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[MATH] there exists a constant [MATH] , such that the probability to reach an optimum of problem [MATH] within [MATH] iterations is [MATH] |
To prove this corollary, first we will obtain a lower bound on [MATH] for [MATH] , using Theorem and the stationary distribution of the associated Markov chain [MATH] After that, analogously to the proof of Corollary , we will compute a lower bound on [MATH] for finite [MATH] , using Theorem |
Proof of Corollary The Markov chain associated to the set of lower bounds [MATH] defined above has the following nonzero transition probabilities |
[EQUATION] [EQUATION] All other elements of matrix are equal to zero. The stationary distribution of the associated Markov chain may be found from the well-known model for diffusion of P. Ehrenfest and T. Ehrenfest. Consider [MATH] molecules in a rectangular container divided into two equal parts A and B. At any time [... |
in container A. The corresponding random walk has transition probabilities [EQUATION] [EQUATION] The stationary distribution in Ehrenfests model (see e.g Feller, 1957 , chapter. 15, § 6) is given by [MATH] Grouping each couple of symmetric states (i.e. the state where A contains [MATH] molecules, B contains [MATH] mole... |
molecules and B contains [MATH] molecules, [MATH] ) into one state we conclude that the Markov chain with transition matrix has the stationary distribution [MATH] for any [MATH] . So by Theorem , vector [MATH] is the limiting lower bound for [MATH] as [MATH] |
We are interested in transient behavior of the EA, so we will obtain a lower bound for the expected population vector [MATH] given a finite [MATH] , using Theorem . Consider the matrix norm [MATH] which is associated to the vector norm [MATH] in the case of left-hand side multiplication of matrices by vectors. For the ... |
Let us find the vector [MATH] which is the limit of the right-hand side in inequality ( as [MATH] . To this end, it suffices to solve the system of equations |
[EQUATION] [EQUATION] Recall that the right-hand sides in inequalities ( ) and ) of Theorems and are equal, given equal matrices . This suggests to put [MATH] , i. e. |
[EQUATION] Again let [MATH] . By properties of the norms under consideration, [MATH] , so by Theorem [EQUATION] for any [MATH] . With [MATH] the average proportion of feasible genotypes is lower-bounded by [MATH] since |
[MATH] Using ( 25 ) and Stirling’s inequality [MATH] we conclude that [MATH] . Now assuming that a constant [MATH] is so lagre that [MATH] , for [MATH] we have |
[EQUATION] so [MATH] and [MATH] By assumption the initial population consists of all-zero strings. Therefore the presence of at least one individual from [MATH] in the current population implies that an optimal solution to a problem [MATH] was already found at least once. Thus, in view of Proposition , after [MATH] ite... |
If the EA is restarted with [MATH] every [MATH] iterations, then by Markov inequality the overall runtime of this iterated EA is [MATH] for any [MATH] |
The tools for the non-elitist EA analysis from Corus et al., 2014 Dang and Lehre, 2016 Eremeev, 2016 can be adjusted to upper-bound the runtime of the EA on [MATH] but in such a case, a non-zero selection pressure would be required with a sufficiently large [MATH] and the results would hold only for [MATH] |
5.5 Upper Bound on Proportion of Optimal Genotypes in Case of OneMax The upper bounds on vector [MATH] obtained in Proposition are not likely to be suitable for obtaining the lower bounds on runtime of the EA in absolute terms due to nonlinearity in the right-hand side of ( 13 ). There are other methods for finding suc... |
Badkobeh et al., 2014 Lehre, 2010 Sudholt, 2013 . The upper bounds on vector [MATH] however may be used for comparison of the EA to the ( [MATH] [MATH] ) EA and the ( [MATH] [MATH] ) EA as it was suggested in Proposition |
To illustrate such a comparison let us consider the EA with bitwise mutation operator [MATH] in the case of OneMax fitness function and assume that [MATH] |
Analogously to the notation form Section [MATH] and [MATH] will stand for the probability to have an optimal current individual on iteration [MATH] of [MATH] [MATH] ) EA and on iteration [MATH] of the ( [MATH] [MATH] ) EA, respectively. In these algorithms we assume that the bitwise mutation operator [MATH] is used and... |
Corollary 4 Suppose that the fitness function is OneMax and the initial population of the EA consists of [MATH] copies of the same solution, chosen uniformly from [MATH] , and the EA uses the bitwise mutation operator with [MATH] Then for any [MATH] holds |
[EQUATION] In particular, if the tournament size [MATH] then [MATH] and [MATH] Proof. In the case of OneMax fitness function the bitwise mutation operator with [MATH] is monotone Borisovsky and Eremeev, 2008 Application of Proposition yields |
[MATH] for the cumulative transition probabilities [MATH] associated with this monotone mutation operator. It is easy to see that [MATH] |
and [MATH] since [MATH] Thus, for the ( [MATH] [MATH] ) EA [EQUATION] as required. In the case of [MATH] this inequality implies that |
[MATH] The result for [MATH] [MATH] ) EA follows analogously. [MATH] A superiority of the ( [MATH] [MATH] ) EA over other evolutionary algorithms in the case of OneMax fitness function and bitwise mutation with [MATH] is well-known from Borisovsky, 2001 Borisovsky and Eremeev, 2008 Sudholt, 2013 Corollary allows to mea... |
