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\begin{document}
\title{An insertion into the Chomsky hierarchy?}
\author{Robert McNaughton \\
Department of Computer Science \\
Rensselaer Polytechnic Institute \\
Troy, NY 12180-3590, U.S.A. \\
mcnaught@cs.rpi.edu}
\date{January, 1999}
\maketitle
\vspace{.2in}
\noindent
{\bf Abstract.} This review paper will report on some recent
discoveries in the area of Formal Languages, chiefly by F. Otto,
G. Buntrock and G. Niemann. These discoveries have pointed out certain
break-throughs connected with the concept of growing context-sensitive
languages, which originated in the 1980's with a paper by E. Dahlhaus
and M.K. Warmuth. One important result is that the deterministic growing
context-sensitive languages turn out to be identical to an
interesting family of formal languages definable in a certain way by
confluent reduction systems.
\vspace{.2in}
\noindent
{\large\bf 1. Growing context-sensitive languages.}
There are several reasons for proposing that the family of GrCSL's (growing
context-sensitive languages) be considered as a new level in the Chomsky hierarchy of
languages. The insertion would be between the CFL's (context-free languages) and the CSL's
(context-sensitive languages), in effect making a new family designated as the ``type
one-and-a-half languages''. This family began to receive attention in 1986, when Dahlhaus
and Warmuth published their result \cite{dw} that the complexity of its membership problem
had a polynomial-time algorithm, in contrast to the P-space-complete membership problem for
the larger family of context-sensitive languages.
A GrCSG (growing context-sensitive grammar) is one in which $|\alpha | < |\beta |$
for every rule $\alpha \rightarrow \beta$; a GrCSL is the language of a GrCSG. This
class of grammars is a proper subclass of the class of CSG's (context-sensitive grammars)
where rules are also permitted in which $|\alpha | = |\beta |$.
The GrCSL's are just one of many studied families properly between the CFL's and the
CSL's. Another was the family of CSL's with linear-bounded derivations
(see, e.g., Ron Book's dissertation \cite{b69}),
i.e., languages having a CSG with a bound $B$ such that every $w \in \Sigma ^*$ derived
in the grammar has a derivation whose length is $\leq B|w|$. In 1964 Gladkij proved in
\cite{gl} that the CSL
\[ \{ wcw^Rcw|w \in \{ a,b \} ^* \} \]
does not have linear-bounded derivations (see also the appendix to \cite{b69}).
($w^R$ means $w$ written backwards.)
More recently it has been proved that the GrCSL's are a proper subfamily of the family
of CSL's with linear bounded derivations (see \cite{b96}, and also \cite{buo},
Corollary 5.4). However, the latter family will no longer be of concern in this paper.
From the work of Lautemann \cite{la} and Buntrock \cite{b96} it follows that
the language
$\ \{ ww|w \in \{ a,b \} ^* \}$ is not a GrCSL. (See also \cite{buo},
especially the penultimate paragraph of Section~1.) Thus we have an improvement on the
Gladkij language for a paradigm CSL that is not a GrCSL.
All CFL's consisting of words of length $\geq 2$ are GrCSL's. The easy proof is by a
constructive modification of the Chomsky normal form. But not all GrCSL's are
context-free, e.g, $\{ ba^{2^n} |n \geq 1 \}$, a GrCSG for which is
\[ ~~~~~~S \rightarrow SK|baa \]
\[ baaK \rightarrow baaaa ~~ \]
\[ aK \rightarrow Kaa ~ \]
The concept GrCSG is based on the length of strings. It is convenient to allow as
GrCSG's the grammars that satisfy a variant of the definition based on the weighted length of
strings. A {\em weighting function\/} $\phi$ on the words over an alphabet $\Sigma$ maps each
word to an integer satisfying the following: (1)~$\phi (x) > 0$ for all $x \in \Sigma$,
(2)~$\phi (\lambda ) = 0$ ($\lambda$ is the null string) and
(3)~$\phi (xy) = \phi(x) + \phi (y)$ for all words $x$ and $y$. We define a grammar to
be a {\em GrCSG in the new sense\/} if there is a weighting function $\phi$ on the words
over the total alphabet such that $\phi (\alpha ) < \phi (\beta )$ for every rule
$\alpha \rightarrow \beta$. Following \cite{bl1} and \cite{bl2} we can prove that,
if $G$ is a GrCSG in the new sense then there is a $G'$ that is a GrCSG in the original
sense such that $L(G') = L(G) \cap \Sigma \Sigma \Sigma ^*$.
