Fair Testing

In this paper we present a solution to the long-standing problem of characterising the coarsest liveness-preserving pre-congruence with respect to a full (TCSP-inspired) process algebra. In fact, we present two distinct characterisations, which give rise to the same relation: an operational one based on a De Nicola-Hennessy-like testing modality which we call should-testing, and a denotational one based on a refined notion of failures. One of the distinguishing characteristics of the should-testing pre-congruence is that it abstracts from divergences in the same way as Milner¿s observation congruence, and as a consequence is strictly coarser than observation congruence. In other words, should-testing has a built-in fairness assumption. This is in itself a property long sought-after; it is in notable contrast to the well-known must-testing of De Nicola and Hennessy (denotationally characterised by a combination of failures and divergences), which treats divergence as catrastrophic and hence is incompatible with observation congruence. Due to these characteristics, should-testing supports modular reasoning and allows to use the proof techniques of observation congruence, but also supports additional laws and techniques. Moreover, we show decidability of should-testing (on the basis of the denotational characterisation). Finally, we demonstrate its advantages by the application to a number of examples, including a scheduling problem, a version of the Alternating Bit-protocol, and fair lossy communication channel

Step-Indexed Relational Reasoning for Countable Nondeterminism

Programming languages with countable nondeterministic choice are computationally interesting since countable nondeterminism arises when modeling fairness for concurrent systems. Because countable choice introduces non-continuous behaviour, it is well-known that developing semantic models for programming languages with countable nondeterminism is challenging. We present a step-indexed logical relations model of a higher-order functional programming language with countable nondeterminism and demonstrate how it can be used to reason about contextually defined may- and must-equivalence. In earlier step-indexed models, the indices have been drawn from {\omega}. Here the step-indexed relations for must-equivalence are indexed over an ordinal greater than {\omega}

Towards a Convenient Category of Topological Domains

We propose a category of topological spaces that promises to be convenient for the purposes of domain theory as a mathematical theory for modelling computation. Our notion of convenience presupposes the usual properties of domain theory, e.g. modelling the basic type constructors, fixed points, recursive types, etc. In addition, we seek to model parametric polymorphism, and also to provide a flexible toolkit for modelling computational effects as free algebras for algebraic theories. Our convenient category is obtained as an application of recent work on the remarkable closure conditions of the category of quotients of countably-based topological spaces. Its convenience is a consequence of a connection with realizability models

A Convenient Category of Domains

We motivate and define a category of "topological domains", whose objects are certain topological spaces, generalising the usual $omega$-continuous dcppos of domain theory. Our category supports all the standard constructions of domain theory, including the solution of recursive domain equations. It also supports the construction of free algebras for (in)equational theories, provides a model of parametric polymorphism, and can be used as the basis for a theory of computability. This answers a question of Gordon Plotkin, who asked whether it was possible to construct a category of domains combining such properties

How to Build an Infinite Lottery Machine

An infinite lottery machine is used as a foil for testing the reach of inductive inference, since inferences concerning it require novel extensions of probability. Its use is defensible if there is some sense in which the lottery is physically possible, even if exotic physics is needed. I argue that exotic physics is needed and describe several proposals that fail and at least one that succeeds well enough

How to Build an Infinite Lottery Machine

An infinite lottery machine is used as a foil for testing the reach of inductive inference, since inferences concerning it require novel extensions of probability. Its use is defensible if there is some sense in which the lottery is physically possible, even if exotic physics is needed. I argue that exotic physics is needed and describe several proposals that fail and at least one that succeeds well enough

