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  1. Can a CFG over large alphabet describe a language, where each terminal appears even number of times?

  2. If yes, would the Chomsky Normal Form be polynomial in |Σ| ?

EDIT: What about a language where each terminal appears 0 or 2 times?

EDIT: Solving this may give interesting extension to "Restricted read twice BDDs and context free grammars"

Find a context-free grammar (CFG) for the language L of all words such that each terminal in a word occurs even number of times over a possibly large alphabet Σ

My long aproach is (the only nonterminal is S):

S ⟶ ε | SS

x ∈ Σ : S ⟶ xSx

x,y ∈ Σ : S ⟶ xxSyy | yySxx | xySxy | xySyx | yxSyx | yxSxy

Second try, incremental approach.

S ⟶ ε | SS
Terminal productions (suboptimal): x,y ∈ Σ : S ⟶ xx | yy |xxyy | yyxx | xyxy | xyyx | yxyx | yxxy
Incremental: x ∈ Σ : S ⟶ SxSxS

By the way, the homework is over now.

Technically it wasn't a homework assigned to me. I don't know the answer and thought the question was easy. Best.

EDIT: I didn't award the bounty :-)

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1  
abcabc isn't in your language, I think! –  yatima2975 Dec 14 '10 at 20:03
3  
Whoops! Have I messed up? The proof outline I had in mind went as follows: none of the second and third productions matches up (as a != c and ab uses different letters than bc) so the starting production would have to be S -> SS. But each S will eventually produce each terminal an even number of times. Chopping abcabc up as ab/cabc or abca/bc won't work either, for obvious reasons. What sequence of productions did your parser find? –  yatima2975 Dec 14 '10 at 21:07
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Sorry, I can't make much sense out of that... For the alphabet {a,b,c} you need the following grammar, if you follow Carl's construction: S -> eps | aT | bU | cV [], T -> aS | bW | cX [a], U -> aW | bS | cY [b], V -> aX | bY | cS [c], W -> aU | bT | cZ [ab], X -> aV | bZ | cT [ac], Y -> aZ | bV | cU [bc], Z -> aY | bX | cZ [abc] In brackets behind each non-terminal is the set of terminals that has occurred an odd number of times. –  yatima2975 Dec 15 '10 at 10:39
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Alas! Your second try is also not working, again with abcabc as counterexample. The second production can't be used since there are no substrings matching those patterns. For the third, x is either a but then S needs to produce bc twice independently! , b but then S needs to produce a which can't be done, or c which has the same problem as a. My gut feeling is it can't be done, but I can't back that up... –  yatima2975 Dec 15 '10 at 18:20
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Of course, if you are just looking for a parser you can as well use full power (TM/RM) and recognize the language in time $\mathcal{O}(n)$ with $k$ bits in memory. –  Raphael Dec 16 '10 at 9:59

5 Answers 5

There is a deterministic finite state machine that accepts a word if and only if each terminal appears an even number of times. Assume that there are $k$ symbols in the alphabet. The machine has $2^k$ states, each of which is associated with a subset of the alphabet, and we identify the states with the subsets. The machine is defined so that:

  • The initial state corresponds to the empty set, and this is the only accepting state.
  • The transition rules are defined as follows. From state $A$, on input $l$, if $l \in A$ then let $B = A - \{l\}$ and move to state $B$. Otherwise let $B = A \cup \{l\}$ and move to state $B$.

The idea is that a state $B$ corresponds to the set of terminals that have appeared an odd number of times so far. Being in state $\emptyset$ corresponds to every terminal appearing an even number of times.

Since the language is accepted by a DFSM, the language is regular. This means that there is, trivially, a CFG that accepts the language. The grammar has one nonterminal for each state of the machine, and the rules of the grammar match up with transitions of the machine. I don't know yet whether there is a context free grammar for this language that is polynomial size compared to the size of the alphabet, though.

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2  
Exactly what I had in mind. –  Raphael Dec 15 '10 at 10:58
    
@Carl: Thank you!! Do you have an opinion or intuition about polynomial size CNF - need this for a graph problem. –  jerr18 Dec 15 '10 at 12:15
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I don't know about a polynomial size CNF but I did work out that you can't do it with a polynomial size DFSM, in fact if there are $k$ letters then you need at least $2^k$ states for any correct DFSM. So my answer is optimal for DFSMs that accept the language. The problem for context free grammars seems much more difficult. –  Carl Mummert Dec 15 '10 at 15:05
    
Thank you again. I hope the second edit is marginally better. –  jerr18 Dec 15 '10 at 17:34
    
Solving this may give interesting extension to "Restricted read twice IBDDs and context free grammars" cstheory.stackexchange.com/questions/3816/… –  jerr18 Dec 16 '10 at 10:36

Edit: Here is a more verbose proof. The original proof is retained below.

