Quantitative verification techniques have been developed for the formal analysis of a variety of probabilistic models, such as Markov chains, Markov decision process and their variants. They can be used to produce guarantees on quantitative aspects of system behaviour, for example safety, reliability and performance, or to help synthesise controllers that ensure such guarantees are met. We propose the model of turn-based probabilistic timed multi-player games, which incorporates probabilistic choice, real-time clocks and nondeterministic behaviour across multiple players. Building on the digital clocks approach for the simpler model of probabilistic timed automata, we show how to compute the key measures that underlie quantitative verification, namely the probability and expected cumulative price to reach a target. We illustrate this on case studies from computer security and task scheduling.
Essential tasks for the verification of probabilistic programs include bounding expected outcomes and proving termination in finite expected runtime. We contribute a simple yet effective inductive synthesis approach for proving such quantitative reachability properties by generating inductive invariants on source-code level. Our implementation shows promise: It finds invariants for (in)finite-state programs, can beat state-of-the-art probabilistic model checkers, and is competitive with modern tools dedicated to invariant synthesis and expected runtime reasoning.
Probabilistic game structures combine both nondeterminism and stochasticity, where players repeatedly take actions simultaneously to move to the next state of the concurrent game. Probabilistic alternating simulation is an important tool to compare the behaviour of different probabilistic game structures. In this paper, we present a sound and complete modal characterisation of this simulation relation by proposing a new logic based on probabilistic distributions. The logic enables a player to enforce a property in the next state or distribution. Its extension with fixpoints, which also charact…
Concurrent stochastic games (CSGs) are an ideal formalism for modelling probabilistic systems that feature multiple players or components with distinct objectives making concurrent, rational decisions. Examples include communication or security protocols and multi-robot navigation. Verification methods for CSGs exist but are limited to scenarios where agents or players are grouped into two coalitions, with those in the same coalition sharing an identical objective. In this paper, we propose multi-coalitional verification techniques for CSGs. We use subgame-perfect social welfare (or social cos…
Probabilistic model checking for stochastic games enables formal verification of systems that comprise competing or collaborating entities operating in a stochastic environment. Despite good progress in the area, existing approaches focus on zero-sum goals and cannot reason about scenarios where entities are endowed with different objectives. In this paper, we propose probabilistic model checking techniques for concurrent stochastic games based on Nash equilibria. We extend the temporal logic rPATL (probabilistic alternating-time temporal logic with rewards) to allow reasoning about players wi…
Automated verification techniques for stochastic games allow formal reasoning about systems that feature competitive or collaborative behaviour among rational agents in uncertain or probabilistic settings. Existing tools and techniques focus on turn-based games, where each state of the game is controlled by a single player, and on zero-sum properties, where two players or coalitions have directly opposing objectives. In this paper, we present automated verification techniques for concurrent stochastic games (CSGs), which provide a more natural model of concurrent decision making and interactio…
Probabilistic programming provides a convenient lingua franca for writing succinct and rigorous descriptions of probabilistic models and inference tasks. Several probabilistic programming languages, including Anglican, Church or Hakaru, derive their expressiveness from a powerful combination of continuous distributions, conditioning, and higher-order functions. Although very important for practical applications, these combined features raise fundamental challenges for program semantics and verification. Several recent works offer promising answers to these challenges, but their primary focus i…
There is a scalability gap between probabilistic and non-probabilistic verification. Probabilistic model checking tools are based either on explicit engines or on (Multi-Terminal) Binary Decision Diagrams. These structures are complemented in areas of non-probabilistic verification by more scalable techniques, such as IC3. We present a symbolic probabilistic model checking algorithm based on IC3-like incremental construction of inductive clauses to partition the state space, interleaved with incremental construction of a system of linear inequalities. This paper compares our implementation to …
Stochastic games are a convenient formalism for modelling systems that comprise rational agents competing or collaborating within uncertain environments. Probabilistic model checking techniques for this class of models allow us to formally specify quantitative specifications of either collective or individual behaviour and then automatically synthesise strategies for the agents under which these specifications are guaranteed to be satisfied. Although good progress has been made on algorithms and tool support, efficiency and scalability remain a challenge. In this paper, we investigate a symbol…
We extend the open games framework for compositional game theory to encompass also mixed strategies, making essential use of the discrete probability distribution monad. We show that the resulting games form a symmetric monoidal category, which can be used to compose probabilistic games in parallel and sequentially. We also consider morphisms between games, and show that intuitive constructions give rise to functors and adjunctions between pure and probabilistic open games.
