Searcharxiv⌕ Search

SEARCH · Searcharxiv

Search Searcharxiv

Search indexed arXiv papers on artificial intelligence, large language models, computer vision and robotics. Read source abstracts and follow links to arXiv.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

100 records · Page 6Linked to original sources
arXiv2026

Unconditional $V^0_1$-independence of a certified hitting-set principle

We show that a certified formalization of the hitting-set-existence axiom of Atserias and Tzameret, instantiated on the parity-based Nisan-Wigderson compression class of Khaniki, is independent of the two-sorted theory $V^0_1$ of $\mathrm{AC}^0$-reasoning, unconditionally: $V^0_1$ proves neither it nor its negation. The same holds for the corresponding certified dual weak pigeonhole principle, whose refutation is witnessed by a single seed that certified-computes every string of the model simultaneously. The mechanism is a bounded-arithmetic transfer of Atserias-Tzameret's reduction from hitting sets to the dual weak pigeonhole principle: the amplification half of that reduction, the sole source of its NP-oracle, is unnecessary at the native stretch of the Nisan-Wigderson map, and the compression half becomes a $V^0_1$-provable implication once circuit evaluation is replaced by its certified $Σ^B_0$ unfolding. This is, to our knowledge, the first independence result for a derandomization-flavoured existence principle at the $\mathrm{AC}^0$-reasoning level, and it makes explicit the bridge between the Khaniki Nisan-Wigderson line and the Atserias-Tzameret reverse mathematics of hitting sets.

cs.CC↗
arXiv2026

The $\mathsf{AC}^0$-Complexity Of Visibly Pushdown Languages

We study the question of which visibly pushdown languages (VPLs) are in the complexity class $\mathsf{AC}^0$ and how to effectively decide this question. Our contribution is to introduce a particular subclass of one-turn VPLs, called intermediate VPLs, for which the raised question is entirely unclear: to the best of our knowledge our research community is unaware of containment or non-containment in $\mathsf{AC}^0$ for any language in our newly introduced class. Our main result states that there is an algorithm that, given a visibly pushdown automaton, correctly outputs exactly one of the following: that its language $L$ is in $\mathsf{AC}^0$, some $m\geq 2$ such that $L$ is $\mathsf{ACC}^0(m)$-hard (implying that $L$ is not in $\mathsf{AC}^0$), or a finite disjoint union of intermediate VPLs that $L$ is constant-depth equivalent to. In the latter of the three cases one can moreover effectively compute $k,l\in\mathbb{N}_{>0}$ with $k\not=l$ such that the concrete intermediate VPL $L(S\rightarrow \varepsilon\mid a c^{k-1} S b_1\mid ac^{l-1}Sb_2)$ is constant-depth reducible to the language $L$. Due to their particular nature we conjecture that either all intermediate VPLs are in $\mathsf{AC}^0$ or all are not. As a corollary of our main result we obtain that in case the input language is a visibly counter language our algorithm can effectively determine if it is in $\mathsf{AC}^0$ - hence our main result generalizes a result by Krebs et al. stating that it is decidable if a given visibly counter language is in $\mathsf{AC}^0$ (when restricted to well-matched words). For our proofs we revisit so-called Ext-algebras (introduced by Czarnetzki et al.), which are closely related to forest algebras (introduced by Bojańczyk and Walukiewicz), and use Green's relations.

cs.FL↗
arXiv2026

Quantum Computing: Lecture Notes

This is a set of lecture notes suitable for a Master's course on quantum computation and information from the perspective of theoretical computer science. The first version was written in 2011, with many extensions and improvements in subsequent years. The first 10 chapters cover the circuit model and the main quantum algorithms (Deutsch-Jozsa, Simon, Shor, Hidden Subgroup Problem, Grover, quantum walks, Hamiltonian simulation and HHL). They are followed by 4 chapters about complexity, 4 chapters about distributed ("Alice and Bob") settings, a chapter about quantum machine learning, one about stabilizer states and Clifford circuits, and a final chapter about quantum error correction. Appendices A and B give a brief introduction to the required linear algebra and some other mathematical and computer science background. All chapters come with exercises, with some hints provided in Appendix C.

quant-ph↗
arXiv2026

Goal Staying Makes Sum-of-Costs Anonymous Multi-Agent Path Finding NP-Hard

Anonymous Multi-Agent Path Finding (AMAPF) admits polynomial-time network-flow algorithms for several objectives, including makespan, total distance, and sum-of-costs (SoC) when agents disappear upon reaching goals. We show that standard goal-staying AMAPF is fundamentally different. We first formulate SoC minimization by augmenting the standard time-expanded flow model with goal-settlement constraints and show that the resulting linear programming relaxation is non-integral. We then prove that minimizing SoC in goal-staying AMAPF is NP-hard via a reduction from 3-SAT. Together with the polynomial-time result for the disappearing variant, this establishes a sharp complexity boundary determined by whether completed agents remain at their goals.

