Compiler theory (Tsinghua University Press) - grammar and language - the language of formal grammar and definitions

Rule (rewrite rule, production or formula)

The form α → β or α :: = β 's (α, β) ordered pair, wherein [alpha] is called the right part of the left portion of the rule, the rule known as beta], where the symbol → (:: =) reading " is defined as "A → a, for example, read as" A is defined as a "
Grammar G is defined as a four-tuple (V _N, V _T , P, S)
Wherein V _N non-terminal symbol set (or variable Syntax entity); V _T terminator set; P rule (α → β) is set, ae (V _N ∪V _T ) ^* , and comprising at least one nonterminal , β∈ (V _N∪ V _T ) ^* , V _N , V _T and P are nonempty finite set
S or identifier called the start symbol , which is a non-terminal, left to occur at least as a rule
V _N and V _T does not contain a common element, i.e., V _N ∩ V _T = Ø
Usually represented by V V _N ∪ V _T , V G, called grammar alphabet or vocabulary

= 2.1 Example There grammar G <V _N , V _T , P, S>, where, V _N = {S}, V _T = {0,1}, P = {S → 0S1, S → 01}, where the non- terminator concentrate containing only one element S, a termination symbol set composed of two elements 0,1, there are two production, is the start symbol S

This example can also be written as

G： S→0S1

　　S→01

G[S]：S→0S1

S→01

Example 2.2 have grammar = G (V _N , V _T , P, S), where V _N = {identifier, letters, numbers}, V _T = {A, B, C, ..., X, Y, Z , 0,1, ..., 9}

P = {<identifier> → <letter>

<Letter> → a

<Letter> → b

　　 ...

　　 <Letter> → z

　　 <Number> → 0

　　 <Number> → 1

　　 ...

　　 <Number> → 9

}

S = <identifier>

Language definition is generated by the grammar, but also need to introduce derived concept, definition of V ^* relationships between symbols, is derived directly => a length of n (n≥1) derivation and a length of n (n≥0) derivation

DERIVATION / direct reduction definition

Grammar is disposed α → β = G (V _N , V _T , P, S) of the rules (or a production in P), γ and δ is V ^* of any symbol
If symbol string v, [omega] meet, v = γαδ, ω = γβδ , said that v (apply rule α → β) produced directly [omega], or that [omega] is a direct derivation of v , or that [omega] v directly to the reduction , referred to for v => ω

For example, for Example 2.1 grammar G, can be given some examples

v = 0S1, ω = 0011, is derived directly: 0S1 => 0011, using the rules: S → 01, where γ = 0, δ = 1
v = S, ω = 0S1, directly derived: S => 0S1, using rules: S → 0S1, where [epsilon] = gamma], [delta] = [epsilon], [epsilon] group which is similar to the identity element
v = 0S1, ω = 00S11, deduced directly: 0S1 => 00S11, using rules, S → 0S1, where γ = 0, δ = 1

For Example 2.1 grammar G, the following example is derived directly

V = <identifier>, [omega] = <identifier> <letter> directly derived: <identifier> => <identifier> <letter>, using the rule: <identifier> → <identifier> <letter> , where γ = δ = ε
V = <identifier> <letter> <number>, [omega] = <letter> <letter> <number>, derived directly: <identifier> <letter> <number> => <letter> <letter> <number>, use rules: <identifier> → <letter>, where γ = ε, δ = <letters> <numeric>
v = abc <number>, ω = abc5, deduced directly: ABC <number> => abc5, using the rule: <number> → 5, where γ = abc, δ = ε

Defined sequence deduced

If the presence of the sequence derived directly: v = [omega] ₀ => [omega] _{. 1} => [omega] ₂ => ... => [omega] _n- = [omega] (n-> 0) called v derived (generated) [omega] (derivation of length n) , also known as ω reduces to v , denoted v ω
If V [omega], or v = ω, is referred to as V [omega] Example 2.1 of grammar, there is a direct sequence deduced v = S1 => 00S11 => 000S11 => 00001111 = ω, i.e. 0S1 00001111, also referred to as 0S1 00001111
2.2 Example of grammar, there is a direct sequence deduced V = <identifier> => <identifier> <number> => <letter> <number> => X <number> => x1 = ω, i.e. <identifier> x1

Sentence (deduced from the results), and sentence (sentence consisting only of terminal symbols) is defined

Set G [S] is a grammar, if the symbol string x is derived from the identification code , i.e., there are S x, called x is grammar G [S] sentence
If only the end of the symbols x, i.e., S x, x∈V ^*_T , x is called G [S] sentences
For example, in the embodiment 2.1, S, 0S1,000111 sentence is grammar G of Example 2.1, wherein G is 000,111 sentences
In the embodiment 2.2, <identifier> <letter>, <letter> <number>, Example 2.2 are a1 sentence grammar G, wherein G is a1 sentences

Language definition generated grammar G

Language grammar G is defined as a set of generated X {| S X, wherein S is the recognition grammar symbols and x∈V ^*_T }, available L (G) of the set represents

Language grammar describes grammar all the sentence (sentence consisting only of terminal symbols) set

Example Consider the grammar G 2.1, there are two production (rule): S → 0S1 and S → 01, by using the n-1 times to generate a first type, and then use a formula to generate a second, to give S => = 0S1> 00S11 => ... => 0 ^{n-. 1-} Sl ^{n--. 1} => 0 ^n- . 1 ^n-

L(G)={0ⁿ1ⁿ|n≥1}

Example 2.3

Set = G (V _N , V _T , P, S), V _N = {S, B, E}, V _T = {A, B, E}, P generated by the following composition formula

S→aSBE
S→aBE
EB→BE
aB→ab
bB→bb
bE→be
eE→ee

If L (G1) = L (G2), G1 and G2 are called grammatical equivalent

For example grammar G [A]:

A→0R
A→01
R→A1