Multi-modal Logic Research Articles

Complete proof systems for weighted modal logic

The weighted transition systems (WTS) considered in this paper are transition systems having both states and transitions labeled with real numbers: the state labels denote quantitative resources, while the transition labels denote costs of transitions in terms of resources. Weighted Modal Logic (WML) is a multi-modal logic that expresses qualitative and quantitative properties of WTSs. While WML has been studied in various contexts and for various application domains, no proof system has been developed for it. In this paper we solve this open problem and propose both weak-complete and strong-complete axiomatizations for WML against WTSs. We prove a series of metatheorems including the finite model property and the existence of canonical models. We show how the proof system can be used in the context of a process-algebra semantics for WML to convert a model-checking problem into a theorem-proving problem. This work emphasizes a series of similarities between WML and the probabilistic/stochastic modal logics for Markov processes and Harsanyi type spaces, such as the use of particular infinitary rules to guarantee the strong-completeness.

Kabbalah Logic and Semantic Foundations for a Postmodern Fuzzy Set and Fuzzy Logic Theory

Despite half a century of fuzzy sets and fuzzy logic progress, as fuzzy sets address complex and uncertain information through the lens of human knowledge and subjectivity, more progress is needed in the semantics of fuzzy sets and in exploring the multi-modal aspect of fuzzy logic due to the different cognitive, emotional and behavioral angles of assessing truth. We lay here the foundations of a postmodern fuzzy set and fuzzy logic theory addressing these issues by deconstructing fuzzy truth values and fuzzy set membership functions to re-capture the human knowledge and subjectivity structure in membership function evaluations. We formulate a fractal multi-modal logic of Kabbalah which integrates the cognitive, emotional and behavioral levels of humanistic systems into epistemic and modal, deontic and doxastic and dynamic multi-modal logic. This is done by creating a fractal multi-modal Kabbalah possible worlds semantic frame of Kripke model type. The Kabbalah possible worlds semantic frame integrates together both the multi-modal logic aspects and their Kripke possible worlds model. We will not focus here on modal operators and axiom sets. We constructively define a fractal multi-modal Kabbalistic L-fuzzy set as the central concept of the postmodern fuzzy set theory based on Kabbalah logic and semantics.

On the Computational Interpretation of CKn for Contextual Information Processing

We aim to establish the multi-modal logic CKn as a baseline for a constructive correspondence theory of constructive modal logics. Just like many classical multi-modal logics may be studied as theories of the basic system K obtained by model-theoretic specialisation, we envisage constructive modal logics to be derived as proof-theoretic enrichments of CKn. The system CKn would then act as a core system for constructive contextual reasoning with controlled information flow. In this paper, as a first step towards this goal, we study CKn as a type theory and introduce its computational λ-calculus, λCKn. Extending previous work on CKn, we present a cut-free contextual sequent system in the spirit of Masini's two-dimensional generalisation of natural deduction and Brunnler's nested sequents and give a computational interpretation for CKn following the Curry-Howard Correspondence. The associated modal type theory λCKn permits an interpretation for both the modalities □ and ◊ of CKn as type operators with simple and independent constructors and destructors, which has been missing in the literature. It is shown that the calculus satisfies subject reduction, strong normalisation and confluence. Since normal forms can be characterised by way of a Gentzen-style typing system with sub-formula property, λCKn is suitable for proof search in CKn. At the same time, λCKn enjoys natural deduction style typing which is important for programming applications. In contrast to most existing modal type theories, which are obtained as theories of the constructive modal logic S4, CKn is not bound to a particular contextual interpretation. Thus, λCKn constitutes the core of a functional language which provides static type checking of information processing to support safe contextual navigation in relational structures like those treated by description logics. We review some existing work on modal type theories and discuss their relation to λCKn.

An axiomatization for the linear logic of knowledge and time LTK r with intransitive time relation

We study the problem of the axiomatization of the linear multimodal logic of knowledge and time LTKr with reflexive intransitive time relation. The logic is defined semantically as the set of formulas true on frames of a special kind. The LTKr-frames are linear chains of clusters connected by a reflexive intransitive relation RT which simulates time. Elements inside a cluster are connected by several equivalence relations imitating the knowledge of different agents. The main result of the article is the proof of the fact that the finite set of formulas proposed by the authors is an axiomatization of the logic LTKr with reflexive intransitive time relation.

