A mathematical study of fuzzy logic; an algebraic approach

Fuzzy set theory and Fuzzy logic is studied from a mathematical point of view. The main goal is to investigatecommon mathematical structures in various fuzzy logical inference systems and to establish a general mathematical basis for fuzzy logic when considered as multi-valued logic. The study is composed of six distinct publications. The first paper deals with Mattila'sLPC+Ch Calculus. THis fuzzy inference system is an attempt to introduce linguistic objects to mathematical logic without defining these objects mathematically.LPC+Ch Calculus is analyzed from algebraic point of view and it is demonstratedthat suitable factorization of the set of well formed formulae (in fact, Lindenbaum algebra) leads to a structure called ET-algebra and introduced in the beginning of the paper. On its basis, all the theorems presented by Mattila and many others can be proved in a simple way which is demonstrated in the Lemmas 1 and 2and Propositions 1-3. The conclusion critically discusses some other issues of LPC+Ch Calculus, specially that no formal semantics for it is given.In the second paper the characterization of solvability of the relational equation RoX=T, where R, X, T are fuzzy relations, X the unknown one, and o the minimum-induced composition by Sanchez, is extended to compositions induced by more general products in the general value lattice. Moreover, the procedure also applies to systemsof equations. In the third publication common features in various fuzzy logicalsystems are investigated. It turns out that adjoint couples and residuated lattices are very often present, though not always explicitly expressed. Some minor new results are also proved.The fourth study concerns Novak's paper, in which Novak introduced first-order fuzzy logic and proved, among other things, the semantico-syntactical completeness of this logic. He also demonstrated that the algebra of his logic is a generalized residuated lattice. In proving that the examination of Novak's logic can be reduced to the examination of locally finite MV-algebras.In the fifth paper a multi-valued sentential logic with values of truth in an injective MV-algebra is introduced and the axiomatizability of this logic is proved. The paper developes some ideas of Goguen and generalizes the results of Pavelka on the unit interval. Our proof for the completeness is purely algebraic. A corollary of the Completeness Theorem is that fuzzy logic on the unit interval is semantically complete if, and only if the algebra of the valuesof truth is a complete MV-algebra. The Compactness Theorem holds in our well-defined fuzzy sentential logic, while the Deduction Theorem and the Finiteness Theorem do not. Because of its generality and good-behaviour, MV-valued logic can be regarded as a mathematical basis of fuzzy reasoning. The last paper is a continuation of the fifth study. The semantics and syntax of fuzzy predicate logic with values of truth in ana injective MV-algerba are introduced, and a list of universally valid sentences is established. The system is proved to be semanticallycomplete. This proof is based on an idea utilizing some elementary properties of injective MV-algebras and MV-homomorphisms, and is purely algebraic.

Kvantiteetti ja aika. 2 : Nominaalinen aspekti ja suomen predikatiivin sijanvaihtelu

Abstract: Quantity and time. 2 : Nominal aspect and the case marking of the Finnish predicate nominal

Teaching mathematics and programming : new approaches with empirical evaluation

Programming and mathematics are core areas of computer science (CS) and consequently also important parts of CS education. Introductory instruction in these two topics is, however, not without problems. Studies show that CS students find programming difficult to learn and that teaching mathematical topics to CS novices is challenging. One reason for the latter is the disconnection between mathematics and programming found in many CS curricula, which results in students not seeing the relevance of the subject for their studies. In addition, reports indicate that students' mathematical capability and maturity levels are dropping. The challenges faced when teaching mathematics and programming at CS departments can also be traced back to gaps in students' prior education. In Finland the high school curriculum does not include CS as a subject; instead, focus is on learning to use the computer and its applications as tools. Similarly, many of the mathematics courses emphasize application of formulas, while logic, formalisms and proofs, which are important in CS, are avoided. Consequently, high school graduates are not well prepared for studies in CS. Motivated by these challenges, the goal of the present work is to describe new approaches to teaching mathematics and programming aimed at addressing these issues: Structured derivations is a logic-based approach to teaching mathematics, where formalisms and justifications are made explicit. The aim is to help students become better at communicating their reasoning using mathematical language and logical notation at the same time as they become more confident with formalisms. The Python programming language was originally designed with education in mind, and has a simple syntax compared to many other popular languages. The aim of using it in instruction is to address algorithms and their implementation in a way that allows focus to be put on learning algorithmic thinking and programming instead of on learning a complex syntax. Invariant based programming is a diagrammatic approach to developing programs that are correct by construction. The approach is based on elementary propositional and predicate logic, and makes explicit the underlying mathematical foundations of programming. The aim is also to show how mathematics in general, and logic in particular, can be used to create better programs.

Scheduling of guarded command based models

Formal methods provide a means of reasoning about computer programs in order to prove correctness criteria. One subtype of formal methods is based on the weakest precondition predicate transformer semantics and uses guarded commands as the basic modelling construct. Examples of such formalisms are Action Systems and Event-B. Guarded commands can intuitively be understood as actions that may be triggered when an associated guard condition holds. Guarded commands whose guards hold are nondeterministically chosen for execution, but no further control flow is present by default. Such a modelling approach is convenient for proving correctness, and the Refinement Calculus allows for a stepwise development method. It also has a parallel interpretation facilitating development of concurrent software, and it is suitable for describing event-driven scenarios. However, for many application areas, the execution paradigm traditionally used comprises more explicit control flow, which constitutes an obstacle for using the above mentioned formal methods. In this thesis, we study how guarded command based modelling approaches can be conveniently and efficiently scheduled in different scenarios. We first focus on the modelling of trust for transactions in a social networking setting. Due to the event-based nature of the scenario, the use of guarded commands turns out to be relatively straightforward. We continue by studying modelling of concurrent software, with particular focus on compute-intensive scenarios. We go from theoretical considerations to the feasibility of implementation by evaluating the performance and scalability of executing a case study model in parallel using automatic scheduling performed by a dedicated scheduler. Finally, we propose a more explicit and non-centralised approach in which the flow of each task is controlled by a schedule of its own. The schedules are expressed in a dedicated scheduling language, and patterns assist the developer in proving correctness of the scheduled model with respect to the original one.

