Probabilistic Modal Logic Research Articles

Modal semantics for reasoning with probability and uncertainty

Abstract This paper belongs to the field of probabilistic modal logic, focusing on a comparative analysis of two distinct semantics: one rooted in Kripke semantics and the other in neighbourhood semantics. The primary distinction lies in the following: The latter allows us to adequately express belief functions (lower probabilities) over propositions, whereas the former does not. Thus, neighbourhood semantics is more expressive. The main part of the work is a section in which we study the modal equivalence between probabilistic Kripke models and a subclass of belief neighbourhood models, namely additive ones. We study how to obtain modally equivalent structures.

A probabilistic modal logic for context-aware trust based on evidence

Trust is an extremely helpful construct when reasoning under uncertainty. Thus, being able to logically formalize the concept in a suitable language is important. However, doing so is problematic for three reasons. First, in order to keep track of the contextual nature of trust, situation trackers are required inside the language. Second, in order to produce trust estimations, agents rely on evidence personally gathered or reported by other agents; this requires elements in the language that can track which agents are used as referrals and how much weight is placed on their opinions. Finally, trust is subjective in nature, thus, personal thresholds are needed to track the trust-propensity of different evaluators. In this paper we propose an interpretation of a probabilistic modal language à la Hennessy-Milner in order to capture a context-aware quantitative notion of trust based on evidence. We also provide an axiomatization for the language and prove soundness, completeness, and decidability results.

Coalgebraic Reasoning with Global Assumptions in Arithmetic Modal Logics

We establish a generic upper bound ExpTime for reasoning with global assumptions (also known as TBoxes) in coalgebraic modal logics. Unlike earlier results of this kind, our bound does not require a tractable set of tableau rules for the instance logics, so that the result applies to wider classes of logics. Examples are Presburger modal logic, which extends graded modal logic with linear inequalities over numbers of successors, and probabilistic modal logic with polynomial inequalities over probabilities. We establish the theoretical upper bound using a type elimination algorithm. We also provide a global caching algorithm that potentially avoids building the entire exponential-sized space of candidate states, and thus offers a basis for practical reasoning. This algorithm still involves frequent fixpoint computations; we show how these can be handled efficiently in a concrete algorithm modelled on Liu and Smolka’s linear-time fixpoint algorithm. Finally, we show that the upper complexity bound is preserved under adding nominals to the logic, i.e., in coalgebraic hybrid logic.

Expressiveness of probabilistic modal logics: A gradual approach

Logical characterizations of probabilistic bisimulation and simulation for Labelled Markov Processes were given by Desharnais et al. These results hold for systems defined on analytic state spaces and assume countably many labels in the case of bisimulation and finitely many labels in the case of simulation.We revisit these results by giving simpler and more streamlined proofs. In particular, our proof for simulation has the same structure as the one for bisimulation, relying on a new result of a topological nature. We also propose a new notion of event simulation.Our proofs assume countably many labels, and we show that the logical characterization of bisimulation may fail when there are uncountably many labels. However, with a stronger assumption on the transition functions (continuity instead of just measurability), we regain the logical characterization result for arbitrarily many labels. These results arose from a game-theoretic understanding of probabilistic simulation and bisimulation.

Constructive logical characterizations of bisimilarity for reactive probabilistic systems

Larsen and Skou characterized bisimilarity over reactive probabilistic systems with a logic including negation, conjunction, and a diamond modality decorated with a probabilistic lower bound. Later on, working on continuous state spaces, Desharnais, Edalat, and Panangaden proved that negation is not necessary to characterize the same equivalence. In this paper, we redemonstrate the former result with a simpler proof and the latter result directly on discrete state spaces without resorting to measure-theoretic arguments. Moreover, we show that conjunction can be replaced by disjunction in both logics, still characterizing the same bisimilarity. To these ends, we introduce reactive probabilistic trees, a fully abstract model for reactive probabilistic systems that allows us to uniformly prove expressiveness of the four probabilistic modal logics by means of a compactness argument. Our proofs are constructive, as they induce for each considered logic an algorithm that builds a distinguishing formula in case of two inequivalent reactive probabilistic systems.