Conclusions In this paper, we presented an approximating model of non-elitist mutation-based EA with tournament selection and obtained upper and lower bounds on proportion of sufficiently good genotypes in population using this model. In the special case of monotone mutation operator, the obtained bounds become tight i... |
Applications of the obtained general lower bounds give an exponentially vanishing tail bound for the Randomized Local Search on unimodal functions and new runtime bounds for the EAs on the 2-satisfiability problem and on a family of set covering problems proposed by E. Balas. |
It is expected that the further research will involve applications of the proposed approach to other combinatorial optimization problems, in particular, the problems with regular structure. |
Most of the lower and upper bounds on expected proportions of genotypes, obtained in this paper, do not take the tournament size into account. It remains an open research question of how to construct the tighter bounds w.r.t. the tournament size. The subsequent research might benefit from joining the analysis of expect... |
It is of interest to compare the tail bounds established in Subsections 5.2 and 5.5 to the tail bounds obtainable using other techniques, e.g. Lehre and Witt, 2014 |
Another open question is how to incorporate the crossover operator into the approximating model. For some types of crossover operators, such as those based on solving the optimal recombination problem Eremeev and Kovalenko, 2014 , the lower bounds from this paper may be easily extended, ignoring the improving capacity ... |
Appendix. In this appendix, we reproduce two results from Borisovsky and Eremeev, 2001 and Borisovsky, 2001 which are used in Section and a well-known result on eigenvalues of thridiagonal Toeplitz matrices. |
The algorithms ( [MATH] [MATH] ) EA and ( [MATH] [MATH] ) EA and probabilities [MATH] and [MATH] [MATH] [MATH] are defined as in Section . For the ( [MATH] [MATH] ) EA and for the ( [MATH] [MATH] ) EA we also define the vectors of probabilities: [MATH] |
[MATH] The following Theorem from Borisovsky and Eremeev, 2001 shows a superiority of the ( [MATH] [MATH] ) EA over the ( [MATH] [MATH] ) EA in the case of monotone mutation operator. For a fair comparison of the algorithms ( [MATH] [MATH] ) EA and ( [MATH] [MATH] ) EA here we allow both of them to make the same number... |
Theorem 5 Suppose that the same monotone mutation operator [MATH] is used in the ( [MATH] [MATH] ) EA and in the [MATH] [MATH] ) EA and [MATH] Then |
[MATH] for any [MATH] The following theorem from Borisovsky, 2001 compares the distribution of a fittest individual [MATH] in the EA population [MATH] over Lebesgue subsets compares to such a distribution of the [MATH] [MATH] ) EA. Let us define a vector [MATH] for the EA, analogously to vectors [MATH] and [MATH] |
[EQUATION] Theorem 6 Suppose that the EA and the ( [MATH] [MATH] ) EA use the same monotone mutation operator [MATH] and [MATH] Then for any [MATH] holds [MATH] |
regardless of selection operator used in the EA. The original manuscript Borisovsky, 2001 is hardly accessible, therefore we provide the proof of Theorem below. |
Proof. It is sufficient to consider the case of [MATH] , since the statement for the general case will follow by induction on [MATH] . Let [MATH] denote a genotype with the highest fitness among the first [MATH] offspring of [MATH] and let [MATH] be a genotype with the highest fitness among [MATH] in the EA population ... |
a) Let us first assume that [MATH] and [MATH] for some fixed [MATH] and let a genotype [MATH] be chosen by the selection operator of the EA. Then for arbitrary [MATH] in view of Proposition we have: |
[EQUATION] Note that [MATH] , which may be established by induction on [MATH] using the inequality [EQUATION] [EQUATION] b) Let us prove that [MATH] for arbitrary initial distributions of the ( [MATH] [MATH] ) EA and the EA, assuming |
[MATH] . We use the total probability formula and the conclusion of case a): [EQUATION] [EQUATION] c) In general, when [MATH] let us note that according to Proposition 1 from Borisovsky and Eremeev, 2001 , in the case of monotone mutation for any [MATH] we can consider [MATH] as the following function on vector [MATH] |
[EQUATION] where [MATH] are the cumulative transition probabilities of mutation operator [MATH] . We denote the relationship ( 28 ) by [MATH] for brevity. Then due to nonnegativity of the multipliers of probabilities [MATH] in ( 28 ), we conclude that [MATH] Finally note that the result of case b) may be written as [MA... |
[MATH] The following result on eigenvalues of thridiagonal Toeplitz matrices may be found e.g. in Noschese et al., 2013 Theorem 7 |
Suppose an [MATH] -matrix is composed of zero elements everywhere except for the diagonal elements, which equal [MATH] the superdiagonal elements which equal [MATH] and subdiagonal elements which equal [MATH] . Then all of eigenvalues of are given by |
[EQUATION] Acknowledgements The research presented in Section was supported by Russian Foundation for Basic Research grants 15-01-00785 and 16-01-00740. The author is grateful to Sergey A. Klokov, Boris A. Rogozin and anonymous referees for helpful comments on earlier versions of this work. |