Note that a context-free grammar in
Chomsky normal form whose language does not have the null word is a GrCSG in the new
sense: take $\phi (x) = 1$ for $x$ a variable and $\phi (x) = 2$ for $x$ a terminal.
It will be convenient to adopt this new definition of GrCSG for the remainder of
this paper, yielding the slight change in the definition of GrCSL.
As is well known, a language is context-sensitive if and only if it is recognized
by a nondeterministic LBA (linear bounded automaton). We can get a corresponding result for
GrCSL's by modifying this automaton to one whose tape decreases in its weighted length at every
move. The best way to work out this idea precisely is to
follow Buntrock and Otto in Section 3 of \cite{buo},
stipulating an automaton with two
pushdown tapes, representing the portions of the LBA tape to the left of the head and the
right of the head, respectively. An elaborate weighting function is defined for
configurations of the automaton, satisfying the condition that if one configuration is
followed by another then the weight of the latter is less than the weight of the former.
This weighting function is based on a weighting function of words over the alphabet,
but it is far too complicated to be described here. (One trick is to
get the effect of weighing each character
of $\Sigma$ more heavily on one pushdown than on the other.)
We thus have a nondeterministic shrinking two-pushdown automaton and the result that
a language is a GrCSL if and only if it is recognized by such a device.
The proof given by Buntrock and Otto
is somewhat similar to the proof that a language is a CSL if and only if it is accepted by a
nondeterministic LBA.
The question naturally arises as to which CSL's are GrCSL's and which are not. No
broad answer has been given to this question. Some well known CSG's using
length-preserving rules have languages that turn out to be GrCSL's, an example
being the grammar:
\[ S \rightarrow SABC|dABC \]
\[ BA \rightarrow AB~~~~~~~~~~~~CB \rightarrow BC~~~~~~~~~~CA \rightarrow AC \]
\[ dA \rightarrow da~~~~~~~~~~~~~aA \rightarrow aa~~~~~~~~~~~~~aB \rightarrow ab \]
\[ bB \rightarrow bb~~~~~~~~~~~~~~bC \rightarrow bc~~~~~~~~~~~~~~cC \rightarrow cc \]
Its language $\{ da^nb^nc^n|n \geq 1 \}$ also has the GrCSG:
\[ S \rightarrow SK|SL|dabc \]
\[ cK \rightarrow Kcc~~~~~~~~~~~~bK \rightarrow Kbb \]
\[ aK \rightarrow Kaa~~~~~~~~~~daK \rightarrow daa \]
\[ cL \rightarrow Mccc~~~~~~~~cM \rightarrow Mcc~~~~~~~~~~~~bM \rightarrow Nbbb \]
\[ bN \rightarrow Nbb~~~~~~~~~~aN \rightarrow Kaaa~~~~~~~~~~daN \rightarrow daaa \]
The weighting function is $\phi (x) = 1$ if $x$ is a variable, $\phi (x) = 2$ if $x$ is a
terminal. (To derive $da^{32}b^{32}c^{32}$ in this grammar we would begin by deriving
$dabcK^5$. But to derive $da^{37}b^{37}c^{37}$ we would note that $37 = 2^5 + 2^2 + 2^0$,
and accordingly we would begin by deriving $dabcKKLKL$.)
With a bit more trouble we could get a GrCSG for $\{ da^nb^nc^n \}$ in the original
sense, i.e., one in which the weighting function is $\phi (x) = 1$ for all variables and
terminals $x$. Also, if we wished to get rid of the $d$, which acts as a left-end
marker, we could do so at the expense of further complication in the grammar.
GrCSG's have the advantage over other CSG's in that,
in each such grammar, the length of a derivation
has an upper bound that is linear in the
length of the word derived. Moreover, as mentioned,
the membership problem for every GrCSL has a polynomial-time algorithm.
(The proof in \cite{dw} goes over to our new definition of GrCSL.)
Dahlhaus and Warmuth \cite{dw} prove that every GrCSL is log-tape reducible to some CFL.
Buntrock and Otto \cite{buo} improve on this result by showing that this reduction can be
done as a one-way log-space reduction; that is to say, the GrCSL is
(quoting Section~1 of \cite{buo}) ``accepted by an
auxiliary pushdown automaton with logarithmic space bound and polynomial time bound that
uses its input tape in a one-way fashion.''
