Chapter 4: Context-Free Grammars

Transcription

1 Chapter 4: Context-Free Grammars 4.1 Basics of Context-Free Grammars Definition A context-free grammars, or CFG, G is specified by a quadruple (N, Σ, P, S), where N is the nonterminal or variable alphabet; Σ is the terminal alphabet and N and Σ are disjoint; P N (N Σ) is a finite set of productions or rules; and S in N is the sentence symbol. 1

2 A production is usually written as A α where A is in N and α is in (N Σ). It is said to be an A-production of production for A, and α is the right-hand side of the production. It is convenient to speck of the vocabulary of G which is denoted by V and defined as N Σ. In examples we typically use the BNF symbol to group together all productions for a single nonterminal. 2

3 Notational Conventions Notterminals are represented by uppercase letters A, B, C,..., S, except when stated otherwise. In particular, S denotes the sentence symbol. Terminals are represented by early lowercase letters a, b,.... Late uppercase letters,..., X, Y, Z are used to mean either a nonterminal or a terminal. Terminal words are represented by late lowercase letters..., x, y, z, while lowercase Greek4 letters are used to represent arbitrary words of terminals and nonterminals. In all cases these symbols can appear with subscripts and/or superscripts. 3

4 A CFG is used to generate a language by rewriting, which we now define. Definition Let G = (N, Σ, P, S) be a CFG and β a word in V. If β can be decomposed into β 1 Aβ 2 for some A in N and for some β 1 and β 2 in V, and there is an A-production A α in P, then β can be rewritten as β 1 αβ 2 or β 1 αβ 2 is said to be derived from β. We have replaced A by α or substituted α for A. In this case we write or simply β β 1 αβ 2 in G β β 1 αβ 2 if G is understood. 4

5 Observe that for β to be rewritten it must contain at least one nonterminal, because only nonterminal have productions associated with them. However, there may be many nonterminals in β and/or many appearances of one nonterminal. In this case β can be rewritten in many different ways. Similarly, when a specific appearance of a nonterminal in β is chosen to be rewritten there may be more than one production available for it and, hence, once more, more than one way of rewriting it, In other words, rewriting is a nondeterministic action. The notion of rewriting is comparable to that of a move in an FA, and in the same way that we obtained a configuration or move sequence for FAs we obtain a rewriting or derivation sequence. 5

6 Definition Let G = (N, Σ, P, S) be a CFG and α be a word in V. If there are words α 0,..., α n 1, β 0,..., β n 1, β in V and nonterminals A 0,..., A n 1, for some n 1, such that α 0 A 0 β 0 α 1 A 1 β 1... α n 1 A n 1 β n 1 β where α = α 0 A 0 β 0, then we write α n β We say that β can be derived from α in n rewriting steps, that is, α n is an n-step derivation or, more simply, a deriavtion. Also, the sequence α 0 A 0 β 0,..., α n 1 A n 1 β n 1, β is a derivation sequence, it details how the derivation occurs, the productions that are used, and to which appearances of symbols they are applied. We write α + β if there is an n 1 and an n-step derivatiopn starting from α and leading to β. Also, we write α β if either α = β or α + β, and in the former case we may also write α 0 β, the 0-step derivation. 6

7 Definition Let G = (N, Σ, P, S) be a CFG. A word α in V is a sentential form if there exists a derivation S α. A derivation S α is called a sentential derivation. A sentential form α is terminal if α is in Σ. A terminal sentential form is also called a sentence. The language of G, denoted by L(G), is the set of all sentences, namely, L(G) = {x : S + x and x is in Σ }. A language L Σ is said to be a context-free language, or CFL, if there is a CFG, G, with L = L(G). The family of CFLs, denoted by L CF, is defined by L CF = {L : L is a CFL}. Finally, we say that two CFGs, G 1 and G 2, are equivalent if L(G 1 ) = L(G 2 ), that is, they secify the same language. 7

8 Can we really be sure we have explored all possible derivation sequences, when attempting to generate a proposed sentence. We can avoid this problem by, first, using the syntax or derivation tree and, second, using cannonical derivations. 8

