Conjunctive grammars and alternating pushdown automata

Abstract

In this paper we introduce a variant of alternating pushdown automata, synchronized alternating pushdown automata, which accept the same class of languages as those generated by conjunctive grammars.

Appendix: Proofs of Lemmas 23 and 

1.1 Proof of Lemma 23

The proof is by induction on the length $k$ of the derivation

$$\begin{aligned}{}[q,X,p] \Rightarrow _L^*x [q_1,X_1,q_2] \cdots [q_m,X_m,p]. \end{aligned}$$

Basis: $k = 0$. That is,

$$\begin{aligned}{}[q,X,p] \Rightarrow _L^0 x [q_1,X_1,q_2] \cdots [q_m,X_m,p]. \end{aligned}$$

Then $x = \epsilon$ and $[q_1,X_1,q_2] \cdots [q_m,X_m,p] = [q,X,p]$. Trivially,

$$\begin{aligned} (q,\epsilon ,X) \vdash ^0 (q,\epsilon ,X). \end{aligned}$$

Induction Step: Assume that the implication holds for all derivations of the length at most $k, k \ge 1$, and let

$$\begin{aligned}{}[q,X,p] \Rightarrow _L^{k+1} x [q_1,X_1,q_2] \cdots [q_m,X_m,p]. \end{aligned}$$

That is,

$$\begin{aligned}{}[q,X,p] \Rightarrow \sigma [q_1^\prime ,Y_1,q_2^\prime ] \cdots [q_\ell ^\prime ,Y_\ell ,q_{\ell + 1}^\prime ] \Rightarrow _L^{k} \sigma y [q_1,X_1,q_2] \cdots [q_m,X_m,p], \end{aligned}$$

(21)

where

$$\begin{aligned} x = \sigma y \end{aligned}$$

(22)

for some $\sigma \in \Sigma \cup \{\epsilon \}$ and $q_1^\prime ,\ldots ,q_{\ell +1}^\prime \in Q$.

It follows from the first step of derivation (21) that

$$\begin{aligned}{}[q,X,p] \rightarrow \sigma [q_1^\prime ,Y_1,q_2^\prime ] \cdots [q_\ell ^\prime ,Y_\ell ,q_{\ell + 1}^\prime ] \in P. \end{aligned}$$

Thus, by the definition of $P$,

$$\begin{aligned} (q_1^\prime ,Y_1 Y_2 \cdots Y_\ell ) \in \delta (q,\sigma ,X). \end{aligned}$$

(23)

Let $h$ be the maximal number such that in the derivation

$$\begin{aligned}{}[q_1^\prime ,Y_1,q_2^\prime ] \cdots [q_\ell ^\prime ,Y_\ell ,q_{\ell + 1}^\prime ] \Rightarrow _L^{k} y [q_1,X_1,q_2] \cdots [q_m,X_m,p] \end{aligned}$$

no rule is applied to $[q_{\ell + 1 - h}^\prime ,Y_{\ell + 1 - h},q_{\ell + 2 - h}^\prime ]$. Then

$$\begin{aligned}{}[q_{\ell + 1 - h}^\prime ,Y_{\ell + 1 - h},q_{\ell + 2 - h}^\prime ] \!\!\!&\cdots [q_\ell ^\prime , Y_{\ell },q_{\ell + 1}^\prime ] \nonumber \\&\,= [q_{m + 1 - h},X_{m + 1 - h},q_{m + 2 - h}] \cdots [q_m,X_{m},p] \end{aligned}$$

(24)

and

$$\begin{aligned} \begin{aligned}&[q_1^\prime ,Y_1,q_2^\prime ] \cdots [q_{\ell - h}^\prime ,Y_{\ell - h},q_{\ell + 1 - h}^\prime ]\\&\quad \Rightarrow _L^{{\le } \, k} y [q_1,X_1,q_2] \cdots [q_{m - h},X_{m - h},q_{m + 1 - h}].~~~~~~~~ \end{aligned} \end{aligned}$$