Introduction

Let $\Sigma$ be some alphabet with $n$ symbols. Let $L$ be the language of words in which each symbol appears an even number of times. Suppose $G$ is a grammar for $L$. Our goal is to given an exponential lower bound on the size of $G$ in terms of $n$.

Transition to Chomsky Normal Form

According to Wikipedia (and not difficult to prove), there is a grammar $G'$ for $L$ in Chomsky Normal Form such that $|G'| = O(|G|^2)$. We will show an exponential lower bound on $|G'|$, and this will imply an exponential lower bound on $|G|$ (if $|G'| = \Omega(c^n)$ for $c > 1$ then $|G| = \Omega(\sqrt{c}^n)$).

For our purposes, a grammar is in Chomsky Normal Form (CNF) if the productions are of the following two types: $A \rightarrow BC$ and $A \rightarrow a$, where $A,B,C$ are non-terminals and $a$ is terminal. Some people also place the restriction that $B,C$ cannot equal the starting symbol $S$, but we do not need this condition.

Word derivation in CNF grammars

Let $w \in L$. Choose some derivation of $w$ in $G'$ (there might be several sharing the same derivation tree). The derivation can be thought of as a binary tree (derivation tree) where each non-leaf node is labeled by a terminal, and each leaf is labeled by a non-terminal; a non-leaf node either has two non-leaf children or one leaf child.

As an example, consider the derivation $$S \rightarrow AB \rightarrow ACD \rightarrow^* xyz.$$ The corresponding derivation tree can be described as follows (sorry for the missing graphics):

  1. The root is labeled $S$.
  2. The root has two children labeled $A,B$.
  3. The node labeled $B$ has two children labeled $C,D$.
  4. The node labeled $A$ has one child labeled $x$, which is a leaf.
  5. The node labeled $C$ has one child labeled $y$, which is a leaf.
  6. The node labeled $D$ has one child labeled $z$, which is a leaf.

Note that in general, several nodes can share a label (indeed, this is how one proves the pumping lemma).

Each node in the tree generates a subword of the original word, which is formed from its descendant leaf nodes in order. Continuing our example:

  1. The root generates the entire word $xyz$.
  2. The node labeled $A$ generates the subword $x$.
  3. The node labeled $B$ generates the subword $yz$.
  4. The node labeled $C$ generates the subword $y$.
  5. The node labeled $D$ generates the subword $z$.

A subword lemma

Lemma. Suppose $w \in L$ is of length $m$, and choose some positive $\ell \leq m$. Then there is a node in the derivation tree of $w$ that generates a subword of length $\alpha$ satisfying $\ell \leq \alpha < 2\ell$.

Proof. Run the following algorithm:

  1. Set $v$ to the root of the derivation tree.
  2. While the subword generated by $v$ has size at least $2\ell$, replace $v$ by its child whose generated subword is longest.
  3. Output the subword generated by $v$.

Note that the node $v$ always has two children, since otherwise it generated a subword of length $1$. If the loop in step (2) never runs then certainly $v$ generates a subword of the requisite length, by our assumption that $\ell \leq m$.

Otherwise, consider the last iteration of the loop. Denote by $w(u)$ the subword generated by a node $u$. In the last iteration of the loop, there is a node $v$ with $|w(v)| \geq 2\ell$, with children $v_1,v_2$ such that $|w(v_1)| \geq |w(v_2)|$; we move to $v_1$. Note that $$2\ell \leq |w(v)| = |w(v_1)| + |w(v_2)| \leq 2|w(v_1)|$$ so that $|w(v_1)| \geq \ell$. Since the loop stopped, $|w(v_1)| < 2\ell$ and $w(v_1)$ is of the requisite size.

The proof

We consider the set $P$ of words in $L$ of the form $\pi(\Sigma) \pi(\Sigma)$, where $\pi$ runs over all permutations of $n$ symbols. For example, if $\Sigma = \{0,1\}$ then $$P = \{0101,1010\}.$$ Note that indeed $P \subseteq L$.

