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In this paper, we address the probabilistic error quantification of a general class of prediction methods. We consider a given prediction model and show how to obtain, through a sample-based approach, a probabilistic upper bound on the…
We introduce two new complexity measures for Boolean functions, or more generally for functions of the form f:S->T. We call these measures sumPI and maxPI. The quantity sumPI has been emerging through a line of research on quantum query…
Stochastic Barrier Functions (SBFs) certify the safety of stochastic systems by formulating a functional optimization problem, which state-of-the-art methods solve using Sum-of-Squares (SoS) polynomials. This work focuses on polynomial SBFs…
Gordeev and Haeusler [GH19] claim that each tautology $\rho$ of minimal propositional logic can be proved with a natural deduction of size polynomial in $|\rho|$. This builds on work from Hudelmaier [Hud93] that found a similar result for…
We study the counting version of the Boolean satisfiability problem #SAT using the ZH-calculus, a graphical language originally introduced to reason about quantum circuits. Using this, we generalize #SAT to a weighted variant we call…
First-order logic has been established as an important tool for modeling and verifying intricate systems such as distributed protocols and concurrent systems. These systems are parametric in the number of nodes in the network or the number…
In this paper, we introduce the hybrid query complexity, denoted as $\mathrm{Q}(f;q)$, which is the minimal query number needed to compute $f$, when a classical decision tree is allowed to call $q'$-query quantum subroutines for any $q'\leq…
We propose a new type of quantum computer which is used to prove a spectral representation for a class F of computable sets. When S in F codes the theorems of a formal system, the quantum computer produces through measurement all theorems…
In this article we show how any formula A with a proof in minimal implicational logic that is super-polynomially sized has a polynomially-sized proof in classical implicational propositional logic . This fact provides an argument in favor…
The term proposition usually denotes in quantum mechanics (QM) an element of (standard) quantum logic (QL). Within the orthodox interpretation of QM the propositions of QL cannot be associated with sentences of a language stating properties…
The Polynomial-Time Hierarchy ($\mathsf{PH}$) is a staple of classical complexity theory, with applications spanning randomized computation to circuit lower bounds to ''quantum advantage'' analyses for near-term quantum computers.…
Using proof-theoretic methods in the style of proof mining, we give novel computationally effective limit theorems for the convergence of the Cesaro-means of certain sequences of random variables. These results are intimately related to…
We investigate the structure of quantum proof systems by establishing collapse results that reveal simplifications in their complexity landscape. By extending classical theorems such as the Karp-Lipton theorem to quantum settings and…
In unitary property testing a quantum algorithm, also known as a tester, is given query access to a black-box unitary and has to decide whether it satisfies some property. We propose a new technique for proving lower bounds on the quantum…
The purpose of this paper is to give an easy to understand with step-by-step explanation to allow interested people to fully appreciate the power of natural deduction for first-order logic. Natural deduction as a proof system can be used to…
Representations of Boolean functions by real polynomials play an important role in complexity theory. Typically, one is interested in the least degree of a polynomial p(x_1,...,x_n) that approximates or sign-represents a given Boolean…
A line of work initiated by Fortnow in 1997 has proven model-independent time-space lower bounds for the $\mathsf{SAT}$ problem and related problems within the polynomial-time hierarchy. For example, for the $\mathsf{SAT}$ problem, the…
In this paper we prove lower bounds for sizes of refutations of unsatisfiable vector Subset Sum instances $\overrightarrow{a}_1 x_1 + \dots + \overrightarrow{a}_n x_n = \overrightarrow{b}$ in the proof system Res(lin$_{\mathbb{F}_q}$) where…
Safety filters based on control barrier functions (CBFs) and high-order control barrier functions (HOCBFs) are often implemented through quadratic programs (QPs). In general, especially in the presence of multiple constraints, feasibility…
The CNF formula satisfiability problem (CNF-SAT) has been reduced to many fundamental problems in P to prove tight lower bounds under the Strong Exponential Time Hypothesis (SETH). Recently, the works of Abboud, Hansen, Vassilevska W. and…