We extended our simulation tool Ntccrt for probabilistic ntcc (pntcc) models. In addition, we developed a verification tool for pntcc models. Using this tool we can prove properties such as the system will go to a successful state with probability p under t discrete time- units. Currently, we are facing a few problems. We can only verify pntcc models using a finite domain constraint system and the encoding of cells ( mathematical entities that can update their value ) is experimental. In addition, in order to reduce the states generated during the verification process we need to implement a pr…
This volume contains the proceedings of the 12th International Symposium on Games, Automata, Logic and Formal Verification (GandALF 2021). The aim of GandALF 2021 symposium is to bring together researchers from academia and industry which are actively working in the fields of Games, Automata, Logics, and Formal Verification. The idea is to cover an ample spectrum of themes, ranging from theory to applications, and stimulate cross-fertilization.
This volume contains the proceedings of the 13th International Symposium on Games, Automata, Logic and Formal Verification (GandALF 2022). The aim of GandALF 2022 symposium is to bring together researchers from academia and industry which are actively working in the fields of Games, Automata, Logics, and Formal Verification. The idea is to cover an ample spectrum of themes, ranging from theory to applications, and stimulate cross-fertilization.
Many important properties of cyber-physical systems (CPS) are defined upon the relationship between multiple executions simultaneously in continuous time. Examples include probabilistic fairness and sensitivity to modeling errors (i.e., parameters changes) for real-valued signals. These requirements can only be specified by hyperproperties. In this work, we focus on verifying probabilistic hyperproperties for CPS. To cover a wide range of modeling formalisms, we first propose a general model of probabilistic uncertain systems (PUSs) that unify commonly studied CPS models such as continuous-tim…
This volume contains the proceedings of the Ninth International Symposium on Games, Automata, Logic and Formal Verification (GandALF 2018). The symposium took place in Saarbrücken, Germany, from the 26th to the 28th of September 2018. The GandALF symposium was established by a group of Italian computer scientists interested in mathematical logic, automata theory, game theory, and their applications to the specification, design, and verification of complex systems. Its aim is to provide a forum where people from different areas, and possibly with different backgrounds, can fruitfully interact. …
Proceedings Fourth International Symposium on Games, Automata, Logics and Formal Verification
本卷包含第四届国际游戏、自动机、逻辑和形式化验证研讨会（GandALF 2013）的会议记录。研讨会于2013年8月29日至31日在意大利Borca di Cadore举行。专题讨论会的会议记录载有三次应邀演讲的摘要和17篇论文，这些论文是经过仔细评估后被接受在会议上发表的。被接受的论文的主题范围很广，包括算法和行为博弈论，博弈语义学，形式语言和自动机理论，模态和时序逻辑，软件验证，混合系统。
We introduce a formal notion of masking fault-tolerance between probabilistic transition systems based on a variant of probabilistic bisimulation (named masking simulation). We also provide the corresponding probabilistic game characterization. Even though these games could be infinite, we propose a symbolic way of representing them, such that it can be decided in polynomial time if there is a masking simulation between two probabilistic transition systems. We use this notion of masking to quantify the level of masking fault-tolerance exhibited by almost-sure failing systems, i.e., those syste…
We study a syntax for specifying quantitative "assertions" - functions mapping program states to numbers - for probabilistic program verification. We prove that our syntax is expressive in the following sense: Given any probabilistic program $C$, if a function $f$ is expressible in our syntax, then the function mapping each initial state $σ$ to the expected value of $f$ evaluated in the final states reached after termination of $C$ on $σ$ (also called the weakest preexpectation $\textit{wp} [C](f)$) is also expressible in our syntax. As a consequence, we obtain a relatively complete verificati…
We present a new inductive rule for verifying lower bounds on expected values of random variables after execution of probabilistic loops as well as on their expected runtimes. Our rule is simple in the sense that loop body semantics need to be applied only finitely often in order to verify that the candidates are indeed lower bounds. In particular, it is not necessary to find the limit of a sequence as in many previous rules.
