Formalization, Modeling and Anticipatory Properties in Computational ...

Formalization, Modeling and Anticipatory Properties in Computational Science for Sustainable Development Alexander Makarenko Institute for Applied Systems Analysis of National Technical University of Ukraine "Kiev Politechnic Institute named by Igor Sikorsky", Prospect Pobedy 37, building 35, 03056, Kyiv-56, Ukraine [email protected]

Abstract Some aspects of operational research methods, approaches and formalism concerned sustainable development are considered. General formal definition for sustainable development is proposed. Short description of new society models useful for sustainable development considering is given. Associative memory is background for such models. The role of strong anticipation property for understanding sustainable development is discussed. New possibilities for sustainable development indexes are proposed. New illustrations for sustainable development problem are given. Some examples for sustainable development (local, global and of practical importance) are described. The role of constraints in mathematical considering of sustainable development is stressed. Keywords: sustainable development, operation research, models, formalization, anticipation, networks I. Introduction. Recently operational research very many objects for investigations and for applications. Sustainable development (SD) is one of them. Supported the concept of development although it has a relatively short history, has already become popular enough, it was a real fact and factor in the political and public life is the subject of numerous publications. Perhaps the basic definition is known somewhere in the early 90s. However, there are many modifications and interpretation of this definition, and they depend on the most likely areas of scientific interest to researchers, considering the problems of sustainable development (SD). It should be noted that, as a rule, it is verbal or qualitative determination of SD, or in the best case study of individual systems, processes, or indexes of SD. Recall that when considering the SD as the basic definition of acceptable identification (1987) Brundtland Commission (https://en.wikipedia.org/wiki/Suatainable_development). Remark that the topic ‘sustainable development ‘have been the the subject of many investigations (for example see the millions of references in Google). Of course the operation research (OR) community also made many investigations. Here we remember some examples with mathematical tools from OR. Say G. Chichilnisky proposed some axioms for SD. Rand D.A. investigated some topics in biology. Aubin J.-P. with colleagues had proposed the investigations of viability of some complex systems. Recently some theoretical considerations had been proposed by J. Sheffran with colleagues. Some problems of practical importance related to climate change investigated by J. Heitzig with colleagues. Also very many problems of development had been investigated in economy. Many reference in OR of development are in proceedings of IFORS and EURO conferences and journals. However, after 25 years, it becomes necessary to further development as the foundation of the concept and allows the formalization and use of computational social sciences methods. Discussion of these issues is devoted considerable number of studies on SD. Further formalization of the concepts and models largely depends on the application area and at the same time can also vary greatly depending on the model used, respectively, therefore continues to be still very relevant. Dispate of many investigations of SD in different branches of science the problem of formalizing SD and applications of OR methods and concepts to SD are still far from adequate general solutions. This is because of the difficulties in understanding of large social systems behavior and it formal description. The reasons of such difficulties have the origins in complexity of such systems, heterogenuities, hierarchicity, mentality of individuals and society and many other factors. Therefore, in this paper we attempt to highlight (very briefly) some aspects, first of all, with the formalization of sustainable development issues, and then give a list of the class of models for society investigation that allow us to consider many problems of global society including SD and discuss the anticipatory property accounting. II. Considerations in the development of the concept of formalizing SD.

