Emulation, Reduction, and Emergence in Dynamical Systems Marco Giunti Università di Cagliari, Italy

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 S 2 is emergent with respect to a lower level system S 1 just in case (a) S 2 is made up of S 1 (intuitively, S 1 is the system of the constitutive parts of S 2 taken in isolation, or in relations different from those typical of S 2 ; see Broad 1925) and (b) P is not one of the properties of S 1.

Reduction – received view It is obvious that, if S 2 is reduced to S 1, then all properties of S 2 are properties of S 1 ; 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 DS 1 and DS 2, as well as a property P, such that DS 1 emulates DS 2, DS 2 has P, but DS 1 does not have P.

Example of a continuous Dyn. Syst. The 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

Rule Rule number = Time Time Twelve cells arranged in a circle. The value of each cell is either 0 or 1. Thus, the CA has 2 12 = 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, (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, is either Z, Z + (integers) or R, R + (reals); T is called the time set; 3.(g t ) t  T is a family of functions from M to M; each function g t is called a state transition or a t-advance of the system; 4.for any t and w  T, for any x  M, a.g 0 (x) = x; b.g t+w (x) = g w (g t (x)).

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

Isomorphism between two DSs Definition u is an isomorphism of DS 2 = (N, (h v ) v  V ) in DS 1 = (M, (g t ) t  T ) iff T = V, u: N  M is a bijection and, for any v  V, for any c  N, u(h v (c)) = g v (u(c)). DS 2 is isomorphic to DS 1 iff there is u which is an isomorphism of DS 2 in DS 1 gvgv u hvhv u c MN

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 DS 2 = (N, (h v ) v  V ) in DS 1 = (M, (g t ) t  T ) iff u: N  M is an injection and, for any v  V, for any c  N, there is t  T such that u(h v (c)) = g t (u(c)) DS 1 emulates DS 2 iff there is u which is an emulation of DS 2 in DS 1 gtgt u hvhv u c MN u(N)u(N)

Virtual System Theorem [VST] If u is an emulation of DS 2 = (N, (h v ) v  V ) in DS 1 = (M, (g t ) t  T ), there is a third system DS 3 = (N, (h v ) v  V ) such that (i) u is an isomorphism of DS 2 in DS 3 ; (ii) all states of DS 3 are states of DS 1 [because N = u(N)] ; (iii) any state transition h v of DS 3 is constructed out of state transitions of DS 1. gtgt u hvhv u c M N u(N)u(N) a hvhv u -1 DS 3 is called the virtual u-system DS 2 in DS 1 gtgt

Emulation → constitution and reduction Because of [VST], if a dynamical system DS 1 emulates a second system DS 2, it makes perfect sense to claim that DS 2 is made up of DS 1, as well as that DS 2 is reduced to DS 1. 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 DS 1 = (M, (g t ) t  Z + ) and DS 2 = (N, (h v ) v  Z + ) such that (i) DS 2 is reduced to DS 1 and (ii) the property P of strong irreversibility is an emergent property of DS 2 with respect to DS 1 h1h1 u a MN u c g1g1 b x y z u h1h1 h1h1 g1g1 g1g1 DS 1 emulates DS 2, DS 1 is logically reversible (thus, not strongly irreversible), and DS 2 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, 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,