May 2016
Intermediate to advanced
354 pages
10h 38m
English
book
Discrete Algebraic Methods
by Volker Diekert, Manfred Kufleitner, Gerhard Rosenberger, Ulrich Hertrampf
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For all subsets I ⊆ {1, . . . , n} and nodes b in G, we introduce an arc from b to b'' = (b − AeI )/2, provided b'' ∈ ℤn and ‖b‖1 ≤ ‖A‖1. The label of this arc is 2x + eI and denotes the affine mapping x ↦ 2x + eI.
This construction is “sound”: by following the arcs we cannot transform an unsolvable linear Diophantine system into a solvable one.
Next, we prove “completeness”: if b is any node in G and x ∈ ℕn is a vector solving Ax = b, then there exists a path in the graph G from b to the final node 0 such that h(0) = x, where h is the label of that path, interpreted as composition of mappings. We show this claim by induction on ‖x‖1.
The assertion is trivial for ‖x‖1 = 0, because then we must have x = b = 0. Thus, we may assume that at least ...
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