In this paper we introduce a new feasible notion of Bennett's logical depth based on pebble transducers. This notion is defined based on the difference between the minimal length descriptional complexity of prefixes of infinite sequences from the perspective of finite-state transducers and pebble transducers. Our notion of pebble-depth satisfies the four fundamental properties of depth: i.e. deep sequences exist, trivial sequences are not deep, random sequences are not deep, and the existence of a slow growth law type result. We also compare pebble-depth to other depth notions based on finite-state transducers, pushdown compressors, and the Lempel-Ziv 78 compression algorithm. We first demonstrate how there exists a normal pebble-deep sequence even though there is no normal finite-state-deep sequence. We next build a sequence that has a pebble-depth level of roughly 1, a pushdown-depth level of roughly 1/2 and a finite-state-depth level of roughly 0. We then build a sequence that has a pebble-depth level of roughly 1/2 and a Lempel-Ziv-depth level of roughly 0.

In previously published work (Jordon and Moser, 2020), notions of finite-state-depth and pushdown-depth were presented. These were based on finite-state transducers and information lossless pushdown compressors. Unfortunately, a complete separation between the two notions was not established. This paper introduces a new formulation of pushdown-depth based on restricting how fast a pushdown compressor's stack can grow. This allows us to do a full comparison by demonstrating the existence of sequences with high finite-state-depth and low pushdown-depth, and vice-versa. A new notion based on the Lempel-Ziv 78 algorithm is also presented. Its difference from finite-state-depth is shown by a Lempel-Ziv deep sequence that is not finite-state deep, and vice versa. Lempel-Ziv-depth's difference from pushdown-depth is shown by building sequences that have a pushdown-depth of roughly 1/2 but low Lempel-Ziv depth, and by a sequence with high Lempel-Ziv depth but low pushdown-depth. Properties of all three notions are also studied.

This paper examines Automatic Complexity, a complexity notion introduced by Shallit and Wang in 2001. We demonstrate that there exists a normal sequence $T$ such that $I(T) = 0$ and $S(T) \leq 1/2$, where $I(T)$ and $S(T)$ are the lower and upper automatic complexity rates of $T$ respectively. We furthermore show that there exists a Champernowne sequence $C$, i.e. a sequence formed by concatenating all strings of length $1$ followed by concatenating all strings of length $2$ and so on, such that $S(C) \leq 2/3$.

We introduce the notion of limit-depth, as a notion similar to Bennett depth, but well behaved on Turing degrees, as opposed to truth-table degrees for Bennett depth. We show limit-depth satisfies similar properties to Bennett depth, namely both recursive and sufficiently random sequences are not limit-deep, and limit-depth is preserved over Turing degrees. We show both the halting problem and Chaitin's omega are limit-deep. We show every limit-deep set has DNR wtt-degree, and some limit-cuppable set does not have a limit-deep wtt degree.

We introduce a new strategy for optimal diversification which combines elements of Diversified Risk Parity and Diversification Ratio, with emphasis on positive risk premiums. The Uncorrelated Positive Bets strategy involves the identification of reliable, independent sources of randomness and the quantification of their positive risk premium. We use principal component analysis to identify the most significant sources of randomness contributing to the market and then apply the Randomness Deficiency Coefficient metric and principal portfolio positivity to identify a set of reliable uncorrelated positive bets. Portfolios are then optimized by maximizing their diversified positive risk premium. We contrast the performance of a range of diversification strategies for a portfolio held for a two-year out-of-sample period with a 30 stock constraint. In particular, we introduce the notion of diversification inefficiency to explain why diversification strategies might outperform the market.

Existing risk-to-reward measures, such as the Sharpe ratio [1] or M2 [2], are based on the idea of quantifying the excess return per unit of deviation in an investment. In this preliminary article we introduce a new probabilistic measure for evaluating investment performance. Randomness Deficiency Coefficient (RDC) expresses the likelihood that the observed excess return of an investment has been generated by chance. Some of the advantages of RDC over existing measures are that it can be used with small historical datasets, is time-frame independent, and can be easily adjusted to take into account the familywise error rate which results from selection bias. We argue that RDC captures the fundamental relationship between risk and reward and prove that it converges with Sharpe's ratio.

In this article we describe the use of a multi-objective evolutionary algorithm for portfolio optimisation based on historical data for the S&P 500. Portfolio optimisation seeks to identify manageable investments that provide a high expected return with relatively low risk. We developed a set of metrics for qualifying the risk/return characteristics of a portfolio's historical performance and combined this with an island model genetic algorithm to identify optimised portfolios. The algorithm was successful in selecting investment strategies with high returns and relatively low volatility. However, although these solutions performed well on historical data, they were not predictive of future returns, with optimised portfolios failing to perform above chance. The implications of these findings are discussed.

The pressing need for eficient compression schemes for XML documents has recently been focused on stack computation [6, 9], and in particular calls for a formulation of information-lossless stack or pushdown compressors that allows a formal analysis of their performance and a more ambitious use of the stack in XML compression, where so far it is mainly connected to parsing mechanisms. In this paper we introduce the model of pushdown compressor, based on pushdown transducers that compute a single injective function while keeping the widest generality regarding stack computation. The celebrated Lempel-Ziv algorithm LZ78 [10] was introduced as a general purpose compression algorithm that outperforms finite-state compressors on all sequences. We compare the performance of the Lempel-Ziv algorithm with that of the pushdown compressors, or compression algorithms that can be implemented with a pushdown transducer. This comparison is made without any a priori assumption on the data's source and considering the asymptotic compression ratio for infinite sequences. We prove that Lempel-Ziv is incomparable with pushdown compressors.

We show that if RP has p-measure nonzero then ZPP=EXP. As corollaries, we obtain a zero-one law for RP, and that both probabilistic classes ZPP and RP have the same p-measure. Finally we prove that if NP has p-measure nonzero then NP=AM.