Levels of reality, realization and reduction: A dynamical outlook

Marco GiuntiUniversity of Cagliari, Italygiunti@unica.ithttp://edu.supereva.it/giuntihome.dadacasa

OutlineGeneral thesis – Dynamics is useful to recast

traditional issues of realization, reduction and emergence.

1. Strict relationship between the functionalist idea of physically realizable system and a particular concept of dynamical model (Galilean model).

2. How reduction can be analyzed by means of the relationship of emulation (realization) between dynamical systems.

3. Conditions for the physical reducibility of a physically realizable system.

Distinction between real and mathematical dynamical systems A Real Dynamical System (RDS) is any concrete

system that changes over time. A Mathematical Dynamical System (MDS) 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, which intuitively represents time, is either Z, Z+ (integers) or R, R+ (reals);

3. (gt)tT is a family of functions from M to M such that (i) for any x M, g0(x) = x and (ii) for any x M, for any t and w T, gt+w(x) = gw(gt(x)). Each function gt is called a t-advance, or a state transition of the system.

Intuitive meaning of the definition of mathematical dynamical system

gt+w

x

gw

x

g0

xgt

t0 t0+t

gt(x)

t

gt

Example of MDS (Galilean model of free fall)

Explicit specificationLet F = (M, (gt)tT) such that M = SV and S = V = T = real numbers gt(s, v) = (s + vt + at2/2, v + at)

Implicit specification Let F = (M, (gt)tT) such that M = SV and S = V = T = real numbers ds(t)/dt = v(t), dv(t)/dt = a

Two senses in which a MDS can be said to be a model of a RDS Simulation model (weak)

the MDS supports simulations of certain relevant aspects of the RDS

Galilean model (strong) each component of the state space

corresponds to a magnitude of the real system

the measurements of such magnitudes correspond to the values determined by the model

Simulation models and cognitive science Simulation models still are the dominating

paradigm in cognitive science. In fact: both computational systems and connectionist

networks are special kinds of MDS (Giunti, M. 1997. Computation, Dynamics and Cognition. Oxford Un. Press);

in cognitive science, either kind of system is routinely employed to devise simulations of relevant aspects of the real cognitive systems under examination.

Typical example (exemplar in Kuhn’s sense): Past Tense Acquisition Model (PTA) by Rumelhart and McClelland (1986. In Parallel distributed processing, vol. 2, 216-217. MIT Press).

Galilean models, cognitive science, and functionalism I advocated the construction of Galilean models in

cognitive science in my 1995 contribution to the book edited by Port and van Gelder (Mind as motion. MIT Press); then, more extensively, in my 1997 book (Computation, Dynamics and Cognition. Oxford Un. Press);

as of today, some of the models inspired by the dynamical approach (van Gelder, T. 1998. The Dynamical Hypothesis in Cognitive Science. Behavioral and Brain Sciences 21, 5:615-28) might be Galilean models. However, I haven’t been able to identify a clear exemplar so far.

I am now going to argue that the concept of Galilean model is deeply involved in the functionalist idea of a system with multiple physical realizations – it can in fact be thought as the underlying mathematical basis of such idea.

The argumentative strategy Thesis – the concept of Galilean model is

the underlying mathematical basis of the functionalist idea of a system with multiple physical realizations (physically realizable system).

Argument – show how the paradigmatic example of a physically realizable functional system, i.e., a Turing machine, can in fact be thought as a mathematical dynamical system that turns out to be a Galilean model of each of its physical realizations.

A standard functional characterization of a Turing machine

A physical realization of a Turing machine is any concrete system which satisfies (implements, works according to) the abstract functional scheme below.

Control unit

Internal memory

External memory

Read/write head

Read/write/move head

aj

qi

aj

qi . . : . . . qiaj:akLqm

.. . : . . . .. . : . . .

ak

L

qm

Mathematical description of the functional scheme of a Turing machine The abstract functional scheme of a Turing

machine can be identified with the mathematical dynamical system T = (M, (gt)tT) such that: M = PCS, where P is the set of the possible

positions of the read/write/move head, C is the set of the possible contents of the whole external memory, and S is the set of the possible contents of the internal memory;

T = Z+ (non-negative integers); let g be the function from M to M determined

by the machine table of the functional scheme; then, g0 is the identity function on M and, for any t 0, gt is the t-th iteration of g.

