An Axiomatic Characterization of the Single-Source Shortest Path Problem

This theory is split into two sections. In the first section, we give a formal proof that a well-known axiomatic characterization of the single-source shortest path problem is correct. Namely, we prove that in a directed graph G = (V,E) with a non-negative cost function on the edges the single-source shortest path function μ : V → R ∪ {∞} is the only function that satisfies a set of four axioms. The first axiom states that the distance from the source vertex s to itself should be equal to zero. The second states that the distance from s to a vertex v ∈ V should be infinity if and only if there is no path from s to v. The third axiom is called triangle inequality and states that if there is a path from s to v, and an edge (u, v) ∈ E, the distance from s to v is less than or equal to the distance from s to u plus the cost of (u, v). The last axiom is called justification, it states that for every vertex v other than s, if there is a path p from s to v in G, then there is a predecessor edge (u, v) on p such that the distance from s to v is equal to the distance from s to u plus the cost of (u, v). In the second section, we give a formal proof of the correctness of an axiomatic characterization of the single-source shortest path problem for directed graphs with general cost functions c : E → R. The axioms here are more involved because we have to account for potential negative cycles in the graph. The axioms are summarized in the three isabelle locales.