Constructivisation through Induction and Conservation

The topic of this thesis lies in the intersection between proof theory and alge- braic logic. The main object of discussion, constructive reasoning, was intro- duced at the beginning of the 20th century by Brouwer, who followed Kant’s explanation of human intuition of spacial forms and time points: these are constructed step by step in a finite process by certain rules, mimicking con- structions with straightedge and compass and the construction of natural numbers, respectively. The aim of the present thesis is to show how classical reasoning, which admits some forms of indirect reasoning, can be made more constructive. The central tool that we are using are induction principles, methods that cap- ture infinite collections of objects by considering their process of generation instead of the whole class. We start by studying the interplay between cer- tain structures that satisfy induction and the calculi for some non-classical logics. We then use inductive methods to prove a few conservation theorems, which contribute to answering the question of which parts of classical logic and mathematics can be made constructive.Tämän opinnäytetyön aiheena on todistusteorian ja algebrallisen logiikan leikkauspiste. Keskustelun pääaiheen, rakentavan päättelyn, esitteli 1900-luvun alussa Brouwer, joka seurasi Kantin selitystä ihmisen intuitiosta tilamuodoista ja aikapisteistä: nämä rakennetaan askel askeleelta äärellisessä prosessissa tiettyjen sääntöjen mukaan, jotka jäljittelevät suoran ja kompassin konstruktioita ja luonnollisten lukujen konstruktiota. Tämän opinnäytetyön tavoitteena on osoittaa, kuinka klassista päättelyä, joka mahdollistaa tietyt epäsuoran päättelyn muodot, voidaan tehdä rakentavammaksi. Keskeinen työkalu, jota käytämme, ovat induktioperiaatteet, menetelmät, jotka keräävät äärettömiä objektikokoelmia ottamalla huomioon niiden luomisprosessin koko luokan sijaan. Aloitamme tutkimalla vuorovaikutusta tiettyjen induktiota tyydyttävien rakenteiden ja joidenkin ei-klassisten logiikan laskelmien välillä. Todistamme sitten induktiivisten menetelmien avulla muutamia säilymislauseita, jotka auttavat vastaamaan kysymykseen siitä, mitkä klassisen logiikan ja matematiikan osat voidaan tehdä rakentaviksi

Proceedings of Sinn und Bedeutung 21

TesisSe realizó un análisis clínico-electrocardiográfico integral de los hemibloqueos comprendiendo incidencia, edad, etiología, evaluación cuantitativa de los criterios diagnósticos, relación con los trastornos de conducción aurículo-ventricular, y pronóstico. Con tal motivo se estudiaron 221hemibloqueos encontrados en 7,130 pacientes adultos de sexo masculino en un servicio de cardiología y medicina. Los hemibloqueos fueron diagnosticados mediante los criterios señalados por Rosenbaum, Castellanos, y Prior y Blount

Polynomial Time Calculi

This dissertation deals with type systems which guarantee polynomial time complexity of typed programs. Such algorithms are commonly regarded as being feasible for practical applications, because their runtime grows reasonably fast for bigger inputs. The implicit complexity community has proposed several type systems for polynomial time in the recent years, each with strong, but different structural restrictions on the permissible algorithms which are necessary to control complexity. Comparisons between the various approaches are hard and this has led to a landscape of islands in the literature of expressible algorithms in each calculus, without many known links between them. This work chooses Light Affine Logic (LAL) and Hofmann's LFPL, both linearly typed, and studies the connections between them. It is shown that the light iteration in LAL, the fixed point variant of LAL, is expressive enough to allow a (non-trivial) compositional embedding of LFPL. The pull-out trick of LAL is identified as a technique to type certain non-size-increasing algorithms in such a way that they can be iterated. The System T sibling of LAL is developed which seamlessly integrates this technique as a central feature of the iteration scheme and which is proved again correct and complete for polynomial time. Because -iterations of the same level cannot be nested, is further generalised to , which surprisingly can express the impredicative iteration of LFPL and the light iteration of at the same time. Therefore, it subsumes both systems in one, while still being polynomial time normalisable. Hence, this result gives the first bridge between these two islands of implicit computational complexity

