Full Nonassociative Lambek Calculus with Modalities and Its Applications in Type Grammars

Wydział Matematyki i InformatykiRozprawa jest poświęcona pełnemu niełącznemu rachunkowi Lambeka wzbogaconemu o różne modalności. Te systemy tworzą pewną rodzinę logik substrukturalnych. W rozprawie badamy rachunki NL (niełączny rachunek Lambeka), DFNL (pełny niełączny rachunek Lambeka z prawami dystrybutywności dla operacji kratowych) i BFNL (DFNL z negacją spełniającą prawa algebr Boole’a) oraz ich rozszerzenia o operatory modalne, tworzące parę rezyduacji i spełniające standardowe aksjomaty logik modalnych (T), (4) i (5). Rozważamy też gramatyki typów oparte na tych rachunkach. Główne wyniki: twierdzenie o eliminacji cięć dla modalnych rozszerzeń NL z założeniami, wielomianowa złożoność relacji konsekwencji dla tych systemów, lemat interpolacyjny dla modalnych rozszerzeń DFNL i BFNL z założeniami, silna własność skończonego modelu dla tych systemów, rozstrzygalność relacji konsekwencji dla tyc systemów, PSPACE-zupełność rachunku BFNL, bezkontekstowość języków generowanych przez gramatyki typów oparte na tych rachunkach. Rozprawa kontynuuje wcześniejsze badania W. Buszkowskiego, M. Farulewskiego, M. Moortgata, A.. Plummera, N. Kurtoniny i innych.The thesis is devoted to full nonassociative Lambek calculus enriched with different modalities. These systems form a family of substrutural logics. In this thesis we study systems NL (nonassociative Lambek calculus), DFNL (full nonassociative Lambek calculus with the distributive laws for lattice operations) and BFNL (DFNL with negation satisfying the laws of Boolean algebras) and their extensions by modal operators, being a residuation pair and fulfilling standard axioms of modal logics (T), (4), (5). We also consider the type grammars based on these calculi. Main results: the cut-elimination theorem for modal extensions of NL with assumptions, the polynomial-time complexity of the consequence relations for these systems, an interpolation lemma for modal extensions of DFNL and BFNL with assumptions, the strong finite model property of the latter systems, the decidability of the consequence relations for the latter systems, the PSPACE-completeness of BFNL, the context-freeness of the languages generated by the type grammars based on these systems. The thesis continues some research of W. Buszkowski, M. Farulewski, M. Moortgat, A. Plummer,, N. Kurtonina and others

Dual-Context Calculi for Modal Logic

We present natural deduction systems and associated modal lambda calculi for the necessity fragments of the normal modal logics K, T, K4, GL and S4. These systems are in the dual-context style: they feature two distinct zones of assumptions, one of which can be thought as modal, and the other as intuitionistic. We show that these calculi have their roots in in sequent calculi. We then investigate their metatheory, equip them with a confluent and strongly normalizing notion of reduction, and show that they coincide with the usual Hilbert systems up to provability. Finally, we investigate a categorical semantics which interprets the modality as a product-preserving functor.Comment: Full version of article previously presented at LICS 2017 (see arXiv:1602.04860v4 or doi: 10.1109/LICS.2017.8005089

Sensitivity study of load-dependent Ritz vectors on modal and seismic responses of cable stayed bridges

In the present article, 3D Finite Element Model (FEM) of a bridge structure under load dynamics is performed in order to assess the sensitivity study of Load-Dependant Ritz vectors (LDR) on modal and seismic responses of cable stayed bridges. In this context, two techniques are examined in the present study for solving structural dynamics problems; the Traditional Modal Superposition (TMS) technique and that of Load-Dependent Ritz orthogonal vectors (LDR). The latter is based on a very efficient algorithm allowing the systematic generation of Load-Dependent Ritz orthogonal vectors (LDR), the accuracy of this method is significantly influenced by the selection of LDR vectors used for the modeling of the structural behavior. The cable-stayed bridge connecting two districts in eastern Algeria, characterized by an expected Peak Ground Acceleration (PGA) equal to 0.275g in accordance with Algerian seismic design code is selected in order to perform critical modal properties such as, frequencies, shapes of the required vibration modes and effective mass participation as well as the dynamic response of the cable stayed bridge under earthquake loadings in three orthogonal directions (longitudinal, transversal and vertical). The results of this study reveal that the LDR vectors method which has the important advantages of short Central Processing Unit (CPU) time as compared to traditional modal method is very efficient for modal and seismic analyses of cable stayed bridges

