Modal Operators Research Articles

Modal dependent type theory and dependent right adjoints

AbstractIn recent years, we have seen several new models of dependent type theory extended with some form of modal necessity operator, including nominal type theory, guarded and clocked type theory and spatial and cohesive type theory. In this paper, we study modal dependent type theory: dependent type theory with an operator satisfying (a dependent version of) the K axiom of modal logic. We investigate both semantics and syntax. For the semantics, we introduce categories with families with a dependent right adjoint (CwDRA) and show that the examples above can be presented as such. Indeed, we show that any category with finite limits and an adjunction of endofunctors give rise to a CwDRA via the local universe construction. For the syntax, we introduce a dependently typed extension of Fitch-style modal λ-calculus, show that it can be interpreted in any CwDRA, and build a term model. We extend the syntax and semantics with universes.

On Modal Logics of Model-Theoretic Relations

Given a class $$\mathcal {C}$$ of models, a binary relation $$\mathcal {R}$$ between models, and a model-theoretic language L, we consider the modal logic and the modal algebra of the theory of $$\mathcal {C}$$ in L where the modal operator is interpreted via $$\mathcal {R}$$ . We discuss how modal theories of $$\mathcal {C}$$ and $$\mathcal {R}$$ depend on the model-theoretic language, their Kripke completeness, and expressibility of the modality inside L. We calculate such theories for the submodel and the quotient relations. We prove a downward Lowenheim–Skolem theorem for first-order language expanded with the modal operator for the extension relation between models.

A new symbolic-based continuous (infinite) modal approach for systems control and operation using computational mathematics

This paper presents the new concept of symbolic-based continuous (infinite) modal approach for systems control and operation in varying parameters environment. In this approach, the system is run under varying parameters form where the variations are sensed and fed back in a continuous manner for determining using symbolic-based expressions the corresponding control functions. The approach permits the simultaneous manipulation of system control and operation with the online changes of system parameters. The generic symbolic approach enables high system operation flexibility, while the continuous (infinite) modal operation ensures complete smoothing system behavior and full operation modal compatibility all over various system varying parameters regions. The realization of the continuous (infinite) modal design is carried out through symbolic-based embedded control expressions and using computational mathematics. The proof of concept of the presented scheme is demonstrated through both an illustrative example and an application representing the implementation of control of inverted pendulum system with varying cart mass operation. It is successfully elucidated that the exact generic tracking of changing parameters could effectively be achieved in very smoothly and with complete continuous modal compatibility and full auto or self-modality. Moreover, the proposed scheme enables providing more performance flexibility compared to the known unimodal and multimodal operation approaches and adds a new dimension for future system control and operation specially in the fields of mechatronics and robotics where smooth and flexible system behavior are to be maintained all over the modes of system operation. Finally, the applications of the suggested approach to examples of parameters varying direct and networked cyber-physical systems are addressed such that the varying parameters are monitored online and continuously fed to the symbolic-control function to yield at the end the exact/generic flexible control scheme.

Algebraic semantics of the $$\left\{ \rightarrow ,\square \right\} $$-fragment of Propositional Lax Logic

In this paper, we will study a particular subvariety of Hilbert algebras with a modal operator $$\square $$, called Lax Hilbert algebras. These algebras are the algebraic semantic of the $$\left\{ \square ,\rightarrow \right\} $$-fragment of a particular intuitionistic modal logic, called Propositional Lax Logic ($$\mathcal {PLL}$$), which has applications to the formal verification of computer hardware. These algebras turn to be a generalization of the variety of Heyting algebras with a modal operator studied, under different names, by Macnab (Algebra Univ 12:5–29, 1981), Goldblatt (Math Logic Q 27(31–35):495–529, 1981; J Logic Comput 21(6):1035–1063, 2010) and by Bezhanishvili and Ghilardi (Ann Pure Appl Logic 147:84–100, 2007). We shall prove that the set of fixpoints of a Lax Hilbert algebra $$\left\langle A,\square \right\rangle $$ is a Hilbert algebra such that its dual space is homeomorphic to the subspace of reflexive elements of the dual space of A. We will define the notion of subframe of a Hilbert space $$\left\langle X,{\mathcal {K}}\right\rangle $$, and we will prove that there is a 1–1 correspondence between subframes of $$\left\langle X,{\mathcal {K}}\right\rangle $$ and binary relations $$Q\subseteq X\times X$$ such that $$\left\langle X,{\mathcal {K}},Q\right\rangle $$ is a Lax Hilbert space. In addition, we will define the notion of subframe variety and we will prove that any variety of Hilbert algebras is a subframe variety.

