Witnessed Symmetric Choice and Interpretations in Fixed-Point Logic with Counting

TL;DR

This paper explores extensions of fixed-point logic with counting, focusing on witnessed symmetric choice and interpretation operators, and investigates their impact on expressiveness and graph canonization.

Contribution

It demonstrates that adding interpretation operators to IFPC+WSC does not increase expressiveness and shows nesting WSC-operators enhances logic power using CFI graphs.

Findings

IFPC+WSC is not closed under FO-interpretations.

Nesting WSC-operators increases expressiveness.

IFPC+WSC+I can canonize certain base graphs and CFI graphs.

Abstract

At the core of the quest for a logic for PTime is a mismatch between algorithms making arbitrary choices and isomorphism-invariant logics. One approach to overcome this problem is witnessed symmetric choice. It allows for choices from definable orbits which are certified by definable witnessing automorphisms. We consider the extension of fixed-point logic with counting (IFPC) with witnessed symmetric choice (IFPC+WSC) and a further extension with an interpretation operator (IFPC+WSC+I). The latter operator evaluates a subformula in the structure defined by an interpretation. This structure possibly has other automorphisms exploitable by the WSC-operator. For similar extensions of pure fixed-point logic (IFP) it is known that IFP+WSCI simulates counting which IFP+WSC fails to do. For IFPC+WSC it is unknown whether the interpretation operator increases expressiveness and thus allows studying the relation between WSC and interpretations beyond counting. We separate IFPC+WSC from IFPC+WSCI by showing that IFPC+WSC is not closed under FO-interpretations. Additionally, we prove that nesting WSC-operators increases the expressiveness using the so-called CFI graphs. We show that if IFPC+WSC+I canonizes a particular class of base graphs, then it also canonizes the corresponding CFI graphs. This differs from various other logics, where CFI graphs provide difficult instances.