This volume contains the proceedings of the 11th International Symposium on Games, Automata, Logic and Formal Verification (GandALF 2020). The symposium took place as a fully online event on September 21-22, 2020. The GandALF symposium was established by a group of Italian computer scientists interested in mathematical logic, automata theory, game theory, and their applications to the specification, design, and verification of complex systems. Its aim is to provide a forum where people from different areas, and possibly with different backgrounds, can fruitfully interact. GandALF has a truly i…
This volume contains the proceedings of the Tenth International Symposium on Games, Automata, Logic and Formal Verification (GandALF 2019). The symposium took place in Bordeaux, France, from the 2nd to the 3rd of September 2010. The GandALF symposium was established by a group of Italian computer scientists interested in mathematical logic, automata theory, game theory, and their applications to the specification, design, and verification of complex systems. Its aim is to provide a forum where people from different areas, and possibly with different backgrounds, can fruitfully interact. GandAL…
In this work, we consider the almost-sure termination problem for probabilistic programs that asks whether a given probabilistic program terminates with probability 1. Scalable approaches for program analysis often rely on modularity as their theoretical basis. In non-probabilistic programs, the classical variant rule (V-rule) of Floyd-Hoare logic provides the foundation for modular analysis. Extension of this rule to almost-sure termination of probabilistic programs is quite tricky, and a probabilistic variant was proposed in [Fioriti and Hermanns 2015]. While the proposed probabilistic varia…
Monash University（莫纳什大学）；University of Oxford（牛津大学）；University of Kaiserslautern（凯泽斯劳滕大学）
Rational verification is the problem of determining which temporal logic properties will hold in a multi-agent system, under the assumption that agents in the system act rationally, by choosing strategies that collectively form a game-theoretic equilibrium. Previous work in this area has largely focussed on deterministic systems. In this paper, we develop the theory and algorithms for rational verification in probabilistic systems. We focus on concurrent stochastic games (CSGs), which can be used to model uncertainty and randomness in complex multi-agent environments. We study the rational ver…
Temporal logics for hyperproperties like HyperLTL use trace quantifiers to express properties that relate multiple system runs. In practice, the verification of such specifications is mostly limited to formulas without quantifier alternation, where verification can be reduced to checking a trace property over the self-composition of the system. Quantifier alternations like $\forall π. \exists π'. ϕ$, can either be solved by complementation or with an interpretation as a two-person game between a $\forall$-player, who incrementally constructs the trace $π$, and an $\exists$-player, who construc…
Monash University（莫纳什大学）；University of Kaiserslautern（凯泽斯劳滕大学）；Sapienza University of Rome（罗马大学）；University of Oxford（牛津大学）
In the context of multi-agent systems, the rational verification problem is concerned with checking which temporal logic properties will hold in a system when its constituent agents are assumed to behave rationally and strategically in pursuit of individual objectives. Typically, those objectives are expressed as temporal logic formulae which the relevant agent desires to see satisfied. Unfortunately, rational verification is computationally complex, and requires specialised techniques in order to obtain practically useable implementations. In this paper, we present such a technique. This tech…
