By Romer

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The arcs are represented by curves drawn between the vertices. An arrow on an arc indicates the order of initial vertex to terminal vertex. Thus A has ﬁve elements: a1 , a2 , a3 , a4 , and L 1 . The arc a1 has as initial vertex the vertex v1 and as terminal vertex the vertex v2 . The arc L 1 is a loop with v2 as both initial −−−−→ and terminal vertices. The function a → ι(a)τ (a) assigns an initial vertex and a → v− terminal vertex to the arc a. On arc a1 , a1 → − 1 v2 , indicating that the initial vertex of a1 is v1 and the terminal vertex of a1 is v2 .

Length is an indicator of the time required for a superposition to compute a given function. Bounds on the lengths of 18 F Networks 19 superpositions are explored in Chapter 6. For a given function and algorithm for computing it, the expression of that algorithm as a fan-in is determined mainly by the original digraph and the time t. The principal graph theoretic advantage of working with a fan-in is that the time t required to compute the value of a function is also the length of the longest path from a leaf, a vertex where a variable is introduced, to its root, the vertex where the value of the computation is read.

A cycle in G is a chain that has endpoints that coincide and is such that no edge appears twice in the sequence. An empty arc, that is, the edge associated with an empty arc, is not a cycle. 1 is a cycle. However, the chain ({L 1 , L 1 }) is not a cycle because the arc L 1 occurs twice in the chain. A path P of length q in a digraph G is a sequence of q arcs and empty arcs a1 , . . 2 The path P is denoted by a1 , . . , aq . 1 the sequence a1 , L 1 is a path of length 2 whereas the sequence a4 , a3 is not a path because the initial vertex of a3 is not the terminal vertex of a4 .

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