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8.3.6 Behavioral complexity Our ability to describe a system arises from measurements or observations of its behavior. The use of system descriptions to define system complexity does not directly take this into account. The complexity profile brought us closer by acknowledging the observer in the space and time scale of the description. By acknowledging the scale of observation, we obtained a mechanism for distinguishing complex systems from equilibrium systems,and a systematic method for characterizing the complexity of a system. There is another approach to reaching the complexity profile that incorporates the observer and system relationship in a more satisfactory manner. It also enables us to consider directly the interaction of the system with its environment, which was not included previously. To introduce the new approach, we return to the underpinning of descriptive complexity and present the concept of behavioral complexity. In Shannon’s approach to the study of information in communication systems, there were two quantities of fundamental interest. The first was the information content of an individual message, and the second was the average information provided by a particular source. The discussion of algorithmic complexity was based on a consideration of the information provided by a particular message—specifically, how much it could be compressed. This carried over into our discussion of physical systems when we introduced the microscopic complexity of a system as the information contained in a particular microscopic realization of the system. When all messages,or all system states, have the same probability, then the information in the particular message is the same as the average information, and we can write: I({x, p}|(U ,N ,V )) = −logP({x, p}) = −
∑ P({x, p})log(P({x, p}))
{x,p }
(8.3.58)
The expression on the right, however, has a different purpose. It is a quantity that characterizes the ensemble rather than the individual microstate. It is a characterization of the source rather than of any particular message. We can pursue this line of reasoning by considering more carefully how we might characterize the source of the information, rather than the messages.One way to characterize the source is to determine the average amount of information in a message. However, if we want to describe the source to someone, the most essential information is to give a description of the kinds of messages that will be received—the ensemble of possible messages. Thus to characterize the source we need a description of the probability of each kind of message. How much information do we need to describe these probabilities? We call this the behavioral complexity of the source. A few examples in the context of a source of messages will serve to illustrate this concept. Any description of a source must assume a language that is to be used. We assume that the language consists of a list of characters or messages that can be received from the source, along with their probabilities.A delimiter (:) is used to separate the messages from their probability. For convenience, we will write probabilities in decimal notation. A second delimiter (,) is used to separate different members of the list.A source that gives zeros and ones at random with equal probability would be described by {1:0.5,0:0.5}. It is convenient to include the length of a message in our
# 29412 Cust: AddisonWesley Au: Bar-Yam Title: Dynamics Complex Systems
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