TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06

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Gap and union theorems for abstract complexity bounds

Statement

Let $Φ$ be a Blum complexity measure. (Gap) for every total recursive $r$ satisfying $r (m) > m$ for every $m$ , there is a total recursive, monotonically increasing bound $t$ such that $C_{t} = C_{r \circ t}$ . (Union) for every recursively enumerable family of recursive bounds $(f_{i})$ satisfying $f_{i} (n) < f_{i + 1} (n)$ for every $i, n$ , there is a recursive $t$ with $C_{t} = ⋃_{i} C_{f_{i}}$ , where $C_{h}$ denotes the programs whose $Φ$ -complexity is eventually at most $h$ .

Facts & Assumptions

Given: a Blum measure, the stated recursive data, and eventual-bound class convention.

Proof

technique · application of the classical recursive constructions

1.1

The Gap Theorem for abstract measures (Hartmanis--Hopcroft, Theorem 7) applies to the given Blum measure and recursive $r > i d$ . Its finite-stage construction uses only bounded tests of the decidable Blum predicate $Φ_{i} (x) = m$ ; it does not attempt to decide whether a program is eventually below a bound. The resulting recursive nondecreasing $t$ has the stage invariant that every indexed program eventually below $r \circ t$ is also eventually below $t$ . Hence $C_{t} = C_{r \circ t}$ .

givenconstruct

2.1

The Union Theorem for abstract measures (Hartmanis--Hopcroft, Theorem 16) applies to the effective strictly increasing sequence $(f_{i})$ . Its recursive finite-injury construction produces a recursive $t$ with two complementary invariants: every $f_{i}$ is eventually at most $t$ , and each indexed program that is outside every $C_{f_{i}}$ exceeds $t$ infinitely often. These give respectively $⋃_{i} C_{f_{i}} \subseteq C_{t}$ and the reverse inclusion.

givenconstruct∎

Depends on

Computable and partial computable functions

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

2 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

Sebastiaan Terwijn, Complexity Theory, Theorem 5.6.1 (standard reference, not scraped)
Hartmanis and Hopcroft, An Overview of the Theory of Computational Complexity, Theorem 16 (standard reference, not scraped)