# Structural and Symbolic Information in the Context of the General Theory of Information
## 6 Relations and Interactions between Symbolic Information and Structural Information
The loss of energy of one of the interacting systems is accompanied by the equivalent gain of energy by another system, and so, information gained by such an energy transfer is accompanied by an equivalent energy loss at the other side. Transfer of symbolic information, however, does not require that the gain of information by a receiver is necessarily at the cost of information loss at the information source. Newspapers, TV, and the Internet demonstrate that symbolic information, once available, may be \"shared\" by numerous receivers without degrading the original information. Sharing or broadcasting information is possible in ways that conserve the information. However, this process may increase or decrease the value of that information [8]. In this paper, we do not study changes of value in the processes of information transmission. There are different ways to define the meaning of conservation of information. Two such options are measure-related conservation and transformation-related conservation. 1. _Measure-related conservation_ of information is realized if, in a process of conversion of a portion of information \(I\) to a different form \(I^{\prime}\), a certain measure of information, e.g., \(S\), remains the same, i.e., \(S(I)=S(I^{\prime})\). For instance, if \(I\) is the structural information of a macroscopic system at some time \(t\), and \(I^{\prime}\) is the structural information of that system at some later time \(t^{\prime}\), and entropy \(S\) is the chosen measure of information, then information is conserved with respect to \(S\) in reversible processes. Similarly, it is possible to understand conservation of mass, spin, and electric and magnetic charge as the conservation of structural information [63]. 2. _Transformation-related conservation_ of information is realized if in a process of conversion of the form \(F\) of a portion of information \(I\) to a different form \(F^{\prime}\), while both \(F\) and \(F^{\prime}\) have the capacity to cause the same changes in a system \(R\). According to the general theory of information both \(F\) and \(F^{\prime}\) represent the same information. For instance, if \(I\) is the symbolic information in a textbook \(F\) written in a language \(L\) and \(F^{\prime}\) is a contextual translation of the textbook \(F\) to another language, then for a student \(R\) the information is transformation-relatedly conserved if the translation is correct and \(R\) knows both languages to the same extent. The translation may be formally written as the operation \(T\): \(F\)\(\rightarrow\)\(F^{\prime}\). It is possible to interpret conservation as existence of the inverse operator \(T^{-1}\), a backward translation, such that \(F=T^{-1}(F^{\prime})\) and all translations \(T\) form a mathematical group. Similarly, structural information in quantum mechanics is subject to conservation in processes governed by a group of unitary evolution operators [28]. Independently of the particular way by which conservation of information may be defined in detail, we may state that, quite generally, conservation of information is understood as a binary relation between two portions of information, \(I\) and \(I^{\prime}\). More precisely, mathematically, this relation is a reflexive _equivalence relation_ because it has the properties of _symmetry_, _transitivity_, and _reflexivity_. All portions of information, which pairwise are mutually information-conservative, form an _equivalence class_. If each element of this class can be generated from another, arbitrarily-chosen element by a certain transformation \(T\), then this set of transformations forms a mathematical group. In the context of this article, considering conservation of information is fundamental as it permits discrimination of processes that violate conservation, namely, processes that destroy or produce information. Several information theories, such as the one of [PERSON], aim at suitable formal descriptions of non-conservative processes.