Mechanical synthesis of sorting algorithms for binary trees by logic and combinatorial techniques

Abstract

We develop logic and combinatorial methods for automating the generation of sorting algorithms for binary trees, starting from input-output specifications and producing conditional rewrite rules. The main approach consists in proving (constructively) the existence of an appropriate output from every input. The proof may fail if some necessary sub-algorithms are lacking. Then, their specifications are suggested and their synthesis is performed by the same principles.Our main goal is to avoid the possibly prohibitive cost of pure resolution proofs by using a natural-style proving in which domain-specific strategies and inference steps lead to a significant increase of efficiency. In addition to classical techniques for natural-style proving, we introduce novel ones (priority of certain types of assumptions, transformation of elementary goals into conditions, special criteria for decomposition of the goal and of the assumptions), as well as methods based on the properties of domain-specific relations and functions. In particular, we use combinatorial techniques in order to generate possible witnesses, which in certain cases lead to the discovery of new induction principles. From the proof, the algorithm is extracted by transforming inductive proof steps into recursions, and case-based proof steps into conditionals.The approach is demonstrated in parallel using the Theorema system, by developing the theory, implementing the prover, and performing the proofs of the necessary properties and synthesis conjectures. It is also validated in the Coq system, which allows to compare the facilities of the two systems from the point of view of our application.

Proof–Based Synthesis of Sorting Algorithms Using Multisets in Theorema

“…Each conjecture for the synthesis of a target function is a quantified statement over some main universal variable. A cover set is a set of universal terms 4 which represent the domain of the main universal variable, as described in [15].…”

“…We follow the proof-based approach to automated synthesis: first one proves automatically a synthesis conjecture which is based on the specification (input and output conditions) of the desired function, then the algorithm is extracted automatically from the proof, in form of conditional rewrite rules. The theoretical basis and the correctness of this scheme is well-known [6] and we used earlier in [11,15].…”

“…However, in that pioneering work, the starting point of the synthesis (besides the specification of the desired function) is a specific algorithm scheme, while in our approach we use general Noetherian induction and cover-set decomposition. In our previous work we study proof-based algorithm synthesis in the theories of lists [9], sets [10] and binary trees [13] separately [7], [8], [14], [11], [15].…”

“…Namely, during the proof of a statement P[t], for any t ′ (also ground term) which represents an object which is strictly smaller than the object represented by t in the Noetherian ordering, P[t ′ ] can be added to the current assumptions. (The soundness of this technique is presented in detail in [15] and it allows to discover concrete induction principles based on the general Noetherian induction.) In our approach we use the Noetherian ordering induced by the strict inclusion of the corresponding multisets, which conveniently extends to a meta-ordering between terms, induced by the strict inclusion of the constants occurring in the respective terms.…”

SPIKE, an automatic theorem prover — revisited

“…Some of the original features are still missing, like the graphical interface and the procedures for checking the completeness and ground convergence properties. In the future, an interesting application of SPIKE would be to the automation of the induction reasoning during the algorithm synthesis process [23]- [25].…”