In this section we assume discuss some aspects that should be remembered before clarified the formalizing SD, and even more modeling. First of all we give some usual description of SD processes. Also we give description of some non-usual aspects of SD which should be considered further. In the presentation we will try to follow the inductive method, moving from a clearer and simple to more complex tasks, so that the resulting reach some understanding of the aspects related to the most common socio- economic and natural- technical and political systems. We call such systems SNET - integrated systems. 2.1. Three aspects of complex SNET. In view of the increasingly voiced ideas SD, SD concept now appears in many projects on administrative management - management systems - environmental, urban, cultural, economic, etc. In general, we note that the idea of sustainability and sustainable development is about tens of millions of links by search engines (example is Google). Often when this is considered in the first place, the idea of conservation and somewhere in implicit form, after all, the second part of the definition of the Brundtland’s Commission solving future generations, that is actually for many ordinary managers consider sustainability for a small time interval up to 10 -15 years (actually this part coincides with the practice of planning and management units). Second, consideration of issues related to SD, it is clear that play an important role as the spatial extent of systems (earth → countries→ countries regions of the territory → cities → landscapes and individual elements of the subsystem and temporal scales, for example, time orders geological epochs to periods of landscape change and climate change, for annual intervals (or less, if necessary). Also there exist clearly observed different time scales in other aspects of SNET: example - civilizations → historic periods → cyclic phenomena in cultural processes → annual scale and etc. 2.2. Geometric illustration of the problems of SD. Ever since the first works by J. Forrester, laid the foundations for mainstreaming SD, played a big role to present the results of calculations and presentation of concepts in the form of visual graphs. When considering such a graph, it usually leads dependence of a quantity (e.g. number of population, or oil, or dirt), etc. from time to time, or all the quantities result in a single graph, but it obscured the notion of SD, at least those aspects that are related to the depletion of resources for future generations. Usually natural resources still exhausted, so the situation should look like Fig. 1.

Fig. 1. Exhaustion of resources case.

S corresponds to the point of impossibility of functioning systems after depleted resources. Note that the trajectory of two possible fixed with the best control method. Note that the graphics data is assumed that the path cannot overcome SNET restriction crisis occurs (at point C) or catastrophe (point S). For example, if the limit - it's oil for transport, running only on gasoline. In fact, of course, history teaches SNET that evolution occurs over time, not so clearly pessimistic, but because of the need for a restructuring of systems that can allow the system to exist and develop). The Fig. 1 is posed here as the illustration of one of the key components of SD concept - namely, constrains. III. Formalization of sustainable development description. First of all we should introduce two cases of SD problems: local and global (in some sense introduced below). First, consider what we call the local circuit (when considering only one generation), which only solves their local time tasks, again starting from the local criteria. First, we point out the structure and concepts that should be considered in the problem of the local SD. Review of recent theoretical investigations, practical problems of SD management and considerations above follows to the needs in formalized SD definition. The case of global SD problem corresponds to considering global systems, large times, strongly evolved environment and non-strictly defined goals.

