Proof-Based Synthesis of Sorting Algorithms Using Multisets in Theorema

TL;DR

This paper presents a formal method for synthesizing sorting algorithms using multisets and proof techniques within the Theorema system, enabling automatic generation and verification of recursive algorithms.

Contribution

It introduces a novel proof-based synthesis approach leveraging multisets and domain-specific inference rules in Theorema for recursive sorting algorithms.

Findings

Successfully synthesized sorting algorithms with multisets

Automated generation of auxiliary function specifications

Powerful strategies for recursive algorithm synthesis

Abstract

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.