On the Comparison of Context-Free Grammars

TL;DR

This paper presents a method for comparing context-free grammars by reducing the problem to solving nonlinear matrix equations, laying the groundwork for automated, probabilistic assessment tools in computer science education.

Contribution

It introduces a novel approach that connects grammar comparison to numerical solutions of matrix equations, enabling probabilistic and automated analysis.

Findings

Reduction of grammar comparison to matrix equations

Foundation for probabilistic comparison algorithms

Potential application in automated student answer assessment

Abstract

In this paper we consider the problem of context-free grammars comparison from the analysis point of view. We show that the problem can be reduced to numerical solution of systems of nonlinear matrix equations. The approach presented here forms a basis for probabilistic comparison algorithms oriented to automatic assessment of of student's answers in computer science.

Equations

$$S=c+ab+acb+accb+acccb+\dots$$

$$S=aAb+c$$

$$A=cA+ϵ$$

$$S(\mu )={\mu }_c+{\mu }_a{\mu }_b+{\mu }_a{\mu }_c{\mu }_b+{\mu }_a{\mu }_c{\mu }_c{\mu }_b+{\mu }_a{\mu }_c{\mu }_c{\mu }_c{\mu }_b+...,$$

$$S(\mu )={\mu }_{a}A(\mu ){\mu }_b+{\mu }_c$$

$$A(\mu )={\mu }_{c}A(\mu )+I$$

$$S_1({\mu }_{a_1},\dots ,{\mu }_{a_n})\ne S_2({\mu }_{a_1},\dots ,{\mu }_{a_n})$$

$$S=\sum _{P\in W(V_T)}\varphi (P)P.$$

$$0\le \varphi (P)\le C{q}^{|P|},$$

$$s(\mu )=\sum _{P\in W(V_T)}\varphi (P)P(\mu )$$

$$S(\mu )=\sum _{P\in W(V_T)}\varphi (P)P(\mu )\le \sum _{N\ge 0}C{q}^N|V_T{|}^N{M}^N<\infty ,$$

$$\varphi (P_k)\ge 1\cdot {\left(\frac{k}{\mu }\right)}^{|P_k|}.$$

$$S(\mu )=\sum _{P\in W(V_T)}\varphi (P)P(\mu )\ge {\left(\frac{k}{\mu }\right)}^{|P_k|}{\mu }^{|P_k|}\ge k.$$

$$\mu_{i}e_{0}=\left\{\begin{array}[]{ll}e_{i},&a_{i}\in\mathcal{S}\\ 0,&\textrm{otherwise}\end{array}\right.$$

$$\mu_{i}e_{i_{1}\ldots i_{l}}=\left\{\begin{array}[]{ll}e_{ii_{1}\ldots i_{l}},&a_{i}a_{i_{1}}\ldots a_{i_{l}}\in\mathcal{S}\\ 0,&\textrm{otherwise}\\ \end{array}\right.$$

$$U(\mu )e_0=\sum _{(a_{i_1}\dots a_{i_N})\in U}{e}_{i_1\dots i_N}\ne \sum _{(a_{i_1}\dots a_{i_N})\in V}{e}_{i_1\dots i_N}=V(\mu )e_0,$$

$$e_0=(1,0,0,0,0),$$

$$e_1=(0,1,0,0,0),$$

$$e_2=(0,0,1,0,0),$$

$$e_{12}=(0,0,0,1,0),$$

$$e_{21}=(0,0,0,0,1),$$

$${\mu }_1{e}_0=e_1,{\mu }_1{e}_1=0,{\mu }_1{e}_2=e_{12},{\mu }_1{e}_{12}=0,{\mu }_1{e}_{21}=0,$$

$${\mu }_2{e}_0=e_2,{\mu }_1{e}_1=e_{21},{\mu }_2{e}_2=0,{\mu }_2{e}_{12}=0,{\mu }_1{e}_{21}=0.$$

$${\mu }_1{\mu }_2{e}_0={\mu }_1{e}_2=e_{12}\ne e_{21}={\mu }_2{e}_1={\mu }_2{\mu }_1{e}_0.$$

$$\Delta (t)=S_1(t\mu )-S_2(t\mu )=t^N(U(\mu )-V(\mu ))+(R_1(t\mu )-R_2(t\mu )).$$

