Capturing Bisimulation-Invariant Exponential-Time Complexity Classes

TL;DR

This paper extends the logical characterizations of bisimulation-invariant complexity classes to the entire exponential time hierarchy using higher-order fixpoint logics, broadening the understanding of their descriptive complexity.

Contribution

It introduces logical frameworks for all exponential time hierarchy classes as extensions of the polyadic mu-calculus with higher-order functions, generalizing previous characterizations.

Findings

Logical counterparts for all classes in the exponential time hierarchy.

Extensions of the polyadic mu-calculus with higher-order functions.

Connection between higher-order logic and exponential time classes.

Abstract

Otto's Theorem characterises the bisimulation-invariant PTIME queries over graphs as exactly those that can be formulated in the polyadic mu-calculus, hinging on the Immerman-Vardi Theorem which characterises PTIME (over ordered structures) by First-Order Logic with least fixpoints. This connection has been extended to characterise bisimulation-invariant EXPTIME by an extension of the polyadic mu-calculus with functions on predicates, making use of Immerman's characterisation of EXPTIME by Second-Order Logic with least fixpoints. In this paper we show that the bisimulation-invariant versions of all classes in the exponential time hierarchy have logical counterparts which arise as extensions of the polyadic mu-calculus by higher-order functions. This makes use of the characterisation of k-EXPTIME by Higher-Order Logic (of order k+1) with least fixpoints, due to Freire and Martins.