Modal Formula Research Articles

Characterising Modal Formulas with Examples

We study the existence of finite characterisations for modal formulas. A finite characterisation of a modal formula φ is a finite collection of positive and negative examples that distinguishes φ from every other, non-equivalent modal formula, where an example is a finite pointed Kripke structure. This definition can be restricted to specific frame classes and to fragments of the modal language: a modal fragment ℒ admits finite characterisations with respect to a frame class ℱ if every formula φ ∈ ℒ has a finite characterisation with respect to ℒ consisting of examples that are based on frames in ℱ. Finite characterisations are useful for illustration, interactive specification and debugging of formal specifications, and their existence is a precondition for exact learnability with membership queries. We show that the full modal language admits finite characterisations with respect to a frame class ℱ only when the modal logic of ℱ is locally tabular. We then study which modal fragments, freely generated by some set of connectives, admit finite characterisations. Our main result is that the positive modal language without the truth-constants ⊤ and ⊥ admits finite characterisations w.r.t. the class of all frames. This result is essentially optimal: finite characterisability fails when the language is extended with the truth constant ⊤ or ⊥ or with all but very limited forms of negation.

Taxonomic revision of Phoxinus minnows (Leuciscidae) from Caucasus, with description of a new narrow-ranged endemic species

Taxonomic revision of Phoxinus from the Caucasus revealed two distinct species. One species, P. colchicus, was known from eastern drainage of Black Sea, but was recorded also in the middle reach of the Kuban (Sea of Azov basin), for the first time. The Kuban population represents a genetically unique sub-lineage of P. colchicus. Its ancestors might have colonized the Kuban system through the event of ancient river capture. Another species inhabits only the Adagum River basin in the lower Kuban and represents a new narrow-ranged endemic species – Phoxinus adagumicus sp. nov. According to mtDNA phylogeny (COI and cytb), P. adagumicus sp. nov. represents deeply divergent and one of the two early branched lineages of the genus Phoxinus being distant to other species (min. p-distance = 0.074) including geographical neighbors – P. chrysoprasius from Crimean Peninsula and P. colchicus from the Caucasus. The new species differs from most Phoxinus species by frequently occurring single-row pharyngeal teeth (modal formula 5–4). The narrow geographic range (ca. 55 km in length and 15–20 km in width) and high anthropogenic load on local water systems suggests the new species is under threat and needs protection.

Variations on the Kripke Trick

AbstractIn the early 1960s, to prove undecidability of monadic fragments of sublogics of the predicate modal logic $$\textbf{QS5}$$ QS 5 that include the classical predicate logic $$\textbf{QCl}$$ QCl , Saul Kripke showed how a classical atomic formula with a binary predicate letter can be simulated by a monadic modal formula. We consider adaptations of Kripke’s simulation, which we call the Kripke trick, to various modal and superintuitionistic predicate logics not considered by Kripke. We also discuss settings where the Kripke trick does not work and where, as a result, decidability of monadic modal predicate logics can be obtained.

Modal parameter calculation and vibration characteristic analysis of transmission line conductor

AbstractStructural damage identification technology based on modal parameters is an extremely effective monitoring method. Because of the structural characteristics of the conductor, with its small diameter and long span, it is not easy to obtain the modal parameters by manual excitation. Here, a method for structural health monitoring of the conductor under wind load excitation is proposed. Firstly, the natural frequency change law of the conductor after broken strand damage is obtained by analyzing the conductor modal parameters calculation formula and IEEE standards. Then, the vibration characteristics are calculated by vibration experiments and finite element methods. Finally, through the broken strand experiment, this paper demonstrates how to determine the damage degree of the conductor using the modal parameters. And it is applied to online monitoring in the field. The results show that this method is convenient, fast, and accurate.

Equivalent static wind loads on canopies of regular railway stations

Wind-induced response, spectrums of the modal generalized force, and equivalent static wind loads (ESWLs) of railway station canopies with single-span and three-span frame are investigated. The results show that the first mode dominates horizontal displacements and frame column stresses, and the second mode dominates vertical displacements and frame beam stresses of the canopy frames. The unfavorable wind directions are determined for the center, near-center and end frames. Based on wind tunnel test results, parameters of simplified mean wind dynamic pressure coefficients and the generalized modal force spectrum formula for the first and second modes are proposed for different canopy span, opening widths and building height ratios of the station building to the canopy. Combining above-mentioned wind-induced response characteristics with the analysis method of universal ESWLs, the ESWLs are derived as the superposition of the peak inertia forces of the two dominating modes, and all nodal displacements and all member stresses under this unique ESWLs are equivalent to the actual dynamic peak responses. The numerical example verifies that all nodal displacements and member stresses calculated by the proposed semi-analytical formula simultaneously agree well with the actual dynamic peak response.