Polynomial Time Calculi

This dissertation deals with type systems which guarantee polynomial time complexity of typed programs. Such algorithms are commonly regarded as being feasible for practical applications, because their runtime grows reasonably fast for bigger inputs. The implicit complexity community has proposed several type systems for polynomial time in the recent years, each with strong, but different structural restrictions on the permissible algorithms which are necessary to control complexity. Comparisons between the various approaches are hard and this has led to a landscape of islands in the literature of expressible algorithms in each calculus, without many known links between them. This work chooses Light Affine Logic (LAL) and Hofmann's LFPL, both linearly typed, and studies the connections between them. It is shown that the light iteration in LAL, the fixed point variant of LAL, is expressive enough to allow a (non-trivial) compositional embedding of LFPL. The pull-out trick of LAL is identified as a technique to type certain non-size-increasing algorithms in such a way that they can be iterated. The System T sibling of LAL is developed which seamlessly integrates this technique as a central feature of the iteration scheme and which is proved again correct and complete for polynomial time. Because -iterations of the same level cannot be nested, is further generalised to , which surprisingly can express the impredicative iteration of LFPL and the light iteration of at the same time. Therefore, it subsumes both systems in one, while still being polynomial time normalisable. Hence, this result gives the first bridge between these two islands of implicit computational complexity

Completeness-via-canonicity in coalgebraic logics

This thesis aims to provide a suite of techniques to generate completeness re- sults for coalgebraic logics with axioms of arbitrary rank. We have chosen to investigate the possibility to generalize what is arguably one of the most suc- cessful methods to prove completeness results in ‘classical’ modal logic, namely completeness-via-canonicity. This technique is particularly well-suited to a coal- gebraic generalization because of its clean and abstract algebraic formalism. In the case of classical modal logic, it can be summarized in two steps, first it isolates the purely algebraic problem of canonicity, i.e. of determining when a variety of boolean Algebras with Operators (BAOs) is closed under canonical extension (i.e. canonical). Secondly, it connects the notion of canonical vari- eties to that of canonical models to explicitly build models, thereby proving completeness. The classical algebraic theory of canonicity is geared towards normal logics, or, in algebraic terms, BAOs (or generalizations thereof). Most coalgebraic log- ics are not normal, and we thus develop the algebraic theory of canonicity for Boolean Algebra with Expansions (BAEs), or more generally for Distributive Lattice Expansions (DLEs). We present new results about a class of expan- sions defined by weaker preservation properties than meet or join preservation, namely (anti)-k-additive and (anti-)k-multiplicative expansions. We show how canonical and Sahlqvist equations can be built from such operations. In order to connect the theory of canonicity in DLEs and BAEs to coalgebraic logic, we choose to work in the abstract formulation of coalgebraic logic. An abstract coalgebraic logic is defined by a functor L : BA → BA, and we can heuristically separate these logics in two classes. In the first class the functor L is relatively simple, and in particular can be interpreted as defining a BAE. This class includes the predicate lifting style of coalgebraic logics. In the second class the functor L can be very complicated and the whole theory requires a different approach. This class includes the nabla style of coalgebraic logics. For simple functors, we develop results on strong completeness and then prove strong completeness-via-canonicity in the presence of canonical frame con- ditions for strongly complete abstract coalgebraic logics. In particular we show coalgebraic completeness-via-canonicity for Graded Modal Logic, Intuitionistic Logic, the distributive full Lambek calculus, and the logic of trees of arbitrary branching degrees defined by the List functor. These results are to the best of our knowledge, new. For a complex functor L we use an indirect approach via the notion of functor presentation. This allows us to represent L as the quotient of a much simpler polynomial functor. Polynomial functors define BAEs and can thus be treated as objects in the first class of functors, in particular we can apply all the above mentioned techniques to the logics defined by such functors. We develop techniques that ensure that results obtained for the simple presenting logic can be transferred back to the complicated presented logic. We can then prove strong-completeness-via-canonicity in the presence of canonical frame conditions for coalgebraic logics which do not define a BAE, such as the nabla coalgebraic logics.Open Acces