Boundedness and Other Theories for Complex Systems

A comparison between the concept of boundedness on the one hand, and the theory of self-organized criticality (SOC) and the deterministic chaos on the other hand, is made. The focus is put on the methodological importance of the general frame through which an enormous class of empirical observations is viewed. The major difference between the concept of boundedness and the theory of self organized criticality is that under boundedness, the response comprises both specific and universal part, and thus a system has well defined “identity,” while SOC assumes response as a global invariant which has only universal properties. Unlike the deterministic chaos, the boundedness is free to explain the sensitivity to initial conditions independently from the mathematical object that generates them. Alongside, it turns out that the traditional approach to the deterministic chaos has its ample understanding under the concept of boundedness.

Is Semantic Physical?!

A critical comparison between the traditional algorithmic approach and the semantic-like one is made. The comparison include topics such as causality, correlations, halting problem, shortest algorithm, intuition, Zipf`s law, and absolute information. The purpose of making this comparison is to delineate neatly the fundamental difference between both approaches and to make clear that, although they are different, they still are counterparts which coexist peacefully. One of the major differences between them turns out to be that whilst the semantic-like approach permits autonomous discrimination between “true” and “false” statement by an intelligent complex system, the traditional algorithmic theory does not allow any autonomous discrimination between a “true” and a “false” statement. On the other hand, their common property turns out to be that it is impossible to acquire absolute knowledge: for example, even the famous “super-minded” Maxwell demon can be deceived.

Invariant Measure

Bounded randomness of mass/energy exchange rates neither presuppose nor selects any specific time scale, thresholds of stability included. Nonetheless, the boundedness of the rates sets certain non-physical correlations among successive increments and thus justifies formation of “sub-walks” on the finest scale. Further, the “U-turns” at the thresholds of stability set certain correlations on the biggest possible scale of a relevant variable. The major question now is how the balance between the universal correlations, set by the “U-turns,” and those of the specific “sub-walks,” set by the bounded randomness, shapes the structure of a BIS that represents the evolutionary pattern of a relevant variable. It is proven that this issue is inherently related to another universal property of complex systems behavior that is power law distributions. It is demonstrated that power law distributions acquire novel understanding in the setting of boundedness: they appear as universal criterion for hierarchical structuring implemented under boundedness.

Time Series Invariants under Boundedness

It is proven that every zero-mean bounded irregular sequence (BIS) has three invariants, i.e. characteristics which stay the same when the environmental statistics changes. The existence of such invariants answers the question how far they ensure certainty of the obtained knowledge and the range of predictability of stable complex systems behavior in a positive way. The certainty of our knowledge is put to test by the lack of global rule for response makes impossible to adjust a priori the corresponding recording equipment to a long run. Then, it is to be expected that the recorded time series does not match the corresponding signal in a uniform way since the record is subject to local distortion which is generally non-linear and acts non-homogeneously on the recording. In turn, this poses the fundamental question whether it is ever possible to establish and/or predict the properties and the future behavior of the complex systems.

Time Series Invariants under Boundedness

One of the basic notions of any type intelligence expressed in a semantic-like manner is the notion of a letter (character). In view of the concept of boundedness, a letter must be implemented as characteristic of a specific natural process. This view sets one of the fundamental demands to every letter to be its autonomity, i.e. to retain its specific characteristics on reoccurrence. Thus, the matter of implementing of a letter turns apparently related to the issue about the robustness of homeostasis to small environmental fluctuations. It is proven that the response of a stable complex system is additively decomposable to a specific steady part and a universal one so that the steady part reoccurs with the same accuracy in an ever-changing environment. This makes the association of the notion of homeostasis with the steady pattern appropriate. In result, the corresponding characteristic of the homeostasis appears as a suitable candidate for a “letter”.

Second Law Viewed as Ban over Perpetuum Mobile

It is proven that under boundedness, the efficiency of a non-mechanical engine never exceeds the efficiency of the corresponding Carnot engine where the engine is free from necessity of a physical coupling to two heat reservoirs. The proof is free from the condition for entropy maximization viewed as condition for reaching equilibrium. Thus the proof substantiates the most ubiquitous formulation of the Second Law to be ban over perpetuum mobile. Further, the ban over the information perpetuum mobile appears a consequence of the most general formulation of the Second Law which asserts that it is impossible to build a non-mechanical “engine,” which works steadily in a cyclic regime without exerting any functional changes of its homeostasis during the working cycle.

Sustainable Evolution in an Ever-Changing Environment

It is suggested that the notion of equation-of-state serves as appropriate common basis for studying the macroscopic behavior of both traditional physical systems and complex systems. The reason is that while the equilibrium systems are characterized both by their energy function and the corresponding equation-of-state, the steady states of out-of-equilibrium systems are defined only by their dynamics, i.e. by their equations-of-state. It is demonstrated that there exists a common measure which generalizes the notion of Gibbs measure so that it acquires two-fold meaning: it appears both as local thermodynamical potential and as probability for robustness to environmental fluctuations. It is proven that the obtained Gibbs measure has very different meaning and role than its traditional counterpart. The first one is that it is derived without prerequisite requirement for simultaneous achieving of any extreme property of the system such as maximization of the entropy.

Hierarchical Order

A generic property of the motion along every allowed trajectory in a semantic state space is that Hiesenberg-like relations hold. Under the condition that the action on each allowed trajectory is stationary, it is possible to define “energy,” “velocity,” and “mass.” The so defined energy is not constant along the trajectory because of permanent mass-energy transformations. Yet, unlike Einstein mass-energy relation, the amount of energy and mass that is transformed is permanently kept bounded along the allowed trajectories. The control over local accumulation of matter/energy in a structured network is implemented through spontaneous emitting of a matter wave. The stationarity of action governs grammar rules, which sets non-random picking of successive semantic units, i.e. word order is a sentence, thus providing non-extensivity of the semantic-like hierarchy.

Hierarchical Order II

The self-organization under boundedness is considered as an operational protocol, which describes the highly non-trivial interplay between the inter-level feedback and the spatio-temporal pattern that describes the corresponding homeostasis. A generic property of the feedback is that it sets metrics in the state space and defines the admissible transitions from any given state. Further, the state space is partitioned into basins-of-attraction so that each of them is tangent to the point called accumulation point. A distinctive property of the basins-of-attraction is that the discrete band of the power spectrum appears as intra-basin invariant and thus turns as appropriate candidate for a “letter.” The accumulation point is associated with the notion of a “space bar.” Then, since the motion in a bounded attractor is orbital, it is appropriate to associate a “word” with an orbit. The latter open the door for assigning a specific non-mechanical engine to each and every orbit. In turn the functional irreversibility of any engine substantiates the sensitivity to permutations of every semantic unit.

Hierarchical Order I

The purpose of the hierarchical organization is to strengthen the response by means of two general implements: (i) specification of multi-level structure so that each level to respond to specific impacts; (ii) the levels cooperate one with another by means of inter-level feedbacks. The role of the inter-level feedbacks is to sustain the response of any given level bounded by means of keeping local amplifications, local damping, and other non-linear effects restrained. The general purpose of the inter-level feedbacks to sustain long-term stability suggests that they must obey boundedness. Further, the ubiquity of the universal properties of the complex systems promptly suggests that the inter-level feedbacks must appear as a bounded “environment” for every hierarchical level. The non-trivial application of the concept of boundedness to quantum phenomena is considered.