# Source: arxiv 1508.06538 # Title: Causality, Information and Biological Computation: An algorithmic software approach to life, disease and the immune system # Sections: all # Downloaded: 2026-03-03T01:55:49.672807+00:00 |
Causality, Information and Biological Computation: An algorithmic software approach to life, disease and the immune system Abstract |
Biology has taken strong steps towards becoming a computer science aiming at reprogramming nature after the realisation that nature herself has reprogrammed organisms by harnessing the power of natural selection and the digital prescriptive nature of replicating DNA. Here we further unpack ideas related to computabilit... |
Introduction Information and computation have transformed the way we look at the world beyond statistical correlations, the way we can perform experiments, through simulations, and the way we can test these hypotheses. In his seminal paper on the question of machine intelligence |
, Turing’s approach consisted in taking a computer to be a black box and evaluating it by way of what one could say about its apparent behaviour. The approach can be seen as a digital version of a cogito, ergo sum dictum, acknowledging that one can only be certain of one’s own intelligence but not of the intelligence o... |
, showed that systems such as arithmetic that have a certain minimal mathematical power to express something, can produce outputs of an infinitely complicated nature |
. Indeed, this means that while one can design software, only trivial programs can be fully understood analytically to the point where one is able to make certain predictions about them. In other words, only by running software can one verify certain of its computational properties, such as whether or not it will crash... |
Turing’s halting problem implies that one cannot in general prove that a machine will ever halt, or that a certain configuration will be reached upon halting; one has therefore to proceed by testing and only by testing. These fundamental theorems and results imply that testing is unavoidable; even under optimal conditi... |
In light of these realities, entire (relatively) new fields designated model checking systems testing and software verification seek to produce and test reliable software based on simple mathematical models. Today’s airplane construction companies and other manufacturers of critical systems such as electronic voting sy... |
As shown in Figs. and , for an extremely simple computer program represented by an elementary cellular automaton (ECA), one needs to perform, in the best case scenario, at least 8 very precise observations at two consecutive times (hence 16) with perfect accuracy in order to hack this computer program, only to unveil i... |
Fig. shows such a minimalistic example that it suggests that the same phenomenon is pervasive in physics and biology, where even the simplest conditions can generate a cascade of apparent randomness. Notice this is of even more basic nature than the phenomenon of chaos where the argument is that arbitrary close initial... |
Multiple sclerosis (MS), for example, is a complex disease in which the insulating shield (the myelin sheath) of nerve cells is damaged and for which there is currently no curative treatment, yet the disease sometimes shows apparent periodic behaviour in the form of relapses. Relapsing eventually becomes progressive me... |
An ideal observer would be able to see the source code directly, but this is virtually impossible in practice, for a number of reasons. First, because it is difficult to separate phenomena from other phenomena in nature, and second because we never have access to first causes, or for that matter, to first causes in com... |
This phenomenon is not exclusive to rule 30 but is pervasive even in the most deterministic algorithmic sciences such as mathematics and computer science, in objects such as the decimal expansion of numbers like [MATH] or the square root of 2, or in the logistic map that leads to chaos behaviour. But notice that these ... |
In other words, as Fig. illustrates, reverse engineering (finding ultimate rules) is extremely difficult, and in biology there is an entire relatively new field devoted to reverse engineering biological networks, also called network reconstruction, where rules involve genes, proteins and metabolites, to name a few inst... |
Natural computation and programmability Sensitivity analysis can be useful for testing the robustness and variability of a system, and it sheds light on the relationship between input and output in a system. This is not hard to identify from a computing perspective, where inputs to computer programs produce an output, ... |
For example, sensitivity measures aim to quantify this uncertainty and its propagation through a system. Among common ways to quantify and study this phenomenon is, for example, the so-called Lyapunov exponent approach |
. This approach consists in looking at the differences that arbitrarily close initial conditions produce in the output of a system, and hence is similar to the generalisation we introduce. Traditionally, if the exponent is large the sensitivity is non-linear, and divergence increases over time. If constant, however, th... |
one has been proposed). Furthermore, measures such as the Lyapunov exponent are meant to detect qualitative changes such as chaotic behaviour, which is not desirable for a programmable system, as is suggested by Fig. . Further research on the connections between programmability and these other dynamical system sensitiv... |
