The author introduced models of linear logic known as ''Interaction Graphs'' which generalise Girard's various geometry of interaction constructions. In this work, we establish how these models essentially rely on a deep connection between zeta functions and the execution of programs, expressed as a cocycle. This is first shown in the simple case of graphs, before begin lifted to dynamical systems. Focussing on probabilistic models, we then explain how the notion of graphings used in Interaction Graphs captures a natural class of sub-Markov processes. We then extend the realisability constructions and the notion of zeta function to provide a realisability model of second-order linear logic over the set of all (discrete-time) sub-Markov processes.
Fuzzy logic is a way to argue with boolean predicates for which we only have a confidence value between 0 and 1 rather than a well defined truth value. It is tempting to interpret such a confidence as a probability. We use Markov kernels, parametrised probability distributions, to do just that. As a consequence we get general fuzzy logic connectives from probabilistic computations on products of the booleans, stressing the importance of joint confidence functions. We discuss binary logic connectives in detail and recover the "classic" fuzzy connectives as bounds for the confidence for general …
Intuitionistic first-order logic extended with a restricted form of Markov's principle is constructive and admits a Curry-Howard correspondence, as shown by Herbelin. We provide a simpler proof of that result and then we study intuitionistic first-order logic extended with unrestricted Markov's principle. Starting from classical natural deduction, we restrict the excluded middle and we obtain a natural deduction system and a parallel Curry-Howard isomorphism for the logic. We show that proof terms for existentially quantified formulas reduce to a list of individual terms representing all possi…
In this paper we show that inference in 2-variable Markov logic networks (MLNs) with cardinality and function constraints is domain-liftable. To obtain this result we use existing domain-lifted algorithms for weighted first-order model counting (Van den Broeck et al, KR 2014) together with discrete Fourier transform of certain distributions associated to MLNs.
A continuous-time Markov chain (CTMC) execution is a continuous class of probability distributions over states. This paper proposes a probabilistic linear-time temporal logic, namely continuous-time linear logic (CLL), to reason about the probability distribution execution of CTMCs. We define the syntax of CLL on the space of probability distributions. The syntax of CLL includes multiphase timed until formulas, and the semantics of CLL allows time reset to study relatively temporal properties. We derive a corresponding model-checking algorithm for CLL formulas. The correctness of the model-che…
A Decidable Probability Logic for Timed Probabilistic Systems
在本文中，我们扩展的谓词逻辑介绍[Beauquier等人。2002]，以处理半马尔可夫过程。我们证明，相对于定性概率性质，模型检验是可判定的这种逻辑应用于半马尔可夫过程。此外，我们将我们的逻辑概率时间自动机考虑经典和紧急语义，并考虑时钟上的谓词。我们证明，半马尔可夫过程的结果也举行概率时间自动机的两个语义考虑。此外，我们证明了马尔可夫过程的结果[Beauquier et al. 2002]是可扩展的概率时间自动机，其中紧急语义被认为是。
We study expressivity of Markov logic networks (MLNs). We introduce complex MLNs, which use complex-valued weights, and we show that, unlike standard MLNs with real-valued weights, complex MLNs are fully expressive. We then observe that discrete Fourier transform can be computed using weighted first order model counting (WFOMC) with complex weights and use this observation to design an algorithm for computing relational marginal polytopes which needs substantially less calls to a WFOMC oracle than a recent algorithm.