Mínimo e máximo em estruturas nominais – uma análise semântica

This paper is concerned with the semantics of the portuguese phrases with the form o mínimo/máximo N (‘the minimum N’) and o mínimo/máximo de N (‘the minimum/maximum of N’). Some nouns may occur in both of these constructions, while others might occur in only one of them, and still other nouns might occur only if accompanied by a modal operator. The proposal is made that these facts can be straightforwardly explained by the hypothesis that the first and the second of these syntactic constructions have, respectively, an extensional and an intensional meaning, together with the fact that some nouns have the same denotation in any possible world, while others denote different sets of entities in different possible worlds.

Grading Modal Judgement

AbstractThis paper proposes a new model of graded modal judgement. It begins by problematizing the phenomenon: given plausible constraints on the logic of epistemic modality, it is impossible to model graded attitudes toward modal claims as judgements of probability targeting epistemically modal propositions. This paper considers two alternative models, on which modal operators are non-proposition-forming: (1) Moss (2015), in which graded attitudes toward modal claims are represented as judgements of probability targeting a ‘proxy’ proposition, belief in which would underwrite belief in the modal claim; (2) a model on which graded attitudes toward modal claims are represented as judgements of credence taking as their objects (non-propositional) modal representations (rather than proxy propositions). The second model, like Moss’s model, is shown to be semantically and mathematically tractable. The second model, however, can be straightforwardly integrated into a plausible model of the role of graded attitudes toward modal claims in cognition and normative epistemology.

Modal Boolean Connexive Logics: Semantics and Tableau Approach

In this paper we investigate Boolean connexive logics in a language with modal operators: □, ◊. In such logics, negation, conjunction, and disjunction behave in a classical, Boolean way. Only implication is non-classical. We construct these logics by mixing relating semantics with possible worlds. This way, we obtain connexive counterparts of basic normal modal logics. However, most of their traditional axioms formulated in terms of modalities and implication do not hold anymore without additional constraints, since our implication is weaker than the material one. In the final section, we present a tableau approach to the discussed modal logics.

Topological and Multi-Topological Frames in the Context of Intuitionistic Modal Logic

We present three examples of topological semantics for intuitionistic modal logic with one modal operator □. We show that it is possible to treat neighborhood models, introduced earlier, as topological or multi-topological. From the neighborhood point of view, our method is based on differences between properties of minimal and maximal neighborhoods. Also we propose transformation of multitopological spaces into the neighborhood structures.

Polarity Semantics for Negation as a Modal Operator

The minimal weakening $${{\textsf {N}}}_0$$ of Belnap-Dunn logic under the polarity semantics for negation as a modal operator is formulated as a sequent system which is characterized by the class of all birelational frames. Some extensions of $${{\textsf {N}}}_0$$ with additional sequents as axioms are introduced. In particular, all three modal negation logics characterized by a frame with a single state are formalized as extensions of $${{\textsf {N}}}_0$$ . These logics have the finite model property and they are decidable.

STCMH with minimal semantic loss

Cross-modal hashing (CMH) has received widespread attention due to high retrieval efficiency, which plays an extremely important role in cross-modal retrieval. Recently, many CMH methods have been proposed to establish the semantic connection of different modalities. However, most of these methods only use a simple quantisation strategy, resulting in large quantisation error, and inferior hash codes. To address this issue, in this study, the authors propose a novel self-taught CMH (STCMH) to minimise the semantic encoding loss. In particular, the common semantic representations across different modalities are first learnt based on collective matrix factorisation. Then, the quantisation procedure based on orthogonal transformation is integrated to encode the semantic representations into discriminative binary codes. Moreover, similarity preservation is imposed to further boost the discriminative power. Finally, hashing functions learning is formulated as a binary classification problem by self-taught scheme. Experimental results on three public datasets demonstrate that STCMH significantly outperforms most state-of-the-art CMH methods.

Some Topological and Modal Operators over the Intuitionistic Fuzzy Sets of Cube Root Type

In this paper, we defined newly topological operators and modal-like operators and negation over Intuitionistic Fuzzy Sets of Cube Root Type (IFSCRT) are proposed and few theorems are proved. In addition, some of the basic properties of the new operators are discussed

Probabilistic divide & congruence: Branching bisimilarity

Since the seminal paper by Bloom, Fokkink and van Glabbeek, the Divide and Congruence technique allows for the derivation of compositional properties of nondeterministic processes from the SOS-based decomposition of their modal properties. In an earlier paper, we extended their technique to deal also with quantitative aspects of process behavior: we proved the (pre)congruence property for strong (bi)simulations on processes with nondeterminism and probability. In this paper we further extend our decomposition method to favor compositional reasoning with respect to probabilistic weak semantics. In detail, we consider probabilistic branching and rooted probabilistic branching bisimilarity, and we propose logical characterizations for them. These are strongly based on the modal operator 〈ε〉 which combines quantitative information and weak semantics by introducing a sort of probabilistic lookahead on process behavior. Our enhanced method will exploit distribution specifications, an SOS-like framework defining the probabilistic behavior of processes, to decompose this particular form of lookahead. We will show how we can apply the proposed decomposition method to derive congruence formats for the considered equivalences from their logical characterizations.

