The reference to the configuration d in this definition is to that part of the object A which explicitly directs S to produce B. To anticipate the discussion of explicit versus implicit encoding of (re-)production processes in the following sections, d is that attribute of A which is an explicit encoding of part of the production process. As the above definition implies, some of the production process may also be implicit in S.
Löfgren points out that the product B of a productive object A in surrounding S may itself be productive, or it may not be. If Bis itself productive, he calls A reproductive. The definition of reproduction is as follows:
Note that these definitions are intentionally phrased in a way that emphasises that an object A which is productive in a surrounding S may rely upon some properties of S to achieve the production.
Following on from the previous two definitions, we can now provide a definition of self-reproduction:
Löfgren points out that the notation `self-reproductive' is actually logically redundant; the term `self-productive' has the same meaning according to our definitions. However, I will stick with the term `self-reproductive' in the following, as this is the terminology commonly used in the literature.
Within this framework, Löfgren examines the concept of self-reproduction from a logical point of view. He points out that ``there are two distinct but natural interpretations. One is to say that A is self-reproductive if, first of all, A is reproductive and, second, A reproduces a copy of itself. The other interpretation is that A is self-reproductive if A is reproductive by itself, that is, A is reproductive in a surrounding S with no properties'' [Löfgren 72] (p.360).
It is worth discussing this distinction a bit further, as it is the source of some confusion. Löfgren refers to the former case as `second-level reproduction', and to the latter as `complete self-reproduction'. It has been argued from a logical-mathematical point of view that complete self-reproduction is paradoxical; for example, Löfgren cites Wittgenstein's argument that no function can be its own argument [Wittgenstein 21] (esp. aphorism no. 3.333), and work by Rosen (e.g. [Rosen 59]).7.6 Second-level reproduction, however, implies no such paradox. Löfgren observes:
``It would seem that many misunderstandings concerning the concept of self-reproduction are due to the different meanings which are commonly attached with it. Von Neumann [von Neumann 66] and Penrose [Penrose 58], for example, use the word self-reproduction for a second-level reproduction, whereas [others, such as] Rosen [Rosen 59], use self-reproduction in a complete sense.
In ordinary biological language the name self-reproduction is mostly used for second-level reproduction, for example, when the mechanisms of cell-division are used to explain the `self-reproducing' properties of the cell. That no logical difficulty arises in connection with this type of `self-reproduction' is well known.'' [Löfgren 68] (pp.423-424).
In the following discussion, I will only be concerned with second-level reproduction, and will use the general term self-reproduction to implicitly refer to this meaning rather than to complete self-reproduction.
A final point to note before delving further into some of the issues involved with systems capable of supporting productive, reproductive and self-reproductive objects, is that some of the examples I will use are logical systems (e.g. cellular automata and Tierra), whereas others are material systems (e.g. biological cell division).7.7 The distinction is unimportant for most of the discussion. However, it is relevant when talking about objects which can direct their own production (auto-reproduction) compared to those which rely on the existence of auxiliary objects in the environment (assisted-reproduction), as I will discuss in Section 7.2.2.