Notes on kAExp(pol) problems for deterministic machines

TL;DR

This paper introduces two natural problems related to deterministic machines to establish lower bounds for the class $k$AExp$_{ ext{pol}}$, extending previous results from the case $k=1$ to arbitrary $k$.

Contribution

It proposes two problems on deterministic machines that serve as tools to prove lower bounds for the class $k$AExp$_{ ext{pol}}$, generalizing known results for $k=1$.

Findings

Defined the $k$AExp$_{ ext{pol}}$-prenex TM problem for deterministic Turing machines.

Introduced the $k$-exp alternating multi-tiling problem analogous to the TM problem.

Extended the $AExp_{pol}$-completeness results from $k=1$ to arbitrary $k$.

Abstract

The complexity of several logics, such as Presburger arithmetic, dependence logics and ambient logics, can only be characterised in terms of alternating Turing machines. Despite quite natural, the presence of alternation can sometimes cause neat ideas to be obfuscated inside heavy technical machinery. In these notes, we propose two problems on deterministic machines that can be used to prove lower bounds with respect to the computational class $k$AExp$_{\text{pol}}$, that is the class of all problems solvable by an alternating Turing machine running in $k$ exponential time and performing a polynomial amount of alternations, with respect to the input size. The first problem, called $k$AExp$_{\text{pol}}$-prenex TM problem, is a problem about deterministic Turing machines. The second problem, called the $k$-exp alternating multi-tiling problem, is analogous to the first one, but on tiling systems. Both problems are natural extensions of the TM alternation problem and the alternating multi-tiling problem proved AExp$_{\text{pol}}$-complete by L. Bozzelli, A. Molinari, A. Montanari and A. Peron in [GandALF, pp. 31-45, 2017]. The proofs presented in these notes follow the elegant exposition in A. Molinari's PhD thesis to extend these results from the case $k = 1$ to the case of arbitrary $k$.