$$\Vert R_1(t\mu )\Vert \le C_1\left((q_1\sigma t{)}^{N+1}+(q_1\sigma t{)}^{N+2}+\dots \right)$$

$$\Vert R_2(t\mu )\Vert \le C_2\left((q_2\sigma t{)}^{N+1}+(q_2\sigma t{)}^{N+2}+\dots \right)$$

$$\Vert R_1(t\mu )-R_2(t\mu )\Vert \le 2C\left((q\sigma t{)}^{N+1}+(q\sigma t{)}^{N+2}+\dots \right)=2C(q\sigma {)}^{N+1}\frac{t^{N+1}}{1-q\sigma t}.$$

$$U=\sum _{\{(k_N^{\prime m},\dots ,k_1^{\prime m})\mid m=\overline{1,M}\}}{a}_{k_N^{\prime m}}\dots a_{k_1^{\prime m}}\mathrm{and}V=\sum _{\{(k_N^{\prime \prime m},\dots ,k_1^{\prime \prime m})\mid m=\overline{1,M}\}}{a}_{k_N^{\prime \prime m}}\dots a_{k_1^{\prime \prime m}}$$

$${\mathcal{P}}_U=\left\{\prod _{j=l+1}^N{u}_{k_j^{\prime m}}{v}_{k_l^{\prime m}}\mid l=\overline{1,N}\right\},\mathrm{and}{\mathcal{P}}_V=\left\{\prod _{j=l+1}^N{u}_{k_j^{\prime \prime m}}{v}_{k_l^{\prime \prime m}}\mid l=\overline{1,N}\right\}.$$

$$\mu_{i}=\left(\begin{array}[]{cc}u_{i}&v_{i}\\ 0&1\end{array}\right).$$

$$\prod_{i=1}^{N}\mu_{k_{i}}=\left(\begin{array}[]{cc}\prod_{i=1}^{N}u_{k_{i}}&\sum_{i=1}^{N}\prod_{j=i+1}^{N}u_{k_{j}}v_{k_{i}}\\ 0&1\end{array}\right).$$

$${\mu }_1{\mu }_1{\mu }_2{\mu }_2{\mu }_1+{\mu }_1{\mu }_2{\mu }_1{\mu }_1{\mu }_2+{\mu }_2{\mu }_1{\mu }_2{\mu }_1{\mu }_1={\mu }_1{\mu }_1{\mu }_2{\mu }_1{\mu }_2+{\mu }_1{\mu }_2{\mu }_2{\mu }_1{\mu }_1+{\mu }_2{\mu }_1{\mu }_1{\mu }_2{\mu }_1.$$

$$\{aabba,abaab,babaa\}\mathrm{and}\{aabab,abbaa,baaba\}$$

Full text

1 Introduction

In this paper we consider a language on an alphabet $V_{T}$, a subset of valid words $W(V_{T})=V_{T}^{*}$. This set being usually infinite, it is necessary to use, for example, grammars as a mechanism for definition of the languages. We are going to consider only context free grammars, covering context free languages. The capability of writing correct grammars is an essential task in computer science (used, for example, in the creation of programming languages, compilers, etc.). Assessment of student’s answers in computer science is a very hard and time-consuming activity. Computer-assisted assessment is a natural way to reduce the time spent by the teachers in their assessment task (see, e.g., [3, 2]). This paper deals with assessment in the theory of context free grammars. Its main objective is to create a theoretical basis for algorithms allowing one to decide if two context free grammars are equivalent or not. It is well-known that the equivalence of two context free grammars is an undecidable problem [11]. The problem of context free grammars equivalence was an object of intensive studies [4, 5]. For example, it was solved when the equivalence is understood in structural sense [10], and some practical algorithms for grammars equivalence checking were developed (see [7, 9], and the references therein).

In order to present the methodology adapt in this paper, let us consider the following simple example. Let the language $L$ be $\{c,ab,acb,accb,acccb,...\}.$ We shall denote the axiom of a grammar by $S$. According to [12] we can write a formal power series

[TABLE]

corresponding to this language. The language $L$ can be generated by the grammar $S\rightarrow aAb\;\mid\;c$ ; $A\rightarrow cA\;\mid\;\epsilon$.

