The synthesis problem asks for the automatic construction of a system from its specification. In the traditional setting, the system is "constructed from scratch" rather than composed from reusable components. However, this is rare in practice, and almost every non-trivial software system relies heavily on the use of libraries of reusable components. Recently, Lustig and Vardi introduced dataflow and controlflow synthesis from libraries of reusable components. They proved that dataflow synthesis is undecidable, while controlflow synthesis is decidable. The problem of controlflow synthesis from libraries of probabilistic components was considered by Nain, Lustig and Vardi, and was shown to be decidable for qualitative analysis (that asks that the specification be satisfied with probability 1). Our main contributions for controlflow synthesis from probabilistic components are to establish better complexity bounds for the qualitative analysis problem, and to show that the more general quantitative problem is undecidable. For the qualitative analysis, we show that the problem (i) is EXPTIME-complete when the specification is given as a deterministic parity word automaton, improving the previously known 2EXPTIME upper bound; and (ii) belongs to UP $\cap$ coUP and is parity-games hard, when the specification is given directly as a parity condition on the components, improving the previously known EXPTIME upper bound.
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.
Synthesis is the automatic construction of a system from its specification. In classical synthesis algorithms, it is always assumed that the system is "constructed from scratch" rather than composed from reusable components. This, of course, rarely happens in real life, where almost every non-trivial commercial software system relies heavily on using libraries of reusable components. Furthermore, other contexts, such as web-service orchestration, can be modeled as synthesis of a system from a library of components. Recently, Lustig and Vardi introduced dataflow and control-flow synthesis from …
We present Tempest, a synthesis tool to automatically create correct-by-construction reactive systems and shields from qualitative or quantitative specifications in probabilistic environments. A shield is a special type of reactive system used for run-time enforcement; i.e., a shield enforces a given qualitative or quantitative specification of a running system while interfering with its operation as little as possible. Shields that enforce a qualitative or quantitative specification are called safety-shields or optimal-shields, respectively. Safety-shields can be implemented as pre-shields or…
We show that probabilistic computable functions, i.e., those functions outputting distributions and computed by probabilistic Turing machines, can be characterized by a natural generalization of Church and Kleene's partial recursive functions. The obtained algebra, following Leivant, can be restricted so as to capture the notion of polytime sampleable distributions, a key concept in average-case complexity and cryptography.
Probabilistic justification logic is a modal logic with two kind of modalities: probability measures and explicit justification terms. We present a tableau procedure that can be used to decide the satisfiability problem for this logic in polynomial space. We show that this upper complexity bound is tight.
We propose a simple technique for verifying probabilistic models whose transition probabilities are parametric. The key is to replace parametric transitions by nondeterministic choices of extremal values. Analysing the resulting parameter-free model using off-the-shelf means yields (refinable) lower and upper bounds on probabilities of regions in the parameter space. The technique outperforms the existing analysis of parametric Markov chains by several orders of magnitude regarding both run-time and scalability. Its beauty is its applicability to various probabilistic models. It in particular …
Boston University（波士顿大学）；Singapore-MIT Alliance for Research and Technology（新加坡-麻省理工学院研究与技术联盟）
In this paper, we present a method for optimal control synthesis of a plant that interacts with a set of agents in a graph-like environment. The control specification is given as a temporal logic statement about some properties that hold at the vertices of the environment. The plant is assumed to be deterministic, while the agents are probabilistic Markov models. The goal is to control the plant such that the probability of satisfying a syntactically co-safe Linear Temporal Logic formula is maximized. We propose a computationally efficient incremental approach based on the fact that temporal l…
Brno University of Technology（布尔诺理工大学）；University of California, Berkeley（加州大学伯克利分校）；RWTH Aachen University（亚琛工业大学）
This paper presents a novel method for the automated synthesis of probabilistic programs. The starting point is a program sketch representing a finite family of finite-state Markov chains with related but distinct topologies, and a PCTL specification. The method builds on a novel inductive oracle that greedily generates counter-examples (CEs) for violating programs and uses them to prune the family. These CEs leverage the semantics of the family in the form of bounds on its best- and worst-case behaviour provided by a deductive oracle using an MDP abstraction. The method further monitors the p…
The model-checking problem for probabilistic systems crucially relies on the translation of LTL to deterministic Rabin automata (DRW). Our recent Safraless translation for the LTL(F,G) fragment produces smaller automata as compared to the traditional approach. In this work, instead of DRW we consider deterministic automata with acceptance condition given as disjunction of generalized Rabin pairs (DGRW). The Safraless translation of LTL(F,G) formulas to DGRW results in smaller automata as compared to DRW. We present algorithms for probabilistic model-checking as well as game solving for DGRW co…
The logic PJ is a probabilistic logic defined by adding (non-iterated) probability operators to the basic justification logic J. In this paper we establish upper and lower bounds for the complexity of the derivability problem in the logic PJ. The main result of the paper is that the complex- ity of the derivability problem in PJ remains the same as the complexity of the derivability problem in the underlying logic J, which is Πp2-complete. This implies hat the probability operators do not increase the complex- ity of the logic, although they arguably enrich the expressiveness of the language.
