False statementConstruction: Literature-sourcedVerification: AI-generatedprecheck passaudited 2026-09-12

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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FALSE: clocked-simulation and time-hierarchy arguments never need constructibility hypotheses

Statement

False claim: every total time bound $t$ with $t (n) \geq n$ can be uniformly materialized and enforced as a same-scale machine clock without a constructibility assumption.

Facts & Assumptions

Given: The false claim above.

[A1]

Every total time bound $t$ with $t (n) \geq n$ can be uniformly materialized and enforced as a same-scale machine clock without a constructibility assumption.

[L1]

A constructible time bound is one whose value can itself be written down within the same asymptotic scale, by Time-constructible and space-constructible functions.

[L2]

A constructible time bound that also satisfies $n \leq t (n)$ for all sufficiently large $n$ yields a uniformly clocked simulator, by A constructible time bound yields a uniformly clocked simulator.

[F1]

There exists a total noncomputable function $b : N \to {0, 1}$ . Indeed, the total computable binary-valued functions form a countable family $(b_{j})_{j \in N}$ because machines have finite descriptions, and the diagonal function $b (n) : = 1 - b_{n} (n)$ differs from every $b_{j}$ at input $j$ .

Refutation

technique · direct

1.1L1F1construct

Define the total bound $t (n) : = 2 n + b (n)$ using [F1]. Then $t (n) \geq n$ . If a uniform constructor could materialize $t (n)$ on input $1^{n}$ , subtracting $2 n$ from its output would compute $b (n)$ , contradicting [F1]. Hence this total bound is not time-constructible in the sense of [L1].

2.1A1L2step 1.1∎

The construction in [L2] works precisely by first computing the bound and then enforcing it as a timeout. Step 1.1 exhibits a total bound satisfying the size condition for which that first operation is impossible. Therefore [A1] is false: a uniform same-scale clock cannot be obtained for every total bound without a constructibility hypothesis.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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

John Watrous, Introduction to the Theory of Computing, Lecture 19: Time-bounded computations (standard reference, not scraped)
Sanjeev Arora and Boaz Barak, Computational Complexity: A Modern Approach (standard reference, not scraped)