Related papers: Pebble-Depth
This paper expands upon existing and introduces new formulations of Bennett's logical depth. In previously published work by Jordon and Moser, notions of finite-state depth and pushdown depth were examined and compared. These were based on…
This paper introduces two complexity-theoretic formulations of Bennett's logical depth: finite-state depth and polynomial-time depth. It is shown that for both formulations, trivial and random infinite sequences are shallow, and a slow…
Pebble transducers are nested two-way transducers which can drop marks (named "pebbles") on their input word. Such machines can compute functions whose output size is polynomial in the size of their input. They can be seen as simple…
In this article, we study the relationship between notions of depth for sequences, namely, Bennett's notions of strong and weak depth, and deep $\Pi^0_1$ classes, introduced by the authors and motivated by previous work of Levin. For the…
Effective complexity measures the information content of the regularities of an object. It has been introduced by M. Gell-Mann and S. Lloyd to avoid some of the disadvantages of Kolmogorov complexity, also known as algorithmic information…
Deterministic two-way transducers with pebbles (aka pebble transducers) capture the class of polyregular functions, which extend the string-to-string regular functions allowing polynomial growth instead of linear growth. One of the most…
In analogy to the well-known notion of finite--state compressibility of individual sequences, due to Lempel and Ziv, we define a similar notion of "finite-state encryptability" of an individual plaintext sequence, as the minimum asymptotic…
Bennett's notion of depth is usually considered to describe the usefulness and internal organization of the information encoded into an object such as an infinite binary sequence. We consider a natural way to relativize the notion of depth…
We derive upper and lower bounds on the overall compression ratio of the 1978 Lempel-Ziv (LZ78) algorithm, applied independently to $k$-blocks of a finite individual sequence. Both bounds are given in terms of normalized empirical entropies…
Polyregular functions are the class of string-to-string functions definable by pebble transducers, an extension of finite-state automata with outputs and multiple two-way reading heads (pebbles) with a stack discipline. If a polyregular…
An infinite binary sequence is Bennett deep if, for any computable time bound, the difference between the time-bounded prefix-free Kolmogorov complexity and the prefix-free Kolmogorov complexity of its initial segments is eventually…
Standard Transformers have a fixed computational depth, fundamentally limiting their ability to generalize to tasks requiring variable-depth reasoning, such as multi-hop graph traversal or nested logic. We propose a depth-recurrent…
We introduce the notion of the depth of a finite group $G$, defined as the minimal length of an unrefinable chain of subgroups from $G$ to the trivial subgroup. In this paper we investigate the depth of (non-abelian) finite simple groups.…
A key attribute that drives the unprecedented success of modern Recurrent Neural Networks (RNNs) on learning tasks which involve sequential data, is their ability to model intricate long-term temporal dependencies. However, a well…
Depth of an object concerns a tradeoff between computation time and excess of program length over the shortest program length required to obtain the object. It gives an unconditional lower bound on the computation time from a given program…
In this paper we introduce a variant of pushdown dimension called bounded pushdown (BPD) dimension, that measures the density of information contained in a sequence, relative to a BPD automata, i.e. a finite state machine equipped with an…
We extend Ziv and Lempel's model of finite-state encoders to the realm of lossy compression of individual sequences. In particular, the model of the encoder includes a finite-state reconstruction codebook followed by an information lossless…
Ribbons are long narrow strips possessing three distinct material length scales (thickness, width, and length) which allow them to produce unique shapes unobtainable by wires or filaments. For example when a ribbon has half a twist and is…
This paper proposes new notions of polynomial depth (called monotone poly depth), based on a polynomial version of monotone Kolmogorov complexity. We show that monotone poly depth satisfies all desirable properties of depth notions i.e.,…
We show that a polyregular word-to-word function is regular if and only if its output size is at most linear in its input size. Moreover a polyregular function can be realized by: a transducer with two pebbles if and only if its output has…