Chapter 8 State Complexity

In the discussion of the comparison of logical complexity to algorithmic complexity, we concluded that logical complexity represents a relaxed interpretation of algorithmic complexity because it ignores patterns which can not be ignored in a rigorous application of algorithmic complexity. Logical complexity accounts for state changes, but does not differentiate between the actual states and therefore can not recognize patterns. State complexity does identify the number of distinct system states and represents a simple form of pattern recognition that is therefore closer to the definition of algorithmic complexity. However, we theorized that a complexity measure in line with a strict interpretation of algorithmic complexity may not be beneficial in optimal searches, based on the shape of the qualitative complexity/performance relationship in Figure 7.52. For conventional elevator (transportation) systems with bidirectional, dedicated pathways, evolutions with all carriages in operation with the fewest interactions are optimal with respect to performance. These evolutions have a constant number of phase lags and therefore the greatest number of state changes per system cycle and therefore the greatest logical complexity. However, the number of distinct states entered by these evolutions can vary over a wide range despite few carriage interactions, depending on the variety of carriage destinations. The algorithmic complexity of an evolution with a high throughput may therefore be little different than the algorithmic complexity of an evolution with a low throughput. A definition of complexity that results in a wide range of throughputs for a given complexity and a wide range of complexities for a given throughput has limited value in searches for optimal configurations.