9 Definition Let G = (N, Σ, P, S) be a CFG and let T be an ordered, directed tree whose nodes are labeled with either symbols from V or the empty word. Then T is a syntax tree (with respect to G) if every node u in T satisfies the following condition: If u is an external node, then it is labeled with a nonterminal symbol, a terminal symbol, or with λ, denoting the empty word, and in this latter case it is the only child of its parent, otherwise u is an internal node labeled with a nonterminal symbol A, and if it has k children, k 1, labeled with X 1,..., X k, from left to right, then the production A X 1... X k is in P. 9

10 The frontier of a syntax tree read in left to right order gives a word, the yield of the tree, denoted by y(t ). Lemma Let G(N, Σ, P, S) be a CFG and T be a syntex tree, with respect to G, having A in N as its root label and α = y(t ). Then A α in G with a derivation sequence that uses the same productions applied to the same appearances of the same symbols that appear in T. Lemma demonstrates that to every syntex tree there corresponds a derivation sequence. In other words, derivation sequences giv rise to all syntex trees; none are omitted. 10

11 Given a syntax tree any traversal of it produces a derivation sequence. However, there are two specific traversals that are of particular interest: preorder and reverse preorder traversals. These correspond, as we shall see, to the following canonical derivation sequences. Definition Let G(N, Σ, P, S) be a CFG. Let β and γ be in V such that β = β 1 Aβ 2, γ = β 1 αβ 2 and A α is in P, that is, β 1 Aβ 2 β 1 αβ 2 in G. If β 1 is in Σ we write β L γ; β is rewritten in the leftmost position to give γ. Similarly, if β 2 is in Σ we write β R γ; β is rewritten in the rightmost position to give γ. 11

12 We extend L and R to L n and R n L, + and R +, and L and R, respectively. We speck of leftmost and rightmost derivations, respectively. We will to prove that whenever there is a sentential derivation of some terminal word in a CFG, then there is always a leftmost and rightmost sentential derivation of the same terminal word. 12

13 Definition Let G(N, Σ, P, S) be a CFG and let β and γ be in V such that D : β = β 0... β n = γ is a derivation in G for some n 0. A nonterminal in γ is clearly not rewritten in D we say it is passive. A nonterminal which is rewritten in D is said to be active. To generalize the notion of a leftmost derivation we require that the derivation is leftmost with respect to the active nonterminals. 13

14 We say that D is a pre-leftmost derivation if at each derivation step the leftmost active nonterminal is rewritten. In this case we write β l n γ, β l + γ, or β l γ. Pre-rightmost derivations are defined similarly, giving rise to the notation β r n γ, β r + γ, or β r γ. A pre-leftmost derivation is a leftmost derivation if the derived word is terminal, since in this case all nonterminals are written, that is, they are all active. Lemma Let G(N, Σ, P, S) be a CFG and T be a syntex tree, with respect to G, having A in N as its root label and α = y(t ). Then A l α in G (A r α in G) with a pre-leftmost (pre-rightmost) derivation that uses the same productions applied to the same appearances of the same symbols that appear in T. 14

15 Theorem Let G(N, Σ, P, S) be a CFG. Then for all x in Σ, the following three conditions are equivalent 1. S + x in G; 2. S L + x in G; 3. S R + x in G. This theorem means that we need consider either only leftmost derivations or only rightmost derivations when generating sentences. 15

16 Definition Let G = (N, Σ, P, S) be a CFG. Let x be a sentence with two distinct syntax trees (leftmost derivations or rightmost derivations). Then x is said to be ambiguous with respect to G. G is ambiguous if there is at least one ambiguous sentence derived from S. We say a CFL, L, is (inherently) ambiguous if for all CFGs, G, with L(G) = L, G is ambiguous. Ambiguity is important, in practice, because it indicates that a sentence may have more than one meaning. In natural languages this causes difficulty in communication, and in the legal arena causes lawsuits. In programming languages ambiguity indicates the possibility that a program which is compiled with different compilers can then be excuted to give different results with the same data. 16

