arXiv subjects

Hunter Monroe

Publications and source records attributed to Hunter Monroe.

11 recordsLinked to original sources

arXiv2026

Hardness as an Information Constraint: A Unifying Meta-Complexity Assumption

Monroe (2026) shows that, if no optimal proof system exists, then every sound arithmetic theory S extending S^1_2 with polynomial-time decidable axioms fails, for all sufficiently large k, to simulate S^1_2+phi_BB(k), where phi_BB(k) asserts the exact k-state Busy Beaver value. This gives the nonexistence hypothesis an information-constraint interpretation through canonical hard instances. If the best-known route to simulation is also necessary--namely, if simulation requires a relative-consistency explanation over a weak base--then the same constraint applies to inaccessible Kolmogorov-randomness facts. We call this conjecture Kolmogorov Hardness (KH). Finite-scale and hierarchy-level forms of KH yield, conditionally, dense families of small hard tautologies, PH noncollapse with explicit dense separators, and SAT notin P/poly. Under separately stated assumptions, variants yield P-inseparability of a disjoint NP pair, no mutual help between mutually conditionally random axioms, one-way functions via Liu--Pass, derandomization, and Feige-style random-refutation hardness. Allender et al. provide a complementary calibration: full access to conventional random-string oracles supports every PSPACE computation, whereas any fixed sound computably axiomatized theory can certify only finitely many positive randomness instances. The framework organizes complexity conjectures around canonical information constraints. It seems self-evident that efficient proofs should not leverage true randomness facts unavailable to the proving theory. Yet KH may be independent of standard metatheories: it resembles reflection, can fail internally in nonstandard models even when externally true, and may constrain the metatheories themselves. We propose a research program on extensions, self-evidence, formal independence, and possible new axioms.

cs.CC↗

arXiv2026

Toward a Characterization of Simulation Between Arithmetic Theories

We study when a sound arithmetic theory $\mathcal S\supseteq S^1_2$ with polynomial-time decidable axioms efficiently proves the bounded consistency statements $Con_{\mathcal S+\phi}(n)$ for a true sentence $\phi$. Equivalently, we ask when $\mathcal S$, viewed as a proof system, simulates $\mathcal S+\phi$. The paper gives two unconditional constraints on possible characterizations. First, for finitely axiomatized sequential $\mathcal S$, if $EA\vdash Con_{\mathcal S}\rightarrow Con_{\mathcal S+\phi}$, then $\mathcal S$ interprets $\mathcal S+\phi$, implying $\mathcal S\vdash^{n^{O(1)}}Con_{\mathcal S}(p(n))\rightarrow Con_{\mathcal S+\phi}(n)$ for some polynomial $p$, and hence $\mathcal S\vdash^{n^{O(1)}}Con_{\mathcal S+\phi}(n)$. Second, if $\mathcal S$ fails to simulate $\mathcal S+\phi$ for some true $\phi$, then for all sufficiently large $k$ it also fails to simulate $S^1_2+\phi_{BB}(k)$, where $\phi_{BB}(k)$ asserts the exact value of the $k$-state Busy Beaver function. Thus any hard true extension yields a canonical Busy Beaver witness to nonsimulation. $\mathcal B$-certified simulation of a target $\mathcal U$ yields $\mathcal B{\vdash}Con_{\mathcal S}{\rightarrow}Con_{\mathcal U}$, giving certification barriers rather than external lower bounds. The paper's central conjectural proposal is: for sound, finitely axiomatized sequential $\mathcal S$, if $EA\not\vdash Con_{\mathcal S}\rightarrow Con_{\mathcal S+\phi}$, then for every constant $c>0$, $\mathcal S\not\vdash^{n^c}Con_{\mathcal S+\phi}(n)$. Under this proposal, hardness follows when $\phi$ is $Con_{\mathcal S}$ or a Kolmogorov-randomness axiom. The latter yields further conjectural consequences and extensions.