Hyperproperties are properties that describe the correctness of a system as a relation between multiple executions. Hyperproperties generalize trace properties and include information-flow security requirements, like noninterference, as well as requirements like symmetry, partial observation, robustness, and fault tolerance. We initiate the study of the specification and verification of hyperproperties of Markov decision processes (MDPs). We introduce the temporal logic PHL (Probabilistic Hyper Logic), which extends classic probabilistic logics with quantification over schedulers and traces. P…
Cardiff University（卡迪夫大学）；University of Bremen（不来梅大学）；KU Leuven（鲁汶大学）
Markov Logic Networks (MLNs) are well-suited for expressing statistics such as "with high probability a smoker knows another smoker" but not for expressing statements such as "there is a smoker who knows most other smokers", which is necessary for modeling, e.g. influencers in social networks. To overcome this shortcoming, we study quantified MLNs which generalize MLNs by introducing statistical universal quantifiers, allowing to express also the latter type of statistics in a principled way. Our main technical contribution is to show that the standard reasoning tasks in quantified MLNs, maxim…
Temporal logic control of general Markov decision processes by approximate policy refinement
California Institute of Technology（加州理工学院）；School of Computing, Newcastle University（纽卡斯尔大学计算机学院）；Computer Science Department, Oxford University（牛津大学计算机科学系）
School of Mathematics, Institute for Research in Fundamental Sciences (IPM)（基础科学研究院数学学院）；Department of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic)（德黑兰理工大学（德黑兰理工学院）数学与计算机科学系）
Independence and conditional independence are fundamental concepts for reasoning about groups of random variables in probabilistic programs. Verification methods for independence are still nascent, and existing methods cannot handle conditional independence. We extend the logic of bunched implications (BI) with a non-commutative conjunction and provide a model based on Markov kernels; conditional independence can be directly captured as a logical formula in this model. Noting that Markov kernels are Kleisli arrows for the distribution monad, we then introduce a second model based on the powers…
Much work has been done to give semantics to probabilistic programming languages. In recent years, most of the semantics used to reason about probabilistic programs fall in two categories: semantics based on Markov kernels and semantics based on linear operators. Both styles of semantics have found numerous applications in reasoning about probabilistic programs, but they each have their strengths and weaknesses. Though it is believed that there is a connection between them there are no languages that can handle both styles of programming. In this work we address these questions by defining a t…
We study the synthesis of policies for multi-agent systems to implement spatial-temporal tasks. We formalize the problem as a factored Markov decision process subject to so-called graph temporal logic specifications. The transition function and the spatial-temporal task of each agent depend on the agent itself and its neighboring agents. The structure in the model and the specifications enable to develop a distributed algorithm that, given a factored Markov decision process and a graph temporal logic formula, decomposes the synthesis problem into a set of smaller synthesis problems, one for ea…
In this paper we propose a (sub)distribution-based bisimulation for labelled Markov processes and compare it with earlier definitions of state and event bisimulation, which both only compare states. In contrast to those state-based bisimulations, our distribution bisimulation is weaker, but corresponds more closely to linear properties. We construct a logic and a metric to describe our distribution bisimulation and discuss linearity, continuity and compositional properties.
TU Dortmund University（多特蒙德工业大学）；University of Bonn（波恩大学）
我们提出了关系线性规划，一个简单的框架结合线性程序（LP）和逻辑程序。关系线性规划（relational linear program，RLP）是一个声明性的LP模板，通过对象、关系和量化变量的逻辑概念定义目标和约束。这使得人们可以表达LP目标和约束关系的不同数量的个人和他们之间的关系，而不列举他们。连同逻辑知识库，有效地由逻辑事实和规则组成的逻辑程序，它归纳出一个基础LP。使用提升线性规划来求解该基础LP。也就是说，如果可能的话，采用地面LP内的对称性来降低其维数，并且使用任何现成的LP求解器来求解简化的程序。与主流LP模板语言（如AMPL）相比，它具有声明式和命令式编程风格的混合，RLP的关系性质允许在关系域上更直观地表示优化问题。我们说明了这一经验的实验近似推理在马尔可夫逻辑网络使用LP松弛，解决马尔可夫决策过程中，集体推理使用LP支持向量机。
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…
We develop a method for computing policies in Markov decision processes with risk-sensitive measures subject to temporal logic constraints. Specifically, we use a particular risk-sensitive measure from cumulative prospect theory, which has been previously adopted in psychology and economics. The nonlinear transformation of the probabilities and utility functions yields a nonlinear programming problem, which makes computation of optimal policies typically challenging. We show that this nonlinear weighting function can be accurately approximated by the difference of two convex functions. This ob…
Effectively combining logic reasoning and probabilistic inference has been a long-standing goal of machine learning: the former has the ability to generalize with small training data, while the latter provides a principled framework for dealing with noisy data. However, existing methods for combining the best of both worlds are typically computationally intensive. In this paper, we focus on Markov Logic Networks and explore the use of graph neural networks (GNNs) for representing probabilistic logic inference. It is revealed from our analysis that the representation power of GNN alone is not e…
We propose a method for efficient training of Q-functions for continuous-state Markov Decision Processes (MDPs) such that the traces of the resulting policies satisfy a given Linear Temporal Logic (LTL) property. LTL, a modal logic, can express a wide range of time-dependent logical properties (including "safety") that are quite similar to patterns in natural language. We convert the LTL property into a limit deterministic Buchi automaton and construct an on-the-fly synchronised product MDP. The control policy is then synthesised by defining an adaptive reward function and by applying a modifi…
