ANTICIPATORY SYSTEMS: MODELS AND APPLICATIONS.

1. ANTICIPATORY SYSTEMS: MODELS AND APPLICATIONS. Alexender Makarenko NTUU “KPI”, Institute for of Applied System Analysis, Kyiv, Ukraine makalex@i.com.ua

2. INTRODUCTION • The presentation is devoted to the description of rather new mathematical objects – namely the system with anticipation. • Mathematically such objects sometimes frequently have the form of advanced equations. • Since the introduction of strong anticipation by D.Dubois the numerous investigations of concrete systems had been proposed. • As concentrated (discrete, ordinary differential equations) as distributed (electromagnetic theory equations, cellular automata) systems with anticipation had been considered earlier. • But further development of the theory of anticipatory systems depends on the investigations of new examples of systems with anticipation and their new applications.

3. So in proposed talk the new examples of distributed and concentrated models with anticipation had been considered – namely the system of hyperbolic differential equations with special boundary conditions, • ordinary differential equations, • discrete dynamical systems etc. • It is proposed the mathematical formulation of problems, possible analytical formulas for solutions and interpretations of presumable distributed solutions. Complex behavior of such solutions is discussed. • Some considerations on symmetry investigations are proposed. • A number of applications of proposed models are described. • A list of further research problems is proposed.

4. STRONG ANTICIPATION • Since the beginning of 90-th in the works by D.Dubois – the idea of strong anticipation had been introduced: “Definition of an incursive discrete strong anticipatory system …: an incursive discrete system is a system which computes its current state at time, as a function of its states at past times, present time,, and even its states at future times • (1) • where the variable x at future times is computed in using the equation itself.

5. WEAK ANTICIPATION • Definition of an incursive discrete weak anticipatory system: an incursive discrete system is a system which computes its current state at time, as a function of its states at past times, present time, , and even its predicted states at future times • (2) • where the variable at future times are computed in using the predictive model of the system” (Dubois D., 2001).

6. I. EXAMPLES OF SYSTEMS WITH ANTICIPATION (CELLULAR AUTOMATA)

7. I. Cellular automata with anticipation • Example: Came ‘Life” with Anticipation

8. Game “Life”: a brief description Rule #1: if a dead cell has 3 living neighbors, it turns to “living”. Rule #2: if a living cell has 2 or 3 living neighbors, it stays alive, otherwise it “dies”. Formalization: Next step function: - state of the k-th cell Dynamics of a N-cell automaton: t – discrete time

9. “LifeA” = “Life” with anticipation Conway’s “Life” “Life” with anticipation weighted additive Dynamics:

10. LifeA: simulations “LifeA”: multiple solutions “Life”: linear dynamics

11. LifeA: simulations • Multivaluedness Multivaluedness Choice Optimal management

12. II. Optimization of Pedestrian Crowd Movement by Models with Anticipation Alexander Makarenko, Dmitry Krushinsky

13. Notes on importance • The movement of large–scale human crowds potentially can result in a variety of unpredictable phenomena: loss of control, loss of correct route and panics, that make groups of pedestrians block, compete and hurt each other. • So: It is evident that special management during such accidents is necessary. Moreover, well-founded plans of evacuation based on realistic scenarios and risk evaluation must be designed. This will either prevent harmful consequences or, at least, alleviate them. • But: Such phenomena are difficult for management because of: - presence of large number of interacting “agents”; - various unpredictable external factors during accidents.

14. Chaotic behavior • hard to control & predict • undesired phenomena: high “pressure”, shock waves, etc. • poor performance (in emergency) optimized infrastructure simulation assessment optimization regulations, direction signs,… • easy to control & predict • evenly distributed pedestrians • good performance (in emergency) Determined behavior the “better” the model for simulation is, the more effective this chain works

15. Why cellular automata? • Simple structure • Complex (realistic?) behavior • Scalability • Effectiveness in computation • Visualization

16. Model description Microscopic Basic assumptions: • Dynamics can be presented by the cellular automata, the model discrete in space and time. • The global route is pre-set. • The irrational behaviour is rare. • Persons are not strongly competitive, i.e. they don’t hurt each other. • Individual differences can be represented by parameters determining the movement behaviour. Stochastic Space- & Time- Discrete

17. Basic model Routing Layer Data Layer P2 P3 P1 P4 • 3 states per cell: • Empty • Obstacle • Pedestrian Cells contain directions that make up shortest exit path Pk– probability of shift in k-th direction (k=1..4)

18. Mentality accounting Artificial intelligence (models of human consciousness and decision-making) Simple statistical rules P1 =0.0 P2 =0.7 P3 =0.0 P4 =0.3

19. Simplest model of anticipating pedestrian Supposition: the pedestrians avoid blocking each other. I.e. a person tries not to move into a particular cell if, as he predicts, it will be occupied by other person at the next step. P2 P3 P1 P4 Pk – probability of shift in direction k (k=1..4) Pk,occ – probability of k-th cell in the neighborhood being occupied (predicted) α– free parameter, expressing influence of anticipation

20. Two basic variants of anticipation accounting were simulated: and All pedestrians have equal rights Fast moving pedestrians have a priority • And two variants of calculation Pk,occ: Observation-based Model-based P2 P3 P1 P2 P4 P3 P1 P4

21. Anticipating pedestrians: simulations E/P – equal rights/priority of fast moving; O/M – observation-/model-based prediction

22. Spatial de-localization Model-based prediction: P2 P4 P3 P1 P3 P2 P4 Cells beyond elementary neighborhood are involved. Thus, the actual (extended) neighborhood has radius R=2.

