Proof–Based Synthesis of Sorting Algorithms for Trees

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.

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“…The results presented in this section have been partially published in (Dramnesc et al, 2016b(Dramnesc et al, , 2016a, see also .…”

Section: Algorithm Synthesis Experimentsmentioning

confidence: 99%

“…Our research on binary trees has been partially presented in some conferences and workshops (Dramnesc et al, 2015c(Dramnesc et al, , 2016b(Dramnesc et al, , 2016a. The present paper constitutes a synthesis of all our results on binary trees but it adds new ideas and improves the presentation of our work as follows.…”

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confidence: 92%

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

Dramnesc

Jebelean

Stratulat

2019

Journal of Symbolic Computation

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“…The prover uses techniques which are specific to the domain of binary trees. Some of these techniques were introduced in [9] and [13] and some of them are introduced in this paper. From the generated proof, the corresponding algorithm is automatically extracted.…”

Section: A Related Workmentioning

confidence: 99%

“…The authors apply in [9] combinatorial techniques in order to discover from proofs auxiliary algorithms, other than Insert, which are versions of an algorithm that merges two sorted binary trees into a sorted one. All these auxiliary algorithms are necessary in the sorting algorithms which were discovered from proofs in [13]. The experiments in the Theorema system are given in [10].…”

Section: A Related Workmentioning

confidence: 99%

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A case study on algorithm discovery from proofs: The insert function on binary trees

Dramnesc

Jebelean

Stratulat

2016

2016 IEEE 11th International Symposium on Applied Computational Intelligence and Informatics (SACI)

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Abstract-We present a proof-based automatic synthesis experiment in the context of sorting binary trees, namely the synthesis of the function which inserts an element in a sorted binary tree at the appropriate place. The algorithm is automatically extracted from the automatically produced proof of the conjecture which expresses the existence of the desired output for each appropriate input of the function. This is a case study with the purpose of finding and illustrating general and domain specific inference rules and strategies for efficiently producing proofs from which the desired algorithms can be extracted.The construction of the necessary theory, of the provers, and the experiment itself are performed in the Theorema system.

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Automatic Synthesis of Merging and Inserting Algorithms on Binary Trees Using Multisets in Theorema

Dramnesc

Jebelean

2020

Mathematical Aspects of Computer and Information Sciences

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Proof–Based Synthesis of Sorting Algorithms for Trees

Cited by 5 publications

References 17 publications

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

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

A case study on algorithm discovery from proofs: The insert function on binary trees

Automatic Synthesis of Merging and Inserting Algorithms on Binary Trees Using Multisets in Theorema

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