Runtime environments formulas

Holds under stack allocation, where every live activation has exactly one record on the stack. Counts the records on the stack at one instant; the maximum over the run is the peak stack depth. Includes the main program's record.

Static scoping with access links. A procedure at nesting depth $n_{\text{current}}$ uses a variable whose nearest enclosing declaration is at depth $d_{\text{var}}\le n_{\text{current}}$. Start at the current record and follow access links this many times; 0 means the variable is local.

Procedure $p$ (depth $n_p$) calls procedure $q$ (depth $n_q$), which is visible from $p$, so $n_q\le n_p+1$. Follow this many access links from $p$'s record; the record reached is the most recent activation of $q$'s lexical parent, and $q$'s access link is set to it. Cases: $n_q=n_p+1$ ($q$ declared inside $p$) gives 0, so the link points to $p$ itself; $n_q=n_p$ gives 1, so the link equals $p$'s own access link; $n_q<n_p$ gives $n_p-n_q+1$ links.