A Probabilistic Formalization of the Appraisal for the OCC Event-Based Emotions

This article presents a logical formalization of the emotional appraisal theory, i.e., it formalizes the cognitive process of evaluation that elicits an emotion. This formalization is psychologically grounded on the OCC cognitive model of emotions. More specifically, we are interested in event-based emotions, i.e., emotions that are elicited by the evaluation of the consequences of an event that either happened or will happen. The formal modelling presented here is based on the AfPL Probabilistic Logic, a BDI-like probabilistic modal logic, which allows our model to verify whether the variables that determine the elicitation of emotions achieved the necessary threshold or not. The proposed logical formalization aims at addressing how the emotions are elicited by the agent cognitive mental states (desires, beliefs and intentions), and how to represent the intensity of the emotions. These are important initial points in the investigation of the dynamic interaction among emotions and other mental states.

Deriving the correctness of quantum protocols in the probabilistic logic for quantum programs

This paper presents a sound axiomatization for a probabilistic modal dynamic logic of quantum programs. The logic can express whether a state is separable or entangled, information that is local to a subsystem of the whole quantum system, and the probability of positive answers to quantum tests of certain properties. The power of this axiomatization is demonstrated with proofs of properties concerning bases of a finite-dimensional Hilbert space, composite systems, entangled and separable states, and with proofs of the correctness of two probabilistic quantum protocols (the quantum leader election protocol and the BB84 quantum key distribution protocol).

PLQP & Company: Decidable Logics for Quantum Algorithms

We introduce a probabilistic modal (dynamic-epistemic) quantum logic PLQP for reasoning about quantum algorithms. We illustrate its expressivity by using it to encode the correctness of the well-known quantum search algorithm, as well as of a quantum protocol known to solve one of the paradigmatic tasks from classical distributed computing (the leader election problem). We also provide a general method (extending an idea employed in the decidability proof in Dunn et al. (J. Symb. Log. 70:353–359, 2005)) for proving the decidability of a range of quantum logics, interpreted on finite-dimensional Hilbert spaces. We give general conditions for the applicability of this method, and in particular we apply it to prove the decidability of PLQP.

Value-passing CCS with noisy channels

Value-passing CCS, a full version of Milner’s CCS, is a process algebra in which actions consist of sending and receiving values through noiseless communication channels. The full calculus is a succinct yet expressive language for the specification and verification of reactive systems. Taking into account the reality of channel noise in reactive systems, in this paper we introduce an extension of value-passing CCS, called value-passing CCS with noisy channels (VCCSN), in which noise is described by a probability distribution over the values. After presenting the reduction operational semantics and labelled operational semantics of VCCSN, we develop the theory of behavioural equivalence by introducing barbed equivalence, barbed congruence, bisimilarity, and full bisimilarity. In particular, we show that barbed equivalence and barbed congruence coincide with bisimilarity and full bisimilarity, respectively. Based upon the labelled operational semantics of VCCSN, we establish a probabilistic modal logic for expressing system properties and show its connection with the notion of bisimilarity. Finally, we use VCCSN to model a communication protocol for ensuring the reliable transmission of data across an error-prone channel.

A Proof of Secrecy by Probabilistic Modal Logic

A Proof of Secrecy by Probabilistic Modal Logic

EXPTIME Tableaux for the Coalgebraic mu-Calculus

The coalgebraic approach to modal logic provides a uniform framework that captures the semantics of a large class of structurally different modal logics, including e.g. graded and probabilistic modal logics and coalition logic. In this paper, we introduce the coalgebraic mu-calculus, an extension of the general (coalgebraic) framework with fixpoint operators. Our main results are completeness of the associated tableau calculus and EXPTIME decidability for guarded formulas. Technically, this is achieved by reducing satisfiability to the existence of non-wellfounded tableaux, which is in turn equivalent to the existence of winning strategies in parity games. Our results are parametric in the underlying class of models and yield, as concrete applications, previously unknown complexity bounds for the probabilistic mu-calculus and for an extension of coalition logic with fixpoints.

