# Remarks on Chu-Ho Fall 2007

Dominique Chu and Wen Kin Ho iterate their previous exercises in misunderstanding and misconstruing of Rosen’s work with their latest paper, “Computational Realizations of Living Systems“, in the Fall 2007 issue of the MIT journal Artificial Life [1]. The abstract:

Robert Rosen’s central theorem states that organisms are fundamentally different from machines, mainly because they are ‘‘closed with respect to efficient causation.” The proof for this theorem rests on two crucial assumptions. The first is that for a certain class of systems (‘‘mechanisms”) analytic modeling is the inverse of synthetic modeling. The second is that aspects of machines can be modeled using relational models and that these relational models are themselves refined by at least one analytic model. We show that both assumptions are unjustified. We conclude that these results cast serious doubts on the validity of Rosen’s proof.

As I will show, both of their complaints are premised on overtly erroneous misstatements of Rosen’s work, rendering their complaints invalid and irrelevant.

Complaint #1:

As to their first complaint about analytic and synthetic models, Chu-Ho begin (Section 2):

In this section we will define the notions of analytic and synthetic models; the objective is to do so as they were introduced by Rosen [in LI,FM]; furthermore, this section will provide some insight into the relationship between these two types of models, as they are relevant for the understanding of the central argument. This will culminate in the formulation of Theorem 1.

We begin by defining analytic models (cf. [LI, Chap. 6C]).

DEFINITION 1: Let f be a mapping from a finite set S to some set U. An analytic model M(S) of the system S is the set of equivalence classes on S induced by f. We say that two elements x, y  $\small{\in}$ S are equivalent and write x  ~f  y if f(x) = f( y). The equivalence class of x on S induced by f is denoted by [x]f and is a subset of S. We denote by Ma(S) the set of all analytic models of S.

Remark 1. The set S represents the system to be modeled and can be thought of as the set of states the system can take (for a more in depth discussion of this assumption see [7, Section 2.2]). If S is a natural system, then it is usually unknown; note that we assume here that S has a finite number of elements; this assumption will simplify the analysis to follow but, as it will turn out, will not affect our overall conclusions. The system can be probed via meters that indicate values of observables; in Definition 1, the observable is the function f, and the measurement results are elements of the set U.

However, this is not Rosen’s definition of analytic model, nor are all the above statements correct. Instead, Rosen defines analytic models thusly [LI p. 162,164]:

In what follows, I shall call any expression of a set S as a Cartesian product of quotient sets an analysis of S. The result of such an analysis is obviously to encode the points of S into a family of values, i.e., into a set of coordinates, which are the values of suitable observables of S. [bold added]

Every such analysis thus gives us a model, in particular, what we have called an analytic model. [bold added]

Plainly, Chu-Ho’s definition does not match Rosen’s. To state it another way, consider the following from Rosen [LI p. 178]:

As I have repeatedly emphasized, the essential feature of a model of S is that it encodes something about S. In an analytic model, we encode an element s of S into a set of observable values

s → {fa(s)}

and thereby encode S itself into a Cartesian product

S → $\small{\prod_{a}}$ fa(S) = $\small{\prod_{a}}$S/∩Rfa

Again, plainly the definitions do not match, and as such, the complaint is invalid and their ensuing argument about analytic and synthetic models, and direct products and direct sums, is irrelevant. If Chu-Ho wish to criticize Rosen, they must first utilize the correct definitions.

Complaint #2:

Chu-Ho’s second complaint is that it is incorrect that “aspects of machines can be modeled using relational models and that these relational models are themselves refined by at least one analytic model”.

Any discussion of Rosennean mechanisms and Rosennean machines must adhere to the definitions of Rosennean mechanism and machine [LI p. 203]:

We shall say that a natural system N is a mechanism if all of its models are simulable.

We shall further say that a natural system N is a machine if and only if it is a mechanism, such that at least one of its models is already a mathematical machine.

Note that both definitions explicitly refer to natural systems (i.e., material systems in the external world), as opposed to formal systems (i.e., linguistic, symbolic systems).

As to this complaint, Chu-Ho again never get off the ground. Their discussion is based upon the relational model of a Turing Machine (TM), and they state “According to Rosen, any TM is a member of the class of mechanisms”. However, Rosen did not make such a statement, nor would he, since TMs are obviously abstract mathematical entities: they are formal systems, not natural systems. Thus their argument is invalid.

Chu-Ho then compound their error by embedding one formal system (a TM) into another formal system (a CA) when they say “We will assume that the TM is implemented in an n x n cellular automaton (CA)…”, and use that construction as the basis for their argument. Their argument is thus further invalidated. Again, if Chu-Ho wish to criticize Rosen, they must first utilize the correct definitions.

As a final aside, Chu-Ho mention a paper[3] by Landauer and Bellman, criticizing Rosen’s work. Landauer-Bellman was answered long ago here.

References:

[1] Chu, D. and Ho, W. 2007. “Computational Realizations of Living Systems“, Artificial Life. Vol. 13, No. 4, Pages 369-381. doi:10.1162/artl.2007.13.4.369

[2] Rosen, R. 1991. Life Itself. Columbia Univ. Press

[3] Landauer, C. and Bellman, K. 2002. “Theoretical Biology: Organisms and Mechanisms”. AIP Conference Proceedings Vol 627(1) pp. 59-70.