*2.2. Boolean Chaos*

In an ideal ABN, the transitions of the signals are arbitrarily fast and the number of transitions increases with time, following a power law. These increasingly fast dynamics result in an unlimited growth of frequency over time, referred to as an ultraviolet catastrophe [30]. However, that phenomenon does not occur in nature because the information-transmitting links and the processing nodes (for instance real logic gates) have a maximum operation frequency, which are physically realized. Hence, they cannot transmit or generate signals above a certain frequency [31]. As a result, the nonideal behaviors of real logic devices are responsible for the origin of chaos in ABNs [30,31]. Those behaviors are (i) Short-pulse rejection (SPR), known as pulse filtering, preventing pulses shorter than a minimum duration from passing through the gate (Theorem 1). (ii) The asymmetry between the logic states, making the propagation delay time through the gate depending on whether the transition is a fall or rise. (iii) The degradation effect triggering a change in the events propagation delay time when they appear in rapid succession. Among them, the degradation effect is the main nonideal behavior source of deterministic chaos in an ABN [32], since Boolean chaos originates from a history-dependent delay [30,31], as defined Lemma 1.

**Theorem 1.** *For a symmetric ABN consisting of a single XOR logic operation with two self-inputs having delays τ*<sup>1</sup> *and τ*1*, and with τspr sufficiently small not collapsing to the always-off state occurs before t* = *τ*2*, the trajectory will never reach the always-off state.*

**Lemma 1.** *For a class of experimental ABN containing at least one XOR connective and feedback loop, deterministic chaos may arise if and only if the degradation effect, which is exhibited at some level in any real ABN, is presented.*

On the other hand, if the ABN has equal delays, the links will produce only regular oscillations. In addition, the fixed points caused by using only symmetric logic functions in the network nodes conduct that the dynamics will always collapse into a low or high logic state, respectively. Theorem 2 and Lemma 2 postulates the conditions. As a reference, the complete proofs of Theorems and Lemmas can be found in [31,35].

**Theorem 2.** *For a symmetric ABN consisting of a single exclusive-OR (XOR) logic operation with two self-inputs having delays τ*<sup>1</sup> *and τ*2*, the attractors are always periodic.*

**Lemma 2.** *The experimentally realized ABNs should not include a Boolean fixed point, for which all Boolean functions are satisfied simultaneously.*