Structural and Computational Existence Results for Multidimensional Subshifts

Symbolic dynamics is a branch of mathematics that studies the structure of infinite sequences of symbols, or in the multidimensional case, infinite grids of symbols. Classes of such sequences and grids defined by collections of forbidden patterns are called subshifts, and subshifts of finite type are defined by finitely many forbidden patterns. The simplest examples of multidimensional subshifts are sets of Wang tilings, infinite arrangements of square tiles with colored edges, where adjacent edges must have the same color. Multidimensional symbolic dynamics has strong connections to computability theory, since most of the basic properties of subshifts cannot be recognized by computer programs, but are instead characterized by some higher-level notion of computability. This dissertation focuses on the structure of multidimensional subshifts, and the ways in which it relates to their computational properties. In the first part, we study the subpattern posets and Cantor-Bendixson ranks of countable subshifts of finite type, which can be seen as measures of their structural complexity. We show, by explicitly constructing subshifts with the desired properties, that both notions are essentially restricted only by computability conditions. In the second part of the dissertation, we study different methods of defining (classes of ) multidimensional subshifts, and how they relate to each other and existing methods. We present definitions that use monadic second-order logic, a more restricted kind of logical quantification called quantifier extension, and multi-headed finite state machines. Two of the definitions give rise to hierarchies of subshift classes, which are a priori infinite, but which we show to collapse into finitely many levels. The quantifier extension provides insight to the somewhat mysterious class of multidimensional sofic subshifts, since we prove a characterization for the class of subshifts that can extend a sofic subshift into a nonsofic one.Symbolidynamiikka on matematiikan ala, joka tutkii äärettömän pituisten symbolijonojen ominaisuuksia, tai moniulotteisessa tapauksessa äärettömän laajoja symbolihiloja. Siirtoavaruudet ovat tällaisten jonojen tai hilojen kokoelmia, jotka on määritelty kieltämällä jokin joukko äärellisen kokoisia kuvioita, ja äärellisen tyypin siirtoavaruudet saadaan kieltämällä vain äärellisen monta kuviota. Wangin tiilitykset ovat yksinkertaisin esimerkki moniulotteisista siirtoavaruuksista. Ne ovat värillisistä neliöistä muodostettuja tiilityksiä, joissa kaikkien vierekkäisten sivujen on oltava samanvärisiä. Moniulotteinen symbolidynamiikka on vahvasti yhteydessä laskettavuuden teoriaan, sillä monia siirtoavaruuksien perusominaisuuksia ei ole mahdollista tunnistaa tietokoneohjelmilla, vaan korkeamman tason laskennallisilla malleilla. Väitöskirjassani tutkin moniulotteisten siirtoavaruuksien rakennetta ja sen suhdetta niiden laskennallisiin ominaisuuksiin. Ensimmäisessä osassa keskityn tiettyihin äärellisen tyypin siirtoavaruuksien rakenteellisiin ominaisuuksiin: äärellisten kuvioiden muodostamaan järjestykseen ja Cantor-Bendixsonin astelukuun. Halutunlaisia siirtoavaruuksia rakentamalla osoitan, että molemmat ominaisuudet ovat olennaisesti laskennallisten ehtojen rajoittamia. Väitöskirjan toisessa osassa tutkin erilaisia tapoja määritellä moniulotteisia siirtoavaruuksia, sekä sitä, miten nämä tavat vertautuvat toisiinsa ja tunnettuihin siirtoavaruuksien luokkiin. Käsittelen määritelmiä, jotka perustuvat toisen kertaluvun logiikkaan, kvanttorilaajennukseksi kutsuttuun rajoitettuun loogiseen kvantifiointiin, sekä monipäisiin äärellisiin automaatteihin. Näistä kolmesta määritelmästä kahteen liittyy erilliset siirtoavaruuksien hierarkiat, joiden todistan romahtavan äärellisen korkuisiksi. Kvanttorilaajennuksen tutkimus valottaa myös niin kutsuttujen sofisten siirtoavaruuksien rakennetta, jota ei vielä tunneta hyvin: kyseisessä luvussa selvitän tarkasti, mitkä siirtoavaruudet voivat laajentaa sofisen avaruuden ei-sofiseksi.Siirretty Doriast

Chaos and quantum-nondemolition measurements

The problem of chaotic behavior in quantum mechanics is investigated against the background of the theory of quantum-nondemolition (QND) measurements. The analysis is based on two relevant features: The outcomes of a sequence of QND measurements are unambiguously predictable, and these measurements actually can be performed on one single system without perturbing its time evolution. Consequently, QND measurements represent an appropriate framework to analyze the conditions for the occurrence of ‘‘deterministic randomness’’ in quantum systems. The general arguments are illustrated by a discussion of a quantum system with a time evolution that possesses nonvanishing algorithmic complexity