# Is model checking PSpace-hard *in formula size*?

Sistla/Clarke proved [SC82] that the LTL model-checking problem is PSpace-complete. Sometimes people write that this problem is "PSpace-hard in $$|\phi|$$" (e.g. [LP85]). What does this mean formally?

How to prove that a decision problem that has several parameters (here, $$\phi$$ and system) is PSpace-hard in one of the parameters? I guess we would need to poly-time reduce some PSpace-complete problem to our problem with the unintereseting parameter being fixed, while the interesting parameter can vary.

But the original proof of Sistla/Clarke [Lemma 4.4, SC82] does not do the thing above! In their proof, both parameters of the resulting LTL model-checking problem instance depend on $$n$$, where $$n$$ is the TM tape bound. Is there a proof that fixes one of the parameters?