The Art of Prolog

ly, this program guesses nondeterministically the correct permutation via permutation(Xs,Ys), and ordered checks that the permutation is actually ordered. Operationally, the behavior is as follows. A query involving sort is reduced to a query involving permutation and ordered. A failure-driven loop ensues. A permutation of the list is generated by permutation and tested by ordered. If the permuted list is not ordered, the execution backtracks to the permutation goal, which generates another permutation to be tested. Eventually an ordered permutation is generated and the computation terminates. Permutation sort is a highly inefficient sorting algorithm, requiring time super-exponential in the size of the list to be sorted. Pushing the tester into the generator, however, leads to a reasonable algorithm. The generator for permutation sort, permutation, selects an arbitrary element and recursively permutes the rest of the list. The tester, ordered, verifies that the first two elements of the permutation are in order, then recursively checks the rest. If we view the combined recursive permutation and ordered goals as a recursive sorting process, we have the basis for insertion sort, Program 3.21. To sort a list, sort the tail of the list and insert the head of the list into its correct place in the order. The arbitrary selection of an element has been replaced by choosing the first element. Another example of the advantage of intertwining generating and testing can be seen with programs solving the N queens problem. The N queens problem requires the placement of N pieces on an Nby-N rectangular board so that no two pieces are on the same line: horizontal, vertical, or diagonal. The original formulation called for 8 queens to be placed on a chessboard, and the criterion of not being on the same line corresponds to two queens not attacking each other under the rules of chess. Hence the problem's name. 253 Nondeterministic Programming