A proposal for a tool that helps handling variability and remains compliant with falsifiability Fr´ ed´ eric Davesne Laboratoire Syst` emes Complexes - CNRS FRE2494 University of Evry 40, Rue du Pelvoux 91020 Evry FRANCE frederic.davesne@iup.univ-evry.fr Abstract throughs give a framework that breaks the complexity and permit to reduce a physical phenomenon to a On one hand, automatics is based on proofs be- model. For example, the trajectory of Earth (which is fore the experiment in order to validate an a pri- a complex system by itself) around the Sun may be ori (partially) known deterministic interaction be- well approximated by using only a few variables (mass, tween the robot and its environment; in return, velocity and position) and by neglecting the other the experimenter may expect the reliability of the planets of the solar system. Those cases, which I call real system based on his model. On another hand, ”favorable cases”, gave birth to scientific theories (like statistical methods are used when the environ- Newton’s theory of gravitation) which are falsifiable ment is supposed unknown; in return, the robot in Popper’sense (Popper, 1968). In particular, these may own adaptation capabilities but its behavior theories enable the prediction of what will happen and is not predictable before the experiment and is what will never happen in reality. not necessary reliable. We believe that the core issue for the first ap- At the opposite, biological systems - even the simplest proach (that we call deterministic approach ) relies - may not be reduced in such a way, mainly because on the fact it cannot handle the variety of all the they are bound to variability . Variability appears when possible situations in an unconstrain environment an entity behaves differently when facing apparently (we call this the variability issue). Oppositely, we equal situations or when apparently equal entities think that the statistical approach is essentially behaves differently when facing the same situation. missing a validating stage before the experiment In these cases, a system or components of a system in order to possibly falsify a proposed model. cannot be easily isolated to model a phenomenon in an The aim of this article is to suggest a methodol- analytical manner. It is interesting to draw a parallel ogy that combines both a validating stage before with the study of the behavior of a mobile robot the experiment and the creation of models that which interacts with a complex and a priori unknown handle unknown environments. To do this, we environment. Even if human have built the robot and suggest models that own a validation statement has programmed it, one must admit that it is not and internal parameters that fix a compromise possible to know precisely before the experiment what between falsifiability and robustness. For simple it will do and will never do: the robot behavior is not cases, we show that it is possible to fix internal really predictable (see (Nehmzow and Walkery, 2003)). parameters in order to meet the two antagonist Hence it is not possible to calculate the reliability of the constraints. As a consequence, we stress that the robot behavior before the experiment. precision of the model has a lower bound and we determine a Heisenberg-like uncertainty principle. The biological and artifact cases share the fact that something in the phenomenon remains unknown by the 1. Context of our study scientist but cannot be neglected; this carries poor or context dependent results. Ad hoc algorithms or learn- 1.1 The variability challenge ing capabilities may be implemented for artifacts to Nowadays, several scientific areas are facing complexity. cope with discovery of a priori unknown characteris- Nature is complex by itself; artifacts made by humans tics of the environment. However, taking a very sim- may be too. However, in some cases, scientific break- ple example from the reinforcement learning domain
(which owns theoretical results), we have shown (see currently working on the implementation of a physics- (Davesne and Barret, 2003)) that good results mainly like relation between effectors and sensors of a robot to depend on the ability of the experimenter to create a improve the reliability of the environment recognition proper context to make the learning algorithm work in process (see (Hazan et al., 2005)). reality, whereas results may not be predictable before 1.3 Purpose of this paper the experiment. It turns to be that, on one side, spe- cific parameters or strategies authorize the fulfillment of In the former paragraph, we have briefly described a task but, on an other side, they disable the possibil- a general method that both includes a falsifiability ity to come up with the variability of the encountered property for proposed models and permits to get ride situations: this may lead to context dependency. of the necessity to model the environment. The toy example exhibited in (Davesne, 2004) shows that this 1.2 General postulate and method - previous method may be carried out successfully. However, the work results are far from being satisfactory for one reason: the set of appropriate environments is too tight to be We postulate that a behavior or a capability of an entity met by real environments. This is due to the purely is the result of an adaptation process in which the in- deterministic tools used to design the constraints on our teraction between the entity and its environment obeys proposed model in (Davesne, 2004). an action/reaction law and the entity fulfills an internal constraint at any time. The entity may be modeled as The core issue we are facing may be expressed as fol- a set of real parameters X i and the internal constraint lows: may be written as follows: • on one side, our method discards the use of statisti- F ( X 1 , X 2 , .., X n ) = 0 (1) cal tools to create a model because the proof steps During the interaction, discovery of new situations have to be performed before any experiment in the tends to break the internal constraint (action of the real environment (data is supposed to be unavailable environment on the entity). The adaptation process when the model is designed). may be seen as the reaction of the entity in order to • on the other side, we need a more flexible mathemati- fulfill the constraint. It implies an internal change of the cal tool than the formulation of equation 1 to express entity (a lot of X i values may vary simultaneously). If the constraints. we suppose that the behavior of the entity only depends In this paper, we develop a proposal to extend the on the knowledge of the X i , the reaction process general formulation of equation 1. Our tool is not implies a modification of the behavior of the entity. supposed to be used exclusively with our methodology Hence the modification of the internal parameters is not but keeps the notion of equality/difference which is task-driven but is due to the fulfillment of an internal mandatory in the context of a falsifiable method. consistency law . The underlying idea suggests that a task cannot be considered independently of the real Section 2. explains the underlying idea of our proposal. robot/real environment interaction . A simple model is analyzed and the flexibility of our ap- proach comparing to the deterministic and correlation If the interaction law is supposed to be known, it methods. Numerical resolution is then proposed. Sec- is possible to determine before any experiment all the tion 3. shows how our proposal might permit to handle reachable internal modifications of the entity. The the- signal processing. Theoretical results are given. oretical framework consists on two separate steps: • proof that it is possible to fulfill the law for every 2. Proposal for an alternative methodol- reachable situation. ogy • determination of the set of environments for which the internal modifications lead to favorable 1 behav- 2.1 Introduction iors of the entity. In this section, we will give a tool example that will il- A model is considered to be suitable if the first item is lustrate our methodology in the rest of this article. Our fulfilled and the resulting set of environments is compat- model considers a linear relation between two real vari- ible with the reality. ables X 1 and X 2 . We have chosen this example because In (Davesne, 2004), we show that a very simple inter- our method may be compared easily with: action law gives rise to reinforcement learning capabil- • the correlation method used in statistics that permits ities for a navigation task of an artificial rat. We are to determine the best line followed by a set of gath- � x i 1 , x i � ered couples . 1 In the experimenter’s point of view. 2
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