# A Category Theoretical Argument against the Possibility of Artificial Life: Robert Rosen's Central Proof Revisited

## Summary (3 min read)

### 1 Introduction

- One of the stated goals of artificial life is the implementation of living systems in silico and their simulation over an entire life cycle [5].
- So far this undertaking has been at best partially successful.

### 2 Models

- In this section the authors will introduce Rosen’s notion of synthetic and analytic model.
- Rosen focused his discussion entirely on models; he largely avoided writing about systems.
- The reason for this is that as observers the authors do not have direct access to real systems in the physical world Q1 Artificial Life Volume 12, Number 12 A Category Theoretical Argument against the Possibility of Artificial LifeD. (the Welt an sich).
- Throughout the remainder of this article, the word ‘‘system’’ will always refer to formal systems unless explicitly stated otherwise.
- An exhaustive discussion of this topic would of course go beyond the scope of this article; the reader who wishes to explore category theory in any depth is referred to textbooks [19, 10].

### 2.1 Categories

- Objects (sets, or anything else; in their case: models).
- For every object X, an identity morphism idX. !.
- In the simplest case, the objects of a category are sets and the morphisms are mappings between the sets.
- Throughout this article the concept of duality will not be explored in any depth, but it will be used to indicate relations between definitions and categories; specifically, the concepts of direct sum and direct product are dual; the reader might compare the definitions in Figures 1 and 3.
- The reason the direct product and sum need to be introduced here is that Rosen essentially identifies them with analytic and synthetic models respectively.

### 2.2 Analytic Models

- While the concepts of analytic and synthetic model are crucial to the entire argument, Rosen only hinted at a mathematical definition.
- Rosen’s notion of model might to some appear somewhat counter-intuitive at first.
- Not only might it be that the observables used are inaccurate, measuring the system only crudely; it might also be that the set of observables used is not sufficient to give a full description of the system.
- The objects of the category are the models, and the morphisms are the refinement relations between them .
- The mutual refinement of two models that are based on unlinked observables is formally the direct product in the category AðSÞ.

### 2.3 Synthetic Models

- The second kind of models are synthetic models.
- The poset has exactly one top element, that is, one element that is not itself included.
- Like analytic models, synthetic models also form a category, YðSÞ .
- The state space of the compound system is fourdimensional, describing the momenta and the positions of both particles.
- Also, the process of taking the direct sum of analytic models is a purely syntactic (or algorithmic) process.

### 2.4 Relation between Analytic and Synthetic Models

- Rosen defines mechanisms as the class of systems of which all analytic and synthetic models are equivalent; according to Rosen, this is the case when direct product and direct sum coincide.
- Artificial Life Volume 12, Number 18 A Category Theoretical Argument against the Possibility of Artificial LifeD. Chu and W. Kin Ho.
- Not every analytic model is a synthetic model.
- Firstly, the authors assume that subtrees in the category of analytic models are always disjoint.
- Replace all linear branches in AðSÞ with the most refined model in the linear branch.

### 2.4.1 Analytic-Synthetic Means S Is Defined on a State Space

- It is straightforward to show that for all state-space-based systems S, analytic and synthetic models will always be equivalent:.
- Thus there are a finite number of unlinked observables that specify the system completely; this in turn means precisely that the systems is defined on a finite state space.
- In Life itself, Rosen makes the connection between the equivalence of synthetic and analytic models and the equivalence of direct sum and direct product.
- Since the concepts of direct product and direct sum are dual to each other, the following is always true:.
- This, however, does not translate directly into a theorem about the equivalence of analytic and synthetic models, because, as the preceding discussion has made clear, the relation between synthetic and analytic models is only partially a relation of duality:.

### 3 Mechanisms and Machines

- In this section the authors will now finally define the notion of mechanism.
- Mechanisms are an important class of systems, because Rosen defines them as the class of systems that are simulable; the question whether life in silico is possible is thus essentially the question whether or not at least some living systems are mechanisms.
- Thus, all mechanisms can be exhaustively defined as state-based systems.
- The best way to convince oneself that Turing machines are indeed mechanisms is to consider implementations of Turing machines in cellular automata (such as in the Game of Life [1]).
- F0; 1g k Rosen points out that models of machines in terms of states are not particularly instructive, and introduces an alternative representation, so-called relational models .

### 4.1 The Proof

- Or more precisely, that the relational model in Figure 7 cannot be reconstructed from a state description.the authors.
- The authors would now like to know whether it is possible to recover the original component f from this state description; more specifically, Artificial Life Volume 12, Number 1 13 A Category Theoretical Argument against the Possibility of Artificial LifeD.
- Let us consider two possibilities: Continuing the argument will lead to no closure.
- According to Rosen this possibility is not compatible with the assumption of a machine.
- In the context of machines, where synthetic models are equivalent to analytic models, one would expect that ‘‘functions are always localized into corresponding organs’’ [23, p. 212].

### 5 The Concept of Implementation

- There are two main possible criticisms of Rosen’s proof.
- This requirement of localization says that in machines the authors can at any time precisely specify which parts of the system implement a particular mapping and which parts do not.
- In the next few paragraphs the authors will explain this in more detail.
- While that may be true, the authors will now show that even in machines those properties are not compatible.
- Because all he has shown is that closed systems share this property with machines; this does not tell us whether or not closed systems are mechanisms; thus the authors still do not know whether artificial life in silico is possible.

### 5.1 Syntax and Semantics

- The discussion in the last section indicated already that there is a problem with the notion of the implementation of a mapping.
- It can thus acquire new properties from the larger system with which it associates.
- From the above quotation it seems that the concept of a mapping and its implementation are semantic; this does not fit very well with the idea of a mechanism as an inherently syntactic system (syntactic in the sense that all aspects of the machine can be derived from the complete description of its parts).
- It is in this context that one often observes that components acquire new properties that also might change according to the context.
- Relational models might be valuable in certain contexts to describe both machines and organisms, yet in the context of Rosen’s ‘‘proof ’’ they are misplaced.

### 6 Discussion and Conclusion

- The authors have mainly concentrated on a comprehensive outline of the mathematical concepts to clarify the mathematical background of the most important notions underlying Rosen’s work.
- The main conclusion of this is that relational models rely on an understanding of the system and its context and function, while state-space-based models only reflect purely formal aspects of systems.
- Biologically important notions, such as ‘‘function’’ (an inherently semantic concept) can therefore trivially never be recovered from such descriptions.
- The authors believe that Rosen’s circles of ideas contain valuable contributions to those big questions, and their reevaluation will help to bring us closer to an understanding of life and artificial life.

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