The following system of formal equations corresponds to this grammar

[TABLE]

Formally applying the iteration method to this system we obtain series (1). Below we define a transform that attributes a matrix meaning to formal power series (1). Namely,

• •

any terminal letter $a,b,c$, is substituted by an $({\cal N}\times{\cal N})$-matrix $\mu_{a},\mu_{b},\mu_{c}$;

• •

the nonterminal symbols $S$ and $A$ are substituted by $({\cal N}\times{\cal N})$-matrix variables $S(\mu)$ and $A(\mu)$;

• •

the formal sum and product are substituted by the matrix ones;

• •

the empty word $\epsilon$ is substituted by the $({\cal N}\times{\cal N})$ identity matrix $I$.

Then a matrix $S(\mu)=S(\mu_{a},\mu_{b},\mu_{c})$ calculated as the sum of the matrix series

[TABLE]

corresponds to $S$. In order to effectively compute this sum we numerically solve the system of matrix equations

[TABLE]

obtained applying the transform to formal system (2) and (3). In the same way, in general case of a grammar with the terminal alphabet $V_{T}=\{a_{1},...,a_{n}\}$, one can calculate the matrix $S(\mu)=S(\mu_{a_{1}},\ldots,\mu_{a_{n}})$.

The main result proved in this paper (Distinguishability Theorem I) shows that if two languages $L_{1}$ and $L_{2}$ generated by context free grammars are different and their ambiguities are bounded in some sense, then there exist a positive integer $\cal N$ and an $({\cal N}\times{\cal N})$-matrix substitution $\mu_{a_{1}},\ldots,\mu_{a_{n}}$ such that

[TABLE]

The condition of ambiguity boundedness excludes, for example, infinite ambiguity.

The solution of an exercise submitted by a computer science student is supposed to be an unambiguous grammar. So it is very important from the point of view of automatic assessment to distinguish between an ambiguous and an unambiguous grammars. Note that the grammars with different ambiguities we consider as different.

This theorem allows us to construct tools for comparison of context free grammars. Namely, we calculate ${S_{1}}(\mu_{a_{1}},\ldots,\mu_{a_{n}})$ and ${S_{2}}(\mu_{a_{1}},\ldots,\mu_{a_{n}})$ for a sufficiently large number of matrix substitutions and if for all substitutions the equality ${S_{1}}(\mu_{a_{1}},\ldots,\mu_{a_{n}})={S_{2}}(\mu_{a_{1}},\ldots,\mu_{a_{n}})$ is satisfied, then we conclude that the grammars are equivalent. In this paper we do not discuss the details of such algorithms, the number of substitutions needed to conclude that the grammars are equivalent with some probability, etc.. This will be the subject of further research. We would like to note that according to our experience one $(2\times 2)$ or $(3\times 3)$ random matrix substitution is enough to distinguish between two different context free grammars.

This paper continues the research started in [1] where we considered distinguishability based on $2\times 2$-matrices. Due to negligence, the distinguishability theorem was not clearly formulated. We remedy this situation below (Theorem 2).

Note also that the idea to use matrices to study formal power series is not new (cf. [13]), but the approach presented in [13] is completely different from the one discussed here.

The paper is organized as follows. In Section 2 we prove the distinguishability theorems. Section 3 contains convergence analysis of the iteration method for nonlinear matrix equations. The limitations of the method are discussed in Section 4. Section 5 contains a brief conclusion.

2 Distinguishability theorems

Any context free language generated by a grammar with finite ambiguity can be defined in terms of a formal power series with associative but not commutative variables [11, 12]. Let $V_{T}$ be the terminal alphabet, $W(V_{T})$ the set of words over $V_{T}$, and $Z_{+}$ be the set of nonnegative integers. A map $\phi:W(V_{T})\rightarrow Z_{+}$ defines a formal power series

[TABLE]

In the sequel, we denote the length of a word $P\subset W(V_{N}\cup V_{T})$ by $|P|$. The cardinality of an alphabet $V$ is denoted by $|V|$. If a context free grammar is unambiguous, then the number of words of length $N$ over an alphabet $V_{T}$ does not exceed $|V_{T}|^{N}$. We shall consider grammars generating formal power series with coefficients satisfying the following growth condition

[TABLE]

where $C$ and $q$ are positive constants. We shall refer to such grammars as grammars of the first class. All other grammars we call grammars of the second class. Notice that in the case of grammars of the first class, we have $\phi(\varepsilon)\leq C$.