Reasoning should be carried out for two cases easier for local and global SD. We here shortly outlook the resume of SD formalization. First, we stress out the structure and concepts that should be considered. 1. System parameters and their description (external, internal, management, etc.) denote the set of parameters {Par}. We're in this subsection shall not consider in detail the properties of these (and the following elements of the description), but only attempt to highlight what the structure should be considered. For example, here we doesn’t consider possible metric and topological structures, relations order, symmetry, and others on the set {Par}. 2. The equations describing the system {Equat}. 3. Set of trajectories of systems {Traj}. 4. Restrictions on the trajectory and the parameters of the system - the set {Ώ}, and set of boundary points (boundaries) constraints {∂ Ώ}. 5. Multiple criteria sustainability {SCrit} or criteria SD {SD Crit}. 6. A plurality of external control parameters {Contr}. 7. Set representing the age structure of populations in the interval [0, T] {Age}. 8. Set of initial conditions {Init}. 9. Structure of the system {StSys}, process structure {StProc} and structure of individuals {StInd}. We assume that such structures do exist (even if precise knowledge on them are abcent). 10. Additional requirements for components (desirable) - additional to the mandatory restrictions {Ώ} and {∂ Ώ}. Denote them {Aux}. 11. Describe the process of decision-making. Denote them {Decis}. If necessary, they can be broken down into separate components. 12. Many uncertainties in the system {NonDef}. If necessary, this set can also be broken down into components. We now describe what is the problem of SD. Definition of Sustainable Development Problem. Find such objects of {St = {StSys} ® {StProc} ® {StInd}}, {Contr}, {Decis}, such that the result is a trajectory in the evolution of the system tr € {Traj}, that is performed t € {SDCrit} ® {Aux}, where t mean value calculation criterion SD at time t on the trajectory tr, and the results of the calculation should be at any given time to lie in {SDCrit} and {Aux}. (Symbol ® means belonging to all parameters to specify the set). Note 1. By 12 the above structures can add one more - 13. Set of models {Models}, if we use the simulation. Note 2. It exists the ambiguity of possible trajectories of the system (which can occur due to various reasons, including a social component of SNET). Therefore, instead of a single trajectory tr can enter the wording with the funnel of trajectories Ptr. Note 3. Perhaps the existence of fluctuations and other uncertainties (by the way, this is essential in risk assessment). Then we can take into account the uncertainty in the objects examined a set of 1-13 Contributed uncertainties. Keeping in mind the structure of definitions of sustainable development problem SD tasks can already move on and expand and at the same time detail definition. Accounting for the presence of many different generations (for simplicity we are talking about two generations), for example, may be that two generations have different criteria for SD, then {SDCrit} = {SDCrit} (Generation 1) ® {SDCrit} (Generation 2). The restrictions on management {RCtrl} = {RCtrl} (Generation 1) ® {RCtrl} (Generation 2) may be different for different generations. In principle, it may be that {SDCrit} = {SDCrit} (Generation 1) ® {MSDCrit} (Generation 2), where {MSDCrit} (Generation 2) - the set of possible criteria of SD for second generation. We may not know exactly {MSDCrit} (Generation 2) by virtue of the fact that it depends on the future of technology, and in fact the future of knowledge {Knowl} (Generation 2) of which we can only speculate (we assume that many of the current knowledge of this generation {Knowl} (Generation 1) in the first approximation is known - or, for example, is a set of technologies implemented in knowledge). The proposed scheme is suitable consideration, it seems, for any variant of SD processes (both for descriptive (verbal)), and for specific practical problems or tasks already). Remark that the mathematical investigations remembered in introductory subsection are particular examples of such models and may be posed in our definition of SD. But mostly such models have considered the particular aspects or subsystems of the general global systems. This followed to lack of general definition of SD in past investigations. Usually implementation of SD depends on systems and tasks considered and include many aspects: description of the system, environment etc. Here we remember only one rather new aspect – namely anticipation. Remark that for considering SD problems (especially global) it is necessary to have the applicable models of society with accounting history, mentality etc. Here below we very shortly described such new models. VI. Model of SD with associative memory Recent understanding and management of SD requires the using of strict definitions of considered problems and also requires using of adequate models. Since the SD is complex multi-aspect phenomenon considering it requires many

different models: environmental, economic, social etc. But also it is necessary to have integral conceptual models which can incorporate all such issues. Recently the less developed is the society modeling. So below we describe the main principles for new class of models of society which were developed by the author as a modification of Hopfield model with associative memory property. Here we present only the distillation of the main ideas of the approach, as well as some of the details of the models and approaches, in the simplest case for example. Imagine a society consisting of N >> 1 personalities and let each individual is characterized by a state i } sli  M li , l  1,..., M i which M il is a set of possible values si . There are vector S i  {s1i ,..., s ki i , s i ki 1 ,... , ..., s M i many opportunities to make the elements in the blocks and levels in such models. In well-developed society personalities have many complex connections. Formalize it. We assume that there are links between i and j individuals. Suppose p Jijpq is the link between p the components of the element i and q components of the element j thus set Q  ({si }, { J ij pq }, i, j  1,2,..., N ) characterizes the state of society. Analysis of the recent models of the sets of media elements and relations shows similarities of these models of society and neural network models. One can similarly describe and hierarchical systems. Links can be very different in nature. The values of bonds may represent a formalization of economic, information, control channels nationality, family, professional interactions and so on. Society - evolution system with dynamic changes over time. Further, for simplicity, we consider only models with discrete time points: 0,1,2, .... , N, .... Following the evolutionary nature of the systems, we believe that naturally regarded as the input of the system at time n values at this time and as the output values in the next time ( n  1) (for n  0,1, 2,... ). Note that in a developing society content element set can vary. For example in the economy a list of companies and corporations are constantly changing bankruptcy and coalition building. Social, political, government networks are also often subjected to changes. This generally leads to a change in the number of elements N (n) and the number of hierarchical levels of M (n) at various time points. A critical step in the creation of new models is to take into account the notion of a global culture of society as a collection of all the material achievements plus spiritual type of morality, ethics, religion, justice, and education. Global culture is also sometimes called the collective memory of society. Global culture - a very stable structure and is the basis of civilization (A. Toynbee, I. Wallerstein). The proposed models have the dynamic principles that allow you to simulate the behavior of a global culture of the time. This is due to the fact that the models have properties of associative memory. Behavior of historical processes resembles the pursuit of a very stable structure and the socalled points of attraction (attractors) in pattern recognition in computer science and neuroscience. Importantly, many of the social subsystem in society also have similar properties, and it allows us to consider the individual submodels. In earlier work Makarenko (1998; 2002); Makarenko and KLestova (1999) the author considered a new class of models of society as a modification of models of neural network. It is known that the dynamics of the Hopfield N

