Final answer:
The question involves constructing a non-deterministic Pushdown Automaton (PDA) with more than 20 states to accept a specific language defined by conditions on strings x, y, and z. A detailed, systematic approach must be taken to account for various cases of comparison, using a non-deterministic method with branching and stack manipulation for checking conditions.
Step-by-step explanation:
The construction of a non-deterministic Pushdown Automaton (PDA) to accept the language \( \{ x\#y\#z \,|\, x, y, z \in \{0, 1\}^+ \text{ with } x \approx y, \text{ or } x \approx z, \text{ or } y \approx z \} \) requires understanding the requirements and a bit of creativity to handle all possible cases.
Since the PDA needs to be non-deterministic and is expected to have more than 20 states, we need to break down the construction into steps and conditions the PDA will check. This task involves creating branches for each comparison between \(x', y', \text{ and } z'\), checking the largest odd positions and comparing the characters at those positions. It should include states for reading and storing the odd-indexed characters of the strings, states for checking the #'s, states for branching to the comparisons, and accepting states if the conditions are met.
It's important to remember that this PDA must differentiate the lengths and elements at odd indexes for \(x', y', \text{ and } z'\). Each branch will initially non-deterministically decide whether to compare \(x\) with \(y\), \(x\) with \(z\), or \(y\) with \(z\). The PDA will push odd-indexed elements of the first string onto the stack and will check against the corresponding odd-indexed elements of the second string during comparison, popping them off if they match until it finds a mismatch or one string ends.