# Space and Time Complexity for Infinite Time Turing Machines

## TL;DR

This paper investigates space complexity notions for Infinite Time Turing Machines, answering open questions, disproving certain equalities, and establishing class separation results.

## Contribution

It provides new insights into space complexity for ITTMs, including disproving equalities and expanding on previous observations with separation results.

## Key findings

- Low space complexity does not imply low time complexity for ITTMs.
- The equalities P=PSPACE, P$_{+}=$PSPACE$_{+}$, and P$_{++}=$PSPACE$_{++}$ do not hold for ITTMs.
- Various separation results between space complexity classes for ITTMs are established.

## Abstract

We consider notions of space complexity for Infinite Time Turing Machines (ITTMs) that were introduced by B. L\"owe and studied further by J. Winter. We answer several open questions about these notions, among them whether low space complexity implies low time complexity (it does not) and whether one of the equalities P=PSPACE, P$_{+}=$PSPACE$_{+}$ and P$_{++}=$PSPACE$_{++}$ holds for ITTMs (all three are false). We also show various separation results between space complexity classes for ITTMs. This considerably expands our earlier observations on the topic in section 7.2.2 of \cite{Ca2}, which appear here as Lemma $6$ up to Corollary $9$.