Let $\mu$ be a map from $V_{T}$ to the set $R^{{\cal N}\times{\cal N}}$ of ${\cal N}\times{\cal N}$-matrices with real components. By $P(\mu)$ we will denote the matrix obtained substituting the letters $a_{i}$ in $P$ by the matrices $\mu_{i}$ and calculating the respective matrix product, i.e., if $P=a_{i_{1}},\ldots,a_{i_{n}}$, then $P(\mu)=\mu_{i_{1}}\ldots\mu_{i_{n}}$. If the series

[TABLE]

converges, its sum is an ${\cal N}\times{\cal N}$-matrix. Observe that the matrix series (5) corresponding to a grammar of the first class converges, if $|\mu_{i}|\leq M<1/(q|V_{T}|)$. Indeed, we have

[TABLE]

whenever $M$ is sufficiently small. On the other hand, the series (5) corresponding to a grammar of the second class diverges for any scalar substitution $\mu_{i}=\mu>0$. Indeed, if $\phi(\varepsilon)=\infty$, then we have $S(\mu)\geq\phi(\varepsilon)=\infty$. Suppose that $\phi(\varepsilon)\neq\infty$. Let $C=1$, $q=k/\mu$, $k>1$, and $\mu>0$. There exists a word $P_{k}\in W(V_{T})$ such that

[TABLE]

Hence we get

[TABLE]

Since $k$ is arbitrary, we obtain $S(\mu)=\infty$. Thus, scalar substitutions allows us to distinguish between grammars of the first and the second classes. We shall show that matrix substitutions permit to distinguish between two different context free grammars of the first class. The distinguishability via matrix substitutions of grammars of the second class is not considered here.

The following distinguishability theorems form a theoretical basis for assessment algorithms.

2.1 General distinguishability theorems

Theorem 1** (distinguishability I)**

Let $S_{1}$ and $S_{2}$ be two different formal power series corresponding to context free grammars of the first class. Then there exist a positive integer $\cal N$ and a matrix substitution $\mu:V_{T}\rightarrow R^{{\cal N}\times{\cal N}}$ such that $S_{1}(\mu)\neq S_{2}(\mu)$.

To prove the theorem we need an auxiliary lemma.

Lemma 1

Let $U$ and $V$ be two different finite formal series composed of words of length $N$. Then there exist a positive integer $\cal N$ and a matrix substitution $\mu:V_{T}\rightarrow R^{{\cal N}\times{\cal N}}$ such that $U(\mu)\neq V(\mu)$.

Proof. Let $a_{j_{1}}\ldots a_{j_{k}}a_{i_{1}}\ldots a_{i_{l}}\in U\cup V$, where $l\leq N$ and $k+l=N$. We say that $a_{i_{1}}\ldots a_{i_{l}}$ is a suffix. Denote the set of suffixes of $U\cup V$ by $\mathcal{S}$. By $\cal N$ we denote the cardinality of $\cal S$ plus one. Let us consider the set of unit orthogonal vectors $\{e_{0}\}\cup\{e_{i_{1}\ldots i_{l}}\mid a_{i_{1}}\ldots a_{i_{l}}\in\mathcal{S}\}$ in an $\cal N$-dimensional space. We define the linear operators $\mu_{i}$, $i=\overline{1,I}$ by:

[TABLE]

[TABLE]

Let $a_{i_{1}}\ldots a_{i_{N}}\in U\cup V$ be a word. The corresponding linear operator has the form $\mu_{i_{1}}\ldots\mu_{i_{N}}\in R^{{\cal N}\times{\cal N}}$. Obviously we have $\mu_{i_{1}}\ldots\mu_{i_{N}}e_{0}=e_{i_{1}\ldots i_{N}}$. Hence we get

[TABLE]

Indeed, the sets of orthogonal vectors in the two sums are different. $\Box$

Consider a simple example that may help the reader to better understand the construction used in the proof of Lemma 1. Let $U=a_{1}a_{2}$ and $V=a_{2}a_{1}$. In this case ${\cal N}=5$. Set

[TABLE]

and put

[TABLE]

Then we have

[TABLE]

Obviously, in this example it is possible to distinguish between $U$ and $V$ with the help of $2\times 2$-matrices. Lemma 1 establishes only the possibility of distinguishability without calculating the minimal needed dimension $\cal N$.

