The Distance Function on a Computable Graph

We apply the techniques of computable model theory to the distance function of a graph. This task leads us to adapt the definitions of several truth-table reducibilities so that they apply to functions as well as to sets, and we prove assorted theorems about the new reducibilities and about functions which have nonincreasing computable approximations. Finally, we show that the spectrum of the distance function can consist of an arbitrary single btt-degree which is approximable from above, or of all such btt-degrees at once, or of the bT-degrees of exactly those functions approximable from above in at most n steps.

💡 Research Summary

The paper investigates the computational complexity of the distance function on computable infinite graphs from the perspective of computable model theory. After introducing the basic setting—connected, symmetric, irreflexive, and computable graphs—the authors focus on the “spectrum” of a distance function, i.e., the set of Turing degrees of all distance functions on graphs isomorphic to a given computable graph. While the spectrum is a standard notion for relations, the distance function is a total function, prompting the authors to adapt several reducibility notions traditionally defined for sets to the functional setting.

First, the authors formalize how a Turing machine can use a function as an oracle. They model a function oracle with three tapes (question, answer, output) and define bounded‑Turing (bT or wtt) and truth‑table (tt, btt) reducibilities for functions. They observe that naïve extensions of m‑reducibility (many‑one) and 1‑reducibility to functions lead to pathological degree structures: constant functions each form their own m‑degree, and simple linear functions can be incomparable. To remedy this, they introduce “augmented m‑reducibility” (≤ₐ), where a function ϕ is ≤ₐ‑reducible to ψ if ϕ ≤ₘ (ι⊕ψ), with ι a trivial “domain‑indicator” function. Under ≤ₐ, all partial computable functions with computable domains collapse into a single degree, eliminating many of the earlier anomalies.

Next, the paper defines the class of functions “approximable from above”: a computable sequence of approximations fₛ(x) that is non‑increasing and eventually stabilizes at the true value f(x). The distance function d_G(x,y) on any computable graph G is naturally of this type, because a systematic search for paths yields a decreasing sequence of path lengths that converges to the shortest one. The authors show that such functions are ∅′‑computable (by the Limit Lemma) but often not computable nor Σ⁰₁‑definable for distances greater than 2. They classify the approximable‑from‑above functions using an Ershov‑style hierarchy based on the number of allowed “mind‑changes” (the number of times the approximation can drop before stabilizing).

The core technical contributions concern the distance‑function spectrum. Three main theorems are proved:

1. Realizing an arbitrary btt‑degree. For any btt‑degree α that contains a function approximable from above, there exists a computable graph G whose distance function d_G has exactly that btt‑degree. Thus any such degree can be “encoded” as a distance function.

2. Realizing all such btt‑degrees simultaneously. There is a computable graph H such that the set of btt‑degrees of its distance function includes every btt‑degree of a function approximable from above. In other words, the spectrum of H is the entire collection of those degrees.

3. Bounding the number of approximation steps. For each fixed n∈ℕ, the authors construct a computable graph K_n whose distance function is approximable from above in at most n steps, and whose spectrum consists precisely of the bT‑degrees of those functions. Consequently, the spectrum can be restricted to a specific level of the Ershov hierarchy.

A crucial auxiliary result (Proposition 2.7) shows that a total function h is truth‑table equivalent to its graph iff h is computably bounded (i.e., there exists a computable bound b such that h(x) ≤ b(x) for all x). This bridges the gap between function‑centric and set‑centric reducibilities and justifies treating distance functions as ordinary functions rather than as binary relations.

Overall, the paper develops a robust framework for analyzing functions—especially natural ones like graph distances—under fine‑grained reducibilities (bT, wtt, tt, btt). By adapting these notions to the functional setting and by exploiting the intrinsic “above‑approximation” property of distances, the authors obtain a detailed picture of which computational degrees can arise as distance functions on computable graphs, and how the complexity of the underlying approximation process controls the resulting degree spectrum. The work opens avenues for further exploration of other graph‑theoretic invariants and for extending the methodology to broader classes of computable structures.