Abstract
Choiceless Polynomial Time (CPT) is one of the candidates in the quest for a logic for polynomial time. It is a strict extension of fixed-point logic with counting (FPC) but to date it is unknown whether it expresses all polynomial-time properties of finite structures. We study the CPT-definability of the isomorphism problem for relational structures of bounded colour class size q (for short, q-bounded structures). Our main result gives a positive answer, and even CPT-definable canonisation procedures, for classes of q-bounded structures with small Abelian groups on the colour classes. Such classes of q-bounded structures with Abelian colours naturally arise in many contexts. For instance, 2-bounded structures have Abelian colours which shows that CPT captures Ptime on 2-bounded structures. In particular, this shows that the isomorphism problem of multipedes is definable in CPT, an open question posed by Blass, Gurevich, and Shelah.
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Abu Zaid, F., Grädel, E., Grohe, M., Pakusa, W. (2014). Choiceless Polynomial Time on Structures with Small Abelian Colour Classes. In: Csuhaj-Varjú, E., Dietzfelbinger, M., Ésik, Z. (eds) Mathematical Foundations of Computer Science 2014. MFCS 2014. Lecture Notes in Computer Science, vol 8634. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-44522-8_5
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