arXiv subjects

Tobias Winkler

Publications and source records attributed to Tobias Winkler.

At least 19 recordsLinked to original sources

arXiv2026

Three non-Hermitian random matrix universality classes of complex edge statistics: Spacing ratios and distributions

The conjectured three generic local bulk statistics amongst all non-Hermitian random matrix symmetry classes have recently been extended to three generic local edge statistics. We study analytically and numerically complex spacing ratios and nearest-neighbour (NN) spacing distributions that characterise such local statistics. We choose the three simplest representatives of these universality classes, given by the Gaussian ensembles of complex Ginibre, complex symmetric and complex self-dual matrices, denoted by class A, AI$^†$ and AII$^†$. In the first part, we analytically study the complex spacing ratio in class A, at finite matrix size $N$. Introducing a conditional point process, we simplify existing expressions and show why an uncontrolled approximation introduced earlier converges well in the large-$N$ limit in the bulk. When specifying to the elliptic Ginibre ensemble, we present a parameter-dependent $N=3$ surmise for the complex spacing ratio, interpolating to that of the Gaussian unitary ensemble (GUE), where such a surmise is very accurate. In the second numerical part, we compare complex spacing ratios, its moments, and NN spacing distributions for all three ensembles with that of uncorrelated points, the two-dimensional (2D) Poisson process, both in the bulk and at the edge. The varying degree of repulsion within these different edge universality classes can be well understood in terms of an effective 2D Coulomb gas description, at different values of inverse temperature $β$. We find indications that the complex spacing ratio does not fully unfold the local statistics at the edge. Finally we verify that for small argument, in all three symmetry classes the NN spacing distributions in the bulk and at the edge are consistent with the universal cubic repulsion.

math-ph↗

arXiv2026

Tractable Hyperproperties for MDPs

Probabilistic hyperproperties describe probabilistic relations between multiple sets of executions in a stochastic system. Prominent examples include information-theoretic characterizations of security and privacy policies. However, model checking for existing probabilistic hyperlogics, such as HyperPCTL and PHL, is undecidable in Markov decision processes (MDPs). In this paper, we study an underexplored problem: the verification of fragments of probabilistic hyperproperties that relate the probabilities of different events to each other, possibly across independent executions of an MDP. Representative verification questions include: Can two different target states be reached from the same initial state with the same probability? (different events), Can a given target state be reached from two different initial states with the same probability? (same event, independent executions), and natural combinations of these forms. Besides reachability, our relational probabilistic properties cover safety, Büchi, and coBüchi objectives. They can also be combined conjunctively, thereby generalizing standard multi-objective MDP properties. We provide efficient algorithms for relevant classes of relational properties, while proving computational hardness and completeness results for others. An implementation of our approach outperforms solvers for more general probabilistic hyperlogics by orders of magnitude on the subset of their benchmarks that lies within our fragment.

cs.LO↗

arXiv2026

Verifying Sampling Algorithms via Distributional Invariants

This paper presents a Hoare-like veri cation framework for discrete probabilistic programs that we apply to two non-trivial sampling algorithms: Lumbroso's Fast Dice Roller and Saad et al.'s Fast Loaded Dice Roller. These algorithms have previously resisted formal veri cation due to their probabilistic nature, intricate loop structure, and parametric input. Our approach complements existing proof rules based on inductive distributional invariants, enabling us to verify both total and partial correctness of the two algorithms.

cs.LO↗

arXiv2026

Generating Functions Meet Occupation Measures: Invariant Synthesis for Probabilistic Loops (Extended Version)

A fundamental computational task in probabilistic programming is to infer a program's output (posterior) distribution from a given initial (prior) distribution. This problem is challenging, especially for expressive languages that feature loops or unbounded recursion. While most of the existing literature focuses on statistical approximation, in this paper we address the problem of mathematically exact inference. To achieve this for programs with loops, we rely on a relatively underexplored type of probabilistic loop invariant, which is linked to a loop's so-called occupation measure. The occupation measure associates program states with their expected number of visits, given the initial distribution. Based on this, we derive the notion of an occupation invariant. Such invariants are essentially dual to probabilistic martingales, the predominant technique for formal probabilistic loop analysis in the literature. A key feature of occupation invariants is that they can take the initial distribution into account and often yield a proof of positive almost sure termination as a by-product. Finally, we present an automatic, template-based invariant synthesis approach for occupation invariants by encoding them as generating functions. The approach is implemented and evaluated on a set of benchmarks.

