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.
Highlights
One of the most important questions in descriptive complexity theory is whether there is a logic capturing Ptime [7, 14, 15].Currently, there are two main branches of research approaching that problem
We introduce a notion of "sequence-like objects" based on the structure of the graphs’ symmetry groups, and we show that no Choiceless Polynomial Time (CPT)-program which only uses sequence-like objects can decide the CFI query over complete graphs, which have linear maximal degree
The first sets off from a seminal result of Immerman and Vardi which shows that fixed-point logic captures Ptime on ordered structures [16, 19]. This precise logical characterisation of polynomial time heavily relies on the presence of a linear order on the input structure: there are many queries, including trivial counting properties, which cannot be defined in fixed-point logic without a linear order
Summary
One of the most important questions in descriptive complexity theory is whether there is a logic capturing Ptime [7, 14, 15]. In the absence of counting, this question was answered by Blass, Gurevich and van den Bussche in [5]: set-like data structures are more powerful than sequence-like data-structures (see [2].) We generalise this result to the case of counting and show that CPT-programs which decide the CFI query over complete graphs have to use set-like objects This yields an interesting lower bound for a fragment of. Interpretation Logic, a characterisation of Choiceless Polynomial Time which was presented in [12]: Interpretation Logic without congruences cannot decide the Cai-Fürer-Immerman query over complete graphs, and is strictly less expressive than the fragment of CPT using only sets of bounded rank.
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