arXiv subjects

Simon Jantsch

Publications and source records attributed to Simon Jantsch.

15 recordsLinked to original sources

arXiv2025

Certificates and Witnesses for Multi-objective {\omega}-regular Queries in Markov Decision Processes

Multi-objective probabilistic model checking is a powerful technique for verifying stochastic systems against multiple (potentially conflicting) properties. To enhance the trustworthiness and explainability of model checking tools, we present independently checkable certificates and witnesses for multi-objective {\omega}-regular queries in Markov decision processes. For the certification, we extend and improve existing certificates for the decomposition of maximal end components and reachability properties. We then derive mixed-integer linear programs (MILPs) for finding minimal witnessing subsystems. For the special case of Markov chains and LTL properties, we use unambiguous B\"uchi automata to find witnesses, resulting in an algorithm that requires single-exponential space. Existing approaches based on deterministic automata require doubly-exponential space in the worst case. Finally, we consider the practical computation of our certificates and witnesses and provide an implementation of the developed techniques, along with an experimental evaluation, demonstrating the efficacy of our techniques.

cs.LO↗

arXiv2023

A Unifying Formal Approach to Importance Values in Boolean Functions

Boolean functions and their representation through logics, circuits, machine learning classifiers, or binary decision diagrams (BDDs) play a central role in the design and analysis of computing systems. Quantifying the relative impact of variables on the truth value by means of importance values can provide useful insights to steer system design and debugging. In this paper, we introduce a uniform framework for reasoning about such values, relying on a generic notion of importance value functions (IVFs). The class of IVFs is defined by axioms motivated from several notions of importance values introduced in the literature, including Ben-Or and Linial's influence and Chockler, Halpern, and Kupferman's notion of responsibility and blame. We establish a connection between IVFs and game-theoretic concepts such as Shapley and Banzhaf values, both of which measure the impact of players on outcomes in cooperative games. Exploiting BDD-based symbolic methods and projected model counting, we devise and evaluate practical computation schemes for IVFs.

cs.GT↗

arXiv2021

Witnessing Subsystems for Probabilistic Systems with Low Tree Width

A standard way of justifying that a certain probabilistic property holds in a system is to provide a witnessing subsystem (also called critical subsystem) for the property. Computing minimal witnessing subsystems is NP-hard already for acyclic Markov chains, but can be done in polynomial time for Markov chains whose underlying graph is a tree. This paper considers the problem for probabilistic systems that are similar to trees or paths. It introduces the parameters directed tree-partition width (dtpw) and directed path-partition width (dppw) and shows that computing minimal witnesses remains NP-hard for Markov chains with bounded dppw (and hence also for Markov chains with bounded dtpw). By observing that graphs of bounded dtpw have bounded width with respect to all known tree similarity measures for directed graphs, the hardness result carries over to these other tree similarity measures. Technically, the reduction proceeds via the conceptually simpler matrix-pair chain problem, which is introduced and shown to be NP-complete for nonnegative matrices of fixed dimension. Furthermore, an algorithm which aims to utilise a given directed tree partition of the system to compute a minimal witnessing subsystem is described. It enumerates partial subsystems for the blocks of the partition along the tree order, and keeps only necessary ones. A preliminary experimental analysis shows that it outperforms other approaches on certain benchmarks which have directed tree partitions of small width.

cs.LO↗

arXiv2021

Witnessing subsystems for probabilistic systems with low tree width

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

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

Determinization and Limit-determinization of Emerson-Lei automata

We study the problem of determinizing $ω$-automata whose acceptance condition is defined on the transitions using Boolean formulas, also known as transition-based Emerson-Lei automata (TELA). The standard approach to determinize TELA first constructs an equivalent generalized Büchi automaton (GBA), which is later determinized. We introduce three new ways of translating TELA to GBA. Furthermore, we give a new determinization construction which determinizes several GBA separately and combines them using a product construction. An experimental evaluation shows that the product approach is competitive when compared with state-of-the-art determinization procedures. We also study limit-determinization of TELA and show that this can be done with a single-exponential blow-up, in contrast to the known double-exponential lower-bound for determinization. Finally, one version of the limit-determinization procedure yields good-for-MDP automata which can be used for quantitative probabilistic model checking.

cs.FL↗

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

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↗

arXiv2020

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↗

arXiv2019

From LTL to Unambiguous Büchi Automata via Disambiguation of Alternating Automata

This paper proposes a new algorithm for the generation of unambiguous Büchi automata (UBA) from LTL formulas. Unlike existing tableau-based LTL-to-UBA translations, our algorithm deals with very weak alternating automata (VWAA) as an intermediate representation. It relies on a new notion of unambiguity for VWAA and a disambiguation procedure for VWAA. We introduce optimizations on the VWAA level and new LTL simplifications targeted at generating small UBA. We report on an implementation of the construction in our tool duggi and discuss experimental results that compare the automata sizes and computation times of duggi with the tableau-based LTL-to-UBA translation of the SPOT tool set. Our experiments also cover the analysis of Markov chains under LTL specifications, which is an important application of UBA.

cs.FL↗