Definability of Cai-Fürer-Immerman Problems in Choiceless Polynomial Time

Pakusa, Wied; Schalthöfer, Svenja; Selman, Erkal

doi:10.4230/lipics.csl.2016.19

Cited by 3 publications

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“…This result was strengthened by Pakusa, Schalthöfer, and Selman to base graphs with logarithmic color class size [26]. The techniques of [6] and [26] are used in [29] to solve the mentioned equation systems.…”

Section: Introductionmentioning

confidence: 99%

Canonization for Bounded and Dihedral Color Classes in Choiceless Polynomial Time

Lichter,

Schweitzer

2020

Preprint

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In the quest for a logic capturing Ptime the next natural classes of structures to consider are those with bounded color class size. We present a canonization procedure for graphs with dihedral color classes of bounded size in the logic of Choiceless Polynomial Time (CPT), which then captures Ptime on this class of structures. This is the first result of this form for non-abelian color classes.The first step proposes a normal form which comprises a "rigid assemblage". This roughly means that the local automorphism groups form 2-injective 3-factor subdirect products. Structures with color classes of bounded size can be reduced canonization preservingly to normal form in CPT.In the second step, we show that for graphs in normal form with dihedral color classes of bounded size, the canonization problem can be solved in CPT. We also show the same statement for general ternary structures in normal form if the dihedral groups are defined over odd domains.

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Section: Introductionmentioning

confidence: 99%

Canonization for Bounded and Dihedral Color Classes in Choiceless Polynomial Time

Lichter,

Schweitzer

2020

Preprint

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Separating Rank Logic from Polynomial Time

Lichter

2023

J. ACM

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In the search for a logic capturing polynomial time the most promising candidates are Choiceless Polynomial Time (CPT) and rank logic. Rank logic extends fixed-point logic with counting by a rank operator over prime fields. We show that the isomorphism problem for CFI graphs over \({\mathbb {Z}}_{2^i} \) cannot be defined in rank logic, even if the base graph is totally ordered. However, CPT can define this isomorphism problem. We thereby separate rank logic from CPT and in particular from polynomial time.

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Choiceless Polynomial Time with Witnessed Symmetric Choice

Lichter,

Schweitzer

2024

J. ACM

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We extend Choiceless Polynomial Time (CPT), the currently only remaining promising candidate in the quest for a logic capturing Ptime , so that this extended logic has the following property: for every class of structures for which isomorphism is definable, the logic automatically captures Ptime . For the construction of this logic we extend CPT by a witnessed symmetric choice operator. This operator allows for choices from definable orbits. But, to ensure polynomial-time evaluation, automorphisms have to be provided to certify that the choice set is indeed an orbit. We argue that, in this logic, definable isomorphism implies definable canonization. Thereby, our construction removes the non-trivial step of extending isomorphism definability results to canonization. This step was a part of proofs that show that CPT or other logics capture Ptime on a particular class of structures. The step typically required substantial extra effort.

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Definability of Cai-Fürer-Immerman Problems in Choiceless Polynomial Time

Cited by 3 publications

References 0 publications

Canonization for Bounded and Dihedral Color Classes in Choiceless Polynomial Time

Canonization for Bounded and Dihedral Color Classes in Choiceless Polynomial Time

Separating Rank Logic from Polynomial Time

Choiceless Polynomial Time with Witnessed Symmetric Choice

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