On cluster approximations of cellular automata

In this thesis I examine one commonly used class of methods for the analytic approximation of cellular automata, the so-called local cluster approximations. This class subsumes the well known mean-field and pair approximations, as well as higher order generalizations of these. While a straightforward method known as Bayesian extension exists for constructing cluster approximations of arbitrary order on one-dimensional lattices (and certain other cases), for higher-dimensional systems the construction of approximations beyond the pair level becomes more complicated due to the presence of loops. In this thesis I describe the one-dimensional construction as well as a number of approximations suggested for higher-dimensional lattices, comparing them against a number of consistency criteria that such approximations could be expected to satisfy. I also outline a general variational principle for constructing consistent cluster approximations of arbitrary order with minimal bias, and show that the one-dimensional construction indeed satisfies this principle. Finally, I apply this variational principle to derive a novel consistent expression for symmetric three cell cluster frequencies as estimated from pair frequencies, and use this expression to construct a quantitatively improved pair approximation of the well-known lattice contact process on a hexagonal lattice.

Describing syntax with star-free regular expressions

Koskenniemen Äärellistilaisen leikkauskieliopin (FSIG) lauseopilliset rajoitteet ovat loogisesti vähemmän kompleksisia kuin mihin niissä käytetty formalismi vittaisi. Osoittautuukin että vaikka Voutilaisen (1994) englannin kielelle laatima FSIG-kuvaus käyttää useita säännöllisten lausekkeiden laajennuksia, kieliopin kuvaus kokonaisuutenaan palautuu äärelliseen yhdistelmään unionia, komplementtia ja peräkkäinasettelua. Tämä on oleellinen parannus ENGFSIG:n descriptiiviseen kompleksisuuteen. Tulos avaa ovia FSIG-kuvauksen loogisten ominaisuuksien syvemmälle analyysille ja FSIG kuvausten mahdolliselle optimoinnillle. Todistus sisältää uuden kaavan, joka kääntää Koskenniemien rajoiteoperaation ilman markkerimerkkejä.

Weak Ehrenfeucht-Fraïssé Games

In this paper we define a game which is played between two players I and II on two mathematical structures A and B. The players choose elements from both structures in moves, and at the end of the game the player II wins if the chosen structures are isomorphic. Thus the difference of this to the ordinary Ehrenfeucht-Fra¨ıss´e game is that the isomorphism can be arbitrary, whereas in the ordinary EF-game it is determined by the moves of the players. We investigate determinacy of the weak EF-game for different (the length of the game) and its relation to the ordinary EF-game.

Almost Stable Matchings by Truncating the Gale–Shapley Algorithm

We show that the ratio of matched individuals to blocking pairs grows linearly with the number of propose–accept rounds executed by the Gale–Shapley algorithm for the stable marriage problem. Consequently, the participants can arrive at an almost stable matching even without full information about the problem instance; for each participant, knowing only its local neighbourhood is enough. In distributed-systems parlance, this means that if each person has only a constant number of acceptable partners, an almost stable matching emerges after a constant number of synchronous communication rounds. We apply our results to give a distributed (2 + ε)-approximation algorithm for maximum-weight matching in bicoloured graphs and a centralised randomised constant-time approximation scheme for estimating the size of a stable matching.

Analysing local algorithms in location-aware quasi-unit-disk graphs

A local algorithm with local horizon r is a distributed algorithm that runs in r synchronous communication rounds; here r is a constant that does not depend on the size of the network. As a consequence, the output of a node in a local algorithm only depends on the input within r hops from the node. We give tight bounds on the local horizon for a class of local algorithms for combinatorial problems on unit-disk graphs (UDGs). Most of our bounds are due to a refined analysis of existing approaches, while others are obtained by suggesting new algorithms. The algorithms we consider are based on network decompositions guided by a rectangular tiling of the plane. The algorithms are applied to matching, independent set, graph colouring, vertex cover, and dominating set. We also study local algorithms on quasi-UDGs, which are a popular generalisation of UDGs, aimed at more realistic modelling of communication between the network nodes. Analysing the local algorithms on quasi-UDGs allows one to assume that the nodes know their coordinates only approximately, up to an additive error. Despite the localisation error, the quality of the solution to problems on quasi-UDGs remains the same as for the case of UDGs with perfect location awareness. We analyse the increase in the local horizon that comes along with moving from UDGs to quasi-UDGs.

Paradoxical reflection in quantum mechanics

This article concerns a phenomenon of elementary quantum mechanics that is quite counter-intuitive, very non-classical, and apparently not widely known: a quantum particle can get reflected at a downward potential step. In contrast, classical particles get reflected only at upward steps. The conditions for this effect are that the wave length is much greater than the width of the potential step and the kinetic energy of the particle is much smaller than the depth of the potential step. This phenomenon is suggested by non-normalizable solutions to the time-independent Schroedinger equation, and we present evidence, numerical and mathematical, that it is also indeed predicted by the time-dependent Schroedinger equation. Furthermore, this paradoxical reflection effect suggests, and we confirm mathematically, that a quantum particle can be trapped for a long time (though not forever) in a region surrounded by downward potential steps, that is, on a plateau.

An Efficient Constraint Grammar Parser based on Inward Deterministic Automata

Pappret conceptualizes parsning med Constraint Grammar på ett nytt sätt som en process med två viktiga representationer. En representation innehåller lokala tvetydighet och den andra sammanfattar egenskaperna hos den lokala tvetydighet klasser. Båda representationer manipuleras med ren finite-state metoder, men deras samtrafik är en ad hoc -tillämpning av rationella potensserier. Den nya tolkningen av parsning systemet har flera praktiska fördelar, bland annat det inåt deterministiska sättet att beräkna, representera och räkna om alla potentiella tillämpningar av reglerna i meningen.

Bounded turning circles are weak-quasicircles

We show that a metric Jordan curve $\Gamma$ is \emph{bounded turning} if and only if there exists a \emph{weak-quasisymmetric} homeomorphism $\phi\colon \mathsf{S}^1 \to \Gamma$.