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AlCons : Deductive Synthesis of Sorting Algorithms in Theorema

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Theoretical Aspects of Computing – ICTAC 2021 (ICTAC 2021)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12819))

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Abstract

We describe the principles and the implementation of AlCons (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.

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Notes

  1. 1.

    This ensures the correctness of algorithms and it is dual to algorithm verification, where the algorithms are first created and then checked.

  2. 2.

    Theory exploration is the generation of interesting statements following from a certain set of axioms and/or for the purpose of developing certain proofs or algorithms.

  3. 3.

    https://www.wolfram.com/mathematica.

  4. 4.

    In other approaches one uses the permutation notion, which must be expressed by specific algorithmic definitions, and whose properties are more difficult to infer.

  5. 5.

    However some fine tuning of the implementation has been necessary, since trees have a more complex structure.

  6. 6.

    Every element of the inductive domain is a ground instance of exactly one term from the cover set.

  7. 7.

    Meta–variables designate terms (witnesses for existential goals) which are unknown at the current stage of the proof.

  8. 8.

    We presented a more detailed survey of the synthesis methods in [11].

  9. 9.

    https://isabelle.in.tum.de/library/HOL/HOL-Library/Sorting_Algorithms.html.

  10. 10.

    Note that this introduces certain exceptions to antisymmetry and transitivity when the empty composite object is involved.

  11. 11.

    In our context, a pattern is a term possibly containing variables, whose ground instantiations define an injective function into the domain.

  12. 12.

    This is just a matter of efficiency, the goal could contain anything as long as the currently synthesized function is not involved.

  13. 13.

    The rule generates proof alternatives for different groupings of the multiset terms.

  14. 14.

    Terms containing universally quantified variables, such that for every element of the domain there exists exactly one term in the set which instantiates to that element.

  15. 15.

    By local convention, here \(x, x', y, y'\) represent any kind of objects.

  16. 16.

    Note that these kind of new assumptions are global: they can be used on any branch of the current proof.

  17. 17.

    https://www.risc.jku.at/research/theorema/software/.

  18. 18.

    https://www.risc.jku.at/people/tjebelea/AlCons.html.

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Author information

Authors and Affiliations

  1. Department of Computer Science, West University, Timişoara, Romania

    Isabela Drămnesc

  2. RISC, Johannes Kepler University, Linz, Austria

    Tudor Jebelean

Authors
  1. Isabela Drămnesc
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Correspondence to Isabela Drămnesc .

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Editors and Affiliations

  1. Nazarbayev University, Nur-Sultan, Kazakhstan

    Antonio Cerone

  2. Department of Informatics, University of Oslo, Oslo, Norway

    Peter Csaba Ölveczky

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Drămnesc, I., Jebelean, T. (2021). AlCons : Deductive Synthesis of Sorting Algorithms in Theorema . In: Cerone, A., Ölveczky, P.C. (eds) Theoretical Aspects of Computing – ICTAC 2021. ICTAC 2021. Lecture Notes in Computer Science(), vol 12819. Springer, Cham. https://doi.org/10.1007/978-3-030-85315-0_18

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  • DOI: https://doi.org/10.1007/978-3-030-85315-0_18

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