Affektiivisuuden laskennallinen määritteleminen

One of the most tangled fields of research is the field of defining and modeling affective concepts, i. e. concepts regarding emotions and feelings. The subject can be approached from many disciplines. The main problem is lack of generally approved definitions. However, e.g. linguists have recently started to check the consistency of their theories with the help of computer simulations. Definitions of affective concepts are needed for performing similar simulations in behavioral sciences. In this thesis, preliminary computational definitions of affects for a simple utility-maximizing agent are given. The definitions have been produced by synthetizing ideas from theories from several fields of research. The class of affects is defined as a superclass of emotions and feelings. Affect is defined as a process, in which a change in an agent's expected utility causes a bodily change. If the process is currently under the attention of the agent (i.e. the agent is conscious of it), the process is a feeling. If it is not, but can in principle be taken into attention (i.e. it is preconscious), the process is an emotion. Thus, affects do not presuppose consciousness, but emotions and affects do. Affects directed at unexpected materialized (i.e. past) events are delight and fright. Delight is the consequence of an unexpected positive event and fright is the consequence of an unexpected negative event. Affects directed at expected materialized (i.e. past) events are happiness (expected positive event materialized), disappointment (expected positive event did not materialize), sadness (expected negative event materialized) and relief (expected negative event did not materialize). Affects directed at expected unrealized (i.e. future) events are fear and hope. Some other affects can be defined as directed towards originators of the events. The affect classification has also been implemented as a computer program, the purpose of which is to ensure the coherence of the definitions and also to illustrate the capabilities of the model. The exact content of bodily changes associated with specific affects is not considered relevant from the point of view of the logical structure of affective phenomena. The utility function need also not be defined, since the target of examination is only its dynamics.

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.

A Possible and Necessary Consistency Proof

After Gödel's incompleteness theorems and the collapse of Hilbert's programme Gerhard Gentzen continued the quest for consistency proofs of Peano arithmetic. He considered a finitistic or constructive proof still possible and necessary for the foundations of mathematics. For a proof to be meaningful, the principles relied on should be considered more reliable than the doubtful elements of the theory concerned. He worked out a total of four proofs between 1934 and 1939. This thesis examines the consistency proofs for arithmetic by Gentzen from different angles. The consistency of Heyting arithmetic is shown both in a sequent calculus notation and in natural deduction. The former proof includes a cut elimination theorem for the calculus and a syntactical study of the purely arithmetical part of the system. The latter consistency proof in standard natural deduction has been an open problem since the publication of Gentzen's proofs. The solution to this problem for an intuitionistic calculus is based on a normalization proof by Howard. The proof is performed in the manner of Gentzen, by giving a reduction procedure for derivations of falsity. In contrast to Gentzen's proof, the procedure contains a vector assignment. The reduction reduces the first component of the vector and this component can be interpreted as an ordinal less than epsilon_0, thus ordering the derivations by complexity and proving termination of the process.