Proof of Theorem 1. Since the series are different, they admit the following representation: $S_{1}=S_{0}+U+R_{1}$ and $S_{2}=S_{0}+V+R_{2},$ where $S_{0}$ is the part of coinciding words of length less than or equal to $N$, $U$ and $V$ are different parts composed of words with length equal to $N$, and $R_{1}$ and $R_{2}$ contain terms with words of length greater than $N$. By Lemma 1 there exist a positive integer $\cal N$ and a matrix substitution $\mu:V_{T}\rightarrow R^{{\cal N}\times{\cal N}}$ such that $U(\mu)\neq V(\mu)$. Let $t>0$. Then we have

[TABLE]

The norms of the matrices $\mu_{i}$, $i=\overline{1,I}$, constructed in Lemma 1, do not exceed some $\sigma>0$. Since the grammars are of the first class, we obtain

[TABLE]

and

[TABLE]

Let $C=\max\{C_{1},C_{2}\}$, $q=\max\{q_{1},q_{2}\}$ and $t<1/(q\sigma)$. Then we get

[TABLE]

From this and (6) we see that $\Delta(t)\neq 0$ whenever $t>0$ is sufficiently small. $\Box$

In many situations it suffices to consider matrix substitutions $\mu:V_{T}\rightarrow R^{2\times 2}$. We associate with the symbols $a_{i}\in V_{T}$, $i=\overline{1,I}$, pairs of variables $u_{i}$ and $v_{i}$, $i=\overline{1,I}$. Let

[TABLE]

be two sets of words. Consider two sets of associated polynomials

[TABLE]

(Here $u_{k^{\prime m}_{N+1}}=u_{k^{\prime\prime m}_{N+1}}=1$.) We say that $U$ and $V$ satisfy condition $({\cal P})$ if ${\cal P}_{U}\neq{\cal P}_{V}$.

Lemma 2

Assume that $U$ and $V$ satisfy condition $({\cal P})$, then there exists a matrix substitution $\mu:V_{T}\rightarrow R^{2\times 2}$ such that $U(\mu)\neq V(\mu)$.

Proof. Consider the matrices

[TABLE]

By induction we easily obtain

[TABLE]

From this representation we see that ${\cal P}_{U}\neq{\cal P}_{V}$ implies $U(\mu)\neq V(\mu)$. $\Box$

Theorem 2** (distinguishability II)**

Let $S_{1}$ and $S_{2}$ be formal power series corresponding to context free grammars of the first class. Assume that the series admit the following representation: $S_{1}=S_{0}+U+R_{1}$ and $S_{2}=S_{0}+V+R_{2},$ where $S_{0}$ is the part of coinciding words of length less than or equal to $N$, $U$ and $V$ are different parts composed of words with length equal to $N$, and $R_{1}$ and $R_{2}$ contain terms with words of length greater than $N$. If $U$ and $V$ satisfy condition $\cal P$, then there exists a matrix substitution $\mu:V_{T}\rightarrow R^{2\times 2}$ such that $S_{1}(\mu)\neq S_{2}(\mu)$.

Proof. Using Lemma 2 and following the proof of Theorem 1 we obtain the result. $\Box$

Examples:

Let $U=\{aab,bab\}$ and $V=\{aba,bba\}$. In this case condition $\cal P$ is satisfied since $u_{1}u_{1}v_{2}\in{\cal P}_{U}$ and $u_{1}u_{1}v_{2}\not\in{\cal P}_{V}$. On the other hand, there exist languages/grammars that cannot be distinguished with the help of $2\times 2$-matrices. For example, using Maxima computer algebra system it is easy to show that for any choice of $2\times 2$-matrices $\mu_{1}$ and $\mu_{2}$ we have

[TABLE]

Therefore the languages

[TABLE]

cannot be distinguished using $2\times 2$-matrices. However substituting $3\times 3$-matrices it is easy show that the languages (7) are different.