17 CFG AMBIGUITY INSTANCE: A CFG, G = (N, Σ, P, S). QUESTION: Is G ambiguous? UNAMBIGUOUS CFG CONSTRUCTION INSTANCE: An ambiguous CFG. G = (N, Σ, P, S). QUESTION: Can an equivalent unambiguous CFG be constructed? CFL AMBIGUITY INSTANCE: An ambigous CFG, G = (N, Σ, P, S). QUESTION: Does there exist an equivalent unambiguous CFG? 17

18 CFG MEMBERSHIP INSTANCE: A CFG, G = (N, Σ, P, S), and a word x in Σ. QUESTION: Is x in L(G)? CFG PARSING INSTANCE: A CFG, G = (N, Σ, P, S), and a word x in Σ. QUESTION: What are the syntax trees of x respect to G? 18

19 4.2 Simplications Redundant Symbols Symbols in a CFG can be redundant because either they do not appear in any sentential form or, even if they do appear in a sentential form, they are unable to derive any terminal words. In the former case such symbols are said to be unreachable and in the latter they are said to be nonterminating. Correspondingly, a symbol that appears in a sentential form is said to be reachable, and one that derives a terminal word is said to be terminating. 19

20 A CFG with neither nonterminating nor unreachable symbols is said to be reduced. We will give two algorithms, one to detect terminating symbols and the second to detect reachable symbols. These form the basis for eliminating redundant symbols from a CFG. The algorithm Terminating Symbols is based on the following idea If a nonterminal A has a production consisting solely of terminating symbols, then A is itself terminating. 20

21 In particular, if G generates a nonempty language, then there must be at least one nonterminal having a production with a terminal right-hand side. Terminating Symbols identifies these nonterminals, treats them as new terminals, and it then repeats the process until no new terminating symbols are found. marking is used to identify the new terminating symbols. 21

22 Algorithm Terminating Symbols. On entry: An arbitrary CFG, G = (N, Σ, P, S). On exit: The set T of terminating symbols. begin {A marking algorithm is used} T := Σ; Mark, in some way, all appearances of terminal symbols in P ; repeat Let Q be the set of unmarked nonterminals which have at least one production whose right-hand side consists solely of marked symbols. (Note that the right-hand side of a λ-production right-hand side of a λ-production consists solely of marked symbols); Mark all appearances in P of the symbols in Q; T := T Q until Q = ; end of Algorithm. 22

23 Algorithm Reachable Symbols. On entry: An arbitrary CFG, G = (N, Σ, P, S). On exit: The set R of reachable symbols. begin {A marking algorithm is used} R := {S}; Q := {S}; {The set of newly marked symbols} repeat Mark all symbols on the right-hand sides of all productions which have left-hand sides in Q; Let Q be the set of newly marked symbols, which are not already in R; R := R Q until Q = ; end of Algorithm. 23

24 Theorem Let G = (N, Σ, P, S) be an arbitrary CFG. Then there is an algorithm which constructs an equivalent CFG, G = (N, Σ, P, S), from G, where N N, P P, and N Σ consists solely of terminating symbols, with the possible exception of S. Theorem Let G = (N, Σ, P, S) be an arbitrary CFG. Then there is an algorithm which constructs an equivalent CFG, G = (N, Σ, P, S), from G, where N N, P P, and N Σ consists solely of reachable symbols. Theorem Let G = (N, Σ, P, S) be an arbitrary CFG. Then there is an algorithm which constructs an equivalent, reduced CFG, G = (N, Σ, P, S), where N N, Σ Σ, and P P. 24

25 4.2.2 Empty Productions An empty production (λ-production or null production) is a production whose right-hand side is the empty word, that is, a production of the form A λ Given a CFG, G = (N, Σ, P, S), if λ is in L(G), then empty productions are necessary; otherwise λ cannot be generated at all. However, when λ is not in L(G) any empty productions in G can be eliminated in such a way that L(G) is unaffected. The elimination procedure only affects productions which contain, on right-hand side, nonterminals that can derive the empty word. We call such nonterminals λ-nonterminals. 25