(25)

Since the derivation (25) is leftmost, there are $y_1,y_2,\ldots ,y_{\ell - h} \in \Sigma ^*$ such that

$$\begin{aligned} y_1 y_2 \cdots y_{\ell - h} = y, \end{aligned}$$

(26)

$$\begin{aligned}{}[q_j^{\prime },Y_j,q_{j+1}^\prime ] \Rightarrow ^{{*}} y_j, \ \ \ \ \ \ \ j = 1,2,\ldots ,\ell - 1 - h, \end{aligned}$$

(27)

and

$$\begin{aligned}{}[q_{\ell - h}^\prime ,Y_{\ell - h},q_{\ell + 1 - h}^\prime ] \Rightarrow ^{{\le }\,k} y_{\ell - h} [q_1,X_1,q_2] \cdots [q_{m - h},X_{m - h},q_{m + 1 - h}]. \end{aligned}$$

(28)

By the “only if” part of [Equivalence (5.3), p. 117], it follows from (27) that

$$\begin{aligned} (q_j^\prime ,y_j,Y_j) \vdash ^*(q_{j+1}^\prime ,\epsilon ,\epsilon ), \ \ \ \ \ \ \ j = 1,2,\ldots ,\ell - 1 - h. \end{aligned}$$

(29)

It follows from (24) that $q_{m + 1 - h} = q_{\ell + 1 - h}^\prime$. Thus, (28) is also

$$\begin{aligned}{}[q_{\ell - h}^\prime ,Y_{\ell - h},q_{\ell + 1 - h}^\prime ] \Rightarrow ^{{\le } \, k} y_{\ell - h} [q_1,X_1,q_2] \cdots [q_{m - h},X_{m - h},q_{\ell + 1 - h}^\prime ] \end{aligned}$$

and, by the induction hypothesis,

$$\begin{aligned} (q_{\ell - h}^\prime ,y_{\ell - h},Y_{\ell - h}) \vdash ^*(q_1,\epsilon , X_1 X_2 \cdots X_{m - h}). \end{aligned}$$

(30)

Now, combining (22), (23), (26), (29), and (30) we obtain

$$\begin{aligned} \begin{array}{rll} (q,x,X) = (q,\sigma y,X) &{} \vdash &{} (q_1^{\prime },y,Y_1 Y_2 \cdots Y_\ell ) = (q_1^\prime ,y_1 y_2 \cdots y_{\ell - h},Y_1 Y_2 \cdots Y_\ell )\\ &{} \vdash ^*&{} (q_2^\prime ,y_2 \cdots y_{\ell - h},Y_2 \cdots Y_\ell )\\ &{} \vdots &{}\\ &{} \vdash ^*&{} (q_{\ell - h}^\prime , y_{\ell - h},Y_{\ell - h} \cdots Y_\ell )\\ &{} \vdash ^*&{} (q_1, \epsilon , X_1 X_2 \cdots X_{m - h}Y_{\ell +1- h} \cdots Y_\ell ). \end{array} \end{aligned}$$

Since, by (24), $Y_{\ell +1- h} \cdots Y_\ell = X_{m +1- h} \cdots X_m$, the proof is complete.

1.2 Proof of Lemma 24

The proof is by induction on the length $k$ of the computation

$$\begin{aligned} (q,x,X) \vdash ^*(q_1,\epsilon ,X_1 \cdots X_m). \end{aligned}$$

Basis: $k = 0$. That is,

$$\begin{aligned} (q,x,X) \vdash ^0 (q_1,\epsilon ,X_1 \cdots X_m). \end{aligned}$$

Then $x = \epsilon$ and $X_1 X_2 \cdots X_m = X$. Trivially,

$$\begin{aligned}{}[q,X,p] \Rightarrow ^0 [q,X,p]. \end{aligned}$$

Induction Step: Assume that the implication holds for all computations of length at most $k, k \ge 1$, and let

$$\begin{aligned} (q,x,X) \vdash ^{k+1} (q_1,\epsilon ,X_1 \cdots X_m). \end{aligned}$$

