AlCons: Deductive Synthesis of Sorting Algorithms in Theorema

We describe the principles and the implementation of AlCons (em Algorithm Constructor), a system for the automatic proof--based synthesis of sorting algorithms on lists and on binary trees, in the frame of the Theorema system. The core of the system is a dedicated prover based on specific inference rules and strategies for constructive proofs over the domains of lists and of binary trees, aimed at the automatic synthesis of sorting algorithms and their auxiliary functions from logical specifications. The specific distinctive feature of our approach is the use of multisets for expressing the fact that two lists (trees) have the same elements. This allows a more natural expression of the properties related to sorting, compared to the classical approach using the permutation relation (a list is a permutation of another). Moreover, the use of multisets leads to special inference rules and strategies which make the proofs more efficient, as for instance: expand/compress multiset terms and solve meta-variables using multiset equalities. Additionally we use a Noetherian induction strategy based on the relation induced by the strict inclusion of multisets, which facilitates the synthesis of arbitrary recursion structures, without having to indicate the recursion schemes in advance. The necessary auxiliary algorithms (like, e.g., for insertion and merging) are generated by the same principles from the synthesis conjectures that are automatically produced during the main proof, using a ``cascading" method, which in fact contributes to the automation of theory exploration. The prover is implemented in the frame of the Theorema system and works in natural style, while the generated algorithms can be immediately tested in the same system.