cs.PL↗

arXiv2025

Probabilistic Programming Meets Automata Theory: Exact Inference using Weighted Automata

Probabilistic programs encode stochastic models as ordinary-looking programs with primitives for sampling numbers from predefined distributions and conditioning. Their applications include, among many others, machine learning and modeling of autonomous systems. The analysis of probabilistic programs is often quantitative - it involves reasoning about numerical properties like probabilities and expectations. A particularly important quantitative property of probabilistic programs is their posterior distribution, i.e., the distribution over possible outputs for a given input (or prior) distribution. Computing the posterior distribution exactly is known as exact inference. We present our current research using weighted automata, a generalization of the well-known finite automata, for performing exact inference in a restricted class of discrete probabilistic programs. This is achieved by encoding distributions over program variables - possibly with infinite support - as certain weighted automata. The semantics of our programming language then corresponds to common automata-theoretic constructions, such as product, concatenation, and others.

cs.FL↗

arXiv2025

Weighted Automata for Exact Inference in Discrete Probabilistic Programs

In probabilistic programming, the inference problem asks to determine a program's posterior distribution conditioned on its "observe" instructions. Inference is challenging, especially when exact rather than approximate results are required. Inspired by recent work on probability generating functions (PGFs), we propose encoding distributions on $\mathbb{N}^k$ as weighted automata over a commutative alphabet with $k$ symbols. Based on this, we map the semantics of various imperative programming statements to automata-theoretic constructions. For a rich class of programs, this results in an effective translation from prior to posterior distribution, both encoded as automata. We prove that our approach is sound with respect to a standard operational program semantics.

cs.FL↗

arXiv2025

Efficient Probabilistic Model Checking for Relational Reachability (Extended Version)

Markov decision processes model systems subject to nondeterministic and probabilistic uncertainty. A plethora of verification techniques addresses variations of reachability properties, such as: Is there a scheduler resolving the nondeterminism such that the probability to reach an error state is above a threshold? We consider an understudied extension that relates different reachability probabilities, such as: Is there a scheduler such that two sets of states are reached with different probabilities? These questions appear naturally in the design of randomized algorithms and in various security applications. We provide a tractable algorithm for many variations of this problem, while proving computational hardness of some others. An implementation of our algorithm beats solvers for more general probabilistic hyperlogics by orders of magnitude, on the subset of their benchmarks that are within our fragment.

cs.LO↗

arXiv2025

Foundations for Deductive Verification of Continuous Probabilistic Programs: From Lebesgue to Riemann and Back

We lay out novel foundations for the computer-aided verification of guaranteed bounds on expected outcomes of imperative probabilistic programs featuring (i) general loops, (ii) continuous distributions, and (iii) conditioning. To handle loops we rely on user-provided quantitative invariants, as is well established. However, in the realm of continuous distributions, invariant verification becomes extremely challenging due to the presence of integrals in expectation-based program semantics. Our key idea is to soundly under- or over-approximate these integrals via Riemann sums. We show that this approach enables the SMT-based invariant verification for programs with a fairly general control flow structure. On the theoretical side, we prove convergence of our Riemann approximations, and establish coRE-completeness of the central verification problems. On the practical side, we show that our approach enables to use existing automated verifiers targeting discrete probabilistic programs for the verification of programs involving continuous sampling. Towards this end, we implement our approach in the recent quantitative verification infrastructure Caesar by encoding Riemann sums in its intermediate verification language. We present several promising case studies.