That is,

$$\begin{aligned} (q,x,X) \vdash (q_1^\prime ,y,Y_1 \cdots Y_\ell ) \vdash ^k (q_1,\epsilon ,X_1 \cdots X_m), \end{aligned}$$

where

$$\begin{aligned} x = \sigma y \end{aligned}$$

(31)

for some $\sigma \in \Sigma \cup \{\epsilon \}$ and

$$\begin{aligned} (q^\prime ,Y_1 \cdots Y_\ell ) \in \delta (q,\sigma ,X). \end{aligned}$$

(32)

Let $h$ be the maximal number such that in the computation

$$\begin{aligned} (q_1^\prime ,y,Y_1 \cdots Y_\ell ) \vdash ^k (q_1,\epsilon ,X_1 \cdots X_m) \end{aligned}$$

the stack height is never less than $h$. Then $Y_{\ell + 2 - h}$ is never exposed as the top symbol, implying

$$\begin{aligned} Y_{\ell + 2 - h} Y_{\ell + 3 - h} \cdots Y_{\ell } = X_{m + 2 - h} X_{m + 3 - h} \cdots X_{m}. \end{aligned}$$

(33)

Thus, there are $y^\prime ,y^{\prime \prime } \in \Sigma ^*$, $q^{\prime \prime } \in Q$, and $Y \in \Gamma$ such that

$$\begin{aligned} y = y^\prime y^{\prime \prime }, \end{aligned}$$

(34)

$$\begin{aligned} (q_1^\prime ,y^\prime ,Y_1 Y_2 \cdots Y_{\ell + 1 -h}) \vdash ^{{\le } \, k} (q^{\prime \prime },\epsilon ,Y) \end{aligned}$$

(35)

and

$$\begin{aligned} (q^{\prime \prime },y^{\prime \prime },Y) \vdash ^{{\le } \, k} (q_1,\epsilon ,X_1 X_2 \cdots X_{m +1 - h}). \end{aligned}$$

(36)

It follows from (35) that there are $y_1, y_2, \ldots , y_{\ell + 1 - h} \in \Sigma ^*$ and there are,

$q_1^\prime ,q_2^\prime ,\ldots ,q_{\ell + 1 -h}^{\prime },q\in Q$ such that

$$\begin{aligned} y^\prime =y_1 y_2 \cdots y_{\ell + 1 - h}, \end{aligned}$$

(37)

$$\begin{aligned} (q_j^\prime ,y_j,Y_j) \vdash ^{*} (q_{j+1}^{\prime },\epsilon ,\epsilon ), \ \ \ \ \ j = 1,2,\ldots ,\ell - h \, \end{aligned}$$

and

$$\begin{aligned} (q_{\ell + 1 - h}^\prime ,y_{\ell + 1 - h},Y_{\ell + 1 - h}) \vdash ^{*} (q^{\prime \prime },\epsilon ,Y). \end{aligned}$$

Therefore, by the “if” part of [Equivalence (5.3), p. 117],

$$\begin{aligned}{}[q_j^\prime , Y_j,q_{j+1}^\prime ] \Rightarrow ^*y_j, \ \ \ \ \ j = 1,2,\ldots ,\ell - h, \end{aligned}$$

(38)

and, by the induction hypothesis, for all $q_{m + 2 - h} \in Q$,

$$\begin{aligned}{}[q_{\ell + 1 - h}^\prime ,Y_{\ell + 1 - h},q_{m + 2 - h}] \Rightarrow ^*y_{\ell + 1 - h}[q^{\prime \prime },Y,q_{m + 2 - h}]. \end{aligned}$$

(39)