neural network model is obtained by considering the functional called "energy" E   J ij si s j ,

wherein {1,  1} -

i j

the possible states of the network elements, N - number of elements, J ij - the connection between i-th and j-th elements. In Hopfild-like system model tends to be one of the few stable states with a minimum of the functional E. Many of the possible initial conditions lead to a small number of such minimal "energy" states, called points of attraction. Potential landscape of "energy" in this case looks something like serfaces of function with has few minimum. It is important the SD problem has the same interpretation but with constrained from Definition of Sustainable Development Problem from subsection II above in the space with much more dimensions. Recall that such a law is correct only in the symmetric case, when J ij  J ji . In general, the models have the form si (t  1)  f i ({si (t )}, {si (t  1)},..., { J ij (t  1)},..., b).

(1)

In the simplest case, the model takes the form of the well-known Hopfield model, presented in many publications and dynamics equations which have the form. In the case of hierarchical systems and symmetric connections between the different elements and different levels, there are also functional - an analogue of the "energy" E. Remark that on of examples of such type models applications was geopolitical problem modeling . 4.1 Models with the accounting of mentality. The internal representation of the external world Accounting mentality requires an examination of the internal structures and including them in the global hierarchical model. There are many approaches to accounting mentality. Most natural way of doing this is to looking for the model for the internal structure also in a class of neural network models. Recall that the original neural network

models were introduced for the study of the brain. Firstly, we can change the basic laws in models (1). At the phenomenological level, this can be achieved by introducing a division of element parameters on external Qije and internal Qiji variables and setting separate laws for two blocks of parameters Ye  f e ( X e , X i , P, E )

(2)

Yi  f i ( X e , X i , P, E )

(3)

X e , Ye , X i , Yi - External and internal input and output parameters. Function f e and f i , may have completely different shape. For example equations for external variables may be in the form of neural network, combined with the differential equations for internal variables. But one of the most promising ways to account for the mentality is search of equation (2), (3) in the neural network class. It offers built-in intelligence to enter the model of the world in the elements that represent the individual or decision-making centers with humans participation. The easiest way is to present an image of the World in the brain of an individual or in the model as the collection of elements and relationships between elements (that is networks). In this image of the World, there is a place for a representation of the individual directly with personal faith, skills, knowledge, preferences.

4.2 One possible method of accounting mentality Laws for elemental behavior should depend on such -representation. Formally, we can introduce the projection operators P to represent the image of the external World in the individual brain: it is very important, but that each individual has their own personal image of the World. Note that the impact of the operator P can be divided into many local projection operators. Then the equation in the previous sections can be replaced by more complex, substituting self-representation of the individual in the right side of the law for the dynamic element to dynamically change the parameters. Some of the simplest options will be presented in the next section, parallel to the description of the anticipatory properties. Of course, there may be a recursion with many levels of recursion in the theory of reflexive systems N. Luhmann, J. Soros, S. Lefebvre and so on. In our scheme, it can be represented as a reciprocal representation of all individuals in the internal representation of each individual. Remark that some such models already had been realized for stock market modeling. IV. Anticipation in models of SD Anticipatory property for some systems had been introduced by R.Rosen and D.Dubois Dubois (2001). Remember from section II that one of very interesting for understanding the social systems property is anticipating. Weak anticipation exists when the system has the model for forecast the future behavior. Strong anticipation takes place when the future state of system isn’t known but is taking into account for evaluating the transition at given moment of time. The example of such model is the next: Si (t  1)  Gi ({si (t )},..., {si (t  g (i ))}, R),