How did Avicenna understand the Barcan formulas?

Abstract In 2003 Zia Movahed pointed to a passage of Avicenna, written probably in 1022, which Movahed claimed anticipated the modal formula of Barcan (that ‘For every $x$ necessarily $\phi $’ entails ‘Necessarily for every $x$$\phi $’), and its converse. Since 2003, examination of early logical writings of Avicenna has clarified how he understood entailments between modal sentences, using his own new temporal language to provide a kind of semantics. In the light of that, Movahed’s claim for the Barcan formula needs some tidying up but is basically correct. But by his semantics Avicenna should not have claimed the converse Barcan formula and probably didn’t claim it.

Modal decompositions and point scatterer approximations near the Minnaert resonance frequencies

AbstractThis paper provides several contributions to the mathematical analysis of subwavelength resonances in a high‐contrast medium containing acoustic obstacles. Our approach is based on an exact decomposition formula which reduces the solution of the sound scattering problem to that of an dimensional linear system, and characterizes resonant frequencies as the solutions to an ‐dimensional nonlinear eigenvalue problem. Under a simplicity assumptions on the eigenvalues of the capacitance matrix, we prove the analyticity of the scattering resonances with respect to the square root of the contrast parameter, and we provide a deterministic algorithm allowing to compute all terms of the corresponding Puiseux series. We then establish a nonlinear modal decomposition formula for the scattered field as well as point scatterer approximations for the far‐field pattern of the sound wave scattered by bodies. As a prerequisite to our analysis, a first part of the work establishes various novel results about the capacitance matrix and its symmetry properties, since qualitative properties of the resonances, such as the leading order of the scattering frequencies or of the corresponding far‐field pattern are closely related to its spectral decomposition.

Tableau-based translation from first-order logic to modal logic

We define a procedure for translation of a given first-order formula to an equivalent modal formula, if such exists, by using tableau-based bisimulation invariance test. Previously developed tableau procedure tests bisimulation invariance of a given first-order formula, and therefore tests whether that formula is equivalent to the standard translation of some modal formula. Using a closed tableau as the starting point, we show how an equivalent modal formula can be effectively obtained.

Modelling and Verification Analysis of Cooperative and Non-Cooperative Games via a Modal Logic Approach

In game theory, a cooperative game (or coalitional game) is a game with competition between groups of players (coalitions) due to the possibility of external enforcement of cooperative behavior (e.g. through contract law). Those are opposed to non-cooperative games in which there is either no possibility to forge alliances or all agreements need to be self-enforcing (e.g. through credible threats). Cooperative games are often analyzed through the framework of cooperative game theory, which focuses on predicting which coalitions will form, the joint actions that groups take and the resulting collective payoffs. It is opposed to the traditional non-cooperative game theory which focuses on predicting individual players’ actions and payoffs and analyzing Nash equilibriums. In this work, the cooperative and non-cooperative game problem is modeled by means of a modal logic formula. Then, using the concept of logic implication, and transforming this logical implication relation into a set of clauses, a modal resolution qualitative method for verification (satisfiability) as well as performance issues, for some queries is applied.

Homeotic change in segment identity derives the human vertebral formula from a chimpanzee‐like one

One of the most contentious issues in paleoanthropology is the nature of the last common ancestor of humans and our closest living relatives, chimpanzees and bonobos (panins). The numerical composition of the vertebral column has featured prominently, with multiple models predicting distinct patterns of evolution and contexts from which bipedalism evolved. Here, we study total numbers of vertebrae from a large sample of hominoids to quantify variation in and patterns of regional and total numbers of vertebrae in hominoids. We compile and study a large sample (N=893) of hominoid vertebral formulae (numbers of cervical, thoracic, lumbar, sacral, caudal segments in each specimen) and analyze full vertebral formulae, total numbers of vertebrae, and super-regional numbers of vertebrae: presacral (cervical, thoracic, lumbar) vertebrae and sacrococcygeal vertebrae. We quantify within- and between-taxon variation using heterogeneity and similarity measures derived from population genetics. We find that humans are most similar to African apes in total and super-regional numbers of vertebrae. Additionally, our analyses demonstrate that selection for bipedalism reduced variation in numbers of vertebrae relative to other hominoids. The only proposed ancestral vertebral configuration for the last common ancestor of hominins and panins that is consistent with our results is the modal formula demonstrated by chimpanzees and bonobos (7 cervical-13 thoracic-4 lumbar-6 sacral-3 coccygeal). Hox gene expression boundaries suggest that a rostral shift in Hox10/Hox11-mediated complexes could produce the human modal formula from the proposal ancestral and panin modal formula.