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arXiv2025

A Proposed Characterization of p-Simulation Between Theories

This paper proposes a characterization of when one axiomatic theory, as a proof system for tautologies, $p$-simulates another, by showing: (i)~if c.e. theory $\mathcal{S}$ efficiently interprets $\mathcal{S}{+}\phi$, then $\mathcal{S}$ $p$-simulates $\mathcal{S}{+}\phi$ (Je\v{r}\'abek in Pudl\'ak17 proved simulation), since the interpretation maps an $\mathcal{S}{+}\phi$-proof whose lines are all theorems into an $\mathcal{S}$-proof; (ii)~$\mathcal{S}$ proves ``$\mathcal{S}$ efficiently interprets $\mathcal{S}{+}\phi$'' iff $\mathcal{S}$ proves ``$\mathcal{S}$ $p$-simulates $\mathcal{S}{+}\phi$'' (if so, $\mathcal{S}$ already proves the $\Pi_1$ theorems of $\mathcal{S}{+}\phi$). To explore whether this framework conceivably resolves other open questions, the paper formulates conjectures stronger than ``no optimal proof system exists'' that imply Feige's Hypothesis, the existence of one-way functions, and circuit lower bounds.

cs.CC↗

arXiv2023

Ruling Out Short Proofs of Unprovable Sentences is Hard

If no optimal propositional proof system exists, we (and independently Pudl\'ak) prove that ruling out length $t$ proofs of any unprovable sentence is hard. This mapping from unprovable to hard-to-prove sentences powerfully translates facts about noncomputability into complexity theory. For instance, because proving string $x$ is Kolmogorov random ($x{\in}R$) is typically impossible, it is typically hard to prove "no length $t$ proof shows $x{\in}R$", or tautologies encoding this. Therefore, a proof system with one family of hard tautologies has these densely in an enumeration of families. The assumption also implies that a natural language is $\textbf{NP}$-intermediate: with $R$ redefined to have a sparse complement, the complement of the language $\{\langle x,1^t\rangle|$ no length $t$ proof exists of $x{\in}R\}$ is also sparse. Efficiently ruling out length $t$ proofs of $x{\in}R$ might violate the constraint on using the fact of $x{\in}R$'s unprovability. We conjecture: any computable predicate on $R$ that might be used in if-then statements (or case-based proofs) does no better than branching at random, because $R$ appears random by any effective test. This constraint could also inhibit the usefulness in circuits and propositional proofs of NOT gates and cancellation -- needed to encode if-then statements. If $R$ defeats if-then logic, exhaustive search is necessary.

cs.CC↗

arXiv2023

Hardness of Ruling Out Short Proofs of Kolmogorov Randomness

A meta-complexity assumption, Feasible Chaitin Incompleteness (FCI), asserts the hardness of ruling out length $t$ proofs that string $x$ is Kolmogorov random (e.g. $x{\in}R$), by analogy to Chaitin's result that proving $x{\in}R$ is typically impossible. By assertion, efficiently ruling out short proofs requires, impossibly, ruling out any proof. FCI has strong implications: (i) randomly chosen $x$ typically yields tautologies hard with high probability for any given proof system, densely witnessing its nonoptimality; (ii) average-case impossibility of proving $x{\in}R$ implies average-case hardness of proving tautologies and Feige's hypothesis; and (iii) a natural language is $\textbf{NP}$-intermediate -- the sparse complement of "$x{\in}R$ lacks a length $t$ proof" (where $R$'s complement is sparse) -- and has $\textbf{P/poly}$ circuits despite not being in $\textbf{P}$. FCI and its variants powerfully assert: (i) noncomputability facts translate to hardness conjectures; (ii) numerous open complexity questions have the expected answers (e.g. non-collapse of $\textbf{PH}$), so one overarching conjecture subsumes many questions; and (iii) an implicit mapping between certain unprovable and hard-to-prove sentences is an isomorphism. Further research could relate FCI to other open questions and hardness hypotheses; consider whether $R$ frustrates conditional program logic, implying FCI; and consider whether an extended isomorphism maps any true unprovable sentence to hard-to-prove sentences.