In our approach to behaviourally evaluate systems for their programmability, a compression algorithm can be used as an interrogator device, where for questions one uses initial conditions of the system, while the answers are the lengths of the compressed outputs. In general, for a system to be reprogrammable, inputs wi... |
These testing ideas are based on whether a system whose source code may never be known is capable of reacting to the environment — the input — as is the case in more formal implementations of this measure of programmability . Such a measure |
would quantify the sensitivity of a system to external stimuli and could be used to define the amenability of a system to being (efficiently) programmed. The basic idea is to replace the observer in Turing’s imitation game with a lossless compression algorithm, which duplicates the relevant subjective qualities of a re... |
The compression algorithm looks at the evolution of a system and determines, by means of feeding the system with different initial conditions (which is analogous to questioning it), whether it reacts to external stimuli. The Kolmogorov complexity of an object is the length of the shortest program that outputs the objec... |
). Then, if the evolution of a program (say a natural phenomenon) is complex and does not react to external stimuli, all compression algorithms will fail at compressing its evolution and no difference between different evolutions for different perturbations will be detected (e.g. by taking the differences between the c... |
For example, as is shown in , certain elementary cellular automata rules that are highly sensitive to initial conditions, and present phase transitions which dramatically change their qualitative behaviour when starting from different initial configurations, can be characterised by these qualitative properties. A furth... |
. Other calculations have been advanced in and The behavioural approach in fact generates a natural classification of objects in terms of their programmability, as sketched in Fig. (bottom). For example, while weather phenomena and Brownian motion have great variability, they are hardly controllable. On the other hand,... |
Information biology One of the aims of our research is to exploit these ideas in order to try to reprogram living systems so as to make them do things we would like them to do, for in the end this is the whole idea behind programming something. Fig. (bottom) is an illustration of an investigation |
into the possibility of mapping the conformational space of a simulation of porphyrin molecules with a view to making them self-organise in different ways. That is, an investigation into what it means to program a nature-like simulated system, to find the inputs for the desired output (that matched the actual behaviour... |
In biology, the greatest challenge is the prediction of behaviour and shape (“shape” determines function in biology). Examples are protein folding or predicting whether immune cells will differentiate in one direction rather than another. In Fig. (bottom), we investigated how we could arrive at certain specific conform... |
This means that with enough data one does not need to perform perturbations but rather to find them as they occur in nature. However, while in recent decades we have come to understand that complex systems such as the immune system cannot be thoroughly analysed and fully understood as regards its components and functio... |
Complex diseases (see Table ) require a complex debugging system, and the immune system can be viewed to play just this role. The immune system is the highly complex and dynamic counterpart of many complex (and simple) diseases, and the most common approach to understanding it has been via evaluating its individual com... |
Interestingly, these two highly complex systems are interconnected — the immune system plays a key role in tumour development: The immune status (“immunoscore”) was found to be the highly predictive for cancer patient survival |
, and evading immune destruction was added to the “hallmarks of cancer” recently . According to the immune surveillance theory, the immune system does not only recognise and combat invading pathogens but also detects host cells that become cancerous, thus eliminating tumor cells as soon as they arise. This requires the... |
Equipped with these ideas that suggest that we can extend concepts that properly belong only to computability and algorithmic information theory, we can devise a software-engineering approach to systems biology. |
3.1 How natural selection programs and reprograms life At some point early in the process of replicating (copying) biological information for cell growth, which is particularly necessary for multicellular organisms, the process reaches the critical juncture of dealing with errors and redundancy that can be quantified b... |
Had the balance not been reached, no copying process would have conveyed the information necessary for organisms to reproduce. The noisy-channel coding theorem (sometimes called Shannon’s theorem), establishes that for any given degree of noise in a communication channel, it is still possible to communicate (digital) d... |
While the nature of the variations (of which some can be identified as errors) can be attributed to noise, the correction is nothing but a mathematical consequence. If errors prevail, both in the primary source and replicants, then the cells and organisms have a greater chance of dying from these variations. On the oth... |
