arXiv subjects

Florian Funke

Publications and source records attributed to Florian Funke.

17 recordsLinked to original sources

arXiv2026

DySkew: Dynamic Data Redistribution for Skew-Resilient Snowpark UDF Execution

Snowflake revolutionized data warehousing with an elastic architecture that decouples compute and storage, enabling scalable solutions for diverse data analytics needs. Building on this foundation, Snowflake has advanced its AI Data Cloud vision by introducing Snowpark, a managed turnkey solution that supports data engineering and AI/ML workloads using Python and other programming languages. While Snowpark's User-Defined Function (UDF) execution model offers high throughput, it is highly vulnerable to performance degradation from data skew, where uneven data partitioning causes straggler tasks and unpredictable latency. The non-uniform computational cost of arbitrary user code further exacerbates this classic challenge. This paper presents DySkew, a novel, data-skew-aware execution strategy for Snowpark UDFs. Built upon Snowflake's new generalized skew handling solution, an adaptive data distribution mechanism utilizing per-link state machines. DySkew addresses the unique challenges of user-defined logic with goals of fine-grained per-row mitigation, dynamic runtime adaptation, and low-overhead, cost-aware redistribution. Specifically, for Snowpark, we introduce crucial optimizations, including an eager redistribution strategy and a Row Size Model to dynamically manage overhead for extremely large rows. This dynamic approach replaces the limitations of the previous static round-robin method. We detail the architecture of this framework and showcase its effectiveness through performance evaluations and real-world case studies, demonstrating significant improvements in the execution time and resource utilization for large-scale Snowpark UDF workloads.

cs.DC↗

arXiv2022

On probability-raising causality in Markov decision processes

The purpose of this paper is to introduce a notion of causality in Markov decision processes based on the probability-raising principle and to analyze its algorithmic properties. The latter includes algorithms for checking cause-effect relationships and the existence of probability-raising causes for given effect scenarios. Inspired by concepts of statistical analysis, we study quality measures (recall, coverage ratio and f-score) for causes and develop algorithms for their computation. Finally, the computational complexity for finding optimal causes with respect to these measures is analyzed.

cs.LO↗

arXiv2021

Causality-Based Game Solving

We present a causality-based algorithm for solving two-player reachability games represented by logical constraints. These games are a useful formalism to model a wide array of problems arising, e.g., in program synthesis. Our technique for solving these games is based on the notion of subgoals, which are slices of the game that the reachability player necessarily needs to pass through in order to reach the goal. We use Craig interpolation to identify these necessary sets of moves and recursively slice the game along these subgoals. Our approach allows us to infer winning strategies that are structured along the subgoals. If the game is won by the reachability player, this is a strategy that progresses through the subgoals towards the final goal; if the game is won by the safety player, it is a permissive strategy that completely avoids a single subgoal. We evaluate our prototype implementation on a range of different games. On multiple benchmark families, our prototype scales dramatically better than previously available tools.

cs.LO↗

arXiv2021

From Verification to Causality-based Explications

In view of the growing complexity of modern software architectures, formal models are increasingly used to understand why a system works the way it does, opposed to simply verifying that it behaves as intended. This paper surveys approaches to formally explicate the observable behavior of reactive systems. We describe how Halpern and Pearl's notion of actual causation inspired verification-oriented studies of cause-effect relationships in the evolution of a system. A second focus lies on applications of the Shapley value to responsibility ascriptions, aimed to measure the influence of an event on an observable effect. Finally, formal approaches to probabilistic causation are collected and connected, and their relevance to the understanding of probabilistic systems is discussed.

cs.LO↗

arXiv2021

A Game-Theoretic Account of Responsibility Allocation

When designing or analyzing multi-agent systems, a fundamental problem is responsibility ascription: to specify which agents are responsible for the joint outcome of their behaviors and to which extent. We model strategic multi-agent interaction as an extensive form game of imperfect information and define notions of forward (prospective) and backward (retrospective) responsibility. Forward responsibility identifies the responsibility of a group of agents for an outcome along all possible plays, whereas backward responsibility identifies the responsibility along a given play. We further distinguish between strategic and causal backward responsibility, where the former captures the epistemic knowledge of players along a play, while the latter formalizes which players -- possibly unknowingly -- caused the outcome. A formal connection between forward and backward notions is established in the case of perfect recall. We further ascribe quantitative responsibility through cooperative game theory. We show through a number of examples that our approach encompasses several prior formal accounts of responsibility attribution.

