Showing papers on "Counting sort published in 1983"

Improving the expressiveness of many sorted logic

Abstract: Many sorted logics can allow an increase in deductive efficiency by eliminating useless branches of the search space, but are usually formulated so that their expressive power is severely limited. The many sorted logic described here is unusual in that the quantifiers are unsorted; the restriction on the range of a quantified variable derives from the argument positions of the function and predicate symbols that it occupies; associated with every non-logical symbol is a sorting function which describes how its sort varies with the sorts of its inputs; polymorphic functions and predicates are thus easily expressible and statements usually requiring several assertions may be compactly expressed by a single assertion. The sort structure may be an arbitrary lattice. Increased expressiveness is obtained by allowing the sort of a term to be a more general sort than the sort of the argument position it occupies. Furthermore, by allowing three boolean sorts (representing 'true', 'false' and 'either true or false'), it is sometimes possible to detect that a formula is contradictory or tautologous without resort to general inference rules. Inference rules for a resolution based system are discussed; these can be proved to be both sound and complete.

Improving the exprfssiwjvess of man-y sorted logic

Abstract: Many sorted logics can allow an increase in deductive efficiency by eliminating useless branches of the search space, but are usually formulated so that their expressive power is severely limited. The many sorted logic described here is unusual in that the quantifiers are unsorted; the restriction on the range of a quantified variable derives from the argument positions of the function and predicate symbols that it occupies; associated with every non-logical symbol is a sorting function which describes how its sort varies with the sorts of its inputs; polymorphic functions and predicates are thus easily expressible and statements usually requiring several assertions may be compactly expressed by a single assertion. The sort structure may be an arbitrary lattice. Increased expressiveness is obtained by allowing the sort of a term to be a more general sort than the sort of the argument position it occupies. Furthermore, by allowing three boolean sorts (representing ‘true’, ‘false’ and ‘either true or false’), it is sometimes possible to detect that a formula is contradictory or tautologous without resort to general inference rules. Inference rules for a resolution based system are discussed; these can be proved to be both sound and complete.

Sorting by Recursive Partitioning

Abstract: : This document presents a new O(n lg lg n) time sort algorithm that is more robust than O(n) distribution sorting algorithms. The algorithm uses a recursive partition-concatenate approach, partitioning each set into a variable number of subsets using information gathered dynamically during execution. Sequences are partitioned using statistical information computed during the sort for each sequence. Space complexity is O(n) and is independent from the order and distribution of the data. If the data is originally in a list, only O(square root of (n) extra space is necessary. The algorithm is insensitive to the initial ordering of the data, and it is much less sensitive to the distribution of the values of the sorting keys than distribution sorting algorithms. Its worst-case time is O(n lg lg n) across all distributions that satisfy a new fractalness criterion. This condition, which is sufficient but not necessary, is satisfied by any set with bounded length keys and bounded repetition of each key. If this condition is not satisfied, its worst case performance degrades gracefully to O(n lg n). In practice, this occurs when the density of the distribution over omega(n) of the keys is a fractal curve (for sets of numbers whose values are bounded), or when the distribution has very heavy tails with arbitrarily long keys (for sets of numbers whose precision is bounded). In some preliminary tests, it was faster than Quicksort for sets of more than 150 elements. The algorithm is practical, works basically in place, can be easily implemented and is particularly well suited both for parallel processing and for external sorting. (Author)