cs.LO↗

arXiv2025

Fixed Point Certificates for Reachability and Expected Rewards in MDPs

The possibility of errors in human-engineered formal verification software, such as model checkers, poses a serious threat to the purpose of these tools. An established approach to mitigate this problem are certificates -- lightweight, easy-to-check proofs of the verification results. In this paper, we develop novel certificates for model checking of Markov decision processes (MDPs) with quantitative reachability and expected reward properties. Our approach is conceptually simple and relies almost exclusively on elementary fixed point theory. Our certificates work for arbitrary finite MDPs and can be readily computed with little overhead using standard algorithms. We formalize the soundness of our certificates in Isabelle/HOL and provide a formally verified certificate checker. Moreover, we augment existing algorithms in the probabilistic model checker Storm with the ability to produce certificates and demonstrate practical applicability by conducting the first formal certification of the reference results in the Quantitative Verification Benchmark Set.

cs.LO↗

arXiv2024

J-P: MDP. FP. PP.: Characterizing Total Expected Rewards in Markov Decision Processes as Least Fixed Points with an Application to Operational Semantics of Probabilistic Programs (Technical Report)

Markov decision processes (MDPs) with rewards are a widespread and well-studied model for systems that make both probabilistic and nondeterministic choices. A fundamental result about MDPs is that their minimal and maximal expected rewards satisfy Bellmann's optimality equations. For various classes of MDPs - notably finite-state MDPs, positive bounded models, and negative models - expected rewards are known to be the least solution of those equations. However, these classes of MDPs are too restrictive for probabilistic program verification. In particular, they assume that all rewards are finite. This is already not the case for the expected runtime of a simple probabilisitic program modeling a 1-dimensional random walk. In this paper, we develop a generalized least fixed point characterization of expected rewards in MDPs without those restrictions. Furthermore, we demonstrate how said characterization can be leveraged to prove weakest-preexpectation-style calculi sound with respect to an operational MDP model.

cs.LO↗

arXiv2024

Markov Decision Processes with Sure Parity and Multiple Reachability Objectives

This paper considers the problem of finding strategies that satisfy a mixture of sure and threshold objectives in Markov decision processes. We focus on a single $ω$-regular objective expressed as parity that must be surely met while satisfying $n$ reachability objectives towards sink states with some probability thresholds too. We consider three variants of the problem: (a) strict and (b) non-strict thresholds on all reachability objectives, and (c) maximizing the thresholds with respect to a lexicographic order. We show that (a) and (c) can be reduced to solving parity games, and (b) can be solved in $\sf{EXPTIME}$. Strategy complexities as well as algorithms are provided for all cases.

cs.GT↗

arXiv2024

Accurately Computing Expected Visiting Times and Stationary Distributions in Markov Chains

We study the accurate and efficient computation of the expected number of times each state is visited in discrete- and continuous-time Markov chains. To obtain sound accuracy guarantees efficiently, we lift interval iteration and topological approaches known from the computation of reachability probabilities and expected rewards. We further study applications of expected visiting times, including the sound computation of the stationary distribution and expected rewards conditioned on reaching multiple goal states. The implementation of our methods in the probabilistic model checker Storm scales to large systems with millions of states. Our experiments on the quantitative verification benchmark set show that the computation of stationary distributions via expected visiting times consistently outperforms existing approaches - sometimes by several orders of magnitude.

cs.LO↗

arXiv2023

Model Checking Temporal Properties of Recursive Probabilistic Programs

Probabilistic pushdown automata (pPDA) are a standard operational model for programming languages involving discrete random choices and recursive procedures. Temporal properties are useful for specifying the chronological order of events during program execution. Existing approaches for model checking pPDA against temporal properties have focused mostly on $ω$-regular and LTL properties. In this paper, we give decidability and complexity results for the model checking problem of pPDA against $ω$-visibly pushdown languages that can be described by specification logics such as CaRet. These logical formulae allow specifying properties that explicitly take the structured computations arising from procedural programs into account. For example, CaRet is able to match procedure calls with their corresponding future returns, and thus allows to express fundamental program properties such as total and partial correctness.

cs.FL↗

arXiv2023

Programmatic Strategy Synthesis: Resolving Nondeterminism in Probabilistic Programs