As pointed out in \cite{bl1}, the family of GrCSL's is an
abstract family of languages, that is to say, this family is closed under union,
concatenation, star iteration, intersection with regular languages, null-word-free
homomorphisms and inverse homomorphisms.
A persistent problem for theoretical computer scientists has been to find a family of
formal languages that will include all programming languages or most programming
languages. The family must be reasonably simple conceptually and must not be so broad as
to include languages that have properties that no reasonable programming language could
have.
This objective is necessarily vague and this is no place to attempt to refine it or even
to discuss it, except to make a brief negative point about the family of GrCSL's: If
the family of formal languages must include a language of programs whose variables
occurring in executable statements must also occur in declaration statements, and if the
family includes variables of unlimited length, then the family of GrCSL's is not a
suitable family for this purpose. The argument for this negative assertion is based on
the piece of evidence that $\{ ww|w \in \{ a,b \} ^* \}$ is not a GrCSL, which indicates
that any programming language in which a defined program may have arbitrarily long
variables, but only those that are declared, is probably not a GrCSL.
The application of ideas from Theoretical Computer Science to actual computing is
difficult to predict with any precision. But it helps to have some general idea of the
possibility of some application, even if the exact nature of this application is vague.
There will be more to say about the applicability of these ideas on GrCSL's when we
investigate the deterministic variety of them in the next section.
\vspace{.5in}
\noindent
{\large\bf 2. The deterministic variety.}
A GrCSL is {\em deterministic\/} if it is recognized by
a deterministic shrinking two-pushdown automaton. The name for this automaton
is rather long; let us call it
a ``D-shrink'' for short. As it is an important concept, it deserves a formal
definition. The following is adapted from the paper by Buntrock and Otto
\cite{buo}:
A {\em D-shrink\/} is a 7-tuple
\[ (Q, \Sigma , \Gamma , \delta ,q_0,B,F). \]
Here $Q$ is the set of states, $\Sigma$ the input alphabet, $\Gamma$ the tape
alphabet ($\Sigma \subset \Gamma$ and $\Gamma \cap Q = \emptyset$),
$q_0$ is the initial state, $B$ is the bottom marker of the two pushdown stores
($B \in \Gamma - \Sigma$), $F$ is the set of accepting states, and $\delta$ is the
transition function:
\[ \delta :Q \times \Gamma ^2 \rightarrow Q \times \Gamma ^* \times \Gamma ^*
~ \cup ~ \{ \emptyset \} . \]
A configuration in a computation of a D-shrink is given as $uqv$ where
$u,v \in \Gamma ^*$ and $q \in Q$. The idea is that $u$ and $v$ are the words on the
left and right pushdown tapes, respectively. Normally $B$ is the leftmost
character of $u$ and the rightmost character of $v$ and occurs nowhere else.
Where $u = u'a$, $v = bv'$, $a,b \in \Gamma$ and $\delta (q,a,b) = (q',w_1,w_2)$,
the word $u'w_1q'w_2v'$ is the next configuration in that computation. If
$\delta (q,a,b) = \emptyset$ then $uqv$ is a halting configuration. An initial
configuration is of the form $Bq_0vB$ where $v$ is the input. The bottom marker
$B$ is never created or destroyed in a computation but may be sensed.
Acceptance of $v$ is either by final state (a configuration $u'qv'$ where $q \in F$)
or by empty store (a configuration $BqB$ for $q \in Q$). The {\em language\/} of
a D-shrink is the set of all accepted inputs.
What makes the D-shrink shrinking is the stipulation that there is a
weighting function $\phi$ on strings over the alphabet $Q \cup \Gamma$ such that, for
$\delta (q,a,b) = (q,w_1,w_2)$, the condition $\phi (w_1qw_2) < \phi (aqb)$ holds.
As mentioned in Section~1, a language is a GrCSL if and only if it is the language of
a nondeterministic shrinking two-pushdown automaton (\cite{buo}, Section~3). If a
GrCSL is accepted by a D-shrink (i.e., a deterministic automaton of the variety) then the
language is said to be a {\em DGrCSL (deterministic growing context-sensitive language).}
I shall argue in the remainder of this paper that the family of DGrCSL's is an important
family of languages, perhaps more important than the larger family of all GrCSL's.