26 A nonterminal A is said a λ-nonterminal if there is a derivation A + λ in G. Note that λ is in L(G) if and only if S is a λ-nonterminal. If G has no λ-nonterminal it is said to be λ-free. 26

27 If G has λ-productions, then in a sentential derivation S α 1 α 2... α n in Σ The lengths of α i may vary tremendously with respect to one another. This is because many nonterminals may be introduced which only contribute the empty word to α n. However, if G has no λ-productions, then we know that S = 1 α 1 α 2... α n The lengths are monotonically increasing. This property is useful when testing whether or not a given words is generated by a CFG. If A + λ, then A α in G with α + λ. 27

28 Algorithm λ-nonterminals. On entry: An arbitrary CFG, G = (N, Σ, P, S). On exit: The set A of λ-nonterminals. begin {A marking algorithm is used} A := ; Mark the right-hand sides of all empty productions in P ; repeat Let Q be the set of unmarked nonterminals which have at least one production whose right-hand side consists solely of marked symbols. Mark all appearances in P of the symbols in Q; A := A Q until Q = ; end of Algorithm. 28

29 To eliminate λ-nonterminals from a CFG we use the following algorithm, which eliminates λ-nonterminals without introducing new ones. The strategy is based on the following idea Let A be a λ-nonterminal in G. Then conceptually we split it into two nonterminals A and A such that A generates all the nonempty terminal words that A does, and A only generates λ. Now in the right-hand side of a production in which A appears once, B αaβ say, we replace it with two productions B αa β and B αa β. Since A only generates λ we replace it by λ giving B αβ and use A in place of A. If B αaβaγ, then we obtain all four combinations of replacements, and so on. 29

30 Let G 1 and G 2 be two CFGs; we say they are λ-equivalent if they generate the same language when the empty word is ignored. In other words, G 1 and G 2 are λ-equivalent if L(G 1 ) {λ} = L(G 2 ) {λ}. Clearly, equivalence always implies λ-equivalence, but the converse implication is not valid. 30

31 Algorithm Eliminate λ-nonterminals. On entry: G = (N, Σ, P, S) is an arbitrary reduced CFG with at least one λ-nonterminal.. On exit: A reduced CFG, G = (N, Σ, P, S), λ-equivalent to G having no λ-nonterminals. begin For all λ-nonterminals A in N, For all productions B β in P with β A = m 1 Let β = β 0 Aβ 1... Aβ m where β 0... β m A = 0; Replace B β in P with 2 m productions {B β 0 X 1 β 1... X m β m : X i is in {λ, A}, 1 i m} Finally, remove all λ-productions from P giving P end of Algorithm. 31

32 Thoerem Let G = (N, Σ, P, S) be an arbitrary reduced CFG. Then there is an algorithm which constructs a reduced, λ-free CFG, G, λ-equivalent to G. In other words, for every CFL, L, there is a λ-free CFG, G with L(G) = L {λ}. 32

33 4.2.3 Unit Productions A production A α in a CFG, G = (N, Σ, P, S), is said to be a unit production if α is in N. A special case of a unit production is the production A A, a circular production. Such a production is similar to the transition (q, λ, q) in a λ-nfa, in that it can be removed immediately without affecting the language of the grammar. A CFG with no unit productions is said to be unit-free. 33

34 Algorithm Eliminate One Unit Production. On entry: G = (N, Σ, P, S) is an arbitrary reduced CFG with no circular productions and at least one unit production. On exit: G is a reduced equivalent CFG with one fewer unit production than G. begin Choose an arbitrary unit production B C in G; there must be at least one; Replace B C in P with B α, for all α in V N, satisfy C C 1... C r α, for some nonterminals C 1,..., C r. end of Algorithm. Theorem Let G = (N, Σ, P, S) be an arbitrary reduced CFG. Then there is an algorithm which constructs an equivalent, unit-free, reduced CFG, G, from G. 34