Scopes and Limits of Modality in Quantum Mechanics

We develop an algebraic frame for the simultaneous treatment of actual and possible properties of quantum systems. We show that, in spite of the fact that the language is enriched with the addition of a modal operator to the orthomodular structure, contextuality remains a central feature of quantum systems.Comment: 9 pages, no figure

On the need for a nonlinear subscale turbulence term in POD models as exemplified for a high Reynolds number flow over an Ahmed body

We investigate a hierarchy of eddy-viscosity terms in POD Galerkin models to account for a large fraction of unresolved fluctuation energy. These Galerkin methods are applied to Large Eddy Simulation data for a flow around the vehicle-like bluff body call Ahmed body. This flow has three challenges for any reduced-order model: a high Reynolds number, coherent structures with broadband frequency dynamics, and meta-stable asymmetric base flow states. The Galerkin models are found to be most accurate with modal eddy viscosities as proposed by Rempfer & Fasel (1994). Robustness of the model solution with respect to initial conditions, eddy viscosity values and model order is only achieved for state-dependent eddy viscosities as proposed by Noack, Morzynski & Tadmor (2011). Only the POD system with state-dependent modal eddy viscosities can address all challenges of the flow characteristics. All parameters are analytically derived from the Navier-Stokes based balance equations with the available data. We arrive at simple general guidelines for robust and accurate POD models which can be expected to hold for a large class of turbulent flows.Comment: Submitted to the Journal of Fluid Mechanic

fMRI Evidence for Modality-Specific Processing of Conceptual Knowledge on Six Modalities

Traditional theories assume that amodal representations, such as feature lists and semantic networks, represent conceptual knowledge about the world. According to this view, the sensory, motor, and introspective states that arise during perception and action are irrelevant to representing knowledge. Instead the conceptual system lies outside modality-specific systems and operates according to different principles. Increasingly, however, researchers report that modality-specific systems become active during purely conceptual tasks, suggesting that these systems play central roles in representing knowledge (for a review, see Martin, 2001, Handbook of Functional Neuroimaging of Cognition). In particular, researchers report that the visual system becomes active while processing visual properties, and that the motor system becomes active while processing action properties. The present study corroborates and extends these findings. During fMRI, subjects verified whether or not properties could potentially be true of concepts (e.g., BLENDER-loud). Subjects received only linguistic stimuli, and nothing was said about using imagery. Highly related false properties were used on false trials to block word association strategies (e.g., BUFFALOwinged). To assess the full extent of the modality-specific hypothesis, properties were verified on each of six modalities. Examples include GEMSTONE-glittering (vision), BLENDER-loud (audition), FAUCET-turned (motor), MARBLE-cool (touch), CUCUMBER-bland (taste), and SOAP-perfumed (smell). Neural activity during property verification was compared to a lexical decision baseline. For all six sets of the modalityspecific properties, significant activation was observed in the respective neural system. Finding modality-specific processing across six modalities contributes to the growing conclusion that knowledge is grounded in modality-specific systems of the brain

Nonlinear normal modes, modal interactions and isolated resonance curves

The objective of the present study is to explore the connection between the nonlinear normal modes of an undamped and unforced nonlinear system and the isolated resonance curves that may appear in the damped response of the forced system. To this end, an energy balancing technique is used to predict the amplitude of the harmonic forcing that is necessary to excite a specific nonlinear normal mode. A cantilever beam with a nonlinear spring at its tip serves to illustrate the developments. The practical implications of isolated resonance curves are also discussed by computing the beam response to sine sweep excitations of increasing amplitudes.Comment: Journal pape

Approximate Dynamic Response of Light Secondary Systems

John I. Parcel Fund