Relationship between T and the physical realizations of the functional scheme Thesis – the MDS (T) corresponding to the

functional scheme of a Turing machine is a Galilean model of each physical realization of such machine.

Argument – First, by the definitions of (i) T and (ii) physical realization of a Turing machine, each component of the state space of T corresponds to a magnitude of each physical realization. Second, by the same definitions, any measurement of such magnitudes must correspond to the values determined by the family of state transitions (gt)tT. Therefore, by the definition of Galilean model, the thesis holds. Q.E.D.

Reduction: from formal theories to semantic models Traditionally, reduction was analyzed in terms of a

deductive relationship between two empirically interpreted formal theories, via bridge principles between the two theories (Nagel E. 1961. The structure of science).

By shifting the attention from formal theories to semantic models, it is natural to think of reduction in terms of some kind of representation relationship (homomorphism) between two models.

As far as MDSs are concerned, there are at least three important relationships to be considered: isomorphism, emulation, and its generalization that I called realization (Giunti 1997, ch. 1, def. 5), not to be confused with physical realization in the functionalist sense.

Isomorphism between two MDSs

1. for any a, b M, for any t T, t V and if gt(a) = b, then ht(u(a)) = u(b);

2. for any c, d N, for any v V, v T and if hv(c) = d, then gv(u

1(c)) = u 1(d).

a

b

gt

u

ht

u

gv

u-1

hv

u-1d

c

MDS1 = (M, (gt)tT) is isomorphic to MDS2 = (N, (hv)vV) iff: there is a bijection u: M  N such that

M M NN

Emulation between two MDSs – Intuition and examples Intuitively, a MDS emulates a second MDS

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 that emulates TM and vice versa; (iii) emulation holds between two simple CAs with radius 1 (binary rule 022 emulates rule 146).

Emulation between two MDSs – Definition

1. for any a, b D, for any t T+, there is v V+ such that, if gt(a) = b, then hv(u(a)) = u(b);

2. for any c, d N, for any v V+, there is t T+ such that, if hv(c) = d, then gt(u

1(c)) = u 1(d).

a

b

gt

u

hv

u

gt

u-1

hv

u-1d

c

MDS1 = (M, (gt)tT) emulates MDS2 = (N, (hv)vV) iff: there is D M , there is a bijection u: D  N such that

M M NND D

Reversible reproduction of irreversible dynamics – Emulation is not enough A MDS is logically irreversible iff it has some non-injective

state transition; MDS is reversible iff its time set is either Z or R.

In a reversible MDS all state transitions are injective. Therefore, by the definition of emulation (condition 2 and

injectivity of u), no reversible MDS can emulate a logically irreversible one.

We thus need to generalize the emulation relation, if we want to account for the exact reproduction of the whole dynamics of a logically irreversible system by a reversible one.

Why do we care? (i) There are reversible systems that are computationally universal (Margolous 1984); (ii) digital computers have computational descriptions that are supposed to be reducible to physical descriptions, which, presumably, are reversible.

Realization between two MDSs

1. for any a, b D, for any t T+, there is v V+ such that, if gt(a) = b, then hv(u(a)) = u(b);

2. for any c, d N, for any v V+, there is t T+ such that, if hv(c) = d, then gt(u

1(c)) u 1(d).

a

b

gt

u

hv

u

gt

u-1

hv

u-1d

c

MDS1 = (M, (gt)tT) emulates MDS2 = (N, (hv)vV) iff: there is D M , there is a surjective function u: D  N such that

M M NND D

Emulation, realization, and reduction Theorem (Giunti 1997, ch. 1, th. 11)

If MDS1 either emulates or realizes MDS2, there is a third system MDS3 such that its states and state-transitions are defined

exclusively in terms of the states and state transitions of MDS1;

MDS3 is isomorphic to MDS2. Because of this theorem, it makes sense to

identify reduction between two MDSs with either emulation or realization. Let us thus stipulate:

MDS2 reduces to MDS1 iff: MDS1 either emulates or realizes MDS2.