Since all words in $P$ are of length $2n$ (recall $n$ is the size of the alphabet $\Sigma$), using the lemma with $\ell = n/3$ (assume for simplicity that $n$ is divisible by $3$) we get that some node in the derivation tree of any $w \in P$ generates a subword $x(w)$ whose size satisfies $$n/3 \leq |x(w)| < 2n/3.$$ Denote by $s(w)$ the non-terminal generating $x(w)$ (it is the label of the node).

Since $|x(w)| \leq n$, all symbols in $x(w)$ are distinct: indeed, any subword of $w$ of length $n$ is a cyclic rotation of the original permutation (think of all subwords of length $3$ in $abcabc$: $abc,bca,cab,abc$).

Our strategy now will be to show that any single non-terminal can generate only a relatively small number of the $x(w)$; since $P$ is big, it will follow that $G'$ must have lots of non-terminals. More explicitly, let $$S = \{ s(w) : w \in P\}.$$ We can find a list of representative, distinct words $w_1,\ldots,w_{|S|}$ such that $$S = \{ s(w_i) : 1 \leq i \leq |S| \}.$$ We will show that for each $w_i$, the set of words $w \in P$ with $s(w) = s(w_i)$ consists of at most $B$ words ($B$ will be determined below). It will follow that the number of distinct non-terminals is at least $$\frac{|P|}{B} = \frac{n!}{B}.$$

So let $w \in P$, and suppose that $s(w) = s(w_i)$ for some fixed $i$. What this means is that we can replace $x(w)$ in $w$ with $x(w_i)$ and still get a word in $L$. The word $w$ has a decomposition $w = lx(w)r$. Every symbol appearing in $x(w)$ appears exactly once in $lr$, and every other symbol appears exactly twice in $lr$. Note that this defines the word $x(w)$ up to permutation, i.e. since we assumed that $lx(w_i)r \in L$, then $x(w_i)$ must be a permutation of $x(w)$ (more accurately, the set of (unique) symbols occurring in $x(w),x(w_i)$ must be the same).

We will now bound the number of words in $P$ that have a subword of length $x(w_i)$ which is a permutation of $x(w_i)$. Any such word $w$ can be broken as $lyzr$ where $yz$ is a permutation of $x(w_i)$ and $|ly|=|zr|=n$; in fact, since $w \in P$ then $ly = zr$. There are $|x(w_i)|!$ possible choices for $yz$. Together, these define $|x(w_i)|$ points of the permutation $\pi(\Sigma) = ly$. For the rest of the points we have $(n-|x(w_i)|)!$ possible choices. Finally, $yz$ can start in at most $n$ "cyclic" locations inside $\pi$, and so $$B \leq n|x(w_i)|!(n-|x(w_i)|)!.$$ When is this expression maximized, as a function of $\alpha = |x(w_i)|$? Taking the logarithm of the bound, $$\log B \leq \log 2n + \log \alpha! + \log (n-\alpha)!.$$ The factorial function (in fact, the Gamma function) is log-convex, and so subject to the condition $n/3 \leq \alpha \leq 2n/3$, this expression is maximized for $\alpha=n/3$ or $\alpha =2n/3$ (the value is the same). That is $$B \leq n(n/3)!(2n/3)!.$$

Plugging the value of $B$ that we have laboriously estimated, we conclude that the number of different non-terminals in $G'$ must be at least $$\frac{|P|}{B} = \frac{n!}{n(n/3)!(2n/3)!} = \frac{1}{n} \binom{n}{n/3}.$$ Now we will use Stirling's approximation $$m! \sim \sqrt{2\pi m} (m/e)^m.$$ Plugging the approximation, we get $$\frac{n!}{2n(n/3)!(2n/3)!} \sim \frac{\sqrt{2\pi n}(n/e)^n}{2n\sqrt{2\pi(n/3)}(n/3e)^{n/3} \sqrt{2\pi(2n/3)}(2n/3e)^{2n/3}}.$$ All powers of $n$ and $e$ cancel, and we're left with the following $$\frac{n!}{n(n/3)!(2n/3)!} \sim \frac{1}{n\sqrt{4\pi n/9}} 3^{n/3} (3/2)^{2n/3} = \Omega\left(n^{-3/2} c^n \right),$$ where the constant $c$ is $$c = 3^{1/3} (3/2)^{2/3} = \frac{3}{2^{2/3}} > 1.$$

In particular, the original grammar $G$ must be of size at least $\Omega(n^{-3/4} \sqrt{c}^n)$.

Exactly 0 or 2 occurrences

Another questioned concerned the language $L'$ of all words which contain exactly $0$ or $2$ occurrences of each symbol in $\Sigma$. The proof given in the preceding section applies equally well to $L'$ since the set of words $P$ we used belongs to $L'$; the rest of the proof uses the fact that certain words are not in $L$, and since $L \supset L'$, they are a fortiori not in $L'$.