(4)

where R is a set of external (control, structural, environmental) parameters. We must emphasize that the right side of (4) depends on the unknown future the values of the state of the element. This form is built opposite to the form of equations with delay. The main essential new property in such case is the possibility of multi-valued solution (that is many values of solution for some moments of time exist for single initial conditions). This may be interpreted as the possibility of many scenarios of development for real social systems. The second key issue is connected to property that the real social system has single realization of historical way (trajectory). So the social system as the whole makes the choice of the own trajectory at any moment of time. Local SD processes usually are the processes with weak anticipation. Global SD processes are strongly anticipative. Scenarios and decisions have background in multivalued solutions and single trajectory choice Makarenko (2002). V. Indexes of stable (supported) development Consideration in subsections I-IV can also help in solving other important problems in a sustainable development namely in the search for sustainable development indices. In a statement provided we have formulated some ideas on sustainable development, its formal definition and possibilities of constructing index sustainable development

which are based on the length of the time evolution of the system. Further development of the theory and practical choice of adequate indices will depend on the use of different mathematical models of the system. Studies have shown particular utility for this purpose models with associative memory, and investigations of the attractors in the dynamics of such systems. According to the analysis of sustainable development there is some hierarchy proposed indexes depending on the chosen granularity of processes in the system. The simplest case corresponds to a small number of essential N

parameters in the system. For example, the parameters may belong to some space X (often X  R , N - not a very large integer, typically less than 100). Then, in the simplest case, we can consider the limit fixed in the space X.    That is s (t )  ,   X , s (t )  X where s (t ) - the state vector of the system for which support the development should be considered. Note that not only the state of the system at time t provided important, but also the evolution of the system should take into account , because the trajectory of the system to be constraints  (where  there is a border region adopted Ω). We introduce the quantity

 J (t )  J (  ( s ( ),  [t , T )); ; ; X ; T )

(5)



as an integral index (or index vector) of supportability degree at time t. In formula (1)  ( s ( ),  [t , T )) is some integral estimate of the distance to the path restrictions on some time interval [t, T). Implicitly in this case we  assume that the trajectory of the system s (t ) may be calculated using some models. Rating  ( s (t )) may also includes derivatives of indices over time and other operators. That is, a method of calculating of the sustainability index corresponds to the weak anticipation. Note that in the special case T  t we come to the case of the existing indexes of supportability. VI. General OR consideration of sustainable development Of course many scientists have proposed some OR investigations of sustainable development including some mathematical approaches (the examples may be proposed and analyzed in full paper). But as we hope our investigations may open the way to further formalization of SD concept. In fact, these considerations allow us to reduce the problem of sustainable development to the problems of modeling and optimization of systems with constraints, which is the prerogative of operation research. But is essentially a new accounting of system descriptions multidimensionality, especially because of mental properties accounting of social actors. Absolutely new is also attracting anticipatory properties for formalization of sustainable development (especially of strong anticipation). Also we describe some experience in application for real problems. References 1. 2. 3. 4.

Dubois D. Incursive and hyperincursive systems, fractal machine and anticipatory logic. Computing Anticipatory Systems: CASYS 2000 - Fourth International Conference. Published by the American Institute of Physics, AIP Conference Proceedings. - 2001. - # 573. Pp. 437 - 451. Makarenko A. Anticipating in modeling of large social systems - neuronets with internal structure and multivaluedness. Int. J. of Computing Anticipatory Systems. - 2002 . – vol.13 , Pp.77 - 92. Makarenko A. and Klestova Z., A new class of global models of associative memory type as a tool for considering global environmental change // Environmental Change, Adaptation, and Security. / Ed. By S.C.Lonergan. NATO ASI Series, 2.Environment. Vol.65. Kluwer Academic Publishers, 1999. Makarenko A. New Neuronet Models of Global Socio- Economical Processes // Gaming /Simulation for Policy Development and Organisational Change / J.Geurts, C.Joldersma, E.Roelofs eds, Tillburg: Tillburg University Press, 1998. Pp. 128- 132.