The language $\{ ba^{2^n}|n \geq 1 \}$ is a DGrCSL. One can design a D-shrink for it
based on the grammar from Section~1:
\[ ~~~~~~S \rightarrow SK|baa \]
\[ baaK \rightarrow baaaa \]
\[ aK \rightarrow Kaa \]
Any word in the language of this grammar will be processed by the D-shrink according to
a rightmost derivation in the grammar, which means that the word is processed from left
to right. For example, the rightmost derivation of the word $ba^{32}$ has the line
$ba^3KaaKK$, to which the rule $aK \rightarrow Kaa$ is applied to the rightmost $aK$,
resulting in the line $ba^3KaKaaK$. The automaton does things in reverse of the order in
the derivation; for that step it might have $Bba^3Ka$ on the left tape and $KB$ on the
right tape, and might be in a state showing that $Kaa$ is between the $ba^3Ka$ and the $K$.
It then would push $aK$ onto the left tape, and
would go into a state showing that the null word is between the $Bba^3KaaK$ on the left
tape and the $KB$ on the right tape.
We shall not verify that this automaton can be made deterministic, and therefore that
the language $\{ ba^{2^n}|n \geq 1 \}$ is a DGrCSL. The language $\{ da^nb^nc^n|n \geq 1 \}$
is also a DGrCSL; details beyond the discussion of this language in Section~1
are omitted.
It is not difficult to prove that the family of
DGrCSL's is closed under complementation.
This observation enables us to prove the existence of CFL's that are
not DGrCSL's. Such a language is $\{a,b \}^* - \{ ww|w \in \{a,b \} ^* \}$. If this
CFL were a DGrCSL then its complement
$\{ ww|w \in \{ a,b \} ^* \}$ would also be a DGrCSL, and hence would be a GrCSL, which (as
mentioned in Section~1) it is not.
And so, although all null-word-free CFL's are GrCSL's, they are not all DGrCSL's.
This may be an unpleasantness that might dissuade some theoreticians from accepting the
family of GrCSL's as a member of the Chomsky hierarchy. As it now stands each family in
the hierarchy is a subclass of the deterministic subclass of the family at the next level.
(Incidentally, as noted in \cite{b96} and \cite{buo}, all null-word-free
deterministic CFL's are DGrCSL's.)
Whether or not it deserves a place in the Chomsky hierarchy,
the family of GrCSL's is an important family. Indeed there is reason to regard
the subclass of
DGrCSL's as being more important than the larger family of GrCSL's. As will be shown in
the next section, the DGrCSL's can be characterized in terms of confluent rewriting
systems, which gives them perhaps even more significance.
(Before going on to Section~3, let us pause to observe that the family of DGrCSL's is
not closed under union or intersection. The simple argument for intersection \cite{no}
is as follows: It is easy to see that the Gladkij language $\{ wcw^Rcw|w \in \{ a,b \} ^* \}$ is
equal to the intersection of two deterministic CFL's, which
are therefore both
DGrCSL's; but the Gladkij language itself is not a DGrCSL. That this
family is also not closed under union follows by the DeMorgan law, since the family is closed
under complementation. Other such results can be found in Section~5 of \cite{no}.)
\vspace{.5in}
\noindent
{\large\bf 3. Confluent string rewriting systems.}
Perhaps the greatest selling
point for the family of GrCSL's is its link with the theory of rewriting
systems as it has been developing since 1970. More specifically, the
selling point is for the family of DGrCSL's, which turns out to be
identical to a family of languages definable in a certain way by confluent
string rewriting systems, as discovered recently by Niemann and Otto
\cite{no}.
Briefly, a {\em string rewriting system\/} is a semi-Thue system. We focus
on systems in which
the application of a rule to a word results in a simplification of the word:
for example, it may be that $|\beta | < |\alpha |$ holds for each rule
$\alpha \rightarrow \beta$. Such systems are
often called {\em reduction systems,\/} since their purpose is to take a
long word and gain some sort of understanding by reducing it to a
shorter word. The length requirement is not a strict requirement, but
one necessary property of a reduction system is that there be no
infinite derivations. Consequently every word can be reduced to an
irreducible word (the Noetherian property).
A further property that is desirable for a reduction system is that
no word can be reduced to two distinct irreducible words (the confluence
property). In reducing a word according to a confluent reduction
system, it is sometimes possible to start reducing in two distinct ways at
the same point in the word. But then the two reduction sequences must
eventually come together.
(For a good exposition of string rewriting systems, see the first two chapters
of \cite{boo}.)
Some languages can be defined by confluent reduction systems; if the
alphabet of the system is the same as the alphabet of the language then
the language is a {\em congruential language,\/}
i.e., the union of some of the congruence classes of a congruence
relation over $\Sigma ^*$. Unfortunately,
most interesting formal languages are not congruential.
