# Is Polynomial Time Choiceless?

@inproceedings{Grdel2015IsPT, title={Is Polynomial Time Choiceless?}, author={Erich Gr{\"a}del and Martin Grohe}, booktitle={Fields of Logic and Computation II}, year={2015} }

A long time ago, Yuri Gurevich made precise the problem of whether there is a logic capturing polynomial-time on arbitrary finite structures, and conjectured that no such logic exists. This conjecture is still open. Nevertheless, together with Andreas Blass and Saharon Shelah, he has also proposed what still seems to be the most promising candidate for a logic for polynomial time, namely Choiceless Polynomial Time (with counting). We survey some recent results on this logic.

## 17 Citations

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

- Computer Science, MathematicsACM Trans. Comput. Log.
- 2018

Dawar, Richerby, and Rossman proved that the CFI-query is CPT-definable, and a notion of “sequencelike objects” based on the structure of the graphs’ symmetry groups is introduced, and it is shown that no C PT-program that only uses sequencelike objects can decide the CPI-query over complete graphs, which have linear maximal degree.

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

- Computer Science, MathematicsCSL
- 2016

A notion of "sequence-like objects" based on the structure of the graphs' symmetry groups is introduced, and it is shown that no CPT-program which only uses sequence- like objects can decide the CFI query over complete graphs, which have linear maximal degree.

Choiceless Computation and Symmetry: Limitations of Definability

- Computer ScienceCSL
- 2021

It is shown that Choiceless Polynomial Time cannot compute a preorder with colour classes of logarithmic size in every hypercube, implying that CFI on unordered hypercubes is a PTIME-problem which provably cannot be tackled with the state-of-the-art choiceless algorithmic techniques.

Choiceless Logarithmic Space

- Computer ScienceMFCS
- 2019

This work proposes here a notion of Choiceless Logarithmic Space which overcomes some of the obstacles posed by Logspace as a less robust complexity class, and addresses the question whether this logic can define all Logspace-queries, and proves that this is not the case.

Deep Weisfeiler Leman

- Computer Science, MathematicsSODA
- 2021

It is proved that, as an abstract computational model, polynomial time DeepWL-algorithms have exactly the same expressiveness as the logic Choiceless Polynomial Time.

On computability and tractability for infinite sets

- Computer Science, MathematicsLICS
- 2018

It is shown that, under suitable assumptions on the underlying structure, a programming language called definable while programs captures exactly the computable functions in a complexity class called fixed-dimension polynomial time, which intuitively speaking describesPolynomial computation on hereditarily definable sets.

Choiceless Polynomial Time, Symmetric Circuits and Cai-Fürer-Immerman Graphs

- Computer Science, MathematicsArXiv
- 2021

This work provides new insights regarding a much more general problem: the existence of a solution to an unordered linear equation system A · x = b over a finite field is CPT-definable if the matrix A has at most logarithmic rank.

Tight Bounds on the Asymptotic Descriptive Complexity of Subgraph Isomorphism

- MathematicsACM Trans. Comput. Log.
- 2019

An infinite family of patterns F such that the existence of a subgraph isomorphic to F is expressible by a first-order sentence of quantifier depth 2/3 v(F) + 1, assuming that the host graph is sufficiently large and connected.

Fully Generic Queries: Open Problems and Some Partial Answers

- Computer ScienceMEDI
- 2019

The class of fully generic queries on complex objects was introduced by Beeri, Milo and Ta-Shma in 1997 and is rather poorly understood, so the big open questions are reviewed and some partial answers are formulated.

The Descriptive Complexity of Subgraph Isomorphism Without Numerics

- MathematicsTheory of Computing Systems
- 2018

It is shown that, for some F, the existence of an F subgraph in sufficiently large connected graphs is definable with quantifier depth ℓ − 3, and the exact values of these descriptive complexity parameters for all connected pattern graphs F on 4 vertices are determined.

## References

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This work proposes a more model-theoretic formalism, called polynomial-time interpretation logic (PIL), that replaces the machinery of hereditarily finite sets and comprehension terms by traditional first-order interpretations, and handles counting by Härtig quantifiers.

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A detailed proof of Shelah's proof of a zero-one law for the complexity class “choiceless polynomial time” is presented, and the extent of its generalizability to other sorts of structures is described.

Limitations of Algebraic Approaches to Graph Isomorphism Testing

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We investigate the power of graph isomorphism algorithms based on algebraic reasoning techniques like Grobner basis computation. The idea of these algorithms is to encode two graphs into a system of…

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