Montague’s Theorem and Modal Logic

In the present piece we defend predicate approaches to modality, that is approaches that conceive of modal notions as predicates applicable to names of sentences or propositions, against the challenges raised by Montague’s theorem. Montague’s theorem is often taken to show that the most intuitive modal principles lead to paradox if we conceive of the modal notion as a predicate. Following Schweizer (J Philos Logic 21:1–31, 1992) and others we show this interpretation of Montague’s theorem to be unwarranted unless a further non trivial assumption is made—an assumption which should not be taken as a given. We then move on to showing, elaborating on work of Gupta (J Philos Logic 11:1–60, 1982), Asher and Kamp (Properties, types, and meaning. Vol. I: foundational issues, Kluwer, Dordrecht, pp 85−158, 1989), and Schweizer (J Philos Logic 21:1–31, 1992), that the unrestricted modal principles can be upheld within the predicate approach and that the predicate approach is an adequate approach to modality from the perspective of modal operator logic. To this end we develop a possible world semantics for multiple modal predicates and show that for a wide class of multimodal operator logics we may find a suitable class of models of the predicate approach which satisfies, modulo translation, precisely the theorems of the modal operator logic at stake.

Multimodal Incompleteness Under Weak Negations

This paper shows that some classes of multimodal paraconsistent logics endowed with weak forms of negation are incompletable with respect to Kripke semantics. The reach of such incompleteness is discussed, and we argue that this shortcoming, more than just a logical predicament, may be relevant for attempts to characterize quantum logics and to handle quantum information and quantum computation.

Special Issue on Multimodal Logics: A Preface

It is a fact that investigations of multimodality are currently an advanced area of inquiry within the immense field of intensional logic. A remarkable aspect of this growth is that important branches of modal logic have been shown to be reconstructable as branches of multimodal logic. To mention only one example, binary conditional operators, as Lennart Aqvist convincingly showed in the 1970s, may be defined inside a logic having two or more primitive monadic modal operators, so that conditional logic may in principle be reconstructed as a branch of multimodal logic. Terms such as “multimodal logic” or “polymodal logic” have become part of common usage since the 1980s, but the history of multimodal logic is longer than what the history of these terms suggests. In 1970, in his “Advice on Modal Logico, Dana Scott wrote:

Modal logics for reasoning about infinite unions and intersections of binary relations

We consider multi-modal logic (with countably infinite number of basic modalities) extended with additional modalities and corresponding to the union and intersection of all basic modalities. We present complete and sound axiomatic systems and polynomial-space terminating tableau-based decision procedures for the basic logic in this language, and its deterministic counterpart . We also show that admits filtration, which can be used independently of our tableaux to establish its decidability.

Continuous Markovian Logics - Axiomatization and Quantified Metatheory

Continuous Markovian Logic (CML) is a multimodal logic that expresses quantitative and qualitative properties of continuous-time labelled Markov processes with arbitrary (analytic) state-spaces, henceforth called continuous Markov processes (CMPs). The modalities of CML evaluate the rates of the exponentially distributed random variables that characterize the duration of the labeled transitions of a CMP. In this paper we present weak and strong complete axiomatizations for CML and prove a series of metaproperties, including the finite model property and the construction of canonical models. CML characterizes stochastic bisimilarity and it supports the definition of a quantified extension of the satisfiability relation that measures the "compatibility" between a model and a property. In this context, the metaproperties allows us to prove two robustness theorems for the logic stating that one can perturb formulas and maintain "approximate satisfaction".

Quantified Multimodal Logics in Simple Type Theory

We present an embedding of quantified multimodal logics into simple type theory and prove its soundness and completeness. A correspondence between QKπ models for quantified multimodal logics and Henkin models is established and exploited. Our embedding supports the application of off-the-shelf higher-order theorem provers for reasoning within and about quantified multimodal logics. Moreover, it provides a starting point for further logic embeddings and their combinations in simple type theory.

