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

07/08/2021
by   Benedikt Pago, et al.
0

Choiceless Polynomial Time (CPT) is currently the only candidate logic for capturing PTIME (that is, it is contained in PTIME and has not been separated from it). A prominent example of a decision problem in PTIME that is not known to be CPT-definable is the isomorphism problem on unordered Cai-Fürer-Immerman graphs (the CFI-query). We study the expressive power of CPT with respect to this problem and develop a partial characterisation of solvable instances in terms of properties of symmetric XOR-circuits over the CFI-graphs: The CFI-query is CPT-definable on a given class of graphs only if: For each graph G, there exists an XOR-circuit C, whose input gates are labelled with edges of G, such that C is sufficiently symmetric with respect to the automorphisms of G and satisfies certain other circuit properties. We also give a sufficient condition for CFI being solvable in CPT and develop a new CPT-algorithm for the CFI-query. It takes as input structures which contain, along with the CFI-graph, an XOR-circuit with suitable properties. The strongest known CPT-algorithm for this problem can solve instances equipped with a preorder with colour classes of logarithmic size. Our result implicitly extends this to preorders with colour classes of polylogarithmic size (plus some unordered additional structure). Finally, our 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 (with respect to the size of the structure that encodes the equation system). This is another example that separates CPT from fixed-point logic with counting.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
02/08/2023

Lower bounds for Choiceless Polynomial Time via Symmetric XOR-circuits

Choiceless Polynomial Time (CPT) is one of the few remaining candidate l...
research
04/09/2018

Symmetric Circuits for Rank Logic

Fixed-point logic with rank (FPR) is an extension of fixed-point logic w...
research
03/29/2018

Capturing Polynomial Time using Modular Decomposition

The question of whether there is a logic that captures polynomial time i...
research
01/23/2019

On the Power of Symmetric Linear Programs

We consider families of symmetric linear programs (LPs) that decide a pr...
research
10/24/2017

On the Polynomial Parity Argument Complexity of the Combinatorial Nullstellensatz

The complexity class PPA consists of NP-search problems which are reduci...
research
07/21/2022

The Two-Stripe Symmetric Circulant TSP is in P

The symmetric circulant TSP is a special case of the traveling salesman ...
research
10/26/2017

Minimum Circuit Size, Graph Isomorphism, and Related Problems

We study the computational power of deciding whether a given truth-table...

Please sign up or login with your details

Forgot password? Click here to reset