arXiv subjects

Antoine Vinciguerra

Publications and source records attributed to Antoine Vinciguerra.

5 recordsLinked to original sources

arXiv2026

An Operator Approach to Register Programs for Catalytic Computing

In a seminal work, Buhrman et al.\ (STOC 2014) introduced catalytic computation and proved that uniform $TC^1$ circuits are computable in catalytic logspace, the class of problems solvable in space $s$ with an additional catalytic tape of size $c$, a tape whose initial content must be restored at the end of the computation. A central ingredient of their proof is the register program model. Namely, they constructed a uniform family of register programs that computes $x^n$ using $n$ registers and four accesses to $x$. Since then, determining the number of registers and input accesses required to compute a polynomial of a given degree has become a central question in the study of catalytic computation. On one hand, we prove that the four-access bound of Buhrman et al.\ is optimal: every passive-output register program computing a polynomial of degree greater than three requires at least four input accesses, independently of the number of registers. On the other hand, we show that their register bound is not optimal. For every $t\geq2$ and every field $K$ of characteristic $0$ or greater than $2t-1$, we construct a register program for $x^{2t-1}$ with four input accesses and $t$ registers. Our proofs rely on derivations and their exponential operators. This approach represents a register program as a series of exponential derivation operators, reducing register restoration to an operator identity. Finally, we use the uniform family of register programs to improve known trade-offs for catalytic streaming algorithms and register programs for matrix powering. The generalization of the lower-bound methods and the construction of the uniform family of register programs were developed with assistance from ChatGPT 5.6.

cs.CC↗

arXiv2026

From the Dirichlet Integral to Lobachevsky's Formula: A Formalization in Lean 4

We formalize the Dirichlet integral and several of its classical applications in the Lean~4 proof assistant. Since the sinc function is not Lebesgue integrable on the positive half-line, the Dirichlet integral must be represented as the limit of integrals over bounded intervals. To avoid the difficulty of removing an exponential factor from a conditionally convergent integral, we instead pass through the absolutely integrable function $\operatorname{sinc}^2$. We evaluate its integral by differentiation under the integral sign and dominated convergence, and then recover the Dirichlet integral from an identity between truncated integrals. Using these results, we formalize the convergence of the Dirichlet cutoff to the Heaviside function and derive several quadratic and bilinear trigonometric integral identities. Finally, we formalize Lobachevsky's integral formula for continuous periodic functions satisfying a reflection symmetry, using the density of cosine polynomials obtained from Mathlib's Fourier analysis on the additive circle.

cs.LO↗

arXiv2026

A Formalization of the Laplace Transform and Its Inversion in Lean 4

We present a Lean 4 formalization of the Laplace transform for complex-valued functions, its fundamental operational rules, and a Bromwich-type inversion theorem proved through real-variable integration and the Dirichlet integral. As an application, we formalize the Laplace-domain solution of the harmonic oscillator and identify its transform with that of $\sin(ωt)$. We also discuss the principal analytic and formalization challenges encountered in the development.

cs.LO↗

arXiv2025

Linear Matroid Intersection is in Catalytic Logspace

Linear matroid intersection is an important problem in combinatorial optimization. Given two linear matroids over the same ground set, the linear matroid intersection problem asks you to find a common independent set of maximum size. The deep interest in linear matroid intersection is due to the fact that it generalises many classical problems in theoretical computer science, such as bipartite matching, edge disjoint spanning trees, rainbow spanning tree, and many more. We study this problem in the model of catalytic computation: space-bounded machines are granted access to \textit{catalytic space}, which is additional working memory that is full with arbitrary data that must be preserved at the end of its computation. Although linear matroid intersection has had a polynomial time algorithm for over 50 years, it remains an important open problem to show that linear matroid intersection belongs to any well studied subclass of $\mathsf{P}$. We address this problem for the class catalytic logspace ($\mathsf{CL}$) with a polynomial time bound ($\mathsf{CLP}$). Recently, Agarwala and Mertz (2025) showed that bipartite maximum matching can be computed in the class $\mathsf{CLP}\subseteq \mathsf{P}$. This was the first subclass of $\mathsf{P}$ shown to contain bipartite matching, and additionally the first problem outside $\mathsf{TC}^1$ shown to be contained in $\mathsf{CL}$. We significantly improve the result of Agarwala and Mertz by showing that linear matroid intersection can be computed in $\mathsf{CLP}$.

cs.CC↗

arXiv2025

Catalytic Computing and Register Programs Beyond Log-Depth

In a seminal work, Buhrman et al. (STOC 2014) defined the class $CSPACE(s,c)$ of problems solvable in space $s$ with an additional catalytic tape of size $c$, which is a tape whose initial content must be restored at the end of the computation. They showed that uniform $TC^1$ circuits are computable in catalytic logspace, i.e., $CL=CSPACE(O(\log{n}), 2^{O(\log{n})})$, thus giving strong evidence that catalytic space gives $L$ strict additional power. Their study focuses on an arithmetic model called register programs, which has been a focal point in development since then. Understanding $CL$ remains a major open problem, as $TC^1$ remains the most powerful containment to date. In this work, we study the power of catalytic space and register programs to compute circuits of larger depth. Using register programs, we show that for every $ε> 0$, $SAC^2 \subseteq CSPACE\left(O\left(\frac{\log^2{n}}{\log\log{n}}\right), 2^{O(\log^{1+ε} n)}\right)$ This is an $O(\log \log n)$ factor improvement on the free space needed to compute $SAC^2$, which can be accomplished with near-polynomial catalytic space. We also exhibit non-trivial register programs for matrix powering, which is a further step towards showing $NC^2 \subseteq CL$.

cs.CC↗