Proof-Based Synthesis of Sorting Algorithms Using Multisets in Theorema
This work addresses the challenge of automated algorithm synthesis for researchers in formal methods and theorem proving, though it appears incremental as it builds on existing proof-based synthesis methods.
The paper tackles the problem of synthesizing list-sorting algorithms by developing novel techniques using multisets to mechanize proofs in the Theorema system, resulting in the automatic generation of specifications and synthesis of auxiliary functions like insert, merge, and split.
Using multisets, we develop novel techniques for mechanizing the proofs of the synthesis conjectures for list-sorting algorithms, and we demonstrate them in the Theorema system. We use the classical principle of extracting the algorithm as a set of rewrite rules based on the witnesses found in the proof of the synthesis conjecture produced from the specification of the desired function (input and output conditions). The proofs are in natural style, using standard rules, but most importantly domain specific inference rules and strategies. In particular the use of multisets allows us to develop powerful strategies for the synthesis of arbitrarily structured recursive algorithms by general Noetherian induction, as well as for the automatic generation of the specifications of all necessary auxiliary functions (insert, merge, split), whose synthesis is performed using the same method.