Bubble sort

Abstract: Many sorted logics can increase deductive efficiency by eliminating useless branches of the search space, but usually this results in reduced expressiveness. The many sorted logic described here has several unusual features which not only increase expressiveness but also can reduce the search space even more than a conventional many sorted logic. The quantifiers are unsorted: the restriction on the range of a variable derives from the argument positions of the nonlogical symbols that it occupies. Polymorphic sort specifications are allowed; thus statements usually requiring several assertions may be compactly expressed by a single assertion. The sort structure may be an arbitrary lattice and the sort of a term can be more general than the sort of the argument position it occupies. It is also shown how it is sometimes possible to use sort information to determine the truth value of a formula without resort normal inference. Inference rules for a resolution based system are discussed; these can be proved to be sound and complete.

Abstract: Empirically, neural networks that attempt to learn programs from data have exhibited poor generalizability. Moreover, it has traditionally been difficult to reason about the behavior of these models beyond a certain level of input complexity. In order to address these issues, we propose augmenting neural architectures with a key abstraction: recursion. As an application, we implement recursion in the Neural Programmer-Interpreter framework on four tasks: grade-school addition, bubble sort, topological sort, and quicksort. We demonstrate superior generalizability and interpretability with small amounts of training data. Recursion divides the problem into smaller pieces and drastically reduces the domain of each neural network component, making it tractable to prove guarantees about the overall system's behavior. Our experience suggests that in order for neural architectures to robustly learn program semantics, it is necessary to incorporate a concept like recursion.

Abstract: The aims of this chapter are to provide an introduction to algorithms and their behaviour. In Computer Science this is normally done using the so called big O notation.