Rich Counter-Examples for Temporal-Epistemic Logic Model Checking

Model checking verifies that a model of a system satisfies a given property, and otherwise produces a counter-example explaining the violation. The verified properties are formally expressed in temporal logics. Some temporal logics, such as CTL, are branching: they allow to express facts about the whole computation tree of the model, rather than on each single linear computation. This branching aspect is even more critical when dealing with multi-modal logics, i.e. logics expressing facts about systems with several transition relations. A prominent example is CTLK, a logic that reasons about temporal and epistemic properties of multi-agent systems. In general, model checkers produce linear counter-examples for failed properties, composed of a single computation path of the model. But some branching properties are only poorly and partially explained by a linear counter-example. This paper proposes richer counter-example structures called tree-like annotated counter-examples (TLACEs), for properties in Action-Restricted CTL (ARCTL), an extension of CTL quantifying paths restricted in terms of actions labeling transitions of the model. These counter-examples have a branching structure that supports more complete description of property violations. Elements of these counter-examples are annotated with parts of the property to give a better understanding of their structure. Visualization and browsing of these richer counter-examples become a critical issue, as the number of branches and states can grow exponentially for deeply-nested properties. This paper formally defines the structure of TLACEs, characterizes adequate counter-examples w.r.t. models and failed properties, and gives a generation algorithm for ARCTL properties. It also illustrates the approach with examples in CTLK, using a reduction of CTLK to ARCTL. The proposed approach has been implemented, first by extending the NuSMV model checker to generate and export branching counter-examples, secondly by providing an interactive graphical interface to visualize and browse them.

Moral hazard resolved by common-knowledge in principal-agent model

This paper investigates the role of common-knowledge in the principal-agent model under uncertainty. We treat the problem: How epistemic conditions will be able to settle a moral hazard in team in the principal-agent model under uncertainty. We shall propose a resolution programme for the moral hazard in the principal-agent model by common-knowledge. Let us assume that the agents have the knowledge structure induced from a binary relation associated with the multi-modal logic S5n or S4n. We show that the moral hazard can be resolved in the principal-agent model under uncertainty if the agents commonly know all their own costs.

反事実条件文推論の動態論理による形式化

It has been common to formalize counterfactuals (or subjunctive conditionals) in natural language in terms of a certain binary sentential connective, as in Stalnaker [12] and D. Lewis [8]. This paper suggests that another formalization by means of unary multi-modal operators is natural and appropriate for some counterfactuals. To see this naturalness and appropriateness, we observe an instance of transitive inference constituted of three counterfactuals in natural language, and formalize it by using expressive power of multi-modal logic, in particular Hennessy-Milner logic(HML) and Dynamic logic (DL). As a result, the instance of transitive inference turns out to be justified by the multi-modalized version of the most fundamental and familiar rules of modal logic, that is, the necessitation rule (NAct) and the axiom (KAct).

Logic-Based Reputation Model in E-Commerce Simulation

We employ a multimodal logic in a decision making mechanism involving trust and reputation. The mechanism is then used in a community of interacting agents which develop cooperative relationships, assess the results against several quality criteria and possibly publish their beliefs inside the group. A new definition is proposed for describing how an agent deals with the common reputation information and with divergent opinions. The definition permits selecting and integrating the knowledge obtained from the peers, based on their perceived trust, as well as on threshold called critical mass. The influence of this parameter and of the number of agents supporting a sentence over its adoption are then investigated.

Partial cut elimination for combinations of propositional multi-modal logics with past time

We consider combinations of nine propositional multi-modal logics with propositional discrete linear time temporal logic with past time. For these combinations, we present sound and complete Gentzen-type sequent calculi with a restricted cut rule.

Higher-order aspects and context in SUMO

This article addresses the automation of higher-order aspects in expressive ontologies such as the suggested upper merged ontology SUMO. Evidence is provided that modern higher-order automated theorem provers like LEO-II can be fruitfully employed for the task. A particular focus is on embedded formulas (formulas as terms), which are used in SUMO, for example, for modeling temporal, epistemic, or doxastic contexts. This modeling is partly in conflict with SUMO’s assumption of a bivalent, classical semantics and it may hence lead to counterintuitive reasoning results with automated theorem provers in practice. A solution is proposed that maps SUMO to quantified multimodal logic which is in turn modeled as a fragment of classical higher-order logic. This way automated higher-order theorem provers can be safely applied for reasoning about modal contexts in SUMO.Our findings are of wider relevance as they analogously apply to other expressive ontologies and knowledge representation formalisms.