Modality, Modal Commitment and Modal Responsibility of the Speeches Delivered by the President of Sri Lanka at the UNO General Assembly: A Systemic Functional Analysis

The main aim of this paper is to unpack the types of modality used, and their level of commitment and responsibility in the selected sample. Also, this study scrutinizes the frequency of occurrence of modal expressions by employing the Systemic Functional Linguistic approach. As was found out, the first case does not reflect a higher frequency of modal expressions, but in the second case, it remains in a higher percentage by showing the commitment of the speaker towards the propositions and proposal of the speeches. Apart from typical modal verbal operators, the study found a considerable number of other modal expressions such as different mood adjuncts and adjectival expressions over the two speeches which also have been used to persuade the listeners of his assembly. The speaker’s commitment to the validity of messages is obviously manifested in both cases of the present study which illustrates how important modality in political discourse.

Pragmatic Hypotheses in the Evolution of Science

This paper introduces pragmatic hypotheses and relates this concept to the spiral of scientific evolution. Previous works determined a characterization of logically consistent statistical hypothesis tests and showed that the modal operators obtained from this test can be represented in the hexagon of oppositions. However, despite the importance of precise hypothesis in science, they cannot be accepted by logically consistent tests. Here, we show that this dilemma can be overcome by the use of pragmatic versions of precise hypotheses. These pragmatic versions allow a level of imprecision in the hypothesis that is small relative to other experimental conditions. The introduction of pragmatic hypotheses allows the evolution of scientific theories based on statistical hypothesis testing to be interpreted using the narratological structure of hexagonal spirals, as defined by Pierre Gallais.

A modal-based approach for estimating electric vehicle energy consumption in transportation networks

Electric vehicles (EVs) will play a central role in future energy-efficient and sustainable transportation systems. Compared to the operation of conventional vehicles, EVs provide significantly reduced energy consumption and lower operating costs. With conventional vehicles, fuel use ties directly to the instantaneous energy consumption required to provide motive power to the wheels. Predicting the energy use can be much more complex for EVs with hybridized powertrains because the onboard vehicle systems are trying to balance the provision of power to the wheels as well as manage the state of charge (SOC) of the battery pack. Traditional modeling methodologies for estimating real-world vehicle energy consumption either depend on numerical analysis of laboratory or on-road vehicle test data or the use of full-system simulation tools. Unfortunately, full-system simulation tools suffer from scaling problems in the context of large transportation network, necessitating the development of approaches that support large transportation network projections of modal EV operations and applicable modal energy use rates (energy use for various on-road modes of operation) to predict EV energy consumption.In this study, a new modal-based approach for estimating EV energy consumption is proposed. The modal-based approach considers the variance of vehicle operating conditions and supports energy estimation for large-scale transportation networks. The Department of Energy’s full-system vehicle simulation tool known as Autonomie® is used to generate energy consumption rates for specific simulations of on-road operating conditions. A sample of EV models was first developed in Autonomie® to simulate a wide range of operating conditions and generate energy use rates for selected EVs. Classification and regression tree (CART) analysis was then applied to the simulation output data to generate energy consumption rates under distinct on-road operating conditions, as represented by combinations of vehicle speed, acceleration rate, and battery state of charge (SOC). A large-scale regional travel demand analysis was performed for the Atlanta, GA metropolitan area, integrating a variety of EV market share scenarios. The CART-derived energy consumption rates are then applied to the model-predicted link-by-link traffic attributes to estimate fleet energy consumption. The modeling framework employs MOVES-embedded driving cycles to represent on-road operations for average speed operating conditions and random initial SOC start levels as model inputs. The model results suggest a 50% PHEV market share can achieve a 30% energy savings without significantly adding to electricity load. This modeling approach can be used to assess network-level energy use for a wide-variety of modeled transportation studies, such as evaluating transportation improvement plans, assessing the net impact on the electric grid, and forecasting the potential benefits of electrifying shared-autonomous vehicles under future scenarios.

