Polyregular Functions on Unordered Trees of Bounded Height

Author:

Bojańczyk Mikołaj1ORCID,Klin Bartek2ORCID

Affiliation:

1. University of Warsaw, Warsaw, Poland

2. University of Oxford, Oxford, UK

Abstract

We consider injective first-order interpretations that input and output trees of bounded height. The corresponding functions have polynomial output size, since a first-order interpretation can use a k -tuple of input nodes to represent a single output node. We prove that the equivalence problem for such functions is decidable, i.e. ‍given two such interpretations, one can decide whether, for every input tree, the two output trees are isomorphic. We also give a calculus of typed functions and combinators which derives exactly injective first-order interpretations for unordered trees of bounded height. The calculus is based on a type system, where the type constructors are products, coproducts and a monad of multisets. Thanks to our results about tree-to-tree interpretations, the equivalence problem is decidable for this calculus. As an application, we show that the equivalence problem is decidable for first-order interpretations between classes of graphs that have bounded tree-depth. In all cases studied in this paper, first-order logic and MSO have the same expressive power, and hence all results apply also to MSO interpretations.

Funder

Narodowe Centrum Nauki

Publisher

Association for Computing Machinery (ACM)

Reference20 articles.

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2. Mikoł aj Bojańczyk. 2018. Polyregular Functions. CoRR, abs/1810.08760 (2018).

3. Mikoł aj Bojańczyk. 2022. Transducers of Polynomial Growth. In Logic in Computer Science (LICS ’22). Association for Computing Machinery, New York, NY, USA.

4. Mikoł aj Bojańczyk. 2023. Folding Interpretations. In Logic in Computer Science (LICS ’22). Association for Computing Machinery.

5. Mikolaj Bojańczyk, Sandra Kiefer, and Nathan Lhote. 2019. String-to-String Interpretations With Polynomial-Size Output. In 46th International Colloquium on Automata, Languages, and Programming, ICALP 2019, July 9-12, 2019, Patras, Greece. 106:1–106:14.

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2. Polyregular Functions Characterisations and Refutations;Lecture Notes in Computer Science;2024

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