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Witnessed Symmetric Choice and Interpretations in Fixed-Point Logic with Counting

Author Moritz Lichter

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ACM Subject Classification
  • Theory of computation → Finite Model Theory
  • Theory of computation → Complexity theory and logic
Keywords
  • witnessed symmetric choice
  • interpretation
  • fixed-point logic
  • counting
  • CFI graphs
  • logic for PTime

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Abstract

At the core of the quest for a logic for Ptime is a mismatch between algorithms making arbitrary choices and isomorphism-invariant logics. One approach to tackle this problem is witnessed symmetric choice. It allows for choices from definable orbits certified by definable witnessing automorphisms.
We consider the extension of fixed-point logic with counting (IFPC) with witnessed symmetric choice (IFPC+WSC) and a further extension with an interpretation operator (IFPC+WSC+I). The latter operator evaluates a subformula in the structure defined by an interpretation. When similarly extending pure fixed-point logic (IFP), IFP+WSC+I simulates counting which IFP+WSC fails to do. For IFPC+WSC, it is unknown whether the interpretation operator increases expressiveness and thus allows studying the relation between WSC and interpretations beyond counting.
In this paper, we separate IFPC+WSC from IFPC+WSC+I by showing that IFPC+WSC is not closed under FO-interpretations. By the same argument, we answer an open question of Dawar and Richerby regarding non-witnessed symmetric choice in IFP. Additionally, we prove that nesting WSC-operators increases the expressiveness using the so-called CFI graphs. We show that if IFPC+WSC+I canonizes a particular class of base graphs, then it also canonizes the corresponding CFI graphs. This differs from various other logics, where CFI graphs provide difficult instances.

Cite As Get BibTex

Moritz Lichter. Witnessed Symmetric Choice and Interpretations in Fixed-Point Logic with Counting. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 133:1-133:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.ICALP.2023.133

Author Details

Moritz Lichter
  • TU Darmstadt, Germany

Funding

This work received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (EngageS: agreement No. 820148).

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