arXiv subjects

Joost-Pieter Katoen

Publications and source records attributed to Joost-Pieter Katoen.

At least 37 records · Page 2Linked to original sources

arXiv2025

A Direct Reduction from Stochastic Parity Games to Simple Stochastic Games

Significant progress has been recently achieved in developing efficient solutions for simple stochastic games (SSGs), focusing on reachability objectives. While reductions from stochastic parity games (SPGs) to SSGs have been presented in the literature through the use of multiple intermediate game models, a direct and simple reduction has been notably absent. This paper introduces a novel and direct polynomial-time reduction from quantitative SPGs to quantitative SSGs. By leveraging a gadget-based transformation that effectively removes the priority function, we construct an SSG that simulates the behavior of a given SPG. We formally establish the correctness of our direct reduction. Furthermore, we demonstrate that under binary encoding this reduction is polynomial, thereby directly corroborating the known $\textbf{NP}\,\mathbf{\cap}\,\textbf{coNP}$ complexity of SPGs and providing new understanding in the relationship between parity and reachability objectives in turn-based stochastic games.

cs.GT↗

arXiv2025

Approximate Probabilistic Bisimulation for Continuous-Time Markov Chains

We introduce $(\varepsilon, δ)$-bisimulation, a novel type of approximate probabilistic bisimulation for continuous-time Markov chains. In contrast to related notions, $(\varepsilon, δ)$-bisimulation allows the use of different tolerances for the transition probabilities ($\varepsilon$, additive) and total exit rates ($δ$, multiplicative) of states. Fundamental properties of the notion, as well as bounds on the absolute difference of time- and reward-bounded reachability probabilities for $(\varepsilon,δ)$-bisimilar states, are established.

cs.LO↗

arXiv2025

Weighted Rewriting: Semiring Semantics for Abstract Reduction Systems

We present novel semiring semantics for abstract reduction systems (ARSs). More precisely, we provide a weighted version of ARSs, where the reduction steps induce weights from a semiring. Inspired by provenance analysis in database theory and logic, we obtain a formalism that can be used for provenance analysis of arbitrary ARSs. Our semantics handle (possibly unbounded) non-determinism and possibly infinite reductions. Moreover, we develop several techniques to prove upper and lower bounds on the weights resulting from our semantics, and show that in this way one obtains a uniform approach to analyze several different properties like termination, derivational complexity, space complexity, safety, as well as combinations of these properties.

cs.LO↗

arXiv2025

Bayesian Inference in Quantum Programs

Conditioning is a key feature in probabilistic programming to enable modeling the influence of data (also known as observations) to the probability distribution described by such programs. Determining the posterior distribution is also known as Bayesian inference. This paper equips a quantum while-language with conditioning, defines its denotational and operational semantics over infinite-dimensional Hilbert spaces, and shows their equivalence. We provide sufficient conditions for the existence of weakest (liberal) precondition-transformers and derive inductive characterizations of these transformers. It is shown how w(l)p-transformers can be used to assess the effect of Bayesian inference on (possibly diverging) quantum programs.

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↗

arXiv2024

Towards Concurrent Quantitative Separation Logic

In this paper, we develop a novel verification technique to reason about programs featuring concurrency, pointers and randomization. While the integration of concurrency and pointers is well studied, little is known about the combination of all three paradigms. To close this gap, we combine two kinds of separation logic -- Quantitative Separation Logic and Concurrent Separation Logic -- into a new separation logic that enables reasoning about lower bounds of the probability to realise a postcondition by executing such a program.

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

A Spectrum of Approximate Probabilistic Bisimulations

This paper studies various notions of approximate probabilistic bisimulation on labeled Markov chains (LMCs). We introduce approximate versions of weak and branching bisimulation, as well as a notion of $\varepsilon$-perturbed bisimulation that relates LMCs that can be made (exactly) probabilistically bisimilar by small perturbations of their transition probabilities. We explore how the notions interrelate and establish their connections to other well-known notions like $\varepsilon$-bisimulation.

cs.LO↗

arXiv2024

Exact Bayesian Inference for Loopy Probabilistic Programs using Generating Functions

We present an exact Bayesian inference method for inferring posterior distributions encoded by probabilistic programs featuring possibly unbounded loops. Our method is built on a denotational semantics represented by probability generating functions, which resolves semantic intricacies induced by intertwining discrete probabilistic loops with conditioning (for encoding posterior observations). We implement our method in a tool called Prodigy; it augments existing computer algebra systems with the theory of generating functions for the (semi-)automatic inference and quantitative verification of conditioned probabilistic programs. Experimental results show that Prodigy can handle various infinite-state loopy programs and exhibits comparable performance to state-of-the-art exact inference tools over loop-free benchmarks.