cs.GT↗

arXiv2021

Probabilistic causes in Markov chains

The paper studies a probabilistic notion of causes in Markov chains that relies on the counterfactuality principle and the probability-raising property. This notion is motivated by the use of causes for monitoring purposes where the aim is to detect faulty or undesired behaviours before they actually occur. A cause is a set of finite executions of the system after which the probability of the effect exceeds a given threshold. We introduce multiple types of costs that capture the consumption of resources from different perspectives, and study the complexity of computing cost-minimal causes.

cs.LO↗

arXiv2021

The Orbit Problem for Parametric Linear Dynamical Systems

We study a parametric version of the Kannan-Lipton Orbit Problem for linear dynamical systems. We show decidability in the case of one parameter and Skolem-hardness with two or more parameters. More precisely, consider a $d$-dimensional square matrix $M$ whose entries are algebraic functions in one or more real variables. Given initial and target vectors $u,v\in \mathbb{Q}^d$, the parametric point-to-point orbit problem asks whether there exist values of the parameters giving rise to a concrete matrix $N \in \mathbb{R}^{d\times d}$, and a positive integer $n\in \mathbb{N}$, such that $N^nu = v$. We show decidability for the case in which $M$ depends only upon a single parameter, and we exhibit a reduction from the well-known Skolem Problem for linear recurrence sequences, suggesting intractability in the case of two or more parameters.

cs.LO↗

arXiv2021

Responsibility and verification: Importance value in temporal logics

We aim at measuring the influence of the nondeterministic choices of a part of a system on its ability to satisfy a specification. For this purpose, we apply the concept of Shapley values to verification as a means to evaluate how important a part of a system is. The importance of a component is measured by giving its control to an adversary, alone or along with other components, and testing whether the system can still fulfill the specification. We study this idea in the framework of model-checking with various classical types of linear-time specification, and propose several ways to transpose it to branching ones. We also provide tight complexity bounds in almost every case.

cs.LO↗

arXiv2020

Reachability in Dynamical Systems with Rounding

We consider reachability in dynamical systems with discrete linear updates, but with fixed digital precision, i.e., such that values of the system are rounded at each step. Given a matrix $M \in \mathbb{Q}^{d \times d}$, an initial vector $x\in\mathbb{Q}^{d}$, a granularity $g\in \mathbb{Q}_+$ and a rounding operation $[\cdot]$ projecting a vector of $\mathbb{Q}^{d}$ onto another vector whose every entry is a multiple of $g$, we are interested in the behaviour of the orbit $\mathcal{O}={<}[x], [M[x]],[M[M[x]]],\dots{>}$, i.e., the trajectory of a linear dynamical system in which the state is rounded after each step. For arbitrary rounding functions with bounded effect, we show that the complexity of deciding point-to-point reachability---whether a given target $y \in\mathbb{Q}^{d}$ belongs to $\mathcal{O}$---is PSPACE-complete for hyperbolic systems (when no eigenvalue of $M$ has modulus one). We also establish decidability without any restrictions on eigenvalues for several natural classes of rounding functions.

cs.CC↗

arXiv2020

SWITSS: Computing Small Witnessing Subsystems

Witnessing subsystems for probabilistic reachability thresholds in discrete Markovian models are an important concept both as diagnostic information on why a property holds, and as input to refinement algorithms. We present SWITSS, a tool for the computation of Small WITnessing SubSystems. SWITSS implements exact and heuristic approaches based on reducing the problem to (mixed integer) linear programming. Returned subsystems can automatically be rendered graphically and are accompanied with a certificate which proves that the subsystem is indeed a witness.

cs.LO↗

arXiv2020

Minimal witnesses for probabilistic timed automata

Witnessing subsystems have proven to be a useful concept in the analysis of probabilistic systems, for example as diagnostic information on why a given property holds or as input to refinement algorithms. This paper introduces witnessing subsystems for reachability problems in probabilistic timed automata (PTA). Using a new operation on difference bounds matrices, it is shown how Farkas certificates of finite-state bisimulation quotients of a PTA can be translated into witnessing subsystems. We present algorithms for the computation of minimal witnessing subsystems under three notions of minimality, which capture the timed behavior from different perspectives, and discuss their complexity.