Higher-Order Aspects and Context in SUMO

This article addresses the automation of higher-order aspects in expressive ontologies such as the Suggested Upper Merged Ontology SUMO. Evidence is provided that modern higher-order automated theorem provers like LEO-II can be fruitfully employed for the task. A particular focus is on embedded formulas (formulas as terms), which are used in SUMO, for example, for modeling temporal, epistemic, or doxastic contexts. This modeling is partly in conflict with SUMO's assumption of a bivalent, classical semantics and it may hence lead to counter-intuitive reasoning results with automated theorem provers in practice. A solution is proposed that maps SUMO to quantified multimodal logic which is in turn modeled as a fragment of classical higher-order logic. This way automated higher-order theorem provers can be safely applied for reasoning about modal contexts in SUMO. Our findings are of wider relevance as they analogously apply to other expressive ontologies and knowledge representation formalisms.

A Tableau Calculus for Minimal Modal Model Generation

Model generation and minimal model generation is useful for fault analysis, verification of systems and validation of data models. Whereas for classical propositional and first-order logic several model minimization approaches have been developed and studied, for non-classical logic the topic has been much less studied. In this paper we introduce a minimal model generation calculus for multi-modal logic K(m) and extensions of K(m) with the axioms T and B. The calculus provides a method to generate all and only minimal modal Herbrand models, and each model is generated exactly once. A novelty of the calculus is a non-standard complement splitting rule designed for minimal model generation. Experiments show the rule has the added benefit of reducing the search space.

Complexity of logics for multi-agent systems with restricted modal context

In this paper we continue our research in computational complexity of fragments of a particular belief-desire-intention (BDI) logic for cooperative problem solving (CPS) called TeamLog. The logic in question combines propositional multi-modal logic to express static aspects of CPS with propositional dynamic logic used for dynamic aspects of CPS. We concentrate here on the static fragment of the logic. The complexity of the satisfiability problem for the static fragment of TeamLog is known to be EXPTIME-complete, even if modal depth is bounded by a constant ≥ 2. We show a restriction on modal context of formulas that makes the problem PSPACE-complete without setting boundary on modal depth. Bounding modal depth, additionally, leads to NPTIME-completeness.

PSPACE Tableau Algorithms for Acyclic Modalized $\boldsymbol{\mathcal{ALC}}$

We study \(\mathcal{ALCK}_m\) and \(\mathcal{ALCS}4_m\), which extend the description logic \(\mathcal{ALC}\) by adding modal operators of the basic multi-modal logics Km and S4m. We develop a sound and complete tableau algorithm \(\Lambda_\mathcal{K}\) for answering \(\mathcal{ALCK}_m\) queries w.r.t. an \(\mathcal{ALCK}_m\) knowledge base with an acyclic TBox. Defining tableau expansion rules in the presence of acyclic definitions by considering only the concept names on the left-hand side of TBox definitions or their negations, allows us to give a PSPACE implementation for \(\Lambda_\mathcal{K}\). We then consider answering \(\mathcal{ALCS}4_m\) queries w.r.t. an \(\mathcal{ALCS}4_m\) knowledge base (with an acyclic TBox) in which the epistemic operators correspond to those of classical multi-modal logic S4m. The expansion rules in the tableau algorithm ΛS4 are designed to syntactically incorporate the epistemic properties. Blocking is corporated into the tableau expansion rules to ensure termination. We also provide a PSPACE implementation for ΛS4. In light of the fact that the satisfiability problem for \(\mathcal{ALCK}_m\) with general TBox and no epistemic properties (i.e., \(\mathbf{K}_{\mathcal{ALC}}\)) is NEXPTIME-complete, we conclude that both \(\mathcal{ALCK}_m\) and \(\mathcal{ALCS}4_m\) offer computationally manageable and practically useful fragments of \(\mathbf{K}_{\mathcal{ALC}}\).