figure 10. the algorithmic calculus.
associative but not commutative. It begins by marking certain nodes of T1 as absorbed and pruning the subtrees below.
This operation is called absorption by analogy with the
absorbing states of a Markov chain: any orbit reaching an
absorbed leaf comes to a halt, broken only after we reattach
a copy of T2 at that leaf. The copy must be properly cropped.
6. 3. Dynamic renormalization
Direct sums model independent subsystems through parallel composition. Direct products model sequential composition. What are the benefits? In pursuit of some form
of contractivity, the generalized flow tracker classifies the
communication graphs by their connectivity properties
and breaks up orbits into sequential segments accordingly
(Figure 10). It partitions the set of stochastic matrices into
classes and decomposes the coding tree T into maximal
subtrees consisting of nodes v with matrices Pv from the
same class. The power of this “renormalization” procedure
is that it can be repeated recursively. We classify the communication graphs by their block-directionality type: G (x) is
of type m → n − m if the agents can be partitioned into A,
B (|A| = m) so that no B-agent ever links to an A-agent; if in
addition, no A-agent links to any B-agent, G (x) is of type m
→ n − m. If we define the renormalization scale wk = | Wtk+ 1| − n + m for k = 1, . . . , l − 1 (where Wt denotes the set of wet
nodes), any path of the coding tree can be expressed as
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The subscripts indicate the lengths of the (underlined)
renormalized subsystems. Varying the shift d may change
the coding tree, so we extend all the previous definitions to
the global coding tree TD with phase space [0, 1]n × D, for a
tiny interval D centered at the origin. We have all the elements in place for the algorithmic proof of Theorem 1 to
proceed: see Chazelle6 for details.
I wish to thank Andrew Appel, Stan Leibler, Ronitt
Rubinfeld, David Walker, and the anonymous referee for
helpful comments and suggestions. This work was supported in part by NSF grants CCF-0832797, CCF-0963825,