Reduction in Dynamical Systems: A Representational View

Marco [email protected]://www.webalice.it/marcogiunti/

Università di CagliariDipartimento di Scienze Pedagogiche e Filosofichevia Is Mirrionis 109123 Cagliari, ITALY

1SILFS 2007 CONFERENCE, Oct 8-10, Milan

Overview (1/4) Traditional view of reduction:

A deductive relationship between two formal theories (Nagel 1961; Schaffner 1967; Churchland 1979, 1985; Hooker 1979, 1981, 2005).

This talk advances an alternative view:Reduction is better conceived as a representational relationship between two mathematical models MS1 and MS2, which grants the retrieval, within the representing model MS1, of an isomorphic image of MS2.

This representational view of reduction is in broad agreement with Suppes’ Reduction Paradigm (1957, 271).

SILFS 2007 CONFERENCE, Oct 8-10, Milan 2

Overview (2/4) The representational theory will be

developed only for dynamical systems,a special kind of mathematical model that captures the intuitive idea of an arbitrary deterministic system (Arnold 1977; Szlensk 1984; Giunti 1997).

The representational relationship between dynamical systems that, I claim, is sufficient for reduction is the one of emulation.

Emulation is typical of computational systems (Wolfram 1983a, 1983b, 1984a, 1984b, 2002), but it more generally holds between two arbitrary dynamical systems when the first one exactly reproduces the whole dynamics of the second one (Giunti 1997, ch. 1).SILFS 2007 CONFERENCE, Oct 8-10, Milan 3

Overview (3/4) The claim that emulation is sufficient for

reduction (in dynamical systems) is supported by a two-step argumentative strategy.

1.A (quite simple) representation theorem (Giunti 1997, ch.1, th. 11), called Virtual System Theorem [VST], according to which:

If DS1 emulates DS2, there is a third system DS3 such that (i) DS2 is isomorphic to DS3; (ii) all states of DS3 are states of DS1; (iii) any state transition of DS3 is constructed out of state transitions of DS1.

2.A general (philosophical) argument to the extent that:

In force of [VST], if DS1 emulates DS2, then DS2 is reduced to DS1.


Overview (4/4)

In this talk, I will not go into the details of either 1. the representation theorem or 2. the general argument.

Rather, I will just try to convey the main intuitions on which they rest, mainly by focusing on simple examples.


Dynamical systems ― Intuition A s intended here, a dynamical system

DS is a kind of mathematical model that formally expresses the notion of an arbitrary deterministic system, either reversible or irreversible, with discrete or continuous time or state space; examples of discrete DS: Turing machines,

cellular automata; examples of continuous DS: iterated

mappings on R, systems specified by ordinary differential equations.


Dynamical systems ― definition Let Z be the integers, Z+ the non-negative

integers, R the reals and R+ the non-negative reals;

DS is a dynamical system iff there is M, T, (gt)tT such that DS = (M, (gt)tT) and

1. M is a non-empty set; M represents all the possible states of the system, and it is called the state space;

2. T is either Z, Z+, R, or R+; T represents the time of the system, and it is called the time set;

3. (gt)tT is a family of functions from M to M; each function gt is called a state transition or a t-advance of the system;

4. for any t, v T, for any x M,i. g0(x) = x ii. gt+v(x) = gv(gt(x)).


Intuitive meaning of the definition of a dynamical system

gt+w

x

gw

x

g0

xgt

t0 t0+t

gt(x)

t

gt


condition 3

condition 4.iicondition

4.i

The transition graph of a DS The best way to represent a DS is by

means of its transition graph (or phase portrait), an oriented and labeled graph that depicts the whole dynamical structure of the system.

Each point (node or vertex) of the graph corresponds to exactly one state of the DS, while each arrow (oriented edge) corresponds to one or more state transitions.

In fact, each arrow is labeled with the name and durations of the corresponding state transitions.


Transition graph ― Examples For ease of exposition, from now on all the

examples will refer to dynamical systems with discrete and non-negative time set T = Z+; finite state space.


h0,1,…

h1,2,…

h1,2,…

h0

h0

g0,1,

…

g1,2,

…g2,3,

…

g0

g0

g1

M N

DS1 = (M, (gt)tZ+) DS2 = (N, (hv)vZ+)

Transition graph of a DS ― Definition An oriented and directed graph is the transition graph of a

DS iff it satisfies the following four conditions:1. For any point, and for any duration t T, there is exactly

one arrow labeled “gt” that departs from that point; thus, in particular, there is no point from which two different

arrows with the same label depart; however, there may be points where two different arrows with

the same label arrive;

2. the departure point of any arrow labeled “go” is always identical to its arrival point;

3. for any two consecutive arrows, the first labeled “gt” and the second labeled “gw”, there is always a third arrow (not necessarily distinct) labeled “gt+w” that connects the departure point of the first arrow with the arrival point of the second one;

4. for any arrow whose departure and arrival points are identical, if it is labeled “gt” , then it is also labeled “gkt” for any k Z+.


Basic structure and categorization of a transition graph

A transition graph can always be thought as being composed of two distinct parts: its basic structure (or ontology), i.e.,

the directed graph without the labels; its categorization (or ideology), i.e.,

each label with the corresponding set of arrows.


