How to prove a problem is EXP hard

Question

Summary of the problem: Given an alternating time turing machine ($M$), a polynomial $p(.)$ and a string ($w$), is it EXPhard to find if $M$ accepts $w$ using not more than $p(|M|+|w|)$ space?

My work: I have a problem I would like to prove is EXPTIME hard. I have a couple of reductions to do so. One is reduction from combinatorial EXP complete games but is hard to read.

The other is a reduction from alternate poly space Turing machine. But instead of taking as input a turing machine ($M$) and a string ($w$) to check for acceptance, it takes as input a turing machine ($M$), a string ($w$) and a polynomial $p(.)$ which gives the amount of space the turing machine is allowed to use. Is this reasonable to do?

Possible solution: Ideally I would like to write a reduction from an exphard problem to alternate turing machine with a polynomial and string. to prove that this problem is hard as well.

Answer 1

This is an immediate consequence of the fact that $\text{APSPACE} = \text{EXPTIME}$ (see, for example, here). Specifically, consider a language in $L \in \text{EXPTIME}$, and let $T$ be an alternating TM that decides $L$ in space $s(n)$, where $s:\mathbb{N} \to \mathbb{N}$ is some polynomial. Then, a reduction from $L$ to your problem operates as follows. Given input $x$ for the reduction, the reduction outputs $\langle T, x, s(|x|)\rangle$. As $T$ decides $L$ with space-complexity $s(n)$, it holds that $x\in L$ iff $T$ accepts $x$ while using at most $s(|x|)$ tape-cells.

Note that $T$ is a constant (does not depend on the input $x$). Therefore, the reduction is computable in polynomial-time as it simply outputs the description of $T$, copies $x$, and computes $s(|x|)$.