On the Use of Causal Rules to Specify How Trust Impacts Change in Knowledge and Belief

We are concerned with models of trust, and the way that trust impacts changes in knowledge and belief. In order to address this problem, we consider transformations on Kripke structures in modal logic. We start with simple trust rules that specify how reported information impacts the truth of a modal formula. Trust rules are defined with respect to a particular source. So an individual trust rule basically indicates when a source’s reported information will cause the truth value of a modal formula to change. In the context of epistemic logic, this means that a trust rule indicates when a report will affect the knowledge or belief of the recipient. We demonstrate how a set of trust rules defines a model transformation in which the underlying accessibility relation is modified to ensure all rules are satisfied. This transformation captures how much the underlying source is trusted to impact the recipients perspective on the world. Model transformations of this kind are commonly used to capture belief change in Dynamic Epistemic Logic. What is distinct in our approach is that we show how simple rules can give a compact and flexible specification of trust in an information source. Our approach is also general, in the sense that we consider arbitrary modal change, which means we can capture different conceptions of knowledge and belief. We compare our approach with related work, particularly on the formaliziation of sensing actions. Future directions and applications are also considered.

Proof-search and countermodel generation for non-normal modal logics: The theorem prover PRONOM

In this work we present PRONOM, a theorem prover and countermodel generator for non-normal modal logics. PRONOM implements some labelled sequent calculi recently introduced for the basic system E and its extensions with axioms M, N, and C based on bi-neighbourhood semantics. PRONOM is inspired by the methodology of leanTAP and is implemented in Prolog. When a modal formula is valid, then PRONOM computes a proof (a closed tree) in the labelled calculi having a sequent with an empty left-hand side and containing only that formula on the right-hand side as a root, otherwise PRONOM is able to extract a model falsifying it from an open, saturated branch. The paper shows some experimental results, witnessing that the performances of PRONOM are promising.

Tableau-based translation from first-order logic to modal logic

We define a procedure for translating a given first-order formula to an equivalent modal formula, if one exists, by using tableau-based bisimulation invariance test. A previously developed tableau procedure tests bisimulation invariance of a given first-order formula, and therefore tests whether that formula is equivalent to the standard translation of some modal formula. Using a closed tableau as the starting point, we show how an equivalent modal formula can be effectively obtained.

Crack Detection in Plate‐Like Structures Using Modal Strain Energy Method considering Various Boundary Conditions

Among vibration‐based damage detection methods, one of the effective approaches for damage localization is the modal strain energy method. In this paper, the modal strain energy method is developed for damage detection in plate‐like structures with various boundary conditions. Firstly, the theory of the modal strain energy method is briefly outlined. In order to overcome the limitation of measuring points, a central difference method is newly employed to compute the partial differential terms in the modal strain energy formula. Finite element analysis is conducted on an aluminum thin plate to obtain the mode shapes before and after the occurrence of damage. Feasibility of the proposed method is verified by investigating plates with different types of boundary conditions. A damage index is presented to identify the location and extent of crack in the plate‐like structures. The analytical results show that the proposed method accurately identifies the crack in the plate structure with various types of boundary conditions by using appropriate mode shapes and damage threshold.

Understanding the material loss of anti-resonant hollow-core fibers.

In this paper, the material loss of anti-resonant hollow-core fiber (AR-HCF) and its properties are studied. We revisit the formula of power attenuation coefficient for the index-guiding optical fiber described by Snyder and Love in the 1980s and derive the modal overlap factor that governs the material loss of hollow-core fibers (HCF). The modal overlap factor formula predicts the material loss of AR-HCF, which agrees with numerical simulations by the finite element method. The optimization of silica-based AR-HCF design for the lowest loss at 4 µm wavelength is numerically discussed where the silica absorption reaches over 800 dB/m. Our work would provide practical guidance to develop low-loss AR-HCF at highly absorptive wavelengths, e.g. in the vacuum UV and mid/far-infrared spectral regions.