35 4.2.4 Binary Form and Chomsky Normal Form A CFG, G = (N, Σ, P, S), is said to be in binary form if for all A α in P, α 2. It is in Chomsky normal form, of CNF, if it is in binary form and there are only two kinds of productions A a; a in Σ, and A BC; B, C in N. 35

36 We prove that for every CFG, G, a λ-equivalent CFG, G, in CNF can be effectively constructed. We do this by first obtaining a binary form equivalent CFG. Letting G = (N, Σ, P, S) be a CFG we define Maxrhs(G) = max{ α : A α is in P }. Clearly, G is in binary form if and only if Maxrhs(G) 2. If Maxrhs(G) > 2, each right-hand side α of length Maxrhs(G) is split into two halves. The second half is replaced by a new nonterminal giving a shorter right-hand side. 36

37 Algorithm Reduce Largest Productions. On entry: An arbitrary reduced CFG, G = (N, Σ, P, S) with Maxrhs(G) 3. On exit: An equivalent reduced CFG, G, with Maxrhs(G ) < Maxrhs(G). begin For all productions A α in P with α = Maxrhs(G) Letting α = α 1 α 2, where α 1 = α /2 1 and α 2 = α /2 2, replace A α in P by A α 1 [Aα] and[aα] α 2, where [Aα] is a new unique nonterminal. {Then 2 α 1 [Aα] ( α + 1)/2 < α } end of Algorithm. 37

38 Theorem Let G = (N, Σ, P, S) be an arbitrary reduced CFG. Then there is an algorithm that constructs an equivalent reduced CFG, G, in binary form from G. Theorem Chomsky Normal Form Theorem Let G = (N, Σ, P, S) be an arbitrary reduced CFG. Then there is an algorithm that constructs a λ-equivalent reduced CFG, G, in CNF form from G. Thus, for every CFL, L, there exists a CFG, G, in CNF with L(G) = L {λ}. 38

39 4.2.5 Greibach Normal Form and Two-Standard Normal Form Let G = (N, Σ, P, S) be a CFG. Then G is said to be in Greibach normal form, or GNF, if each production A α in P satisfies α is in ΣN, that is, its right-hand side begins with a terminal and is followed by a, possibly empty, sequence of nonterminals. 39

40 For m 1, G is said to be in m-standard normal form, or m-snf, if each production A α in P satisfies α is in m i=0 ΣN i, that is, it is in GNF, but there are never more than m nonterminals on the right-hand sides of productions. We are particularly interested in 2-standard normal form. To obtain a Greibach normal form grammar from a given CFG we first remove the left-recursion. 40

41 Let G = (N, Σ, P, S) be a CFG. We say that G is left-recursive if there is a nonterminal A in G and a derivation A L + Aα, for some α in V. We also say that such a nonterminal is left-recursive. If there is a derivation A R + αa we say A and G are right-recursive. If A Aα is in P we say A is directly left-recursive and if A αa is in P we say A is directly right-recursive. Finally, G is non-left- (non-right-) recursive if it is not left- (right-) recursive. 41

42 Theorem Let G = (N, Σ, P, S) be a reduced, λ-free, unit-free CFG. Then there is an algorithm which constructs an equivalent non-left-recursive CFG, G, from G. In other words, for every CFL, L, there exists a non-left-recursive CFG, G, with L(G) = L {λ}. Clearly, similar result holds for non-right-recursive CFGs too. 42

43 For the construction of CFGs in GNF there is a curcial observation, as follows. Observation Let G = (N, Σ, P, S) be a reduced, non-left-recursive CFG in CNF. Then there is a nonterminal A in N whose productions all begin with a terminal, that is A only has terminal left appearances. 43

44 Given a λ-free, non-left-recursive CFG we begin to obtain an equivalent GNF grammar by carrying out the following substitution rule For each nonterminal A having only terminal left appearances, replace each occurrence of A in the leftmost position of a production with its right-hand sides. So if A α 1 α 2 α 3 D and B AC, then B α 1 C α 2 C α 3 DC. After this has been carried out either all nonterminals have only terminal left appearances or there is some nonterminal B that has a production starting with a nonterminal. 44