A logical duality for underspecified probabilistic systems

This paper establishes a Stone-type duality between specifications and infLMPs. An infLMP is a probabilistic process whose transitions satisfy super-additivity instead of additivity. Interestingly, its simple structure can encode a mix of probabilistic and non-deterministic behavior, which, as we show, is strongly related to another well-known such model: probabilistic automata. Our duality puts in relation the category of infLMPs and a category of abstract representations of them based on properties only. We exhibit a Galois connection between these categories and show that we have an adjunct pair of functors when restricted to LMPs only. Our duality also shows that an infLMP can be considered as a demonic representative of a system’s information. Moreover, it carries forward a view where states are less important, and events, or properties, become the main characters, as it should be in probability theory. Along the way, we show that bisimulation and simulation are naturally interpreted in this setting, and we exhibit the interesting relationship between infLMPs and the usual probabilistic modal logics. This paper is an extended version of a Concur ’09 paper [13]; in particular, the comparison of infLMPs with probabilistic automata and the Galois connection are new.

Optimizing Conditional Logic Reasoning within CoLoSS

The generic modal reasoner CoLoSS covers a wide variety of logics ranging from graded and probabilistic modal logic to coalition logic and conditional logics, being based on a broadly applicable coalgebraic semantics and an ensuing general treatment of modal sequent and tableau calculi. Here, we present research into optimisation of the reasoning strategies employed in CoLoSS. Specifically, we discuss strategies of memoisation and dynamic programming that are based on the observation that short sequents play a central role in many of the logics under study. These optimisations seem to be particularly useful for the case of conditional logics, for some of which dynamic programming even improves the theoretical complexity of the algorithm. These strategies have been implemented in CoLoSS; we give a detailed comparison of the different heuristics, observing that in the targeted domain of conditional logics, a substantial speed-up can be achieved.

CoLoSS: The Coalgebraic Logic Satisfiability Solver

CoLoSS, the Coalgebraic Logic Satisfiability Solver, decides satisfiability of modal formulas in a generic and compositional way. It implements a uniform polynomial space algorithm to decide satisfiability for modal logics that are amenable to coalgebraic semantics. This includes e.g. the logics K, KD, Pauly's coalition logic, graded modal logic, and probabilistic modal logic. Logics are easily integrated into CoLoSS by providing a complete axiomatisation of their coalgebraic semantics in a specific format. Moreover, CoLoSS is compositional: it synthesises decision procedures for modular combinations of logics that include the fusion of two modal logics as a special case. One thus automatically obtains reasoning support e.g. for logics interpreted over probabilistic automata that combine non-determinism and probabilities in different ways.

PSPACE bounds for rank-1 modal logics

For lack of general algorithmic methods that apply to wide classes of logics, establishing a complexity bound for a given modal logic is often a laborious task. The present work is a step towards a general theory of the complexity of modal logics. Our main result is that all rank-1 logics enjoy a shallow model property and thus are, under mild assumptions on the format of their axiomatisation, in PSPACE . This leads to a unified derivation of tight PSPACE -bounds for a number of logics, including K , KD , coalition logic, graded modal logic, majority logic, and probabilistic modal logic. Our generic algorithm moreover finds tableau proofs that witness pleasant proof-theoretic properties including a weak subformula property. This generality is made possible by a coalgebraic semantics, which conveniently abstracts from the details of a given model class and thus allows covering a broad range of logics in a uniform way.