23. Growth of the neighbourhood … … and impact on performance

24. Temporal de-localization Example scenarios tree… Temporal de-localization Multi-step prediction Scenarios tree analysis

25. Evolution of the model of pedestrian MP(R,T) – model of pedestrian R – radius of (extended) neighborhood; T – time horizon of anticipation

26. De-localization and performance MP(1,0) MP(2,1) MP(R,1) evacuation time ? MP(R,T) MP(∞, ∞) … … … absolute global minimum Will the ultimate model reach the theoretically optimal behavior ?

27. III. Computer soccer

28. II. EXAMPLES OF SYSTEM WITH ANTICIPATION(SUSTAINABLE DEVELOPMENT AS ANTICIPATORY PROCESSES)

29. Basic description of Sustainable Development (SD) • “Sustainable development is development that meets the needs of the present without compromising the ability of future generations to meet their own needs. • It contains within it two key concepts: • the concept of 'needs', in particular the essential needs of the world's poor, to which overriding priority should be given; • and the idea of limitations imposed by the state of technology and social organization on the environment's ability to meet present and future needs.“ - • -following G.H.Bruntland Commission (1987)

30. The role of strong and weak anticipation in SD processes • But SD processes can be considered from the point of view of anticipatory systems, especially with ‘strong’ and ‘weak’ anticipation introduced by D. Dubois since 90th. Many formal definitions had been described in the literature. Here we remember one of the definitions which is useful for understanding the role of anticipation in SD. • “Dubois (2000) distinguished between weak anticipation, that is, when systems use a model of themselves for computing future states, and strong anticipation, that is, when the system uses itself for the construction of its future states. In the latter case anticipation is no longer similar to prediction” .

31. Trajectories of system in case of fatal restrictions of resources

32. Case of possible change of leading resources

33. Formalization of sustainability and sustainable development • Necessary components for SD: • Resources • Restrictions on resources -- Evolutionary aspects • Goal of the system • Existing of many generations • Indexes of sustainability -- Decision – making • Mental properties of peoples and cultural aspects • Environment • Technology …

34. Simple examples (1)

35. SD integral indexes on time interval • Let us introduce the value • (1) • as the integral index (or vector of indexes) of sustainability at time t. J nn formula (1) is some integral evaluation of the distance of trajectory from the restrictions on some time interval [t, T). Implicitly in this case we suppose that the trajectories of the system can be calculated by some models. • Intrinsically evaluation of may also include using the derivatives of indexes with time derivatives and other operators. That is such case calculation of sustainability indexes corresponds to the weak anticipation. Note then in special case of we receive the case of many recent sustainability indexes.

36. Simple illustration for SD evaluation

37. ANTICIPATORY RESTRICTIONS • The formal constructions of sustainability indexes above can easy formally modified. But the sense of such constructions and derived results may change principally because of principal difference between weak and strong anticipation. • So in our case we cannot predict the trajectories and the restrictions depends on the evolution of considered systems. • Thus in such case we can speak on the ‘anticipatory’ restrictions on accepted domain of space for system variables. • So we cannot calculate the distance from the future unknown restrictions. • This follows to the needs of models with anticipation, their multy-valued solutions and multy-valued restrictions.

38. Multi-valued solutions in anticipatory systems with known restrictions

39. Multi-valued solutions in anticipatory systems with anticipatory restrictions

40. Transitions between attractors with restrictions

41. III. NEURAL NETWORKS WITHANTICIPATION:PROBLEMS AND PROSPECTS

42. INTRODUCTIONNonlinear networks science (problems and effects):stabilitybifurcationschaossynchronizationturbulencechimera states…..

43. MODELS AND SYSTEMS (RECENTLY):coupled oscillatorscoupled mapsneural networkscellular automatao.d.e. system………MAINLY of NEUTRAL orWITH DELAY

44. ANTICIPATION • Neural network learning (Sutton, Barto, 1982) • Control theory (Pyragas, 2000?) • Neuroscience (1970-1980,… , 2009) • Traffic investigations and models (1980, …, 2008) • Biology (R. Rosen, 1950- 60- ….) • Informatics, physics, cellular automata, etc. (D. Dubois, 1982 - ….) • Models of society (Makarenko, 1998 - …)

45. ANTICIPATION • The anticipation property is that the individual makes a decision accounting the future states of the system [1]. • One of the consequences is that the accounting for an anticipatory property leads to advanced mathematical models. Since 1992 starting from cellular automata the incursive relation had been introduced by D. Dubois for the case when • ‘the values of of state X(t+1) at time t+1 depends on values X(t-i) at time t-i, i=1,2,…, the value X(t) at time t and the value X(t+j) at time t+j, j=1,2,… as the function of command vector p’ [1].

46. ANTICIPATION • In the simplest cases of discrete systems this leads to the formal dynamic equations (for the case of discrete time t=0, 1, ..., n, ... and finite number of elements M): • where R is the set of external parameters (environment, control), {si(t)} the state of the system at a moment of time t (i=1, 2, …, M), g(i) horizon of forecasting, {G} set of nonlinear functions for evolution of the elements states.

47. “In the same way, the hyper-incursion is an extension of the hyper recursion in which several different solutions can be generated at each time step” [1, p.98]. According [1] the anticipation may be of ‘weak’ type (with predictive model for future states of system, the case which had been considered by R. Rosen) and of ‘strong’ type when the system cannot make predictions.

48. HOPFIELD TYPE NETWORK WITH ANTICIPATION

49. SOME EXAMPLES OF MODELS

50. EXAMPLE OF ACTIVATION FUNCTION