Generalization

We have seen that the lower bound holds for both $L$ and $L'$. When does the proof hold for a language $\Lambda$? We used only two properties of $\Lambda$:

  1. $\Lambda$ contains $P$.
  2. If $w_1,w_2 \in P$, $x_1,x_2$ are the subwords of $w_1,w_2$ chosen using the subword lemma, and $w_1(x_1:=x_2),w_2(x_2:=x_1) \in \Lambda$ (i.e. the word resulting by replacing $x_1$ with $x_2$ in $w_1$, and the word resulting by replacing $x_2$ with $x_1$ in $w_2$) then $x_2$ is a permutation of $x_1$.

When will the second condition fail? The words $x_1,x_2$ both have no repeated symbols. If they are not permutations of each other then $w_1(x_1:=x_2)$ will have a symbol repeated $1$ or $3$ times. So words containing such symbols must lie out of $\Lambda$. Considering $w_2(x_2:=x_1)$, we see that it is enough to exclude one of the possibilities. We get the following theorem.

Theorem. Suppose $\Lambda$ is a language containing $P$ in which no word contains any symbol exactly once, or in which no word contains any symbol exactly $3$ times. Then any grammar for $\Lambda$ must have size $\Omega(n^{-3/4} C^n)$, where $C = \sqrt{3}/\sqrt[3]{2}$.

As an example, the theorem applies for the language in which the number of occurrences of each symbol lies in $S$ as long as $2 \in S$ and $1,3 \notin S$. For example, $L$ corresponds to $S = \{0,2,4,6,\ldots\}$ and $L'$ corresponds to $S = \{0,2\}$.

We leave the reader the following easy generalizations.

Theorem. Suppose $\Lambda$ is a language containing $\{ \pi(\Sigma)^\ell \}$ for some $\ell > 0$, in which no word contains any symbol exactly $\ell + \epsilon$ times, where $\epsilon$ is either $1$ or $-1$. Then any grammar for $\Lambda$ must have size $\Omega(n^{-3/4} C^n)$, where $C = \sqrt{3}/\sqrt[3]{2}$.


Original, condensed proof

Let $L$ be the language of words in which each symbol appears an even number of times. We give an exponential lower bound (in $n = |\Sigma|$) on the size of a Chomsky Normal Form (CNF) grammar for $L$. This implies an exponential lower bound on the size of any context-free grammar for $L$, since (according to Wikipedia) the blowup to CNF is at most quadratic.

Given any derivation of a word $w$ of length $m$ and any $\ell < m/2$ we can always find a non-terminal symbol in the derivation which derives a subword $x$ of $w$ of length $\ell \leq |x| < 2\ell$: start from the root of the derivation, and always pick the larger branch. The first time we hit a branch smaller than $2\ell$, it must be at least $\ell$ in size.

Consider the set of all $S = n!$ words in $L$ of the form $\pi\pi$ (all of size $2n$), where $\pi$ is a permutation of $\Sigma$. For each such word $w$ we can find a symbol $s_w$ generating a subword $x_w$ of size $n/3 \leq |x_w| < 2n/3$. Note that all symbols in $x_w$ are distinct. Therefore if $s_u = s_v$ then $x_u,x_v$ must be permutations of each other.

Given $w$, for how many words $u$ can it be that $s_w = s_u$? To get from $w$ to $u$ we need to permute $x_w$, to permute the rest of the permutation, and to choose the starting location of $x_u$ within the permutation generating $w_u$. All in all, the number of possibilities is at most $$M = n|x_w|!(n-|x_w|)! \leq n(n/3)!(2n/3)!.$$ Therefore the number of distinct symbols in the grammar is at least $$\frac{S}{M} = \frac{n!}{n(n/3)!(2n/3)!} = O\left(\frac{c^n}{n^{1.5}}\right),$$ where $c = 3/2^{2/3} \approx 1.88988$ (thanks to Wolfram Alpha).

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+1 for just answering. I have no idea if the answer is correct. –  jerr18 Dec 22 '10 at 8:09
    
@jerr18 Maybe the new version is more understandable. If there are still points which you find unclear, I can expound on them. –  Yuval Filmus Dec 24 '10 at 1:58
    
Thank you. I have some irrational doubts about the correctness of your solution. Nothing personal. –  jerr18 Dec 25 '10 at 8:21
    
Unfortunately I lack the knowledge to verify your proof. So far I resort to watching peer review :-) [homework hunters should trust the peer review actors, by the way. There are some blatantly wrong statements on this forum...] –  jerr18 Dec 26 '10 at 8:37

Edit: This solution is wrong, as explained in the comments below. The proof breaks in the italicized phrase.