Ilmoitusmenettely rahanpesun torjumiseksi

Rahanpesu on rikollisesta toiminnasta hankitun varallisuuden siirtoa lailliseen talousjärjestelmään. Rahanpesu on liitännäinen teko ja se edellyttää aina esirikosta. Esirikos voi olla mikä tahansa rikos, josta varallisuutta hankitaan - huumausainerikos tai talousrikos. Tutkielmassa on selvitetty lainopillista menetelmää käyttäen Suomen rahanpesulain ja ilmoitusmenettelyn tavoitteita ja sisältöä, sekä kansainvälisen rahanpesusäännöstön vaikutuksia Suomen säännöstöön. Empiiristä aineistoa kerättiin teemahaastattelujen avulla ja sen avulla selvitettiin, miten Suomen ilmoitusvelvolliset noudattavat laissa ja ilmoitusmenettelyssä asetettuja velvollisuuksia käytännössä. Tutkimustulosten mukaan, ilmoitusvelvollisiin kohdistetaan suora normatiivinen ohjaus. Ilmoitusvelvolliset noudattavat, hyväksyvät sääntelyn ja uskovat siihen. Tämä auttaa Suomen viranomaisia rahanpesutorjuntatyössä. Suomen rahanpesusäännöstöä on kehitetty vuosien aikana vastamaan kansainvälisiä vaatimuksia. Sääntelyssä on vielä puutteita. On olemassa myös muita ulkopuolisia tekijöitä, jotka hankaloittavat rahanpesun tehokasta torjuntaa.

Implementing RDF metadata in Hydra

Workshop at Open Repositories 2014, Helsinki, Finland, June 9-13, 2014

[Kurzrezensionen]

Kirjallisuusarvostelu

Suomen essiivi

Essiivi on suomen kielen sijamuoto, jota käytetään nykykielessä erilaisten olotilojen ilmaisemiseen (Hän on opettajana ~ sairaana). Se on taustaltaan lokatiivi, jonka paikanilmaisutehtävä näkyy edelleen erilaisissa kiteytymissä (kotona, luona). Essiivi kuuluu myös ajankohdanilmaisujärjestelmään (lauantaina, ensi vuotena). Tässä tutkimuksessa kuvaan essiivisijan käyttöä erityisesti olotilan ilmaisemisen näkökulmasta. Kiinnitän huomiota sijan merkitykseen ja pyrin tarkentamaan sen lauseopillista kuvausta hyödyntämällä yleislingvistis-kielitypologisia olotilanilmausten kategoriointitapoja. Tutkimukseni aineisto on kolmijakoinen. Pääaineistonani on Lauseopin arkisto, jota koskevan haun tuloksena olen saanut 9096 essiivisijaisen sanan sisältävää lauseketta virkekonteksteineen. Täydennän tätä aineistoa käytöstä poimituilla esimerkeillä ja intuitioon pohjautuvilla ns. selvillä tapauksilla. Tutkimusmenetelmänä on pääosin aineiston kvalitatiivinen tarkastelu, mutta tarkastelen essiiviä myös teoreettisemmin sääntöjärjestelmän näkökulmasta. Tärkeimpänä lähtökohtanani on tuottaa tutkimustietoa, joka kykenee käymään vuoropuhelua kielitypologisen tutkimuksen kanssa siten, että tutkimus kuitenkin tapahtuu yksittäiskielen ehdoilla. Erityisesti semantiikan kuvauksessa tarkastelutapani on kognitiivisen kielitieteen mukaisesti orientoitunut. Essiivin tehtävänä on pelkistetyssä nominaalilauseessa ilmaista olotilan tilapäisyyttä ja muutoksellisuutta, kun taas merkitystä kantavan verbin yhteydessä essiivisijainen olotilan ilmaus saa usein muita merkityksiä ja voi ilmaista myös vaihtoehtoisen tai tilanteen syynä olevan olotilan. Aspektinäkökulmasta essiivi tuo tilaan toimin¬nallisia piirteitä, ja se myös muodostaa selvästi omanlaisensa reviirin suhteessa muihin olotilaa ja sen lähimerkityksiä ilmaiseviin kieli- ja johto-opin kategorioihin. Syntaktisen luokittelun näkökulmasta essiivisijaiset predikoivat olotilanilmaukset jakautuvat kolmeen pääryhmään sen perusteella, millainen niiden suhde lauseen predikaattiin on. Ne voivat toimia kopulalauseen nominaalipredikaatteina, merkitystä kantavien verbien täydennyksinä ja sekundaarisina predikaatteina, jotka ovat määritteitä. Lisäksi essiivisijaiset olotilan ilmaukset voivat toimia predikoimattomina lauseadverbiaaleina. Tutkimus on essiivisijan laaja aineistopohjainen kuvaus, joka osoittaa, että yleislingvistis-kielitypologinen lähestymistapa sopii suomen essiivin kuvaukseen. Alakategorioinnissa joudutaan kuitenkin turvautumaan semantiikkaan. Myös pääkategorioiden jatkumomaisuus on hyväksyttävä. Tulosten perusteella on mahdollista käydä keskustelua siitä, olisiko predikoivien lauseenjäsenten syntaktista luokittelua mahdollista uudistaa. Lisäksi tutkimus avaa uusia kysymyksiä tutkittavaksi erityisesti sijojen käyttöä vertailevasta näkökulmasta.