Also, by (36) and the induction hypothesis, if $h \le m,$ or , by the “if” part of [Equivalence (5.3), p.117], if $h=m+1,$ for all $q_2,q_3,\dots ,q_{m + 1 - h} \in Q$,

$$\begin{aligned}{}[q^{\prime \prime },Y,q_{m + 2 - h}] \Rightarrow _L^*y^{\prime \prime } [q_1,X_1,q_2] \cdots [q_{m + 1 - h},X_{m + 1 - h},q_{m + 2 - h}]. \end{aligned}$$

(40)

Finally, by the definition of $G_A$, it follows from (32) that

$$\begin{aligned}&[q,X,p] \rightarrow \sigma [q_1^\prime ,Y_1,q_2^\prime ] \cdots [q_{\ell + 1 - h}^\prime ,Y_{\ell + 1 - h},q_{m + 2 - h}]\nonumber \\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad [q_{m + 2 - h},Y_{\ell + 2 - h},q_{m + 3 - h}] \cdots [q_m,Y_\ell ,p] \in P. \end{aligned}$$

Thus, by (33),

$$\begin{aligned}&[q,X,p] \rightarrow \sigma [q_1^\prime ,Y_1,q_2^\prime ] \cdots [q_{\ell + 1 - h}^\prime ,Y_{\ell + 1 - h},q_{m + 2 - h}]\nonumber \\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad [q_{m + 2 - h},X_{m + 2 - h},q_{\ell + 3 - h}] \cdots [q_m,X_m,p] \in P \end{aligned}$$

as well, and, combining the latter rule with (38), (39), (40), (37), (34), and (31), we obtain

$$\begin{aligned}{}[q,X,p] \!\!&\Rightarrow \ \sigma [q_1^\prime ,Y_1,q_2^\prime ] [q_2^\prime ,Y_2,q_3^\prime ] \cdots [q_{\ell + 1 - h}^\prime ,Y_{\ell + 1 - h},q_{m + 2 - h}]\nonumber \\&\quad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad [q_{m + 2 - h},X_{m + 2 - h},q_{\ell + 3 - h}] \cdots [q_m,X_m,p]\\&\Rightarrow ^*\sigma y_1 [q_2^\prime ,Y_2,q_3^\prime ] \cdots [q_{\ell + 1 - h}^\prime ,Y_{\ell + 1 - h},q_{m + 2 - h}]\\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad [q_{m + 2 - h},X_{m + 2 - h},q_{\ell + 3 - h}] \cdots [q_m,X_m,p]\\&\ \vdots \\&\Rightarrow ^*\sigma y_1 \cdots y_{\ell - h} [q_{\ell + 1 - h}^\prime ,Y_{\ell + 1 - h},q_{m + 2 - h}]\\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad [q_{m + 2 - h},X_{m + 2 - h},q_{\ell + 3 - h}] \cdots [q_m,X_m,p]\\&\Rightarrow ^*\sigma y_1 \cdots y_{\ell - h} y_{\ell + 1- h} [q^{\prime \prime },Y,q_{m + 2 - h}]\\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad [q_{m + 2 - h},X_{m + 2 - h},q_{\ell + 3 - h}] \cdots [q_m,X_m,p]\\&\Rightarrow ^*\sigma y_1 \cdots y_{\ell - h} y_{\ell + 1 - h} y^{\prime \prime } [q_1,X_1,q_2] \cdots [q_{m + 1 - h},X_{m + 1 - h},q_{\ell + 2 - h}]\\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad [q_{m + 2 - h},X_{m + 2 - h},q_{\ell + 3 - h}] \cdots [q_m,X_m,p]\\&= \ \, \sigma y^\prime y^{\prime \prime } [q_1,X_1,q_2] \cdots [q_m,X_m,p]\\&= \, \sigma y [q_1,X_1,q_2] \cdots [q_m,X_m,p]\\&= \, x [q_1,X_1,q_2] \cdots [q_m,X_m,p].\\ \end{aligned}$$