An Entertaining Example of Using the Concepts of Context-Free Grammar and Pushdown Automation

A formal-linguistic approach for solving an entertaining task is made in this paper. The well-known task of the Hanoi towers is discussed in relation to some concepts of discrete mathematics. A context-free grammar which generate an algorithm for solving this task is described. A deterministic pushdown automation which in its work imitates the work of monks in solving the task of the Hanoi towers is built.

💡 Research Summary

The paper presents a formal‑linguistic treatment of the classic Tower of Hanoi puzzle, showing how its recursive solution can be expressed both as a context‑free grammar (CFG) and as a deterministic pushdown automaton (DPDA). After a brief introduction to the puzzle—three pegs, N distinct disks initially stacked on the first peg, with the goal of moving the entire stack to the third peg using the second as an auxiliary while obeying the usual size constraints—the author reviews basic notions of formal languages: alphabets, strings, the empty string ε, and the definition of a CFG Γ = ⟨V, W, S, P⟩ (V terminals, W non‑terminals, S start symbol, P production rules).

Task 2 – Constructing a CFG for any N.
For a given positive integer N the grammar Γ_N is defined as follows:

• Terminals V = { p_ij | i, j ∈ {1,2,3}, i ≠ j }. The symbol p_ij denotes the elementary move “move the top disk from peg i to peg j”.
• Non‑terminals W = { h_ij(n) | i, j ∈ {1,2,3}, i ≠ j, n = 1,…,N }. The symbol h_ij(n) stands for the sub‑task “move n disks from peg i to peg j using the remaining peg as auxiliary”.
• Start symbol S = h_13(N), i.e. the whole problem.
• Productions consist of two families:
  1. Base case (n = 1): h_ij(1) → p_ij.
  2. Recursive case (n ≥ 2): h_ij(n) → h_ik(n‑1) p_ij h_kj(n‑1), where k is the third peg distinct from i and j.

These rules are exactly the textbook recursive algorithm: to move n disks from i to j, first move n‑1 disks to the auxiliary peg k, then move the largest disk directly (p_ij), and finally move the n‑1 disks from k onto j. The author proves by induction that for every N the language L(Γ_N) is non‑empty and, more strongly, contains a single word of length 2^N − 1. This word enumerates the moves in the optimal order, establishing a one‑to‑one correspondence between derivations in the grammar and valid Hanoi solutions.

Task 3 – Building a deterministic PDA.
The paper then shifts to automata theory. A deterministic pushdown automaton M_N is constructed with:

• No input alphabet (the automaton works on ε‑input only).
• Stack alphabet W′ = V ∪ { h_ij(n) | 1 ≤ n ≤ N‑1 } ∪ { z_0 }.
• A single state q_0 and no accepting states (acceptance is by empty stack).
• Transition function δ consisting of four ε‑rules:
  1. Replace the initial stack symbol z_0 with h_12(N‑1) p_13 h_23(N‑1).
  2. For any h_ij(1) on top of the stack, replace it with p_ij.
  3. For any h_ij(n) with n ≥ 2, replace it with h_ik(n‑1) p_ij h_kj(n‑1) (the same decomposition as the grammar).
  4. For any terminal p_ij on top, pop it (i.e., replace it with ε).

Because all transitions are ε‑moves, the automaton never reads external symbols; it simply unfolds the stack according to the same recursive pattern used in the grammar. The author demonstrates, again by induction on the stack depth, that starting from the initial configuration the PDA will eventually empty its stack after exactly 2^N − 1 pop operations, each pop corresponding to a legitimate disk move. Consequently, M_N simulates the monks’ solution of the Hanoi puzzle without any external input, confirming that a deterministic PDA can realize the optimal Hanoi algorithm.

Key insights and contributions.

1. The recursive structure of the Tower of Hanoi is naturally captured by a context‑free grammar; the grammar’s derivation length directly yields the optimal move count.
2. A deterministic PDA can be programmed solely via stack rewrites to execute the same recursion, illustrating that pushdown memory suffices for this class of recursive problems.
3. The work bridges discrete mathematics education and formal language theory, providing a concrete example where abstract concepts (CFGs, PDAs) have an immediate algorithmic interpretation.

Overall, the paper offers a clear, mathematically rigorous exposition of how a classic combinatorial puzzle can be modeled both as a language generator and as a stack‑based automaton, reinforcing the deep connections between recursion, formal grammars, and automata theory.