Disjunctive Normal Form for Multi-Agent Modal Logics Based on Logical Separability

Modal logics are primary formalisms for multi-agent systems but major reasoning tasks in such logics are intractable, which impedes applications of multi-agent modal logics such as automatic planning. One technique of tackling the intractability is to identify a fragment called a normal form of multiagent logics such that it is expressive but tractable for reasoning tasks such as entailment checking, bounded conjunction transformation and forgetting. For instance, DNF of propositional logic is tractable for these reasoning tasks. In this paper, we first introduce a notion of logical separability and then define a novel disjunctive normal form SDNF for the multiagent logic Kn, which overcomes some shortcomings of existing approaches. In particular, we show that every modal formula in Kn can be equivalently casted as a formula in SDNF, major reasoning tasks tractable in propositional DNF are also tractable in SDNF, and moreover, formulas in SDNF enjoy the property of logical separability. To demonstrate the usefulness of our approach, we apply SDNF in multi-agent epistemic planning. Finally, we extend these results to three more complex multi-agent logics Dn, K45n and KD45n.

Analysis on circumferential natural frequencies of stator in permanent magnet synchronous motor

Accurate calculation on circumferential natural frequencies of stator structure in permanent magnet synchronous motor (PMSM) is very important for predicting the vibration and noise of PMSM. The analytical modal formula for the stator structure in PMSM is proposed, starting from the mathematical displacement model of cylindrical shell, then deriving the analytical formula for the stator structure. Firstly, finite element method (FEM) is applied to analyze the circumferential natural frequencies of cylindrical shells with different length, wall thickness, mean diameter, diameter-thickness ratio, size and installation position of terminal box. The influences on circumferential natural frequencies of cylindrical shells are studied. Furthermore, a new analytical formula considering diameter-thickness ratio for calculating circumferential natural frequencies of cylindrical shells is developed. Compared with the results of FEM, the correctness of the analytical formula is verified. Then, novel analytical formulas considering stiffness errors are proposed to calculate the circumferential natural frequencies of stator core and motor case. Compared with the results of FEM, the effectiveness of analytical method is validated. Finally, a new analytical formula considering diameter-thickness ratio, stiffness errors, existence of terminal box and mutual constraints is presented to calculate the circumferential natural frequencies of stator structure. Compared with the results of FEM and experimental method, the relative errors are respective within |4.43 %| and |3.38 %|. The validity of analytical formula is confirmed.

Forgetting in multi-agent modal logics

In the past decades, forgetting has been investigated for many logics and has found many applications in knowledge representation and reasoning. In this paper, we study forgetting in multi-agent modal logics. We adopt the semantic definition of existential bisimulation quantifiers as that of forgetting. We resort to canonical formulas of modal logics introduced by Moss. An arbitrary modal formula is equivalent to the disjunction of a unique set of satisfiable canonical formulas. We show that, for the logics of Kn, Dn, Tn, K45n, KD45n and S5n, the result of forgetting an atom from a satisfiable canonical formula can be computed by simply substituting the literals of the atom with ⊤. Thus we show that these logics are closed under forgetting, and hence have uniform interpolation. Finally, we generalize the above results to include common knowledge of propositional formulas.

From Wittgenstein’s N-operator to a New Notation for Some Decidable Modal Logics

Wittgenstein’s N-operator is a ‘primitive sign’ which shows every complex proposition is the result of the truth-functional combination of a finite number of component propositions, and thus provides a mechanical method to determine logical truth. The N-operator can be interpreted as a generalized Sheffer stroke. In this paper, I introduce a new ‘primitive sign’ that is a hybrid of generalized Sheffer stroke and modality, and give a uniform expression for modal formulas. The general form of modal formula in the new notation is [A0···An−1; B0···Bm−1], which is semantically equivalent to ¬A0∨···∨¬ An−1∨◊ (¬B0∨···∨¬Bm−1). Based on this new notation, I propose several analytic axiomatic systems for some decidable modal logics. Every axiom of these analytic systems is an ‘Atomic-Sheffer’, which is the result of the combination of a finite number of component propositions. The inferential rules are analytic in that the set of elementary propositions that are combined in the premiss overlaps the set of elementary propositions combined in the conclusion, in virtue of which every complex proposition can be reduced to an ‘Atomic-Sheffer’ at the ultimate level. The analytic modal systems have the same classical inferential rules. Different modal systems can be built by adding special modal inferential rules. In an analytic system for modal logic L, valid formulas on L-models can be proved by a purely mechanical method.

Logical Time and Space of the Network Intrusion

Nowadays, one of the biggest threats for modern computer networks are the cyber attacks. One of the possible ways how to increase the level of computer networks security is a deployment of a network intrusion detection system. This paper deals with the behavior of the network intrusion detection system during specific network intrusion. We formally describe this network intrusion by the modal linear logic formula. Based on this formula, logical space and logical time is expressed from the attacker, and the network environment point of view in the usage of the Ludics theory.