Adversative versus concessive while-clauses in native and learner English texts: A corpus-based systemic functional description

AbstractDespite the commonly reported underuse of linking adverbials of contrast and concession (such as yet, nevertheless) by English as a Foreign Language (EFL) learners in writing, relatively little is known about the use of structural conjunctions in this regard. The present work uses a corpus approach to investigate the use of while, a polysemous conjunction of contrast and concession, in the writing of Chinese EFL learners as compared with their British native-speaker counterparts. The analysis of while-clauses is informed by clause-complexing and textual descriptions of clause in Systemic Functional Linguistics (Halliday and Matthiessen, 2013). Preference for initial concessive while-clauses by native-speaker students was found, in sharp contrast to the dominant use of final adversative while-clauses by Chinese EFL learners. Analysis of the native-speaker data revealed that initial concessive while-clause is characterized by equivalence or relatedness of topical themes of while-clause and its main clause, confirming the discourse-organizing function of thematic hypotactic clauses. In addition, the pattern of non-human subjects and low-value modal operators (e.g. While this … may …) associated dominantly and exclusively with initial concessive while-clauses in the native corpus serves further evidence of distinctive features of concessive while-clauses. This study adds to a growing body of literature on the SFL approach to second language writing and is among the first to combine corpus-based methodologies and SFL theoretical framework to analyze logico-semantic relations. The study concludes with some pedagogical implications for teaching adversative and concessive while-clauses to EFL learners.

The Modified Modal Operators over the Generalized Interval Valued Intuitionistic Fuzzy Sets

Interval valued intuitionistic fuzzy set (IVFS) as an extension of intuitionistic fuzzy sets is described by two parameters, namely membership degree and non-membership degree which are expressed in terms of intervals rather than crisp numbers. IVFS can be used to handle uncertainty and vagueness in real world decision making problems and operators of IVFSs have a key role in this filed. Thus, in this work we define newly defined modal operators over generalized interval valued intuitionistic fuzzy sets by modifying the existing operators. The new proposed operators are the integrity and comprehensive. Then, we describe the desirable properties of the proposed operators and discuss the special cases of them in details. Furthermore, the relationship between operators is examined. Finally, an illustrative example is provided for comparison.

Epistemic logic without closure

All standard epistemic logics legitimate something akin to the principle of closure, according to which knowledge is closed under competent deductive inference. And yet the principle of closure, particularly in its multiple premise guise, has a somewhat ambivalent status within epistemology. One might think that serious concerns about closure point us away from epistemic logic altogether—away from the very idea that the knowledge relation could be fruitfully treated as a kind of modal operator. This, however, need not be so. The abandonment of closure may yet leave in place plenty of formal structure amenable to systematic logical treatment. In this paper we describe a family of weak epistemic logics in which closure fails, and describe two alternative semantic frameworks in which these logics can be modelled. One of these—which we term plurality semantics—is relatively unfamiliar. We explore under what conditions plurality frames validate certain much-discussed principles of epistemic logic. It turns out that plurality frames can be interpreted in a very natural way in light of one motivation for rejecting closure, adding to the significance of our technical work. The second framework that we employ—neighbourhood semantics—is much better known. But we show that it too can be interpreted in a way that comports with a certain motivation for rejecting closure.

Hybrid Logic as extension of Modal and Temporal Logic

La lógica temporal fue creada por Arthur Prior para representar información temporal en un sistema lógico mediante operadores modales-temporales como P, F, H o G. Intuitivamente tales operadores pueden entenderse respectivamente como “fue alguna vez en el pasado...”, “será alguna vez en el futuro...”, “ha sido siempre en el pasado...” y “será siempre en el futuro...”. La evaluación de las fórmulas construidas a partir de ellos se lleva a cabo en semánticas kripkeanas y, de este modo, la lógica modal y la temporal están relacionadas. Sin embargo, aunque sus mecanismos permiten formalizar la información modal-temporal con cierta precisión, ambas lógicas adolecen de un problema de expresividad que la lógica híbrida es capaz de solventar. En efecto, uno de los problemas de la lógica modal reside en su incapacidad para nombrar puntos concretos dentro de un modelo. La lógica temporal, al basarse en ella, tampoco puede hacerlo. Pero la lógica de primer orden sí es capaz gracias a las constantes y a la relación de identidad. La lógica híbrida, que resulta de combinar la lógica modal con la lógica de primer orden, sería una solución a este problema. El principal objetivo de este artículo consiste en explicar el origen de la lógica híbrida a partir de la modal-temporal para mostrar qué añade a ambos sistemas en la representación de información, porqué es más expresiva que ellos y qué relación guarda con el lenguaje de correspondencia de la lógica de primer orden.

Modal Operators and Personal Pronouns in Roosevelt’s Inaugural Addresses

This paper tries to apply the interpersonal function of Halliday’s systemic functional grammar to analyze the interpersonal meanings of modal operators and personal pronouns in Roosevelt’s four inaugural addresses, that is, the dynamic and close relationship between the speaker and the hearers. By means of statistic method and stylistic analysis, this paper takes Roosevelt’s four inaugural speeches as objects of study, and mainly explores the interpersonal meanings of modal operators or linguistic forms so as to bring home Roosevelt’s linguistic techniques and stylistic effects. Roosevelt shows great interest in modal operators and personal pronouns to appeal to his audience’s emotional responses, to strengthen his in-group intimacy, and to convince his audience of his political purposes.