We consider imperative programs that involve both randomization and pure nondeterminism. The central question is how to find a strategy resolving the pure nondeterminism such that the so-obtained determinized program satisfies a given quantitative specification, i.e., bounds on expected outcomes such as the expected final value of a program variable or the probability to terminate in a given set of states. We show how memoryless and deterministic (MD) strategies can be obtained in a semi-automatic fashion using deductive verification techniques. For loop-free programs, the MD strategies resulting from our weakest precondition-style framework are correct by construction. This extends to loopy programs, provided the loops are equipped with suitable loop invariants - just like in program verification. We show how our technique relates to the well-studied problem of obtaining strategies in countably infinite Markov decision processes with reachability-reward objectives. Finally, we apply our technique to several case studies.

cs.LO↗

arXiv2023

On Certificates, Expected Runtimes, and Termination in Probabilistic Pushdown Automata

Probabilistic pushdown automata (pPDA) are a natural operational model for a variety of recursive discrete stochastic processes. In this paper, we study certificates - succinct and easily verifiable proofs - for upper and lower bounds on various quantitative properties of a given pPDA. We reveal an intimate, yet surprisingly simple connection between the existence of such certificates and the expected time to termination of the pPDA at hand. This is established by showing that certain intrinsic properties, like the spectral radius of the Jacobian of the pPDA's underlying polynomial equation system, are directly related to expected runtimes. As a consequence, we obtain that there always exist easy-to-check proofs for positive almost-sure termination: does a pPDA terminate in finite expected time?

cs.FL↗

arXiv2023

Certificates for Probabilistic Pushdown Automata via Optimistic Value Iteration

Probabilistic pushdown automata (pPDA) are a standard model for discrete probabilistic programs with procedures and recursion. In pPDA, many quantitative properties are characterized as least fixpoints of polynomial equation systems. In this paper, we study the problem of certifying that these quantities lie within certain bounds. To this end, we first characterize the polynomial systems that admit easy-to-check certificates for validating bounds on their least fixpoint. Second, we present a sound and complete Optimistic Value Iteration algorithm for computing such certificates. Third, we show how certificates for polynomial systems can be transferred to certificates for various quantitative pPDA properties. Experiments demonstrate that our algorithm computes succinct certificates for several intricate example programs as well as stochastic context-free grammars with $> 10^4$ production rules.

cs.FL↗

arXiv2023

Exact Probabilistic Inference Using Generating Functions

Probabilistic programs are typically normal-looking programs describing posterior probability distributions. They intrinsically code up randomized algorithms and have long been at the heart of modern machine learning and approximate computing. We explore the theory of generating functions [19] and investigate its usage in the exact quantitative reasoning of probabilistic programs. Important topics include the exact representation of program semantics [13], proving exact program equivalence [5], and -- as our main focus in this extended abstract -- exact probabilistic inference. In probabilistic programming, inference aims to derive a program's posterior distribution. In contrast to approximate inference, inferring exact distributions comes with several benefits [8], e.g., no loss of precision, natural support for symbolic parameters, and efficiency on models with certain structures. Exact probabilistic inference, however, is a notoriously hard task [6,12,17,18]. The challenges mainly arise from three program constructs: (1) unbounded while-loops and/or recursion, (2) infinite-support distributions, and (3) conditioning (via posterior observations). We present our ongoing research in addressing these challenges (with a focus on conditioning) leveraging generating functions and show their potential in facilitating exact probabilistic inference for discrete probabilistic programs.

cs.PL↗

arXiv2022

Does a Program Yield the Right Distribution? Verifying Probabilistic Programs via Generating Functions

We study discrete probabilistic programs with potentially unbounded looping behaviors over an infinite state space. We present, to the best of our knowledge, the first decidability result for the problem of determining whether such a program generates exactly a specified distribution over its outputs (provided the program terminates almost surely). The class of distributions that can be specified in our formalism consists of standard distributions (geometric, uniform, etc.) and finite convolutions thereof. Our method relies on representing these (possibly infinite-support) distributions as probability generating functions which admit effective arithmetic operations. We have automated our techniques in a tool called prodigy, which supports automatic invariance checking, compositional reasoning of nested loops, and efficient queries on various quantities of to the output distribution, as demonstrated by experiments.

cs.LO↗