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

Explanation Closure, Action Closure, and the Sandewall Test Suite for Reasoning about Change

Explanation closure (EC) axioms were previously introduced as a means of solving the frame problem. This paper provides a thorough demonstration of the power of EC combined with action closure (AC) for reasoning about dynamic worlds, by way of Sandewall's test suite of 12-or-so problems [Sandewall 1991; 1992]. Sandewall's problems range from the "Yale turkey shoot" (and variants) to the "stuffy room" problem, and were intended as a test and challenge for nonmonotonic logics of action. The EC/AC-based solutions for the most part do not resort to nonmonotonic reasoning at all, yet yield the intuitively warranted inferences in a direct, transparent fashion. While there are good reasons for ultimately employing nonmonotonic or probabilistic logics---e.g., pervasive uncertainty and the qualification problem---this does show that the scope of monotonic methods has been underestimated. Subsidiary purposes of the paper are to clarify the intuitive status of EC axioms in relation to action effect axioms; and to show how EC, previously formulated within the situation calculus, can be applied within the framework of a temporal logic similar to Sandewall's "discrete fluent logic," with some gains in clarity