We investigate logics and equivalence relations that capture the qualitative behavior of Markov Decision Processes (MDPs). We present Qualitative Randomized CTL (QRCTL): formulas of this logic can express the fact that certain temporal properties hold over all paths, or with probability 0 or 1, but they do not distinguish among intermediate probability values. We present a symbolic, polynomial time model-checking algorithm for QRCTL on MDPs.
The logic QRCTL induces an equivalence relation over states of an MDP that we call qualitative equivalence: informally, two states are qualitatively equivalent if the sets of formulas that hold with probability 0 or 1 at the two states are the same. We show that for finite alternating MDPs, where nondeterministic and probabilistic choices occur in different states, qualitative equivalence coincides with alternating bisimulation, and can thus be computed via efficient partition-refinement algorithms. On the other hand, in non-alternating MDPs the equivalence relations cannot be computed via partition-refinement algorithms, but rather, they require non-local computation. Finally, we consider QRCTL*, that extends QRCTL with nested temporal operators in the same manner in which CTL* extends CTL. We show that QRCTL and QRCTL* induce the same qualitative equivalence on alternating MDPs, while on non-alternating MDPs, the equivalence arising from QRCTL* can be strictly finer. We also provide a full characterization of the relation between qualitative equivalence, bisimulation, and alternating bisimulation, according to whether the MDPs are finite, and to whether their transition relations are finitely-branching.
National and Kapodistrian University of Athens（雅典国立卡波迪斯特里安大学）；University of Ioannina（约阿尼纳大学）
Logic Programs with Ordered Disjunction (LPODs) extend classical logic programs with the capability of expressing preferential disjunctions in the heads of program rules. The initial semantics of LPODs, although simple and quite intuitive, is not purely model-theoretic. A consequence of this is that certain properties of programs appear non-trivial to formalize in purely logical terms. An example of this state of affairs is the characterization of the notion of strong equivalence for LPODs. Although the results of Faber et al. (2008) are accurately developed, they fall short of characterizing …
In answer set programming, two groups of rules are considered strongly equivalent if they have the same meaning in any context. In some cases, strong equivalence of programs in the input language of the grounder gringo can be established by deriving rules of each program from rules of the other. The possibility of such proofs has been demonstrated for a subset of that language that includes comparisons, arithmetic operations, and simple choice rules, but not aggregates. This method is extended here to a class of programs in which some uses of the #count aggregate are allowed. This paper is und…
On Equivalence of Infinitary Formulas under the Stable Model Semantics
在直觉主义逻辑中等价的命题公式，或者在其扩展中被称为这里和那里的逻辑，具有相同的稳定模型。我们将这个定理扩展到具有无限长合取和析取的命题公式，并展示如何将此推广应用于证明答案集编程中聚合的性质。逻辑程序设计理论与实践（Theory and Practice of Logic Programming，TPLP）
The two major systems of formal verification are model checking and algebraic model-based testing. Model checking is based on some form of temporal logic such as linear temporal logic (LTL) or computation tree logic (CTL). One powerful and realistic logic being used is CTL, which is capable of expressing most interesting properties of processes such as liveness and safety. Model-based testing is based on some operational semantics of processes (such as traces, failures, or both) and its associated preorders. The most fine-grained preorder beside bisimulation (mostly of theoretical importance) …
In the field of Answer Set Programming (ASP), two logic programs are strongly equivalent if they are ordinarily equivalent under any extensions. This property provides a theoretical foundation for studying many aspects of logic programs such as program simplification and transformation etc. Therefore, strong equivalence has been investigated extensively for ASP and its extensions such as LPMLN. In this paper, we present a syntactic approach to studying the strong equivalence of logic programs, which provides several interesting results and would help us understand the strong equivalence from a…
Dynamics and uncertainty are essential features of real-life argumentation, and many recent studies have focused on integrating both aspects into Dung's well-known abstract Argumentation Frameworks (AFs). This paper proposes a combination of the two lines of research through a well-behaved logical tool: Dynamic Logic of Propositional Assignments (DL-PA). Our results show that the main reasoning tasks of virtually all existing formalisms qualitatively representing uncertainty about AFs are encodable in DL-PA. Moreover, the same tool is also useful for capturing dynamic structures, such as contr…
We study coalgebraic modal logic to characterise behavioural equivalence in the presence of side effects, i.e., when coalgebras live in a (co)Kleisli or an Eilenberg-Moore category. Our aim is to develop a general framework based on indexed categories/fibrations that is common to the aforementioned categories. In particular, we show how the coalgebraic notion of behavioural equivalence arises from a relation lifting (a special kind of indexed morphism) and we give a general recipe to construct such liftings in the above three cases. Lastly, we apply this framework to derive logical characteris…
Bisimulations and Modal Logics for Higher Dimensional Automata
Norwegian University of Science and Technology（挪威科技大学）；University of Edinburgh（爱丁堡大学）；University of New South Wales（新南威尔士大学）；University of Warsaw（华沙大学）
Answer set programming is a prominent declarative programming paradigm used in formulating combinatorial search problems and implementing different knowledge representation formalisms. Frequently, several related and yet substantially different answer set programs exist for a given problem. Sometimes these encodings may display significantly different performance. Uncovering precise formal links between these programs is often important and yet far from trivial. This paper presents formal results carefully relating a number of interesting program rewritings. It also provides the proof of corre…
We introduce a general and compositional, yet simple, framework that allows us to derive soundness and expressiveness results for modal logics characterizing behavioural equivalences or metrics (also known as Hennessy-Milner theorems). It is based on Galois connections between sets of (real-valued) predicates on the one hand and equivalence relations/metrics on the other hand and covers a part of the linear-time-branching-time spectrum, both for the qualitative case (behavioural equivalences) and the quantitative case (behavioural metrics). We derive behaviour functions from a given logic and …
State-based models of concurrent systems are traditionally considered under a variety of notions of process equivalence. In the particular case of labelled transition systems, these equivalences range from trace equivalence to (strong) bisimilarity, and are organized in what is known as the linear time -- branching time spectrum. A combination of universal coalgebra and graded monads provides a generic framework in which the semantics of concurrency can be parametrized both over the branching type of the underlying transition systems and over the granularity of process equivalence. We show in …
