Emulation, Reduction, and Emergence in Dynamical Systems

Marco GiuntiUniversità di Cagliari, Italygiunti@unica.ithttp://edu.supereva.it/giuntihome.dadacasa

Outline The received view about emergence and

reduction is that they are incompatible categories. (Beckermann 1992; Kim 1992)

Contrary to the received view, I argue that emergence and reduction can hold together.

In dynamical systems, emulation is sufficient for reduction;

this representational view of reduction, contrary to the standard deductivist one, is compatible with the existence of structural properties of the reduced system that are not also properties of the reducing one. Thus, under this view, reduction and emergence are not incompatible.

A classic definition of emergence Intuitively, a property of a high level system is

said to be emergent if it is not one of the properties of more basic parts, which, together, make up the system. More precisely:

A property P of a high level system S2 is emergent with respect to a lower level system S1 just in case (a) S2 is made up of S1 (intuitively, S1 is the system of the constitutive parts of S2 taken in isolation, or in relations different from those typical of S2; see Broad 1925) and (b) P is not one of the properties of S1.

Reduction – received view It is obvious that, if S2 is reduced to S1,

then all properties of S2 are properties of S1;

thus, by condition (b) of the definition of emergence, it follows that emergence and reduction are incompatible.

My view: by no means is the above principle obvious; in fact, it is false. It thus follows that emergence and reduction may hold together.

The argumentative strategy To support this thesis, I will focus attention on

dynamical systems, and on the emulation relationship between them;

in virtue of a general representation theorem, I will argue that, for any two dynamical systems, the emulation relationship is sufficient for both reduction and constitution (i.e., the being made up of relationship);

therefore, to show that both reduction and emergence can hold together, it will suffice to exhibit two dynamical systems DS1 and DS2, as well as a property P, such that DS1 emulates DS2, DS2 has P, but DS1 does not have P.

Example of a continuous Dyn. Syst. The Galilean model of free fall Explicit specification

Let F = (M, (gt)tT) such that

M = SV and S = V = T = real numbersgt(s, v) = (s + vt + at2/2, v + at)

Implicit specification

Let F = (M, (gt)tT) such that

M = SV and S = V = T = real numbersds(t)/dt = v(t), dv(t)/dt = a

Rule 111 110 101 100 011 010 001 000

0 1 0 1 1 0 1 0

Rule number 010110102 = 9010

Time 0 0 1 0 0 1 0 1 0 1 1 1 1

Time 1 0 0 1 1 0 0 0 0 1 0 0 1

Twelve cells arranged in a circle. The value of each cell is either 0 or 1. Thus, the CA has 212 = 4096 possible states.

Example of a discrete Dyn. Syst. A finite Cellular Automaton

A Dynamical System (DS) is a mathematical model that expresses the idea of a deterministic system (discrete/continuous, revers./irrevers.) A Dynamical System (DS) is a set theoretical structure

(M, (gt)tT) such that:

1. the set M is not empty; M is called the state-space of the system;

2. the set T, is either Z, Z+ (integers) or R, R+ (reals); T 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 and w T, for any x M,

a. g0(x) = x;

b. gt+w(x) = gw(gt(x)).

Intuitive meaning of the definition of dynamical system

gt+w

x

gw

x

g0

xgt

t0 t0+t

gt(x)

t

gt

Isomorphism between two DSs Definition

u is an isomorphism of DS2 = (N, (hv)vV) in DS1 = (M, (gt)tT) iff T = V, u: N  M is a bijection and, for any v V, for any c N, u(hv(c)) = gv(u(c)).DS2 is isomorphic to DS1 iff there is u which is an isomorphism of DS2 in DS1

gv

u

hv

u

c

M N

Emulation between two DSs Intuition and examples Intuitively, a DS emulates a second DS

when the first one exactly reproduces the whole dynamics of the second one.

Examples (i) a universal Turing machine emulates all TMs; (ii) for any TM there is a cellular automaton CA that emulates TM, and vice versa; (iii) emulation holds between two binary CAs with neighborhood of radius 1 (Wolfram’s rule 22 emulates rule 146).

Emulation between two DSs Definition

u is an emulation of DS2 = (N, (hv)vV) in DS1 = (M, (gt)tT) iff u: N  M is an injection and, for any v V, for any c N, there is t T such that u(hv(c)) = gt(u(c))DS1emulates DS2 iff there is u which is an emulation of DS2 in DS1

gt

u

hv

u

c

M Nu(N)

Virtual System Theorem [VST]If u is an emulation of DS2 = (N, (hv)vV) in DS1 = (M, (gt)tT),

there is a third system DS3 = (N, (hv)vV) such that

(i) u is an isomorphism of DS2 in DS3;

(ii) all states of DS3 are states of DS1 [because N = u(N)] ;

(iii) any state transition hv of DS3 is constructed out of state

transitions of DS1.

gt

u

hv

u

c

M Nu(N)

a

hv

u-1

DS3 is called the virtual u-system DS2 in DS1

gt

Emulation →constitution and reduction Because of [VST], if a dynamical system

DS1 emulates a second system DS2, it

makes perfect sense to claim that DS2 is

made up of DS1, as well as that DS2 is

reduced to DS1.

In other words, I maintain that, in virtue of [VST], emulation is sufficient for both constitution and reduction.

Emergence and reduction hold together: example a pair of cascades DS1 = (M, (gt)tZ+) and DS2 =

(N, (hv)vZ+) such that (i) DS2 is reduced to DS1

and (ii) the property P of strong irreversibility is

an emergent property of DS2 with respect to DS1

h1

u

a

M N

u

c

g1

b

x

y

z

u

h1

h1g1

g1

DS1 emulates DS2, DS1 is logically reversible (thus, not strongly irreversible), and DS2 is strongly irreversible

That’s all

Thank you

References

Beckermann, Ansgar (1992), “Supervenience, Emergence and Reduction”, in Ansgar Beckermann, Tommaso Toffoli, and Jaegwon Kim (eds.), Emergence or Reduction? Essays on the Prospects of Nonreductive Physicalism. Berlin: Walter de Gruyter, 94-118.

Broad, Charlie Dunbar (1925), The Mind and its Place in Nature. London: Routledge and Kegan Paul.

Kim, Jaegwon (1992), “Downward Causation in Emergentism and Non-reductive Physicalism”, in Ansgar Beckermann, Tommaso Toffoli, and Jaegwon Kim (eds.), Emergence or Reduction? Essays on the Prospects of Nonreductive Physicalism. Berlin: Walter de Gruyter, 119-138.