cs.LO↗

arXiv2019

Farkas certificates and minimal witnesses for probabilistic reachability constraints

This paper introduces Farkas certificates for lower and upper bounds on minimal and maximal reachability probabilities in Markov decision processes (MDP), which we derive using an MDP-variant of Farkas' Lemma. The set of all such certificates is shown to form a polytope whose points correspond to witnessing subsystems of the model and the property. Using this correspondence we can translate the problem of finding minimal witnesses to the problem of finding vertices with a maximal number of zeros. While computing such vertices is computationally hard in general, we derive new heuristics from our formulations that exhibit competitive performance compared to state-of-the-art techniques. As an argument that asymptotically better algorithms cannot be hoped for, we show that the decision version of finding minimal witnesses is NP-complete even for acyclic Markov chains.

cs.LO↗

arXiv2017

The $L^2$-torsion polytope of amenable groups

We introduce the notion of groups of polytope class and show that torsion-free amenable groups satisfying the Atiyah Conjecture possess this property. A direct consequence is the homotopy invariance of the $L^2$-torsion polytope among $G$-CW-complexes for these groups. As another application we prove that the $L^2$-torsion polytope of an amenable group vanishes provided that it contains a non-abelian elementary amenable normal subgroup.

math.GR↗

arXiv2016

Alexander and Thurston norms, and the Bieri-Neumann-Strebel invariants for free-by-cyclic groups

We investigate Friedl-L\"uck's universal $L^2$-torsion for descending HNN extensions of finitely generated free groups, and so in particular for $F_n$-by-$\mathbb{Z}$ groups. This invariant induces a semi-norm on the first cohomology of the group which is an analogue of the Thurston norm for $3$-manifold groups. We prove that this Thurston semi-norm is an upper bound for the Alexander semi-norm defined by McMullen, as well as for the higher Alexander semi-norms defined by Harvey. The same inequalities are known to hold for $3$-manifold groups. We also prove that the Newton polytopes of the universal $L^2$-torsion of a descending HNN extension of $F_2$ locally determine the Bieri-Neumann-Strebel invariant of the group. We give an explicit means of computing the BNS invariant for such groups. As a corollary, we prove that the Bieri-Neumann-Strebel invariant of a descending HNN extension of $F_2$ has finitely many connected components. When the HNN extension is taken over $F_n$ along a polynomially growing automorphism with unipotent image in $GL(n, \mathbb{Z})$, we show that the Newton polytope of the universal $L^2$-torsion and the BNS invariant completely determine one another. We also show that in this case the Alexander norm, its higher incarnations, and the Thurston norm all coincide.

math.GR↗

arXiv2016

The integral polytope group

We show that the Grothendieck group associated to integral polytopes in $\mathbb{R}^n$ is free-abelian by providing an explicit basis. Moreover, we identify the involution on this polytope group given by reflection about the origin as a sum of Euler characteristic type. We also compute the kernel of the norm map sending a polytope to its induced seminorm on the dual of $\mathbb{R}^n$.

math.MG↗

arXiv2015

The Grothendieck group of polytopes and norms

Polytopes in R^n with integral vertices form a monoid under the Minkowski sum, and the Grothendieck construction gives rise to a group. We show that every symmetric polytope is a norm in this group for every n.

math.GT↗

arXiv2012

Compacting Transactional Data in Hybrid OLTP & OLAP Databases

Growing main memory sizes have facilitated database management systems that keep the entire database in main memory. The drastic performance improvements that came along with these in-memory systems have made it possible to reunite the two areas of online transaction processing (OLTP) and online analytical processing (OLAP): An emerging class of hybrid OLTP and OLAP database systems allows to process analytical queries directly on the transactional data. By offering arbitrarily current snapshots of the transactional data for OLAP, these systems enable real-time business intelligence. Despite memory sizes of several Terabytes in a single commodity server, RAM is still a precious resource: Since free memory can be used for intermediate results in query processing, the amount of memory determines query performance to a large extent. Consequently, we propose the compaction of memory-resident databases. Compaction consists of two tasks: First, separating the mutable working set from the immutable "frozen" data. Second, compressing the immutable data and optimizing it for efficient, memory-consumption-friendly snapshotting. Our approach reorganizes and compresses transactional data online and yet hardly affects the mission-critical OLTP throughput. This is achieved by unburdening the OLTP threads from all additional processing and performing these tasks asynchronously.

cs.DB↗