Varieties of Cost Functions.

Regular cost functions were introduced as a quantitative generalisation of regular languages, retaining many of their equivalent characterisations and decidability properties. For instance, stabilisation monoids play the same role for cost functions as monoids do for regular languages. The purpose of this article is to further extend this algebraic approach by generalising two results on regular languages to cost functions: Eilenberg's varieties theorem and profinite equational characterisations of lattices of regular languages. This opens interesting new perspectives, but the specificities of cost functions introduce difficulties that prevent these generalisations to be straightforward. In contrast, although syntactic algebras can be defined for formal power series over a commutative ring, no such notion is known for series over semirings and in particular over the tropical semiring

Varieties of Cost Functions

Regular cost functions were introduced as a quantitative generalisation of regular languages, retaining many of their equivalent characterisations and decidability properties. For instance, stabilisation monoids play the same role for cost functions as monoids do for regular languages. The purpose of this article is to further extend this algebraic approach by generalising two results on regular languages to cost functions: Eilenberg\u27s varieties theorem and profinite equational characterisations of lattices of regular languages. This opens interesting new perspectives, but the specificities of cost functions introduce difficulties that prevent these generalisations to be straightforward. In contrast, although syntactic algebras can be defined for formal power series over a commutative ring, no such notion is known for series over semirings and in particular over the tropical semiring

Eilenberg Theorems for Free

Eilenberg-type correspondences, relating varieties of languages (e.g. of finite words, infinite words, or trees) to pseudovarieties of finite algebras, form the backbone of algebraic language theory. Numerous such correspondences are known in the literature. We demonstrate that they all arise from the same recipe: one models languages and the algebras recognizing them by monads on an algebraic category, and applies a Stone-type duality. Our main contribution is a variety theorem that covers e.g. Wilke's and Pin's work on $\infty$-languages, the variety theorem for cost functions of Daviaud, Kuperberg, and Pin, and unifies the two previous categorical approaches of Boja\'nczyk and of Ad\'amek et al. In addition we derive a number of new results, including an extension of the local variety theorem of Gehrke, Grigorieff, and Pin from finite to infinite words

Performance Evaluation of Selected Cost Functions in Non Negative Matrix Factorization Based Decomposition of Acoustic Mixture

Interaction of acoustic signals when several audio sources are active simultaneously results in the disturbance of estimation of an individual source by co-occurring sounds. Data decomposition therefore constitutes one of the core tasks in monaural source separation.  Particularly, in semi-supervised learning approach, viable means of achieving this is through the application of Non-negative Matrix Factorization (NMF). Owing to a paucity of information on the application of this method, especially in a speech system, evaluation of some cost functions in NMF-based monaural speech decomposition was investigated in this study. A generalized gradient descent algorithm is derived for the minimization while three cost functions: Euclidean Distance, Kullback-Leibler Divergence and Itakura-Saito divergences are applied to the derived separation NMF algorithm.  These divergences are evaluated using experimental data while the performance of each of these is evaluated based on the cost values and convergence rate. Itakura-Saito divergence yields optimal performance over the other two divergences for given number of iterations and number of channels. Keywords— Cost functions, non-negative matrix factorization, speech separation, evaluatio

Two Tank Level Control Systems Using Dynamic Matrix Control and Study of Its Tuning Parameter

Liquid level has a major role in the process industries especially in chemical plants, pharmaceutical industries, etc and the controlling of this parameter is a big task. Every time controlling manually is not possible therefore by using simulation methods, these targets are achieved. This project deals with the Two Tank Systems on which Dynamic Matrix Controller algorithm has been applied for the simulation work. Effect of tuning parameters such as prediction horizon (p), control horizon (m) and model length (n) are studied and data were collected. Different performances i.e. rise time, settling time etc. were seen for different varying tuning parameters. Also the Empirical Formula has been derived, based on the observations for optimal performance of the system. Process parameters have been changed and best performance has been observed with respect to the time constant. This project work manages the study and examination of DMC for the given Tank System under MATLAB simulation windo

Álgebras de estabilização

Tese de mestrado, Matemática, Universidade de Lisboa, Faculdade de Ciências, 2018Este trabalho explora uma generalização da teoria algébrica das Linguagens Formais. Tendo os trabalhos de Thomas Colcombet e de Laure Daviaud, Denis Kuperberg e Jean-Éric Pin sobre funções de custo como ponto de partida, apresentamos os conceitos de ideal de ordem, álgebra de estabilização e autómato de estabilização. Obtemos generalizações de resultados conhecidos no âmbito das Linguagens Formais, como por exemplo do Teorema de Eilenberg e do Teorema de Schützenberger sobre identidades associadas a variedades, e também obtemos uma resposta para o Problema da Igualdade. Provamos também que o conceito de ideal de ordem reconhecível é uma generalização do conceito de função custo reconhecível pelo que a teoria desenvolvida se aplica também no estudo das funções de custo.This thesis explores a generalisation of the algebraic theory of Formal Languages. Having the work of Thomas Colcombet and of Laure Daviaud, Denis Kuperberg and Jean-Éric Pin about cost functions as a starting point, we present the concepts of ordered ideal, stabilisation algebra and stabilisation automata. We generalize various known results in Formal Languages, for example, the Eilenberg Theorem and the Schützenberger Theorem about identities associated to varieties, besides we answer to the Equality Problem. We also prove that the concept of recognisable ordered ideal is a generalisation of the concept of recognizable cost function, whence the theory developed in this thesis also applies to the study of cost functions

Dualities in modal logic

Categorical dualities are an important tool in the study of (modal) logics. They offer conceptual understanding and enable the transfer of results between the different semantics of a logic. As such, they play a central role in the proofs of completeness theorems, Sahlqvist theorems and Goldblatt-Thomason theorems. A common way to obtain dualities is by extending existing ones. For example, Jonsson-Tarski duality is an extension of Stone duality. A convenient formalism to carry out such extensions is given by the dual categorical notions of algebras and coalgebras. Intuitively, these allow one to isolate the new part of a duality from the existing part. In this thesis we will derive both existing and new dualities via this route, and we show how to use the dualities to investigate logics. However, not all (modal logical) paradigms fit the (co)algebraic perspective. In particular, modal intuitionistic logics do not enjoy a coalgebraic treatment, and there is a general lack of duality results for them. To remedy this, we use a generalisation of both algebras and coalgebras called dialgebras. Guided by the research field of coalgebraic logic, we introduce the framework of dialgebraic logic. We show how a large class of modal intuitionistic logics can be modelled as dialgebraic logics and we prove dualities for them. We use the dialgebraic framework to prove general completeness, Hennessy-Milner, representation and Goldblatt-Thomason theorems, and instantiate this to a wide variety of modal intuitionistic logics. Additionally, we use the dialgebraic perspective to investigate modal extensions of the meet-implication fragment of intuitionistic logic. We instantiate general dialgebraic results, and describe how modal meet-implication logics relate to modal intuitionistic logics