2.2 Comparison of short words

In many situations the difference between two grammars can be detected comparing short words trough substitution of $2\times 2$-matrices. We fulfilled about $47\cdot 10^{6}$ tests to analyze all finite languages over terminal alphabet $\{a,b,c\}$ containing no more than three words of length less than or equal to five. We found that only the following languages cannot be distinguished using $2\times 2$-matrix substitutions. Namely, the pair of the languages

[TABLE]

and other pairs obtained as the result of permutation of the letters $\{a,b,c\}$ and/or substitution of $c$ by $a$ or $b$. We shall denote this set of pairs of languages by $L$

This proves the following theorem.

Theorem 3** (distinguishability III)**

Let $S_{1}$ and $S_{2}$ be formal power series corresponding to context free grammars of the first class over the terminal alphabet $\{a,b,c\}$ and let $N\leq 5$. Assume that the series admit the following representation: $S_{1}=S_{0}+U+R_{1}$ and $S_{2}=S_{0}+V+R_{2},$ where $S_{0}$ is the part of coinciding words of length less than or equal to $N$, $U$ and $V$ are different parts composed of no more than three words with length equal to $N$, and $R_{1}$ and $R_{2}$ contain terms with words of length greater than $N$. If the pair $U$ and $V$ does not coincide with one of the pairs from $L$, then there exists a matrix substitution $\mu:V_{T}\rightarrow R^{2\times 2}$ such that $S_{1}(\mu)\neq S_{2}(\mu)$.

Proof. Following the proof of Theorem 1 we obtain the result. $\Box$

Note that the pair of languages (8) can be used to construct examples of infinite languages that cannot be distinguished using $2\times 2$-matrices. One of such examples is the pair

[TABLE]

and

[TABLE]

Note also that in all the grammars of several programming languages and of educational exercises that we analyzed, the distinguishability was always possible using $(2\times 2)$-matrices.

3 Systems of nonlinear matrix equations

It is well-known [11, 12] that to any context free grammar there corresponds a system of nonlinear equations that allows one to obtain the respective formal power series via successive iterations. The terms of the series are the words of the respective language. This correspondence between series and systems of nonlinear equations makes it possible to effectively compute the sums of the series for any ${\cal N}\times{\cal N}$-matrix substitution. Let $X_{i}$, $i=\overline{0,m}$, be the nonterminals of a context free grammar and let $P^{i}_{j}$, $i=\overline{0,m}$, $j=\overline{1,l_{i}}$, be the words that appear in the right-hand sides of productions with the left-hand sides $X_{i}$. The system of equations corresponding to the grammar has the form

[TABLE]

Substituting the symbols of the terminal alphabet $a_{k}\in P^{i}_{j}$ by matrices $\mu_{k}$, we obtain a system of nonlinear matrix equations with unknowns $X_{i}$. This system, $X=F(X)$, can be solved using the iterative process $X^{k+1}=F(X^{k})$, $X^{0}=0$, or using the Newton method. As we shall see in the sequel the convergence of the method of successive iterations can be guaranteed for a large class of grammars, for example for the grammars in Chomsky and Greibach normal forms. Note that for regular grammars system (9) is linear.

3.1 Convergence of successive iterations

Let us consider a context free grammar with productions

[TABLE]

where $\mathcal{P}_{j}^{0,i}\in W(V_{N}\cup V_{T})\setminus W(V_{T})$ and $p_{j}^{i}\in W(V_{T})$, $p_{j}^{i}\neq\varepsilon$. We assume that the words ${\cal P}_{j}^{i}$ contain more than one symbol. (For example, any grammar without renaming and $\varepsilon$-rules is of this type.) The structure of these words can be described in the following manner: ${\cal P}_{j}^{l,i}=q^{l+1,i}_{j}X_{k^{l+1,i}_{j}}{\cal P}_{j}^{l+1,i},\;\;\;l=\overline{1,L_{j}^{i}},$ where ${\cal P}_{j}^{l,i}\in W(V_{N}\cup V_{T})\cup\{\varepsilon\}$ and $q_{j}^{l,i}\in W(V_{T})\cup\{\varepsilon\}$. The corresponding system of equations has the following structure:

[TABLE]

To simplify the notations we denote by $P$ the matrix $P(\mu)$ obtained substituting the symbols $a_{i}$ and $X_{i}\in P$ by the matrices $\mu_{i}$ and $\xi_{i}$, respectively. We use the notation $\tilde{P}$ when the symbols $X_{i}\in P$ are substituted by the matrices $\tilde{\xi}_{i}$. Let $X=(X_{1},\ldots,X_{n})$ and $\tilde{X}=(\tilde{X}_{1},\ldots,\tilde{X}_{n})$ be two collections of $n$ matrices ${\cal N}\times{\cal N}$. Then, using our notations, we have

[TABLE]

[TABLE]

[TABLE]

where $Y_{j}^{l,i},Z_{j}^{l,i}\in W(V_{N}\cup V_{T})$ and $n_{j}^{i}$ is the number of nonterminals in the word ${\cal P}_{j}^{0,i}$. Assume that the norms of all matrices $a_{i}$, $X_{i}$, and $\tilde{X}_{i}$ do not exceed $\delta>0$ and that $\bar{n}\delta<1$, where $\bar{n}=\sum_{i,j}n_{j}^{i}$. Then from the representation (11) we obtain

[TABLE]

Set $\tilde{X}=0$. Then if $\|X_{i}\|<\delta$, we get

[TABLE]

Let us consider a closed ball ${\cal B}=\{X\mid\max_{j=\overline{1,n}}\|X_{j}\|\leq\delta\}$ in the space of matrices $X=(X_{1},\ldots,X_{n})$. We have proved that $F({\cal B})\subset{\cal B}$ and that $F$ is a contracting map. Hence there exists a unique fixed point $\hat{X}=F(\hat{X})\in{\cal B}$. This fixed point is the limit of the sequence of iterations

[TABLE]

Thus we have the following result.

Theorem 4** (Convergence)**

Assume that the system of nonlinear equations corresponding to a context free grammar has form (10) and that the words with nonterminal symbols, ${\cal P}_{j}^{i}$, contain more than one symbol. Then substituting the terminal symbols by matrices with a sufficiently small norm (less than $\delta$), we can guarantee the convergence of the sequence (13) to a unique solution of the matrix system (10).

Example:

Let us consider the grammar

[TABLE]

The respective system of equations is

[TABLE]

The conditions of Theorem 4 are satisfied (the words $SaA$ and $cSd$ contain more than one symbol). Therefore the system can be solved using the iteration method whenever the symbols $a$, $b$, $c$, and $d$ are replaced by matrices with a sufficiently small norm.

3.2 Other cases where the iteration method can be used

In many cases the grammar may have productions of the form $A\rightarrow B$. The iteration method can be applied also to the corresponding system of equations after some transformation. Namely, assume that the system has the form

[TABLE]

where $F$ has form (10) considered above and $\Lambda:R^{({\cal N}\times{\cal N})n}\rightarrow R^{({\cal N}\times{\cal N})n}$ is a linear operator such that there exists the inverse $(I-\Lambda)^{-1}$. Then system (16) is equivalent with the system $X=(I-\Lambda)^{-1}F(X).$ Obviously the map $X\rightarrow(I-\Lambda)^{-1}F(X)$ is contracting and transforms ${\cal B}=\{X\mid\max_{j=\overline{1,n}}\|X_{j}\|\leq\delta\}$ into $\cal B$, whenever the terminal symbols are replaced by matrices with a sufficiently small norms.

Example:

Let us consider the grammar

[TABLE]

which is a correct solution of an exercise. (This example is taken from [1].) The corresponding system of equations reads:

[TABLE]

Below we present three other possible answers.

