Levels of reality, realization and reduction: A dynamical outlook Marco Giunti University of Cagliari, Italy

Outline General 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, (g t ) tT ) 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.(g t ) tT is a family of functions from M to M such that (i) for any x M, g 0 (x) = x and (ii) for any x M, for any t and w T, g t+w (x) = g w (g t (x)). Each function g t is called a t-advance, or a state transition of the system.

Intuitive meaning of the definition of mathematical dynamical system gt+wgt+w x gwgw x g0g0 x gtgt t0t0 t0+tt0+t gt(x)gt(x) t gtgt

Example of MDS (Galilean model of free fall)  Explicit specification Let F = (M, (g t ) tT ) such that M = SV and S = V = T = real numbers g t (s, v) = (s + vt + at 2 /2, v + at)  Implicit specification Let F = (M, (g t ) tT ) such that M = SV 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 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, 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 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 ajaj qiqi ajaj qiqi.. :... q i a j :a k Lq m... : :... akLakL qmqm

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, (g t ) tT ) such that: M = PCS, 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, g 0 is the identity function on M and, for any t 0, g t 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 (g t ) tT. 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 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 g t (a) = b, then h t (u(a)) = u(b); 2.for any c, d  N, for any v  V, v  T and if h v (c) = d, then g v (u ­1 (c)) = u ­1 (d). a b gtgt u htht u gvgv u -1 hvhv d c MDS 1 = (M, (g t ) tT ) is isomorphic to MDS 2 = (N, (h v ) vV ) iff: there is a bijection u: M  N such that M MNN

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 g t (a) = b, then h v (u(a)) = u(b); 2.for any c, d  N, for any v  V +, there is t  T + such that, if h v (c) = d, then g t (u ­ 1 (c)) = u ­1 (d). a b gtgt u hvhv u gtgt u -1 hvhv d c MDS 1 = (M, (g t ) tT ) emulates MDS 2 = (N, (h v ) vV ) iff: there is D M, there is a bijection u: D  N such that M MNN DD

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 g t (a) = b, then h v (u(a)) = u(b); 2.for any c, d  N, for any v  V +, there is t  T + such that, if h v (c) = d, then g t (u ­ 1 (c))  u ­1 (d). a b gtgt u hvhv u gtgt u -1 hvhv d c MDS 1 = (M, (g t ) tT ) emulates MDS 2 = (N, (h v ) vV ) iff: there is D M, there is a surjective function u: D  N such that M MNN DD

Emulation, realization, and reduction  Theorem (Giunti 1997, ch. 1, th. 11) If MDS 1 either emulates or realizes MDS 2, there is a third system MDS 3 such that its states and state-transitions are defined exclusively in terms of the states and state transitions of MDS 1 ; MDS 3 is isomorphic to MDS 2.  Because of this theorem, it makes sense to identify reduction between two MDSs with either emulation or realization. Let us thus stipulate:  MDS 2 reduces to MDS 1 iff: MDS 1 either emulates or realizes MDS 2.

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  MDS 2 = (N, (h v ) vV ) is partially physically reducible at level λ iff: there is RDS, there is MDS 1 = (M, (g t ) tT ) such that MDS 2 is a Galilean model of RDS at λ, MDS 1 is a Galilean model of RDS at the strictly physical level, and MDS 1 either emulates or realizes MDS 2.  MDS 2 = (N, (h v ) vV ) is physically reducible at level λ iff: there is RDS such that MDS 2 is a Galilean model of RDS at λ and, for any RDS, if MDS 2 is a Galilean model of RDS at λ, then there is MDS 1 = (M, (g t ) tT ) such that MDS 1 is a Galilean model of RDS at the strictly physical level, and MDS 1 either emulates or realizes MDS 2.  MDS 2 = (N, (h v ) vV ) is physically irreducible at level λ iff: there is RDS such that MDS 2 is a Galilean model of RDS at λ and MDS 2 is not partially physically reducible at level λ.

Physical irreducibility of MDS 2 at level λ physical λ RDS 1 RDS 2 RDS 3 MDS 2 is a Galilean model of

Partial physical reducibility of MDS 2 at level λ physical λ RDS 1 RDS 2 RDS 3 MDS 1,1 MDS 2 realizes or emulates is a Galilean model of

Physical reducibility of MDS 2 at level λ physical λ RDS 1 RDS 2 RDS 3 MDS 1,1 MDS 1,3 MDS 1,2 MDS 2 realizes or emulates is a Galilean model of

That’s all Thank you

Isomorphism between two MDSs  MDS 1 = (M, (g t ) tT ) is isomorphic to MDS 2 = (N, (h v ) vV ) iff: there is a bijection u: M  N such that 1.for any a, b  M, for any t  T, t  V and if g t (a) = b, then h t (u(a)) = u(b); 2.for any c, d  N, for any v  V, v  T and if h v (c) = d, then g v (u ­1 (c)) = u ­1 (d).

Emulation between two MDSs – Definition  MDS 1 = (M, (g t ) tT ) emulates MDS 2 = (N, (h v ) vV ) 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 g t (a) = b, then h v (u(a)) = u(b); 2.for any c, d  N, for any v  V +, there is t  T + such that, if h v (c) = d, then g t (u ­ 1 (c)) = u ­1 (d).

Realization between two MDSs MDS 1 = (M, (g t ) tT ) realizes MDS 2 = (N, (h v ) vV ) 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 g t (a) = b, then h v (u(a)) = u(b); 2.for any c, d  N, for any v  V +, there is t  T + such that, if h v (c) = d, then g t (u ­ 1 (c)) u ­1 (d).