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

Abstract

Choiceless Polynomial Time (CPT) is one of the most promising candidates in the search for a logic capturing P time . The question whether there is a logic that expresses exactly the polynomial-time computable properties of finite structures, which has been open for more than 30 years, is one of the most important and challenging problems in finite model theory. The strength of Choiceless Polynomial Time is its ability to perform isomorphism-invariant computations over structures, using hereditarily finite sets as data structures. But, because of isomorphism-invariance, it is choiceless in the sense that it cannot select an arbitrary element of a set—an operation that is crucial for many classical algorithms. CPT can define many interesting P time queries, including (a certain version of) the Cai-Fürer-Immerman (CFI) query. The CFI-query is particularly interesting, because it separates fixed-point logic with counting from P time and has since remained the main benchmark for the expressibility of logics within P time . The CFI-construction associates with each connected graph a set of CFI-graphs that can be partitioned into exactly two isomorphism classes called odd and even CFI-graphs. The problem is to decide, given a CFI-graph, whether it is odd or even. For the case where the CFI-graphs arise from ordered graphs, Dawar, Richerby, and Rossman proved that the CFI-query is CPT-definable. However, definability of the CFI-query over general graphs remains open. Our first contribution generalises the result by Dawar, Richerby, and Rossman to the variant of the CFI-query derived from graphs with colour classes of logarithmic size, instead of colour class size one. Second, we consider the CFI-query over graph classes where the maximal degree is linear in the size of the graphs. For the latter, we establish CPT-definability using only sets of small, constant rank, which is known to be impossible for the general case. In our CFI-recognising procedures we strongly make use of the ability of CPT to create sets, rather than tuples only, and we further prove that, if CPT worked over tuples instead, then no such procedure would be definable. We introduce a notion of “sequencelike objects” based on the structure of the graphs’ symmetry groups, and we show that no CPT-program that only uses sequencelike objects can decide the CFI-query over complete graphs, which have linear maximal degree. From a broader perspective, this generalises a result by Blass, Gurevich, and van den Bussche about the power of isomorphism-invariant machine models (for polynomial time) to a setting with counting.

Separating Rank Logic from Polynomial Time

“…The already mentioned result by Dawar, Richerby, and Rossman [10] uses deeply nested sets and is restricted to totally ordered base graphs. This result was strengthened by Pakusa, Schalthöfer, and Selman [28] to base graphs with logarithmic color class size. Recently, the result for bounded abelian color classes by Abu Zaid, Grädel, Grohe, and Pakusa [29] was extended by Lichter and Schweitzer [23] to graphs with bounded color classes with dihedral colors and also for certain structures of arity 3.…”

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

“…While the development of tools for inexpressibility results is our main objective in this work, we also use our circuit framework to obtain a new CPT-algorithm for the CFI-query. This algorithm is stronger than the previously best one from [16] in the sense that it can solve a slightly "less ordered" version of the CFI-problem (which is still far from the unordered version). Concretely, we consider a setting where along with the (unordered) CFI-graph in question, we are given -as a sort of auxiliary structure -an XOR-circuit whose input gates are associated with the vertices of the underlying graph.…”

“…Choiceless Polynomial Time is strictly stronger than FPC because it can solve variants of the CFIquery on ordered or preordered base graphs (see [7], [16]), which is not possible in FPC. More generally, CPT can canonise preordered structures with small Abelian colour classes [19].…”

“…In [14], we already demonstrated the usefulness of these tools in the context of the CFI-problem. We showed that a particular CPT-algorithm for a preordered version of the CFI-problem [16] cannot be used for the unordered CFI-problem because the required preorders are not CPT-definable in all graphs (for symmetry reasons). This of course does not imply that there is no CPT-algorithm at all for the unordered CFI-problem; it just rules out the one from [16].…”

“…We showed that a particular CPT-algorithm for a preordered version of the CFI-problem [16] cannot be used for the unordered CFI-problem because the required preorders are not CPT-definable in all graphs (for symmetry reasons). This of course does not imply that there is no CPT-algorithm at all for the unordered CFI-problem; it just rules out the one from [16]. With the new results in this paper, we can now say more generally what kind of structures we need to analyse with respect to symmetry in order to actually rule out all CPT-algorithms for the CFI-problem: The mentioned XOR-circuits over the underlying graphs.…”