Example ― Basic structure and categorization of DS2 For instance, let us

consider again DS2. Its basic structure is as

shown by the animation;

while its categorization consists in assigning: the three circular arrows

to the label h0; the two straight arrows

and the top circular arrow to any other label.


h0,1,…

h1,2,…

h1,2,…

h0

h0

N

DS2 = (N, (hv)vZ+)

Reduction in dynamical systems ― Main intuition

The central idea of the representational view of reduction is that a dynamical system DS2 is reduced to a dynamical system DS1 if an isomorphic image of the basic structure of DS2 is embedded in the basic structure of DS1.



h0,1,…

h1,2,…

h1,2,…

h0

h0

g0,1,

…

g1,2,

…g2,3,

…

g0

g0

g1

M N

DS1 = (M, (gt)tZ+) DS2 = (N, (hv)vZ+)

Reduction in dynamical systems ― Example DS2 is reduced to DS1. Let us see why.


M N

DS1 = (M, (gt)tZ+) DS2 = (N, (hv)vZ+)

Reduction in dynamical systems ― Example The basic structures of both DS1 and DS2


M N

DS1 = (M, (gt)tZ+) DS2 = (N, (hv)vZ+)

Reduction in dynamical systems ― Example An isomorphic image of the basic structure of

DS2 is embedded in the basic structure of DS1.

u

u

u


M N

DS1 = (M, (gt)tZ+) DS2 = (N, (hv)vZ+)

Reduction in dynamical systems ― Example Therefore, we can transport the categorization of DS2 to DS1;

so that, in DS1, we get an isomorphic image of DS2 without adding anything to the basic structure (the ontology) of DS1.

u

u

u

h0,1,…

h1,2,…

h1,2,…

h0

h0

h0,1,…

h1,2,…

h1,2,…

h0

h0

Final remark What all this has to do with emulation? In fact, in the previous example, the function u

is an emulation of DS2 in DS1, and it is the existence of this function that allows us to retrieve in DS1, embedded in its basic structure, an

isomorphic image of the basic structure of DS2;

transport the categorization of DS2 to DS1, and thus obtain, within DS1, an isomorphic image of DS2 itself (this system is called the u-virtual DS2 in DS1).

This is why I take emulation to be sufficient for reduction in dynamical systems.


Further developments As presented here, the representational view of

reduction is limited to dynamical systems as purely mathematical models with no empirical interpretation. However, this view can be extended to empirically interpreted

dynamical systems (DS, IH), where an empirical interpretation IH links the purely mathematical model DS to a real phenomenon H that DS describes.

Also, this talk has been exclusively concerned with the simplest form of the emulation relationship, which may very well be the basis for a representational account of total and exact reduction. We need a more sophisticated version of emulation for

dealing with cases of asymptotic, partial and approximate reduction (Hooker 2004).

It is possible to define such a version of emulation, and then employ it for a treatment of asymptotic, partial and approximate reduction in empirically interpreted dynamical systems.



ReferencesArnold, Vladimir I. (1977), Ordinary Differential Equations. Cambridge, MA: The MIT Press.

Churchland, Paul M. (1979), Scientific Realism and the Plasticity of Mind. Cambridge: Cambridge University Press.

——— (1985), “Reduction, Qualia, and the Direct Introspection of Brain States”, Journal of Philosophy, 82, 1:8-28.

Giunti, Marco (1997), Computation, Dynamics, and Cognition. New York: Oxford University Press.

Hooker, Clifford Alan (1979), “Critical Notice: R. M. Yoshida’s Reduction in the Physical Sciences”, Dialogue 18:81-99.

——— (1981), “Towards a General Theory of Reduction”, Dialogue 20:38-59, 201-236, 496-529.

——— (2004), “Asymptotics, Reduction and Emergence”, British Journal for the Philosophy of Science 55:435-479.

——— (2005), “Reduction as Cognitive Strategy”, in Keeley, Brian L. (ed.), Paul Churchland. New York: Cambridge University Press, 154-174.

Nagel, Ernest (1961), The Structure of Science. New York: Harcourt, Brace & World.

Schaffner, Kenneth F. (1967), “Approaches to Reduction”, Philosophy of Science 34, 2:137-147.

Suppes, Patrick (1957), Introduction to Logic. New York: D. Van Nostrand Company.

Szlensk, Wieslaw (1984), An Introduction to the Theory of Smooth Dynamical Systems. Chichister, England: John Wiley and Sons.

Wolfram, Stephen (1983a), “Statistical Mechanics of Cellular Automata”, Reviews of Modern Physics 55, 3:601-644.

——— (1983b), “Cellular Automata”, Los Alamos Science 9:2-21.

——— (1984a), “Computer Software in Science and Mathematics”, Scientific American 56:188-203.

——— (1984b), “Universality and Complexity in Cellular Automata”, in Doyne Farmer, Tommaso Toffoli, and Stephen Wolfram (eds.), Cellular Automata. Amsterdam: North Holland Publishing Company, 1-35.

——— (2002), A New Kind of Science. Champaign, IL: Wolfram Media, Inc.SILFS 2007 CONFERENCE, Oct 8-10, Milan 22

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Reduction in Dynamical Systems: A Representational View