Design and control of autonomous systems that operate in uncertain or adversarial environments can be facilitated by formal modelling and analysis. Probabilistic model checking is a technique to automatically verify, for a given temporal logic specification, that a system model satisfies the specification, as well as to synthesise an optimal strategy for its control. This method has recently been extended to multi-agent systems that exhibit competitive or cooperative behaviour modelled via stochastic games and synthesis of equilibria strategies. In this paper, we provide an overview of probabi…
Rice University（莱斯大学）；University of Colorado, Boulder（科罗拉多大学博尔德分校）
Many systems are naturally modeled as Markov Decision Processes (MDPs), combining probabilities and strategic actions. Given a model of a system as an MDP and some logical specification of system behavior, the goal of synthesis is to find a policy that maximizes the probability of achieving this behavior. A popular choice for defining behaviors is Linear Temporal Logic (LTL). Policy synthesis on MDPs for properties specified in LTL has been well studied. LTL, however, is defined over infinite traces, while many properties of interest are inherently finite. Linear Temporal Logic over finite tra…
We model randomized complexity classes in the style of Implicit Computational Complexity. We introduce PSTA, a probabilistic version of STA, the type-theoretical counterpart of Soft Linear Logic. PSTA is a type assignment for an extension of Simpson's Linear Lambda Calculus and its surface reduction, where Linear additives express random choice. Linear additives are weaker than the usual ones; they allow for duplications harmlessly affecting the computational cost of normalization. PSTA is sound and complete w.r.t. probabilistic polynomial time functions and characterizes the probabilistic com…
We are concerned with the average case runtime complexity analysis of a prototypical imperative language endowed with primitives for sampling and probabilistic choice. Taking inspiration from known approaches from to the modular resource analysis of non-probabilistic programs, we investigate how a modular runtime analysis is obtained for probabilistic programs.
We present iSAPP (Imperative Static Analyser for Probabilistic Polynomial Time), a complexity verifier tool that is sound and extensionally complete for the Probabilistic Polynomial Time (PP) complexity class. iSAPP works on an imperative programming language for stack machines. The certificate of polynomiality can be built in polytime, with respect to the number of stacks used.