Approached as computer programs, one can explain how certain patterns, e.g. the information content in molecules, such as DNA or RNA, may have been produced in the first place |
. There is also the problem of the uncaused first computer program, given that the process generating information and eventually computation required computation in the first place (e.g. the very first laws in our universe). In fact, we have explored these ideas, taking seriously the possibility that nature computes, a... |
of the property of being “computer-” or “algorithm-like”, the property of algorithmicity . In we undertook a search for statistical evidence, with interesting results. |
Hacking strategies have been in place in biological systems since the beginning, viruses for example, are unable to replicate by themselves, but they trespass the cell membrane and release their DNA content into the cell nucleus to be replicated. Cells are the basic units of life because of this property, they are the ... |
Viruses are clearly (in consensus among biologists) non-living structures of encapsulated DNA that evolve by undergoing a process of natural selection and are evidence of non-living matter subject to the same process. Though they may appear so, viruses have no self-purpose or will of their own, neither as individuals o... |
3.2 A hacker view of cancer and the immune system Equipped by these mechanisms of evolution by natural selection and the way in which self-replicating cells can be hacked a hack to fight diseases has been devised in the efforts to treat cancer. Traditional drug and radiation therapies have not been very successful so f... |
). There is therefore a potential to develop these ideas into a more systematic information-theoretic and software-engineering view of life, cancer and immune-related diseases based on these amazing purely computational mechanisms in biology. While it is clear that information processing is in a very fundamental sense ... |
Cancer, for example, is like a computer programming bug that does not serve the purpose of the multicellular organism. Cancer cells are ultimately cells that grow uncontrollably and do not fulfill their contract to die or stop proliferating at the rate at which they must if the multicellular host is to remain stable an... |
Cancer can be seen as a purely information-theoretic problem: the information dictating the way in which a cell replicates is compromised, either because it, as it were, reneges on a contract with the multicellular organism, resulting in the cell behaving selfishly and replicating with no controls, or else because nois... |
Likewise, the immune system can be seen as an error-correcting code. One key aspect of the immune system is diversity which is largely contributed by T and B lymphocytes, cell types of the adaptive immune system. By gene rearrangement of segments in their antigen receptor genes, a highly complex repertoire of different... |
When B cells and T cells are activated and begin to replicate, some of their progeny become long-lived memory cells, remembering each specific pathogen encountered, and can mount a strong, faster response if the pathogen is detected again. This means the memory can take the form of either passive short-term memory or a... |
The memory of the immune system stores all the information relating to all the pathogens we have encountered in our lives . When B cells and T cells are activated some will become memory cells. Throughout the lifetime of an animal these memory cells effectively form a “database” of effective B and T lymphocytes |
. Upon interaction with a previously encountered antigen, the appropriate memory cells are selected and activated. The major functions of the acquired immune system that involve information include: |
Pattern recognition:“non-self” antigens in the presence of “self”, during the process of antigen presentation. Communication: Generation of responses that are tailored to maximally eliminate specific pathogens or pathogen-infected cells. |
Storage (memory): Development of immunological memory, in which pathogens are “remembered” through memory cells. Yet another illustration of how the immune system can be seen from a purely informational perspective is the fact that newborn infants have no immune memory, as they have not been exposed to microbes, and ar... |
, so human babies have high levels of antibodies even at birth, with the same range of antigen specificities as their mother. Protective passive immunity can also be transferred artificially from one individual to another |
This brief account reveals the extent of the role played by information in the immune system. Immune-related diseases are related to informational dysfunction, such as wrong signaling, defects in immune tolerance or misguided pattern recognition. This can cause the immune system to fail to properly distinguish between ... |
A model that a team created shows that the stability of a gene network stems from several major factors. These factors include “effective” genome size, proteome turnover, and DNA repair rate but also gene network connectivity. The researchers concluded that hacking any of these parameters one could increase an organism... |
Cellular death has a strong information-theoretic component, evident in the way evolution has programmed multicellular organisms. Cancer, and indeed laboratory cell lines (the cells used in labs) are immortal |
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