An Arithmetization of Logical Oppositions

An arithmetic theory of oppositions is devised by comparing expressions, Boolean bitstrings, and integers. This leads to a set of correspondences between three domains of investigation, namely: logic, geometry, and arithmetic. The structural properties of each area are investigated in turn, before justifying the procedure as a whole. Io finish, I show how this helps to improve the logical calculus of oppositions, through the consideration of corresponding operations between integers

Complex Clause of Bahasa Indonesia From the Point of View of Systemic Functional Linguistic

This paper discusses the meaning of the complex clause by a series of processes that are combined in a logical of the two clauses. Complex clause can be combined through one of two logical-semantic relationships that is expansion or projection. Systemic Functional Linguistics approach is used to describe the five texts taken at random. The five texts are: (1) Gembala dan malaikat, (2) Pendahuluan, (3) Melakukan Studi Gender dalam Bahasa, (4) Korporasi, Kerja dan Kultur, dan (5) Gara-gara Dilarang Bertemu (1) Shepherd and angels, (2) Introduction, (3) Doing Gender in Language Studies, (4) Corporations, Employment and Culture, and (5) Due to Divorce, No Meet Pets. The analysis of the five texts is not opposed but complementary to one another. By using a qualitative descriptive method, it was found two types of logical-semantic relations: (1) Expansion, and (2) projection. Integrating through one of the logical-semantic relations: expansion or projection is realized through a system of mutual dependence or taxis that can be divided into paratactic and hypotactic. Key words: Clause complex, expansion, projectio

Codensity Lifting of Monads and its Dual

We introduce a method to lift monads on the base category of a fibration to its total category. This method, which we call codensity lifting, is applicable to various fibrations which were not supported by its precursor, categorical TT-lifting. After introducing the codensity lifting, we illustrate some examples of codensity liftings of monads along the fibrations from the category of preorders, topological spaces and extended pseudometric spaces to the category of sets, and also the fibration from the category of binary relations between measurable spaces. We also introduce the dual method called density lifting of comonads. We next study the liftings of algebraic operations to the codensity liftings of monads. We also give a characterisation of the class of liftings of monads along posetal fibrations with fibred small meets as a limit of a certain large diagram.Comment: Extended version of the paper presented at CALCO 2015, accepted for publication in LMC

Automata Minimization: a Functorial Approach

In this paper we regard languages and their acceptors - such as deterministic or weighted automata, transducers, or monoids - as functors from input categories that specify the type of the languages and of the machines to categories that specify the type of outputs. Our results are as follows: A) We provide sufficient conditions on the output category so that minimization of the corresponding automata is guaranteed. B) We show how to lift adjunctions between the categories for output values to adjunctions between categories of automata. C) We show how this framework can be instantiated to unify several phenomena in automata theory, starting with determinization, minimization and syntactic algebras. We provide explanations of Choffrut's minimization algorithm for subsequential transducers and of Brzozowski's minimization algorithm in this setting.Comment: journal version of the CALCO 2017 paper arXiv:1711.0306

Feedback and generalized logic

Although the distinction between software and hardware is a posteriori, there is an a priori distinction that masquerades as the software—hardware distinction. This is the distinction between procedure interconnection, the semantics of flow chart diagrams, which is known to be described by the regular expression calculus; and system interconnection, the semantics of network diagrams, which is described by a certain logical calculus, dual to a calculus of regular expressions. This paper presents a proof of the duality in a special case, and gives the interpretation of the logical calculus for sequential machine interconnection. A minimal realization theorem for feedback systems is proved, which specializes to known open loop minimal realization theorems

Strongly Complete Logics for Coalgebras

Coalgebras for a functor model different types of transition systems in a uniform way. This paper focuses on a uniform account of finitary logics for set-based coalgebras. In particular, a general construction of a logic from an arbitrary set-functor is given and proven to be strongly complete under additional assumptions. We proceed in three parts. Part I argues that sifted colimit preserving functors are those functors that preserve universal algebraic structure. Our main theorem here states that a functor preserves sifted colimits if and only if it has a finitary presentation by operations and equations. Moreover, the presentation of the category of algebras for the functor is obtained compositionally from the presentations of the underlying category and of the functor. Part II investigates algebras for a functor over ind-completions and extends the theorem of J{\'o}nsson and Tarski on canonical extensions of Boolean algebras with operators to this setting. Part III shows, based on Part I, how to associate a finitary logic to any finite-sets preserving functor T. Based on Part II we prove the logic to be strongly complete under a reasonable condition on T