Asymptotic Proportion of Hard Instances of the Halting Problem

Although the halting problem is undecidable, imperfect testers that fail on some instances are possible. Such instances are called hard for the tester. One variant of imperfect testers replies "I don't know" on hard instances, another variant fails to halt, and yet another replies incorrectly "yes" or "no". Also the halting problem has three variants: does a given program halt on the empty input, does a given program halt when given itself as its input, or does a given program halt on a given input. The failure rate of a tester for some size is the proportion of hard instances among all instances of that size. This publication investigates the behaviour of the failure rate as the size grows without limit. Earlier results are surveyed and new results are proven. Some of them use C++ on Linux as the computational model. It turns out that the behaviour is sensitive to the details of the programming language or computational model, but in many cases it is possible to prove that the proportion of hard instances does not vanish.Comment: 18 pages. The differences between this version and arXiv:1307.7066v1 are significant. They have been listed in the last paragraph of Section 1. Excluding layout, this arXiv version is essentially identical to the Acta Cybernetica versio

All Linear-Time Congruences for Familiar Operators

The detailed behaviour of a system is often represented as a labelled transition system (LTS) and the abstract behaviour as a stuttering-insensitive semantic congruence. Numerous congruences have been presented in the literature. On the other hand, there have not been many results proving the absence of more congruences. This publication fully analyses the linear-time (in a well-defined sense) region with respect to action prefix, hiding, relational renaming, and parallel composition. It contains 40 congruences. They are built from the alphabet, two kinds of traces, two kinds of divergence traces, five kinds of failures, and four kinds of infinite traces. In the case of finite LTSs, infinite traces lose their role and the number of congruences drops to 20. The publication concentrates on the hardest and most novel part of the result, that is, proving the absence of more congruences

Stop It, and Be Stubborn!

A system is AG EF terminating, if and only if from every reachable state, a terminal state is reachable. This publication argues that it is beneficial for both catching non-progress errors and stubborn set state space reduction to try to make verification models AG EF terminating. An incorrect mutual exclusion algorithm is used as an example. The error does not manifest itself, unless the first action of the customers is modelled differently from other actions. An appropriate method is to add an alternative first action that models the customer stopping for good. This method typically makes the model AG EF terminating. If the model is AG EF terminating, then the basic strong stubborn set method preserves safety and some progress properties without any additional condition for solving the ignoring problem. Furthermore, whether the model is AG EF terminating can be checked efficiently from the reduced state space

Rehuarvon määritys Lapin koeasemalla

vokKirjasto Aj-

Competition and access price regulation in the broadband market

We construct a model for differentiated Cournot competition between service-based and infrastructure-based firms, out of which one infrastructure-based firm (the incumbent) supplies to the service-based firms. We seek for and compare the socially optimal and the incumbent’s profit maximizing access price in two scenarios: (i) service-based firms and incumbent supply homogeneous services (partial differentiation), and (ii) all services are horizontally differentiated (uniform differentiation). We show that in both cases the incumbent never forecloses service-based firms if infrastructure-based competition is present or if services are somewhat differentiated. Under uniform differentiation the welfare optimizing access price is below marginal cost, hence the incumbent subsidizes the production of service-based firms and makes zero profit. In the case of partial differentiation, the same result obtains when both markets are concentrated. However, if markets are not concentrated, the socially optimal access fee exceeds the marginal cost.

Roudan kehittymisen tilastollinen malli

vokKirjasto Aj-

A Simple Character String Proof of the "True but Unprovable" Version of G\"odel's First Incompleteness Theorem

A rather easy yet rigorous proof of a version of G\"odel's first incompleteness theorem is presented. The version is "each recursively enumerable theory of natural numbers with 0, 1, +, *, =, logical and, logical not, and the universal quantifier either proves a false sentence or fails to prove a true sentence". The proof proceeds by first showing a similar result on theories of finite character strings, and then transporting it to natural numbers, by using them to model strings and their concatenation. Proof systems are expressed via Turing machines that halt if and only if their input string is a theorem. This approach makes it possible to present all but one parts of the proof rather briefly with simple and straightforward constructions. The details require some care, but do not require significant background knowledge. The missing part is the widely known fact that Turing machines can perform complicated computational tasks.Comment: In Proceedings AFL 2014, arXiv:1405.527

Kolmen pääravinteen vaikutus satoon ajanjaksoina 1926-39, 1940-54 ja 1955-64. Yleislannoituskokeiden tuloksia

vokKirjasto Aj-