cs.MA↗
arXiv2026

Orientations without transitive arcs for cubic graphs and phylogenetic networks

An $st$-orientation of an undirected graph $G$ is an acyclic digraph with a single source $s$ and a single sink $t$ that can be obtained from $G$ by assigning a direction to each edge. The classical problem of deciding if an undirected graph $G$ has an $st$-orientation can be solved efficiently. On the other hand, deciding if an $st$-orientation of $G$ exists that does not have any transitive arc is NP-complete, even if each vertex of $G$ has degree at most four. Here we show that this last decision problem remains NP-complete if $G$ is cubic, which settles an open question by Binucci et al. (2025). We obtain NP-completeness for two variants of the problem: (i) $s$ and $t$ are fixed and given as part of the input and (ii) $s$ and $t$ can be chosen freely. We then use these results to investigate the computational complexity of a problem that arises in computational evolution. Specifically, we show that the problem of deciding if an unrooted binary phylogenetic network has an orientation as a rooted binary phylogenetic network without any shortcuts (the analog of a transitive arcs in phylogenetics) is NP-complete. Our results connect the two (mostly) distinct research areas of orienting undirected graphs and orienting unrooted phylogenetic networks.

cs.CC↗
arXiv2026

Edge Geography is XNLP-hard for Pathwidth and in XP for Tree-Partition Width

Directed Edge Geography and Undirected Edge Geography are classical PSPACE-complete two-player graph games in which players alternately make moves along edges, deleting each one after use; the first player unable to move loses. We prove that both problems are XNLP-hard when parameterized by pathwidth, addressing a question raised by Bodlaender over 30 years ago. On the positive side, we observe that Directed Edge Geography is fixed-parameter tractable when parameterized by treewidth and maximum degree. We also prove that both problems are in XP on simple graphs when parameterized by tree-partition width. These results develop modern lower-bound and decomposition-based algorithmic methods for width-based questions in PSPACE-complete graph games.

cs.CC↗
arXiv2026

Separating QMA from QCMA with a classical oracle

We construct a classical oracle proving that, in a relativized setting, the set of languages decidable by an efficient quantum verifier with a quantum witness (QMA) is strictly bigger than those decidable with access only to a classical witness (QCMA). The separating classical oracle we construct is for a decision problem we coin spectral Forrelation -- the oracle describes two subsets of the boolean hypercube, and the computational task is to decide if there exists a quantum state whose standard basis measurement distribution is well supported on one subset while its Fourier basis measurement distribution is well supported on the other subset. This is equivalent to estimating the spectral norm of a "Forrelation" matrix between two sets that are accessible through membership queries. Our lower bound derives from a simple observation that a query algorithm with a classical witness can be run multiple times to generate many samples from a distribution, while a quantum witness is a "use once" object. This observation allows us to reduce proving a QCMA lower bound to proving a sampling hardness result which does not simultaneously prove a QMA lower bound. To prove said sampling hardness result for QCMA, we observe that quantum access to the oracle can be compressed by expressing the problem in terms of bosons -- a novel "second quantization" perspective on compressed oracle techniques, which may be of independent interest. Using this compressed perspective on the sampling problem, we prove the sampling hardness result, completing the proof.

quant-ph↗
arXiv2025

Turing complete Navier-Stokes steady states via cosymplectic geometry

In this article, we construct stationary solutions to the Navier-Stokes equations on certain Riemannian $3$-manifolds that exhibit Turing completeness, in the sense that they are capable of performing universal computation. This universality arises on manifolds admitting nonvanishing harmonic 1-forms, thus showing that computational universality is not obstructed by viscosity, provided the underlying geometry satisfies a mild cohomological condition. The proof makes use of a correspondence between nonvanishing harmonic $1$-forms and cosymplectic geometry, which extends the classical correspondence between Beltrami fields and Reeb flows on contact manifolds.

math.DG↗
arXiv2025

Reduction of the group isomorphism problem to the group automorphism problem

It is well known that the graph isomorphism problem is polynomial-time reducible to the graph automorphism problem (in fact these two problems are polynomial-time equivalent). We show that, analogously, the group isomorphism problem is polynomial-time reducible to the group automorphism problem. Reductions to other relevant problems like automorphism counting are also given.

cs.CC↗
arXiv2023

Local minima in quantum systems

Finding ground states of quantum many-body systems is known to be hard for both classical and quantum computers. As a result, when Nature cools a quantum system in a low-temperature thermal bath, the ground state cannot always be found efficiently. Instead, Nature finds a local minimum of the energy. In this work, we study the problem of finding local minima in quantum systems under thermal perturbations. While local minima are much easier to find than ground states, we show that finding a local minimum is computationally hard for classical computers, even when the task is to output a single-qubit observable at any local minimum. In contrast, we prove that a quantum computer can always find a local minimum efficiently using a thermal gradient descent algorithm that mimics the cooling process in Nature. To establish the classical hardness of finding local minima, we consider a family of two-dimensional Hamiltonians such that any problem solvable by polynomial-time quantum algorithms can be reduced to finding ground states of these Hamiltonians. We prove that for such Hamiltonians, all local minima are global minima. Therefore, assuming quantum computation is more powerful than classical computation, finding local minima is classically hard and quantumly easy.

quant-ph↗