... representationA.1
For those comfortable with quantum mechanics, think of a patch as a quantum mechanical operator, and the representation as the basis set. The analogy breaks down pretty quickly, however, since an operator could be described in any complete basis set, while a patch modifying the file foo can only be described in the rather small set of contexts which have a file foo to be modified.
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... treeA.2
This is very similar to the second-quantized picture, in which any state is seen as the result of a number of creation operators acting on the vacuum, and provides a similar set of simplifications--in particular, the exclusion principle is very elegantly enforced by the properties of the anti-hermitian fermion creation operators.
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...$P_2$A.3
This notation is inspired by the notation of matrix multiplication or the application of operators upon a Hilbert space. In the algebra of patches, there is multiplication (i.e. composition), which is associative but not commutative, but no addition or subtraction.
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... tree).A.4
The fact that commutation can fail makes a huge difference in the whole patch formalism. It may be possible to create a formalism in which commutation always succeeds, with the result of what would otherwise be a commutation that fails being something like a virtual particle (which can violate conservation of energy), and it may be that such a formalism would allow strict mathematical proofs (whereas those used in the current formalism are mostly only hand waving ``physicist'' proofs). However, I'm not sure how you'd deal with a request to delete a file that has not yet been created, for example. Obviously you'd need to create some kind of antifile, which would annihilate with the file when that file finally got created, but I'm not entirely sure how I'd go about doing this. $\ddot\frown$ So I'm sticking with my hand waving formalism.
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... consideredA.5
Alas, I don't know how to prove that the two constraints even can be satisfied. The best I have been able to do is to believe that they can be satisfied, and to be unable to find an case in which my implementation fails to satisfy them. These two requirements are the foundation of the entire theory of patches (have you been counting how many foundations it has?).
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