DefinitionDefinition: Literature-sourcedProof: Not applicable

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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The bounded reachability recursion for directed paths of length at most $2^{i}$

Definition

Let $G = (V, A)$ be a finite digraph in the sense of Multigraphs, loops and directed graphs as variants distinct from the default finite simple graph, let $u, v \in V$ , and let $i \in N$ . Define the predicate ${Reach}_{i}^{G} (u, v)$ recursively by

${Reach}_{0}^{G} (u, v) ⟺ (u = v or (u, v) \in A),$

and, for $i \geq 1$ ,

${Reach}_{i}^{G} (u, v) ⟺ \exists z \in V ({Reach}_{i - 1}^{G} (u, z) \land {Reach}_{i - 1}^{G} (z, v)) .$

This is the bounded reachability recursion. Its intended meaning is that ${Reach}_{i}^{G} (u, v)$ asks whether there is a directed path from $u$ to $v$ of length at most $2^{i}$ .

Remarks

The level $i = 0$ allows equality as well as one legal arc, so it captures paths of lengths $0$ and $1$ exactly.
The recursive step halves the permitted path-length scale.

Depends on

Multigraphs, loops and directed graphs as variants distinct from the default finite simple graph

Used by

Dependency tree · two levels

3 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Eric Blais, Models of Computation, 17. Space Complexity (standard reference, not scraped)
Sanjeev Arora and Boaz Barak, Computational Complexity: A Modern Approach (standard reference, not scraped)

The bounded reachability recursion for directed paths of length at most 2i2i