MATH] be an infinite ultraproduct of members of [MATH] . In section we show that exact pseudofinite [MATH] -expansions of [MATH] always exist. In particular, pseudofinite [MATH] -expansions of pseudofinite fields do exist. |
Section deals with definable groups in [MATH] -structures. Our motivation is to classify definable groups in [MATH] -expansions of pseudofinite fields. There are some results about definable groups in [MATH] -structures when the base theory is superstable in |
using the group configuration theorem. The problem to generalise these results is that in simple (even in supersimple) theories, there is no nice version of the group configuration theorem available in general. However, pseudofinite fields are exceptional: the group configuration theorem for pseudofinite fields has ess... |
. We can easily deduce that definable groups in [MATH] -expansions of pseudofinite fields are virtually isogenous to algebraic groups. |
However, this is not very satisfactory. It is of course the best one could get when one compares definable groups in [MATH] -expansions of pseudofinite fields with algebraic groups. But as has been noticed in |
, “since the geometry on [MATH] is trivial, we expected adding [MATH] should not introduce new definable groups”. With the help of the group chunk theorem in simple theories (see Fact 23 ) we give a more satisfactory answer, namely, there are no new definable groups in [MATH] -structures when the base theory is supersi... |
Pseudofinite [MATH] -structures This section deals with pseudofinite [MATH] -structures built from one-dimensional asymptotic classes. |
One-dimensional asymptotic classes are classes of finite structures with a nicely behaved dimension and counting measure on all families of uniformly definable sets. They are introduced in |
inspired by the class of finite fields. We recall the definition of a one-dimensional asymptotic class and list some examples. Definition 6 |
Fix a language [MATH] . A class [MATH] of finite [MATH] -structures is called a one-dimensional asymptotic class if the following holds for every [MATH] and every formula [MATH] with [MATH] |
1. There is a positive constant [MATH] and a finite set [MATH] such that for any [MATH] and [MATH] , either [MATH] or there is [MATH] with |
[EQUATION] 2. For every [MATH] there is an [MATH] -formula [MATH] such that for any [MATH] and [MATH] [MATH] if and only if [MATH] |
Examples of one-dimensional asymptotic classes are: The class of finite fields. The class of finite-dimensional vector spaces over finite fields. |
The class of finite cyclic groups. Fact 7 , Lemma 4.1] Let [MATH] be a one-dimensional asymptotic class and [MATH] an infinite ultraproduct of members of [MATH] . Then [MATH] is supersimple of [MATH] -rank 1. |
In particular, the theory of any infinite ultraproduct of members of a one-dimensional asymptotic class is a model of a geometric theory, and we will show that it always allows an exact pseudofinite [MATH] -expansion. |
Definition 8 Let [MATH] be a one-dimensional asymptotic class in a language [MATH] . Let [MATH] [MATH] non-empty) be an [MATH] -formula and [MATH] be as in Definition . Put |
[EQUATION] For a structure [MATH] and a subset [MATH] , we say [MATH] covers [MATH] in [MATH] if the following holds: [EQUATION] |
Let [MATH] be a formula. Suppose [MATH] is algebraic ( [MATH] can be empty) over any [MATH] . For a structure [MATH] and a linearly-ordered subset [MATH] , we say that [MATH] avoids [MATH] in [MATH] if there is no [MATH] such that [MATH] and |
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