Combinatorial Spaces And Order Topologies

An archetypal problem discussed in computer science is the problem of searching for a given number in a given set of numbers. Other than sequential search, the classic solution is to sort the list of numbers and then apply binary search. The binary search problem has a complexity of O(logN) for a list of N numbers while the sorting problem cannot be better than O(N) on any sequential computer following the usual assumptions. Whenever the problem of deciding partial order can be done in O(1), a variation of the problem on some bounded list of numbers is to apply binary search without resorting to sort. The overall complexity of the problem is then O(logR) for some radius R. The following upper-bound for finite encodings is shown: O(log|X|∞ log log N) Also, the topology of orderings can provide efficient algorithms for search problems in combinatorial spaces. The main characteristic of those spaces is that they have typical space complexities of O(2N ), O(N !) and O(NN ). The factorial case describes an order topology that can be illustrated using the combinatorial polytope . When a known order topology can be combined to a given formulation of a search problem, the resulting search problem has a polylogarithmic complexity. This logarithmic complexity can then become useful in combinatorial search by providing a logarithmic break-down. These algorithms can be termed as the class of search algorithms that do not require read and are equivalent to the class of logarithmically recursive functions. Also, the notion of order invariance is discussed. 1. Computable Structures A few terminological remarks can be made on the algebra of computable structures. Remark 1.1. A computable space is defined using the distinguishability problem. Remark 1.2. A computable space is said to be complete if it can represent a Turing-complete space. Furthermore, the following can be noted: Remark 1.3. The equality operator is defined using the cut metric of Equation 3.0.11. This is denoted = N .