Slowdown and saturation of internal time according to the statistics of information input: a minimal model of response systems

TL;DR

The paper investigates how an information-processing system updates its internal state when external inputs arrive according to statistics. By defining internal time as the number of observed code kinds, it derives that, under Poisson input from a finite code set, the mean internal time slows down over time and saturates at the total number of codes, making internal time negligible compared to physical time on long horizons. In the uniform case, it provides a closed-form nonlinear mapping between physical time and internal time and shows the physical time needed to advance one internal unit grows near saturation; it also introduces a weighted internal time to account for information content. An ancillary entropy measure quantifies what remains uncertain about the observed codes given only internal time, with unimodal behavior and explicit results in the uniform case and informative bounds in the non-uniform case. The work further discusses heavy-tailed inputs (Zipf) and outlines extensions to more general input statistics and richer internal-time definitions, highlighting the statistic-driven nature of internal-time slowdown and its potential implications for cognitive and information-processing systems.

Abstract

We consider a response system that updates its internal state in accordance with information input arriving from outside. In this paper, we define as internal time the ``number of kinds'' of codes that have been observed at least once up to a given time, and analyze how the way internal time advances is determined by the statistics of information input (arrival rate and code distribution). When arrivals follow a Poisson process, the average advancing speed of internal time decreases monotonically with time, and if the number of kinds of codes is finite, it eventually approaches an upper limit and saturates. As a result, on long time scales, internal time becomes relatively shorter than physical time. For a uniform code distribution, we provide a closed form for the correspondence between internal time and physical time, and show that the physical time required to ``advance internal time by one step'' increases in later stages. As an ancillary quantity, we quantify by conditional entropy the remaining uncertainty of ``which codes have been observed'' when only internal time is known, and we give unimodality and the maximization time in the uniform case, and upper bounds, equality conditions, and expressions of the difference from the upper bound in the non-uniform case. Finally, we also present a generalization that assigns weights (description lengths) to each code so that internal time is ticked according to the amount of information in the input.

Figures (3)

• Figure 1: Correspondence between the mean internal time $\mathbb{E}[\tau(t)]$ and physical time $t$ in the uniform case. It increases rapidly in an early stage, and saturates in a later stage so that the advance of internal time is strongly slowed down.
• Figure 2: Physical time $\Delta t_{\mathrm{phys}}(\tau)$ required to obtain "one unit of internal time" in the uniform case (Eq. \ref{['eq:dt_phys']}). It increases sharply in later stages, showing that the advance of internal time is strongly slowed down.
• Figure 3: Mean internal time $\mathbb{E}[\tau(t)]$ in Zipf-law input (Eq. \ref{['eq:zipf']}) (Eq. \ref{['eq:Etau_zipf']}). Due to the heavy tail, the later-stage tail toward saturation tends to become long.

Theorems & Definitions (9)

• Theorem 1: Slowdown and saturation (general $p$)
• proof
• Theorem 2: Identity
• proof
• Theorem 3: Conditional equiprobability under the uniform distribution
• Lemma 1: Symmetry
• Theorem 4: Unimodality and maximization time
• Theorem 5: Equality condition: the uniform distribution is the unique achiever (Poisson arrivals)
• Theorem 6: Decomposition identity