Probabilistic team semantics is a framework for logical analysis of probabilistic dependencies. Our focus is on the axiomatizability, complexity, and expressivity of probabilistic inclusion logic and its extensions. We identify a natural fragment of existential second-order logic with additive real arithmetic that captures exactly the expressivity of probabilistic inclusion logic. We furthermore relate these formalisms to linear programming, and doing so obtain PTIME data complexity for the logics. Moreover, on finite structures, we show that the full existential second-order logic with additi…
We study probabilistic complexity classes and questions of derandomisation from a logical point of view. For each logic L we introduce a new logic BPL, bounded error probabilistic L, which is defined from L in a similar way as the complexity class BPP, bounded error probabilistic polynomial time, is defined from PTIME. Our main focus lies on questions of derandomisation, and we prove that there is a query which is definable in BPFO, the probabilistic version of first-order logic, but not in Cinf, finite variable infinitary logic with counting. This implies that many of the standard logics of f…
Given a Markov decision process (MDP) $M$ and a formula $Φ$, the strategy synthesis problem asks if there exists a strategy $σ$ s.t. the resulting Markov chain $M[σ]$ satisfies $Φ$. This problem is known to be undecidable for the probabilistic temporal logic PCTL. We study a class of formulae that can be seen as a fragment of PCTL where a local, bounded horizon property is enforced all along an execution. Moreover, we allow for linear expressions in the probabilistic inequalities. This logic is at the frontier of decidability, depending on the type of strategies considered. In particular, stra…
Permissive Controller Synthesis for Probabilistic Systems
Faculty of Informatics, Masaryk University（马萨里克大学信息学院）；EECS, Queen Mary, University of London（伦敦大学玛丽女王学院电气电子与计算机科学与工程系）；Department of Computer Science, University of Oxford（牛津大学计算机科学系）；School of Computer Science, University of Birmingham（伯明翰大学计算机科学学院）
Let L be some extension of classical propositional logic. The non-iterated probabilistic logic over L, is the logic PL that is defined by adding non-nested probabilistic operators in the language of L. For example in PL we can express a statement like "the probability of truthfulness of A is at 0.3" where A is a formula of L. The iterated probabilistic logic over L is the logic PPL, where the probabilistic operators may be iterated (nested). For example, in PPL we can express a statement like "this coin is counterfeit with probability 0.6". In this paper we investigate the influence of probabi…
In a recent paper, the author has shown how Interaction Graphs models for linear logic can be used to obtain implicit characterisations of non-deterministic complexity classes. In this paper, we show how this semantic approach to Implicit Complexity Theory (ICC) can be used to characterise deterministic and probabilistic models of computation. In doing so, we obtain correspondences between group actions and both deterministic and probabilistic hierarchies of complexity classes. As a particular case, we provide the first implicit characterisations of the classes PLogspace (un-bounded error prob…
In this paper, we investigate the rational synthesis problem for concurrent game structure for a variety of objectives ranging from reachability to Muller condition. We propose a new algorithm that establishes the decidability of the non cooperative rational synthesis problem that relies solely on game theoretic technique as opposed to previous approaches that are logic based. Thanks to this approach, we construct a zero-sum turn-based game that can be adapted to each one of the afore mentioned objectives thus obtain new complexity results. In particular, we show that reachability, safety, Büc…
University of Oxford（牛津大学）；University of Pennsylvania（宾夕法尼亚大学）
Reinforcement Learning (RL) has emerged as an efficient method of choice for solving complex sequential decision making problems in automatic control, computer science, economics, and biology. In this paper we present a model-free RL algorithm to synthesize control policies that maximize the probability of satisfying high-level control objectives given as Linear Temporal Logic (LTL) formulas. Uncertainty is considered in the workspace properties, the structure of the workspace, and the agent actions, giving rise to a Probabilistically-Labeled Markov Decision Process (PL-MDP) with unknown graph…
Epistemic protocol specifications allow programs, for settings in which multiple agents act with incomplete information, to be described in terms of how actions are related to what the agents know. They are a variant of the knowledge-based programs of Fagin et al [Distributed Computing, 1997], motivated by the complexity of synthesizing implementations in that framework. The paper proposes an approach to the synthesis of implementations of epistemic protocol specifications, that reduces the problem of finding an implementation to a sequence of model checking problems in approximations of the u…
We consider the synthesis of control policies for probabilistic systems, modeled by Markov decision processes, operating in partially known environments with temporal logic specifications. The environment is modeled by a set of Markov chains. Each Markov chain describes the behavior of the environment in each mode. The mode of the environment, however, is not known to the system. Two control objectives are considered: maximizing the expected probability and maximizing the worst-case probability that the system satisfies a given specification.
We introduce a novel variant of BSS machines called Separate Branching BSS machines (S-BSS in short) and develop a Fagin-type logical characterisation for languages decidable in non-deterministic polynomial time by S-BSS machines. We show that NP on S-BSS machines is strictly included in NP on BSS machines and that every NP language on S-BSS machines is a countable union of closed sets in the usual topology of R^n. Moreover, we establish that on Boolean inputs NP on S-BSS machines without real constants characterises a natural fragment of the complexity class existsR (a class of problems polyn…