However, if we allow ourselves to supplement the alphabet of the
reduction system to include some control characters along with the
alphabet of the language, we get something that is more fruitful.
The paper \cite{mc} investigated the question of
which formal languages could be defined in this way. The results and
conceptual development of that paper have recently been surpassed in a
remarkable way by Niemann and Otto \cite{no}, whom the present
exposition will follow.
Another desirable property that our reduction systems must have is
that they be weight reducing in the sense defined in Section~1; that is
to say, there is a weight function $\phi$ such that, for every rule
$\alpha \rightarrow \beta$, $\phi (\beta ) < \phi (\alpha )$. A reduction system
has the {\em generalized Church-Rosser property \/} if it is confluent and
weight reducing. (It has the {\em strict Church-Rosser property\/} if it
is confluent and length-reducing. Except for a few isolated remarks, we
shall generally ignore the strict property for the remainder of this paper
in favor of the more general property.)
A language $L \subseteq \Sigma ^*$ is a {\em GenCRL (generalized
Church-Rosser language)\/} if there exists a reduction system $S$ with
the generalized Church-Rosser property satisfying the following
conditions:
\begin{quote}
(1) The alphabet $\Gamma$ of $S$ contains $\Sigma$ as a proper subset; \\
(2) There are $Y, t_1$ and $t_2$, where $t_1,t_2 \in (\Gamma - \Sigma)^*$,
$Y \in \Gamma - \Sigma$ and $Y$ is irreducible in $S$, such that, for
all $w \in \Sigma ^*$, $w \in L$ if and only if
$t_1wt_2 \rightarrow ^* Y$ (viz., $Y$ is derivable from $t_1wt_2$ in $S$)
\end{quote}
Notice that if we have a language $L$ that is of interest to us then
such a system $S$ would be a nice thing to have for testing membership in
$L$. Given any $w \in \Sigma ^*$, we would form the word $t_1wt_2$ and
reduce it modulo $S$. If the reduced word is $Y$ then $w \in L$; if not
then it is not. Since every rule of $S$ is weight-reducing, the length of
the reduction of $t_1wt_2$ is linear in $|\phi (w)|$, and hence linear in
$|w|$. Each step of the reduction is rather easy; we simply scan the
word that we have for a subword that is the left side of a rule. (Things
can be done so that the
total amount of time spent in scanning during the entire reduction is
insignificant.) When we find such a subword we reduce the word
accordingly. If we find that there is no such subword and the word at
that point is not simply $Y$ then we know that the original word $w$ is
not in $L$.
As proved in \cite{buo} and \cite{no}, a language $L$ is a GenCRL if and
only if it is a DGrCSL. In effect, the D-shrink is a suitable mechanism
for reduction of the word in the reduction system; in fact, precisely
suitable. This automaton is similar to the automaton conceived by Ron Book
\cite{b82} to reduce a word according to a reduction system with the
Church-Rosser property.
In \cite{no} it is demonstrated that every GenCRL is
also a CRL; in other words the reduction system can be modified so as to
allow the length function as the weighting function (i.e., for each
$x \in \Gamma$, $\phi (x) = 1$). In the same paper, it is also
demonstrated that the reduction system can be modified so that, for
every $w \in \Sigma ^*$, $t_1wt_2$ reduces either to $Y$ (indicating
``yes,'' that $w \in L$), or to $N$ (indicating ``no,'' that it is not);
both $N$and $Y$ are in $\Gamma - \Sigma$. These results settled
questions left open in \cite{mc}. Furthermore, they make the
characterization of DGrCSL's in terms of rewriting systems even more
significant than they appear at first. They also strengthen our feeling
that the DGrCSL's constitute an important family of languages.
An interesting open question concerns the language
$\{ ww^R|w \in \{ a,b \} ^* \}$, which is clearly a GrCSL, since it is
context-free. I conjecture, however, that it is not a DGrCSL. In \cite{mc}
there is a plausibility argument that it is not a CRL.
In conclusion, I suspect there will probably be few theoreticians who
will press for any modification of the Chomsky hierarchy. Nevertheless, I
hope many will come to realize that both the family of GrCSL's and the
family of DGrCSL's will play important roles in the future of computer
science. I am especially convinced of the importance of the DGrCSL's,
since they have such a solid link with the contemporary theory of
rewriting systems, and, in particular, with string rewriting systems having
the confluence property.
\vspace{.2in}
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\end{document}