Alternative correct solution

The following grammar is different but generates the same language:

[TABLE]

The corresponding system of equations is

[TABLE]

Wrong answer

The following grammar does not generate the same language (does not generate the word cbabd):

[TABLE]

The corresponding system is

[TABLE]

Ambiguous grammar

The following grammar generates the same language but is ambiguous (the word $baba$ can be generated by different ways):

[TABLE]

The corresponding system has the form

[TABLE]

System (18) corresponding to grammar (17) is equivalent to the system

[TABLE]

Replacing the symbols $a$, $b$, $c$, and $d$ by $(2\times 2)$-matrices with sufficiently small norm we get a system that can be solved using the iteration method. In the same way we can transform and solve other systems. Starting the iterative process with $S=A=0$ and solving systems (18), (20), (22), and (24) we see that the difference between $S$ components of solution of systems (18) and (20) is zero, while for the pair (18) and (22) or (18) and (24) the difference is not zero. This allows one to clearly distinguish between right and wrong answers.

Note that the interval where the components of the matrices are generated must be (a) sufficiently small in order to guarantee the convergence of the iterative process, (b) big enough to distinguish between two different languages.

Another important case deals with the grammars having productions of the form $A\rightarrow\varepsilon$. In this case some equations have the form

[TABLE]

Introducing new variables $Y_{i}=X_{i}-I$ in many situations it is possible to transform the system to a system satisfying conditions of Theorem 4.

Example:

Let us consider the grammar

[TABLE]

The corresponding system of nonlinear equations is

[TABLE]

Introducing new variable $B=A-I$ we obtain the system

[TABLE]

which can be solved by iteration method.

4 Comparison with a simple heuristic method

One natural heuristic method to compare two context free grammars is:

• •

Explicitly check with a parsing algorithm (for example, the classic CYK algorithm [8] or its modification [6]) all words up to length $n$;

• •

If difference is found, output ”grammars are different”;

• •

If no difference is found, output ”grammars could be the same”.

Such an approach does not allow us to distinguish between two grammars with different ambiguities and cannot be applied to grammars with distinguishing words of big length.

For example, let us consider two grammars:

[TABLE]

and

[TABLE]

The productions $S\textrightarrow\varepsilon\mid I\;I\;S$ and $I\textrightarrow a\mid b\mid c\mid d$ generate all the words of even length containing $a$, $b$, $c$ and $d$, while $A$ generates the word $a^{15}$ and $B$ generates the word $b^{15}$.

Then distinguishing words exist, but have length 15 or more. This means that any generation and testing based algorithm must test about $4^{15}\approx 10^{9}$ words. This is would be a very time-consuming procedure.

The algorithm presented here can easily distinguish between these to grammars within a fraction of a second.

5 Limitations of iteration method

In some cases the iteration method cannot distinguish between two different grammars. This happens when we consider grammars with very long words from $W(V_{T})$. The point is that the computer precision is not sufficient to correctly compute products of many small numbers. Consider the following grammar

[TABLE]

The corresponding system of matrix equations reads

[TABLE]

Let the second grammar be

[TABLE]

with the corresponding system of equations

[TABLE]

Systems (26) and (28) are equivalent to the equations

[TABLE]

and

[TABLE]

respectively. To guarantee the convergence of iterations we have to impose the restriction $\|a_{1}\|+\|a_{2}\|+\ldots+\|a_{n}\|=\alpha<1.$ From the inequality of arithmetic and geometric means we get

[TABLE]

Hence for sufficiently large $n$ we have $\|a_{1}\|\ldots\|a_{n}\|\leq\alpha^{n}/n^{n}<\delta,$ where $\delta>0$ is the computer precision. Thus the computer interprets the iterative processes

[TABLE]

and

[TABLE]

as the same process

[TABLE]

and, therefore, the method does not allow to distinguish between two grammars.

However, the method can be applied to real size grammars, for example, simplified C with 44 NT symbols and 104 production rules, with a satisfactory result. The comparison of such grammars takes about 81ms of CPU-time111On a 5 years old i5 Linux machine with 4 Gbytes RAM..

6 Conclusion

In this paper we addressed the problem of context free grammars comparison from the mathematical analysis point of view. A substitution of terminal letters by matrices allows us to reduce the comparison problem to numerical solution of systems of nonlinear matrix equations. Besides the elegance of the process, this method constitutes a solid base for construction of algorithms and tools for comparison of context free grammars. Our experiments with a built prototype show that the use of this comparison method with $(2\times 2)$ and $(3\times 3)$-matrices in problems appearing in e-learning framework and even in cases of large grammars is very efficient.