# Proof of the pumping lemma for context-free languages using pushdown automata

The pumping lemma for regular languages can be proved by considering a finite state automaton which recognizes the language studied, picking a string with a length greater than its number of states, and applying the pigeonhole principle. The pumping lemma for context-free languages (as well as Ogden's lemma which is slightly more general), however, is proved by considering a context-free grammar of the language studied, picking a sufficiently long string, and looking at the parse tree.

Given the similarity of the two pumping lemmas, you would expect that the context-free one can also be proved in a similar way to the regular one by considering a pushdown automaton which recognizes the language, rather than a grammar. However, I didn't manage to find any reference to such a proof.

Hence my question: is there a proof of the pumping lemma for context-free languages which only involves pushdown automata and not grammars?

I thought about this problem again, and I think I have a full proof. It is a bit more tricky than what I anticipated. Comments are very welcome! Update: I submitted this proof on arXiv, in case this is useful to someone: http://arxiv.org/abs/1207.2819

$\DeclareMathOperator{\fp}{fp}$ $\DeclareMathOperator{\lp}{lp}$ $\newcommand{\fpp}[1]{\widehat{\fp{#1}}}$ $\newcommand{\lpp}[1]{\widehat{\lp{#1}}}$

Let $L$ be a context-free language over an alphabet $\Sigma$. Let $A$ be a pushdown automaton which recognizes $L$, with stack alphabet $\Gamma$. We denote by $|A|$ the number of states of $A$. Without loss of generality, we can assume that transitions of $A$ pop the topmost symbol of the stack and either push no symbol on the stack or push on the stack the previous topmost symbol and some other symbol.

We define $p' = |A|^2 |\Gamma|$ and $p = |A| (|\Gamma|+1)^{p'}$ the pumping length, and will show that all $w \in L$ such that $|w| > p$ have a decomposition of the form $w = u v x y z$ such that $|vxy| \leq p$, $|vy| \geq 1$ and $\forall n \geq 0, u v^n x y^n z \in L$.

Let $w \in L$ such that $|w| > p$. Let $\pi$ be an accepting path of minimal length for $w$ (represented as a sequence of transitions of $A$), we denote its length by $|\pi|$. We can define, for $0 \leq i < |\pi|$, $s_i$ the size of the stack at position $i$ of the accepting path. For all $N > 0$, we define an $N$-level over $\pi$ as a set of three indices $i, j, k$ with $0 \leq i < j < k \leq p$ such that:

1. $s_i = s_k, s_j = s_i + N$
2. for all $n$ such that $i \leq n \leq j$, $s_i \leq s_n \leq s_j$
3. for all $n$ such that $j \leq n \leq k$, $s_k \leq s_n \leq s_k$.

(For an example of this, see the picture for case 2 below which illustrates an $N$-level.)

We define the level $l$ of $\pi$ as the maximal $N$ such that $\pi$ has an $N$-level. This definition is motivated by the following property: if the size of the stack over a path $\pi$ becomes larger than its level $l$, then the stack symbols more than $l$ levels deep will never be popped. We will now distinguish two cases: either $l < p'$, in which case we know that the same configuration for the automaton state and the topmost $l$ symbols of the stack is encountered twice in the first $p+1$ steps of $\pi$, or $l \geq p'$, and there must be a stacking and unstacking position that can be repeated an arbitrary number of times, from which we construct $v$ and $y$.

Case 1. $l < p'$. We define the configurations of $A$ as the couples of a state of $A$ and a sequence of $l$ stack symbols (where stacks of size less than $l$ with be represented by padding them to $l$ with a special blank symbol, which is why we use $|\Gamma| + 1$ when defining $p$). By definition, there are $|A| (|\Gamma| + 1)^l$ such configurations, which is less than $p$. Hence, in the $p+1$ first steps of $\pi$, the same configuration is encountered twice at two different positions, say $i < j$. Denote by $\widehat{i}$ (resp. $\widehat{j}$) the position of the last letter of $w$ read at step $i$ (resp. $j$) of $\pi$. We have $\widehat{i} \leq \widehat{j}$. Hence, we can factor $w = u v x y z$ with $y z = \epsilon$, $u = w_{0 \cdots \widehat{i}}$, $v = w_{\widehat{i} \cdots \widehat{j}}$, $x = w_{\widehat{j} \cdots |w|}$. (By $w_{x \cdots y}$ we denote the letters of $w$ from $x$ inclusive to $y$ exclusive.) By construction, $|vxy| \leq p$.

We also have to show that $\forall n \geq 0, u v^n x y^n z = u v^n x \in L$, but this follows from our observation above: stack symbols deeper than $l$ are never popped, so there is no way to distinguish configurations which are equal according to our definition, and an accepting path for $u v^n x$ is built from that of $w$ by repeating the steps between $i$ and $j$, $n$ times.

Finally, we also have $|v| > 0$, because if $v = \epsilon$, then, because we have the same configuration at steps $i$ and $j$ in $\pi$, $\pi' = \pi_{0 \cdots i} \pi_{j \cdots |\pi|}$ would be an accepting path for $w$, contradicting the minimality of $\pi$.