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arXiv2022

Average-Case Hardness of Proving Tautologies and Theorems

We consolidate two widely believed conjectures about tautologies -- no optimal proof system exists, and most require superpolynomial size proofs in any system -- into a $p$-isomorphism-invariant condition satisfied by all paddable $\textbf{coNP}$-complete languages or none. The condition is: for any Turing machine (TM) $M$ accepting the language, $\textbf{P}$-uniform input families requiring superpolynomial time by $M$ exist (equivalent to the first conjecture) and appear with positive upper density in an enumeration of input families (implies the second). In that case, no such language is easy on average (in $\textbf{AvgP}$) for a distribution applying non-negligible weight to the hard families. The hardness of proving tautologies and theorems is likely related. Motivated by the fact that arithmetic sentences encoding "string $x$ is Kolmogorov random" are true but unprovable with positive density in a finitely axiomatized theory $\mathcal{T}$ (Calude and J{\"u}rgensen), we conjecture that any propositional proof system requires superpolynomial size proofs for a dense set of $\textbf{P}$-uniform families of tautologies encoding "there is no $\mathcal{T}$ proof of size $\leq t$ showing that string $x$ is Kolmogorov random". This implies the above condition. The conjecture suggests that there is no optimal proof system because undecidable theories help prove tautologies and do so more efficiently as axioms are added, and that constructing hard tautologies seems difficult because it is impossible to construct Kolmogorov random strings. Similar conjectures that computational blind spots are manifestations of noncomputability would resolve other open problems.

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arXiv2012

Speedup for Natural Problems and Noncomputability

A resource-bounded version of the statement "no algorithm recognizes all non-halting Turing machines" is equivalent to an infinitely often (i.o.) superpolynomial speedup for the time required to accept any coNP-complete language and also equivalent to a superpolynomial speedup in proof length in propositional proof systems for tautologies, each of which implies P!=NP. This suggests a correspondence between the properties 'has no algorithm at all' and 'has no best algorithm' which seems relevant to open problems in computational and proof complexity.

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arXiv2010

Are Causality Violations Undesirable?

Causality violations are typically seen as unrealistic and undesirable features of a physical model. The following points out three reasons why causality violations, which Bonnor and Steadman identified even in solutions to the Einstein equation referring to ordinary laboratory situations, are not necessarily undesirable. First, a space-time in which every causal curve can be extended into a closed causal curve is singularity free--a necessary property of a globally applicable physical theory. Second, a causality-violating space-time exhibits a nontrivial topology--no closed timelike curve (CTC) can be homotopic among CTCs to a point, or that point would not be causally well behaved--and nontrivial topology has been explored as a model of particles. Finally, if every causal curve in a given space-time passes through an event horizon, a property which can be called "causal censorship", then that space-time with event horizons excised would still be causally well behaved.

gr-qc↗

arXiv2006

Topology and Closed Timelike Curves II: Causal structure

Because no closed timelike curve (CTC) on a Lorentzian manifold can be deformed to a point, any such manifold containing a CTC must have a topological feature, to be called a timelike wormhole, that prevents the CTC from being deformed to a point. If all wormholes have horizons, which typically seems to be the case in space-times without exotic matter, then each CTC must transit some timelike wormhole's horizon. Therefore, a Lorentzian manifold containing a CTC may nevertheless be causally well behaving once its horizon's are deleted. For instance, there may be a Cauchy-like surface through which every timelike curve passes one and only once before crossing a horizon.

gr-qc↗

arXiv2006

The Duality of Time Dilation and Velocity

Time dilation $\frac{1}{\sqrt{1-v^2}}$ and relative velocity $v$ are observationally indistinguishable in the special theory of relativity, a duality that carries over into the general theory under Fermi coordinates along a curve (in coordinate-independent language, in the tangent Minkowski space along the curve). For example, on a clock stationary at radius $r$, a distant observer sees time dilation of $\frac{1}{\sqrt{1-v^2}}=\frac{1}{\sqrt{1-2M/r}}$ under the Schwarzschild metric and sees the clock receding with a relative velocity of $v=\sqrt{2M/r}$ under the Painlev{é}-Gullstrand free fall metric. Duality implies that during gravitational collapse, the intensifying time dilation observed at the star's center from a fixed radius $r>0$ is indistinguishable (along a curve) from an increasing relative velocity at which the center recedes as seen from any direction, implying a local inflation.

gr-qc↗

arXiv2005

Singularity-Free Collapse through Local Inflation

In common relativistic models of gravitational collapse, the interior experiences a bubble-like local inflation, allowing radii to diverge rather than converge toward a singularity. This proves a conjecture of Shatskiy. At the horizon, the solution locally resembles the neck of a wormhole from the exterior to the interior. The implied limiting curvature is at the horizon not the Planck level.

astro-ph↗