45 Observation Let G = (N, Σ, P, S) be a reduced, non-left-recursive CFG. Then there is a nonterminal B with B-productions having left appearances of either terminals or nonterminals which have only terminal left appearances. Observation and are both special cases of a more general result, which we prefer to prove. Observation shows the existence of nonterminals which in one leftmost derivation step produce a terminal on the left, while Observation captures those that require two leftmost derivation steps. 45

46 Definition Let G = (N, Σ, P, S) be a non-left-recursive CFG. For all A in N, define int(a) by int(a) is the largest integer k such that A L k 1 Bα L aβ, for some a in Σ, B in N, and α, β in (N Σ). Let intmax(g) = max({int(a) : A is in N}). 46

47 Lemma Let G = (N, Σ, P, S) be a reduced, non-left-recursive CFG. Then 1. int is well-defined with 1 intmax #N; 2. for all i, 1 i intmax, there is an A with int(a) = i; and 3. if A Bα is in P then int(a) > int(b), that is int provides a partial ordering of N. Theorem Greibach Normal Form Theorem Given an arbitrary CFG, G = (N, Σ, P, S), there is an algorithm that constructs a λ-equivalent CFG, G, in CNF from it. In other words, for every CFL, L, there exists a CFG, G, in CNF with L(G) = L {λ}. 47

48 Algorithm GNF Conversion. On entry: A reduced, λ-free, non-left-recursive CFG, G = (N, Σ, P, S). On exit: A reduced, equivalent CFG, G = (N, Σ, P, S) in Greibach normal form. begin T := {A : A is in N and int(a) = 1}; P := {A α : A α is in P and int(a) = 1}; P := P P ; for i = 2 to intmax do begin for all A in T do replace all left appearances of A in P by the right-hand sides of all A-productions in P ; T := {A : A is in N and int(a) = i}; P := P {A α : A α is in P and int(a) = i}; P := P P end; end of Algorithm. 48

49 Lemma The Nonterminal Substitution Lemma Let G = (N, Σ, P, S) be an arbitrary CFG, A α 1 Bα 2 be a production in P, and B β 1... β r be the B-productions in P. Let G = (N, Σ, P, S) be a new CFG in which P = P except that A α 1 Bα 2 is replaced by A α 1 β 1 α 2... α 1 β r α 2. Then L(G ) = L(G). 49

50 It is possible to prove a stronger version of the GNF theorem, namely, it is sufficient to allow only productions of the following types 1. A α; 2. A αb; 3. A αbc; where B and C are not necessarily distinct from A or each other. In other words, 2-SNF is sufficient to capture all CFLs. 50

51 Theorem Greibach 2-Standard Normal Form Theorem Let G be an arbitrary reduced CFG. Then there is an algorithm that constructs a reduced λ-equivalent CFG, G, in 2-SNF from G. In other words, for every CFL, L, there exists a 2-SNF CFG, G, with L(G) = L {λ}. 51

52 4.3 Linear and Regular Grammars We show that 1-standard normal form grammars generate L DF A and, hence, L REG. Theorem For every reduced CFG, G, in 1-SNF there exists an NFA, M, with L(M) {λ} = L(G), and conversely. 1-standard normal form grammars are sometimes called regular grammars, or REGG. 52

53 Let G = (N, Σ, P, S) be a reduced CFG; then it is said to be linear, or a LING, if all productions in P are either of forma x or A xby, for some terminal words x and y. G is said to be right linear, or a RLING, if its productions are either of the form A x or A xb. Left linear CFGs, or LLINGs, are defined similarly. Clearly, a CFG in 1-SNF is a RLING and a LING. A CFL, L, is said to be a linear language or LINL if L = L(G), for some LING, G. 53

54 Linear grammars, on the other hand, are more powerful than FAs and less powerful than unrestricted CFGs. The family of linear languages is denoted by L LIN, and defined by L LIN = {L : L = L(G) for some LING, G}, has attained a central place in the theory of CFLs. 4.4 Extended Context-Free Grammars 54