(Note that this case amounts to applying the pumping lemma for regular languages by hardcoding the topmost $l$ stack symbols in the automaton state, which is adequate because $l$ is small enough to ensure that $|w|$ is larger than the number of states of this automaton. The main trick is that we must adjust for $\epsilon$-transitions.)

Case 2. $l \geq p'$. Let $i, j, k$ be a $p'$-level. To any stack size $h$, $s_i \leq h \leq s_j$, we associate the last push $\lp(h) = \max(\{y \leq j | s_y = h\})$ and the first pop $\fp(h) = \min(\{y \geq j | s_y = h\})$. By definition, $i \leq \lp(h) \leq j$ and $j \leq \fp(h) \leq k$. Here is an illustration of this construction. To simplify the drawing, I omit the distinction between the path positions and word positions which we will have to do later.

We say that the full state of a stack size $h$ is the triple formed by:

1. the automaton state at position $\lp(h)$
2. the topmost stack symbol at position $\lp(h)$
3. the automaton state at position $\fp(h)$

There are $p'$ possible full states, and $p' + 1$ stack sizes between $s_i$ and $s_j$, so, by the pidgeonhole principle, there exist two stack sizes $g, h$ with $s_i \leq g < h \leq s_j$ such that the full states at $g$ and $h$ are the same. Like in Case 1, we define by $\lpp(g)$, $\lpp(h)$, $\fpp(h)$ and $\fpp(g)$ the positions of the last letters of $w$ read at the corresponding positions in $\pi$. We factor $w = u v x y z$ where $u = w_{0 \cdots \lpp(g)}$, $v = w_{\lpp(g) \cdots \lpp(h)}$, $x = w_{\lpp(h) \cdots \fpp(h)}$, $y = w_{\fpp(h) \cdots \fpp(g)}$, and $z = w_{\fpp(g) \cdots |w|}$.

This factorization ensures that $|vxy| \leq p$ (because $k \leq p$ by our definition of levels).

We also have to show that $\forall n \geq 0, u v^n x y^n z \in L$. To do so, observe that each time that we repeat $v$, we start from the same state and the same stack top and we do not pop below our current position in the stack (otherwise we would have to push again at the current position, violating the maximality of $\lp(g)$), so we can follow the same path in $A$ and push the same symbol sequence on the stack. By the maximality of $\lp(h)$ and the minimality of $\fp(h)$, while reading $x$, we do not pop below our current position in the stack, so the path followed in the automaton is the same regardless of the number of times we repeated $v$. Now, if we repeat $w$ as many times as we repeat $v$, since we start from the same state, since we have pushed the same symbol sequence on the stack with our repeats of $v$, and since we do not pop more than what $v$ has stacked by minimality of $\fp(g)$, we can follow the same path in $A$ and pop the same symbol sequence from the stack. Hence, an accepting path from $u v^n x y^n z$ can be constructed from the accepting path for $w$.

Finally, we also have $|vy| > 1$, because like in case 1, if $v = \epsilon$ and $y = \epsilon$, we can build a shorter accepting path for $w$ by removing $\pi_{\lp(g)\cdots\lp(h)}$ and $\pi_{\fp(h)\cdots\fp(g)}$.

Hence, we have an adequate factorization in both cases, and the result is proved.

(Credit goes to Marc Jeanmougin for helping me with this proof.)

Yes it is possible. We could use the notion of surface configurations; they were introduced by Cook a long time back. With this it should be quite easy to get a version of pumping lemma out.

As to surface configurations, almost any paper on LogCFL should carry its definition. Here is a recent paper and here is a thesis

Maybe someone more energetic can spell out the details!

• Thanks for answering! Yes, it is pretty natural to look at the combination of automaton state and topmost stack symbol. I am still thinking about this problem, though, and I can't manage to figure out the details... Help is appreciated. :-) – a3nm Jul 26 '11 at 18:05

For completeness a reference to a proof in this direction.

A.Ehrenfeucht, H.J.Hoogeboom, G.Rozenberg: Coordinated pair systems. I: Dyck words and classical pumping RAIRO, Inf. Théor. Appl. 20, 405-424 (1986)

Abstract. The notion of a coordinated pair system [...] corresponds very closely to (is another formulation of) the notion of a push-down automaton. In this paper we [...] investigate the possibility of obtaining pumping properties of context-free languages via the analysis of computations in cp systems. In order to do this we analyze the combinatorial structure of Dyck words. The properties of Dyck words we investigate stem from the combinatorial analysis of computations in cp systems. We demonstrate how this correspondence can be used for proving the classical pumping lemma.

When discussing this problem with Géraud Sénizergues, he pointed me this paper by Sakarovitch that already proves this result. The proof seems to date back to this paper by Ogden.

References:

• Sakarovitch, Jacques. Sur une propriété d’itération des langages algébriques déterministes. (French. English summary). Math. Systems Theory 14 (1981), no. 3, 247–288.
• William F. Ogden. 1969. Intercalation theorems for stack languages. In Proceedings of the first annual ACM symposium on Theory of computing (STOC '69).