Probabilistic Logic over Paths

We introduce a probabilistic modal logic PPL extending the work of [Ronald Fagin, Joseph Y. Halpern, and Nimrod Megiddo. A logic for reasoning about probabilities. Information and Computation, 87(1,2):78–128, 1990; Ronald Fagin and Joseph Y. Halpern. Reasoning about knowledge and probability. Journal of the ACM, 41(2):340–367, 1994] by allowing arbitrary nesting of a path probabilistic operator and we prove its completeness. We prove that our logic is strictly more expressive than other logics such as the logics cited above. By considering a probabilistic extension of CTL we show that this additional expressive power is really needed in some applications.

Specifying and reasoning about uncertain agents

Logical formalisation of agent behaviour is desirable, not only in order to provide a clear semantics of agent-based systems, but also to provide the foundation for sophisticated reasoning techniques to be used on, and by, the agents themselves. The possible worlds semantics offered by modal logic has proved to be a successful framework in which to model mental attitudes of agents such as beliefs, desires and intentions. The most popular choices for modeling the informational attitudes involves annotating the agent with an S5-like logic for knowledge, or a KD45-like logic for belief. However, using these logics in their standard form, an agent cannot distinguish situations in which the evidence for a certain fact is ‘equally distributed’ over its alternatives, from situations in which there is only one, almost negligible, counterexample to a ‘fact’. Probabilistic modal logics are a way to address this, but they easily end up being both computationally and conceptually complex, for example often lacking the property of compactness. In this paper, we propose a probabilistic modal logic P F KD45, in which the probabilities of the possible worlds range over a finite domain of values, while still allowing the agent to reason about infinitely many options. In this way, the logic remains compact, implying that the agent still has to consider only finitely many possibilities for probability distributions during a reasoning task. We demonstrate a sound, compact and complete axiomatization for P F KD45 and show that it has several appealing features. Then, we discuss an implemented decision procedure for the logic, and provide a small example.

A finite model construction for coalgebraic modal logic

In recent years, a tight connection has emerged between modal logic on the one hand and coalgebras, understood as generic transition systems, on the other hand. Here, we prove that (finitary) coalgebraic modal logic has the finite model property. This fact not only reproves known completeness results for coalgebraic modal logic, which we push further by establishing that every coalgebraic modal logic admits a complete axiomatisation in rank 1; it also enables us to establish a generic decidability result and a first complexity bound. Examples covered by these general results include, besides standard Hennessy–Milner logic, graded modal logic and probabilistic modal logic.

Probabilistic temporal logics via the modal mu-calculus

This paper presents a mu-calculus-based modal logic for describing properties of reactive probabilistic labeled transition systems (RPLTSs) and develops a model-checking algorithm for determining whether or not states in finite-state RPLTSs satisfy formulas in the logic. The logic is based on the distinction between (probabilistic) “systems” and (nonprobabilistic) “observations”: using the modal mu-calculus, one may specify sets of observations, and the semantics of our logic then enable statements to be made about the measures of such sets at various system states. The logic may be used to encode a variety of probabilistic modal and temporal logics; in addition, the model-checking problem for it may be reduced to the calculation of solutions to systems of non-linear equations. Finally, the logic induces an equivalence on RPLTSs that coincides with accepted notions of probabilistic bisimulation in the literature.

Fuzzy unification and first order probabilistic logic

The notion of unification for fuzzy sets in fuzzy logic programming is explored in this article from the standpoint of a theoretical framework based on first-order probabilistic modal logic. The fundamental difference between the latter perspective and other approaches described in the literature lies in the characterization of fuzzy sets, which are construed in the logical framework as derived entities having a probabilistic deep structure. The analysis reveals that a few semantic unification techniques are sound even in this probabilistic approach, provided that some specific conditions are enforced. In addition, the approach helps to clarify some important aspects of the informal interpretation for the unification process, including a rationale for the applicability requisites of computational techniques. In the discussion, the reference model adopted for the notion of fuzzy unification is that of evidential logic programming, that is, the one underlying the programming language FRIL.5 © 2002 Wiley Periodicals, Inc.