Levels of reality and the physical reducibility of MDSs – Analysis (1/2)

1. There are different levels of reality that provide physical realizations (in the functionalist sense) of mathematical dynamical systems. For, in effect, we can have Galilean models of a RDS at the strictly physical level, at the chemical level, at the information processing level, etc.

2. However, simply showing that a MDS is a Galilean model of some RDS at some level of reality λ (i.e. showing the physical realizability of the MDS at λ) is by no means sufficient for claiming its physical reducibility at λ.

Levels of reality and the physical reducibility of MDSs – Analysis (2/2)

3. For, to support this claim, we should further show that there is a second MDS that (i) is a Galilean model of the same RDS at the strictly physical level and (ii) emulates or realizes the first MDS.

4. Moreover, we should in fact require that the above condition holds for any RDS of which the first MDS is a Galilean model at λ. For, if the above condition only holds for some of these RDS, we can only claim a partial physical reducibility of the first MDS at level λ.

Physical reducibility/irreducibility of a MDS at level λ – Definitions MDS2 = (N, (hv)vV) is partially physically reducible at level λ iff:

there is RDS, there is MDS1 = (M, (gt)tT) such that MDS2 is a

Galilean model of RDS at λ, MDS1 is a Galilean model of RDS at

the strictly physical level, and MDS1 either emulates or realizes

MDS2.

MDS2 = (N, (hv)vV) is physically reducible at level λ iff: there is

RDS such that MDS2 is a Galilean model of RDS at λ and, for any

RDS, if MDS2 is a Galilean model of RDS at λ, then there is MDS1

= (M, (gt)tT) such that MDS1 is a Galilean model of RDS at the

strictly physical level, and MDS1 either emulates or realizes

MDS2.

MDS2 = (N, (hv)vV) is physically irreducible at level λ iff: there is

RDS such that MDS2 is a Galilean model of RDS at λ and MDS2 is

not partially physically reducible at level λ.

Physical irreducibility of MDS2 at level λ

physical

λ

RDS1 RDS2 RDS3

MDS2

is a Galilean model of

Partial physical reducibility of MDS2 at level λ

physical

λ

RDS1 RDS2 RDS3

MDS1,1 MDS2

realizes or emulates

is a Galilean model of

is a Galilean model of

Physical reducibility of MDS2 at level λ

physical

λ

RDS1 RDS2 RDS3

MDS1,1

MDS1,3

MDS1,2

MDS2

realizes or emulates

is a Galilean model of

is a Galilean model of

That’s allThank you

Isomorphism between two MDSs MDS1 = (M, (gt)tT) is isomorphic to MDS2

= (N, (hv)vV) iff: there is a bijection

u: M  N such that

1. for any a, b M, for any t T, t V and if gt(a) = b, then ht(u(a)) = u(b);

2. for any c, d N, for any v V, v T and if hv(c) = d, then gv(u 1(c)) = u 1(d).

Emulation between two MDSs – Definition

MDS1 = (M, (gt)tT) emulates MDS2 =

(N, (hv)vV) iff: there is D M, there is a

bijection u: D N such that

1. for any a, b D, for any t T+, there is

v V+ such that, if gt(a) = b, then

hv(u(a)) = u(b);

2. for any c, d N, for any v V+, there is

t T+ such that, if hv(c) = d, then gt(u

1(c)) = u 1(d).

Realization between two MDSs

MDS1 = (M, (gt)tT) realizes MDS2 = (N, (hv)vV)

iff: there is D M, there is a surjective function u: D N such that

1. for any a, b D, for any t T+, there is v V+ such that, if gt(a) = b, then hv(u(a)) = u(b);

2. for any c, d N, for any v V+, there is t T+ such that, if hv(c) = d, then gt(u

1(c)) u 1(d).