cs.PL↗

arXiv2024

Unknown Biases and Timing Constraints in Timed Automata

Timed automata are the formal model for real-time systems. Extensions with discrete probabilistic branching have been considered in the literature and successfully applied. Probabilistic timed automata (PTA) do require all branching probabilities and clock constraints to be constants. This report investigates PTA in which this constraint is relaxed: both branching probabilities and clock constraints can be parametric. We formally define this PTA variant and define its semantics by an uncountable parametric Markov Decision Process (pMDP). We show that reachability probabilities in parametric L/U-PTA can be reduced to considering PTA with only parametric branching probabilities. This enables the usage of existing techniques from the literature. Finally, we generalize the symbolic backward and digital clock semantics of PTA to the setting with parametric probabilities and constraints.

cs.LO↗

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↗

arXiv2024

Verifying Concurrent Stacks by Divergence-Sensitive Bisimulation

The verification of linearizability -- a key correctness criterion for concurrent objects -- is based on trace refinement whose checking is PSPACE-complete. This paper suggests to use \emph{branching} bisimulation instead. Our approach is based on comparing an abstract specification in which object methods are executed atomically to a real object program. Exploiting divergence sensitivity, this also applies to progress properties such as lock-freedom. These results enable the use of \emph{polynomial-time} divergence-sensitive branching bisimulation checking techniques for verifying linearizability and progress. We conducted the experiment on concurrent lock-free stacks to validate the efficiency and effectiveness of our methods.

cs.PL↗

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

Parameter Synthesis for Markov Models: Covering the Parameter Space

Markov chain analysis is a key technique in formal verification. A practical obstacle is that all probabilities in Markov models need to be known. However, system quantities such as failure rates or packet loss ratios, etc. are often not -- or only partially -- known. This motivates considering parametric models with transitions labeled with functions over parameters. Whereas traditional Markov chain analysis relies on a single, fixed set of probabilities, analysing parametric Markov models focuses on synthesising parameter values that establish a given safety or performance specification $φ$. Examples are: what component failure rates ensure the probability of a system breakdown to be below 0.00000001?, or which failure rates maximise the performance, for instance the throughput, of the system? This paper presents various analysis algorithms for parametric discrete-time Markov chains and Markov decision processes. We focus on three problems: (a) do all parameter values within a given region satisfy $φ$?, (b) which regions satisfy $φ$ and which ones do not?, and (c) an approximate version of (b) focusing on covering a large fraction of all possible parameter values. We give a detailed account of the various algorithms, present a software tool realising these techniques, and report on an extensive experimental evaluation on benchmarks that span a wide range of applications.

cs.LO↗

arXiv2023

Search and Explore: Symbiotic Policy Synthesis in POMDPs

This paper marries two state-of-the-art controller synthesis methods for partially observable Markov decision processes (POMDPs), a prominent model in sequential decision making under uncertainty. A central issue is to find a POMDP controller - that solely decides based on the observations seen so far - to achieve a total expected reward objective. As finding optimal controllers is undecidable, we concentrate on synthesising good finite-state controllers (FSCs). We do so by tightly integrating two modern, orthogonal methods for POMDP controller synthesis: a belief-based and an inductive approach. The former method obtains an FSC from a finite fragment of the so-called belief MDP, an MDP that keeps track of the probabilities of equally observable POMDP states. The latter is an inductive search technique over a set of FSCs, e.g., controllers with a fixed memory size. The key result of this paper is a symbiotic anytime algorithm that tightly integrates both approaches such that each profits from the controllers constructed by the other. Experimental results indicate a substantial improvement in the value of the controllers while significantly reducing the synthesis time and memory footprint.

cs.LO↗

arXiv2023

Finding an $ε$-close Variation of Parameters in Bayesian Networks

This paper addresses the $ε$-close parameter tuning problem for Bayesian Networks (BNs): find a minimal $ε$-close amendment of probability entries in a given set of (rows in) conditional probability tables that make a given quantitative constraint on the BN valid. Based on the state-of-the-art "region verification" techniques for parametric Markov chains, we propose an algorithm whose capabilities go beyond any existing techniques. Our experiments show that $ε$-close tuning of large BN benchmarks with up to 8 parameters is feasible. In particular, by allowing (i) varied parameters in multiple CPTs and (ii) inter-CPT parameter dependencies, we treat subclasses of parametric BNs that have received scant attention so far.

cs.AI↗

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↗