Predictability and Randomness

Algorithmic theories of randomness can be related to theories of probabilistic sequence prediction through the notion of a predictor, defined as a function which supplies lower bounds on initial-segment probabilities of infinite sequences. An infinite binary sequence $z$ is called unpredictable iff its initial-segment "redundancy" $n+\log p(z(n))$ remains sufficiently low relative to every effective predictor $p$. A predictor which maximizes the initial-segment redundancy of a sequence is called optimal for that sequence. It turns out that a sequence is random iff it is unpredictable. More generally, a sequence is random relative to an arbitrary computable distribution iff the distribution is itself an optimal predictor for the sequence. Here "random" can be taken in the sense of Martin-L\"{o}f by using weak criteria of effectiveness, or in the sense of Schnorr by using stronger criteria of effectiveness. Under the weaker criteria of effectiveness it is possible to construct a universal predictor which is optimal for all infinite sequences. This predictor assigns nonvanishing limit probabilities precisely to the recursive sequences. Under the stronger criteria of effectiveness it is possible to establish a law of large numbers for sequences random relative to a computable distribution, which may be useful as a criterion of "rationality" for methods of probabilistic prediction. A remarkable feature of effective predictors is the fact that they are expressible in the special form first proposed by Solomonoff. In this form sequence prediction reduces to assigning high probabilities to initial segments with short and/or numerous encodings. This fact provides the link between theories of randomness and Solomonoff's theory of prediction.Comment: 30 pages + refs. A re-typeset University of Alberta Technical Report, no longer available as suc

The construction of viewpoint aspect: the imperfective revisited

This paper argues for a constructionist approach to viewpoint Aspect by exploring the idea that it does not exert any altering force on the situation-aspect properties of predicates. The proposal is developed by analyzing the syntax and semantics of the imperfective, which has been attributed a coercer role in the literature as a de-telicizer and de-stativizer in the progressive, and as a de-eventivizer in the so-called ability (or attitudinal) and habitual readings. This paper proposes a unified semantics for the imperfective, preserving the properties of eventualities throughout the derivation. The paper argues that the semantics of viewpoint aspect is encoded in a series of functional heads containing interval-ordering predicates and quantifiers. This richer structure allows us to account for a greater amount of phenomena, such as the perfective nature of the individual instantiations of the event within a habitual construction or the nonculminating reading of perfective accomplishments in Spanish. This paper hypothesizes that nonculminating accomplishments have an underlying structure corresponding to the perfective progressive. As a consequence, the progressive becomes disentangled from imperfectivity and is given a novel analysis. The proposed syntax is argued to have a corresponding explicit morphology in languages such as Spanish and a nondifferentiating one in languages such as English; however, the syntax-semantics underlying both of these languages is argued to be the same

Monotonic Solution of the Frame Problem in the Situation Calculus: An Efficient Method for Worlds with Fully Specified Actions

The paper is concerned with the succinct axiomatization and efficient deduction of non-change, within McCarthy and Hayes' Situation Calculus. The idea behind the proposed approach is this: suppose that in a room containing a man, a robot and a cat as the only potential agents, the only action taken by the man within a certain time interval is to walk from one place to another, while the robot's only actions are to pick up a box containing the (inactive) cat and carry it from its initial place to another. We wish to prove that a certain object (such as the cat, or the doormat) did not change color. We reason that the only way it could have changed color is for the man or the robot to have painted or dyed it. But since these are not among the actions which actually occurred, the color of the object is unchanged. Thus we need no frame axioms to the effect that walking and carrying leave colors unchanged (which is in general false in multi-agent worlds), and no default schema that properties change only when we can prove they do (which is in general false in incompletely known worlds). Instead we use explanation-closure axioms specifying all primitive actions which can produce a given type of change within the setting of interest. A method similar to this has been proposed by Andrew Haas for single-agent, serial worlds. The contribution of the present paper lies in showing (1) that such methods do indeed encode non-change succinctly, (2) are independently motivated, (3) can be used to justify highly efficient methods of inferring non-change, specifically the "sleeping dog" strategy of STRIPS, and (4) can be extended to simple multiagent worlds with concurrent actions. An ultimate limitation may lie in the lack of a uniform strategy for deciding what fluents can be affected by what agents in a given domain. In this respect probabilistic methods appear promising