Let $\Sigma = \{ \sigma_i \}$. Take the following grammar of size $O(|\Sigma|)$: $$S \longrightarrow \sigma_i S \sigma_i S | \varepsilon$$ Clearly in every word produced, every terminal appears an even number of times (proof by induction). In order to generate a word $w$ matching this description, suppose that $w = \sigma_i w'$. Then $\sigma_i$ must appear again in $w'$, say $w = \sigma_i z_1 \sigma_i z_2$. Note that $z_1,z_2$ also match the description.

Now in CNF: $$\begin{align*} S &\longrightarrow \varepsilon \\ S &\longrightarrow S_i S_i \\ S_i &\longrightarrow \sigma_i | A_i T \\ A_i &\longrightarrow \sigma_i \\ T &\longrightarrow S_i S_i \end{align*}$$

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Thank you. Before accepting I will need some time. –  jerr18 Dec 21 '10 at 10:06
    
Thank you. I have some irrational doubts about the correctness of your solution (if I understand it correctly). Seems epsillon removal will require some troubling sub-words. I can live with your CNF if the rest is correct... –  jerr18 Dec 21 '10 at 11:22
    
@Yuval Filmus over the alphabet {a,b,c} I don't see how to generate abcabc using your rules and the parser <a href="ncc.up.pt/cgm/">cgm</a>; claims it is not in the language (may be a parser bug). Am I missing something? –  jerr18 Dec 21 '10 at 13:36
    
cgm is at ncc.up.pt/cgm –  jerr18 Dec 21 '10 at 13:37
    
You got me! I'll keep on thinking. –  Yuval Filmus Dec 21 '10 at 18:36

I have no idea how to comment. But I do know how to answer. Sorry about that.

I considered the pumping lemma. And I considered how many words are necessary to pump the language.

Consider the language over {0,1}.

We need the following words: 00, 11, 0101, and 1010. To pump we can combine any words by concatenation or insertion. For this example a grammar would be

S => | S0S0S | S1S1S | S0S1S0S1S | S1S0S1S0S

A quick count shows the number of words necessary for the grammar with n terminals is exponential in n. I guess that shows there is no polynomial sized grammar.

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This is not a proof! Maybe the idea can be turned into a proof, can you try to explain it more verbosely? What do you mean by "how many words are necessary to pump the language"? –  Yuval Filmus Dec 23 '10 at 3:17
    
+1 for just answering. I have no idea if the answer is correct. –  jerr18 Dec 23 '10 at 8:28
    
@user5055 If your goal is just to win 50 pts, a faster way will be to answer other questions on the site. –  jerr18 Dec 23 '10 at 8:48

jerr18 ---

Ignoring the fact that my posted "answer" was clearly indicated to be a "comment" ...

Perhaps you do not understand what a proof is. As a writer: a proof is what makes me believe a result. As a reader whatever you wish to accept is a proof. A counterexample is what is needed to show a proof is defective. Sometimes a believer will use a comment to provide a counterexample. (I am not asking that you make me disbelieve my comment/proof by providing a counter example.)

As for your question, I believe that the empty language is poly for any alphabet and it satisfies your requirement that every terminal appears an even number of times. I suppose your question should be reposed to conform to the effort that Yuval Filmus put forth.

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I think the OP meant the language consisting of <i>all</i> words in which each symbol appears an even number of times (never or twice in the variant). –  Yuval Filmus Dec 26 '10 at 7:09
    
Your concept of proof is very Lakatosian. Working mathematicians believe (perhaps naively) that there are some absolute standard of proof, such that whenever a result is proven, no counterexample is possible. That's the common meaning of "proof" in the mathematical community. –  Yuval Filmus Dec 26 '10 at 7:11
    
+1. Other suggestions for reposing? Technically it wasn't a homwork. At least assigned to me. –  jerr18 Dec 26 '10 at 8:56
    
Thank you. Trying to decrease my crankiness level: If I knew the answer, I wouldn't ask. The negative answer is probably on the right site of the bet (I suppose the answer to this question is negative). The answer is not convincing for my level of expertise and the answerer tried the other side of the bet too. –  jerr18 Dec 27 '10 at 8:24

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