## 7 Extensions  
Our CToE selection principle is applicable to perfect, deterministic, discrete, and complete models \(q\) of our universe. None of the existing sane world models is of this kind. In this section I extend the CToE selection principle to more realistic, partial, approximate, probabilistic, and/or parametric models for finite, infinite and even continuous universes. **Partial theories. ** Not all interesting theories are ToEs. Indeed, most theories are only partial models of aspects of our world. We can reduce the problem of selecting good partial theories to CToE selection as follows: Let \(o_{1:t}^{true}\) be the complete observation, and \((q,\!s)\) be some theory explaining only some observations but not all. For instance, \(q\) might be the heliocentric model and \(s\) be such that all bits in \(o_{1:t}^{true}\) that correspond to planetary positions are predicted correctly. The other bits in \(o_{1:t}^{qs}\) are undefined, e.g. the position of cars. We can augment \(q\) with a (huge) table \(b\) of all bits for which \(o_{i}^{qs}\!\
eq\!o_{i}^{true}\). Together, \((q,\!b,\!s)\) allows to reconstruct \(o_{1:t}^{true}\) exactly. Hence for two different theories, the one with smaller length  
\[\text{Length}(q)+\text{Length}(b)+\text{Length}(s) \tag{4}\]  
should be selected. We can actually spare ourselves from tabling all those bits that are unpredicted by all \(q\) under consideration, since they contribute the same overall constant. So when comparing two theories it is sufficient to consider only those observations that are correctly predicted by one (or both) theories. If two partial theories \((q,s)\) and \((q^{\prime},s^{\prime})\) predict the same phenomena equally well (_i.e._, \(o^{qs}_{1:t}\!=\!o^{q^{\prime}s^{\prime}}_{1:t}\
eq o^{true}_{1:t}\)), then \(b\!=\!b^{\prime}\) and minimizing (4) reduces to minimizing (3). **Approximate theories. ** Most theories are not perfect but only approximate reality, even in their limited domain. The geocentric model is less accurate than the heliocentric model, [PERSON]'s mechanics approximates general relativity, _etc._ Approximate theories can be viewed as a version of partial theories. For example, consider predicting locations of planets with locations being coded by (truncated) real numbers in binary representation, then [PERSON] gets more bits right than [PERSON]. The remaining erroneous bits could be tabled as above. Errors are often more subtle than simple bit errors, in which case correction programs rather than just tables are needed. **Celestial example. ** The ancient celestial models just capture the movement of some celestial bodies, and even those only imperfectly. Nevertheless it is interesting to compare them. Let us take as our corpus of observations \(o^{true}_{1:t}\), say, all astronomical tables available in the year 1600, and ignore all other experience. The geocentric model \(q^{G}\) more or less directly describes the observations, hence \(s^{G}\) is relatively simple. In the heliocentric model \(q^{H}\) it is necessary to include in \(s^{H}\) a non-trivial coordinate transformation to explain the geocentric astronomical data. Assuming both models were perfect, then, if and only if \(q^{H}\) is simpler than \(q^{G}\) by a margin that is larger than the extra complications due to the coordinate transformation (Length\((q^{G})\!-\!\)Length\((q^{H})\!>\!\)Length\((s^{H})\!-\!\)Length\((s^{G})\)), we should regard the heliocentric model as better. If/since the heliocentric model is more accurate, we have to additionally penalize the geocentric model by the number of bits it doesn't predict correctly. This clearly makes the heliocentric model superior. **Probabilistic theories. ** Contrary to a deterministic theory that predicts the future from the past for sure, a probabilistic theory assigns to each future a certain chance that it will occur. Equivalently, a deterministic universe is described by some string \(u\), while a probabilistic universe is described by some probability distribution \(Q(u)\), the a priori probability of \(u\).