Research Paper On Chomsky Normal Form Automata

In formal language theory, a context-free grammarG is said to be in Chomsky normal form (first described by Noam Chomsky)[1] if all of its production rules are of the form:[2]:92–93,106

ABC,   or
Aa,   or
S → ε,

where A, B, and C are nonterminal symbols, a is a terminal symbol (a symbol that represents a constant value), S is the start symbol, and ε denotes the empty string. Also, neither B nor C may be the start symbol, and the third production rule can only appear if ε is in L(G), namely, the language produced by the context-free grammar G.

Every grammar in Chomsky normal form is context-free, and conversely, every context-free grammar can be transformed into an equivalent one[note 1] which is in Chomsky normal form and has a size no larger than the square of the original grammar's size.

Converting a grammar to Chomsky normal form[edit]

To convert a grammar to Chomsky normal form, a sequence of simple transformations is applied in a certain order; this is described in most textbooks on automata theory.[2]:87–94[3][4][5] The presentation here follows Hopcroft, Ullman (1979), but is adapted to use the transformation names from Lange, Leiß (2009).[6][note 2] Each of the following transformations establishes one of the properties required for Chomsky normal form.

START: Eliminate the start symbol from right-hand sides[edit]

Introduce a new start symbol S0, and a new rule

S0S,

where S is the previous start symbol. This doesn't change the grammar's produced language, and S0 won't occur on any rule's right-hand side.

TERM: Eliminate rules with nonsolitary terminals[edit]

To eliminate each rule

AX1 ... a ... Xn

with a terminal symbol a being not the only symbol on the right-hand side, introduce, for every such terminal, a new nonterminal symbol Na, and a new rule

Naa.

Change every rule

AX1 ... a ... Xn

to

AX1 ... Na ... Xn.

If several terminal symbols occur on the right-hand side, simultaneously replace each of them by its associated nonterminal symbol. This doesn't change the grammar's produced language.[2]:92

BIN: Eliminate right-hand sides with more than 2 nonterminals[edit]

Replace each rule

AX1X2 ... Xn

with more than 2 nonterminals X1,...,Xn by rules

AX1A1,
A1X2A2,
... ,
An-2Xn-1Xn,

where Ai are new nonterminal symbols. Again, this doesn't change the grammar's produced language.[2]:93

DEL: Eliminate ε-rules[edit]

An ε-rule is a rule of the form

A → ε,

where A is not S0, the grammar's start symbol.

To eliminate all rules of this form, first determine the set of all nonterminals that derive ε. Hopcroft and Ullman (1979) call such nonterminals nullable, and compute them as follows:

  • If a rule A → ε exists, then A is nullable.
  • If a rule AX1 ... Xn exists, and every single Xi is nullable, then A is nullable, too.

Obtain an intermediate grammar by replacing each rule

AX1 ... Xn

by all versions with some nullable Xi omitted. By deleting in this grammar each ε-rule, unless its left-hand side is the start symbol, the transformed grammar is obtained.[2]:90

For example, in the following grammar, with start symbol S0,

S0AbB | C
BAA | AC
Cb | c
Aa | ε

the nonterminal A, and hence also B, is nullable, while neither C nor S0 is. Hence the following intermediate grammar is obtained:[note 3]

S0AbB | Ab | bB | b   |   C
BAA | A | A | ε   |   AC | C
Cb | c
Aa | ε

In this grammar, all ε-rules have been "inlined at the call site".[note 4] In the next step, they can hence be deleted, yielding the grammar:

S0AbB | Ab | bB | b   |   C
BAA | A   |   AC | C
Cb | c
Aa

This grammar produces the same language as the original example grammar, viz. {ab,aba,abaa,abab,abac,abb,abc,b,bab,bac,bb,bc,c}, but apparently has no ε-rules.

UNIT: Eliminate unit rules[edit]

A unit rule is a rule of the form

AB,

where A, B are nonterminal symbols. To remove it, for each rule

BX1 ... Xn,

where X1 ... Xn is a string of nonterminals and terminals, add rule

AX1 ... Xn

unless this is a unit rule which has already been (or is being) removed.

Order of transformations[edit]

Mutual preservation
of transformation results
Transformation Xalways preserves (✓)
resp. may destroy (✗) the result of Y:
X\YSTARTTERMBINDELUNIT
START
TERM
BIN
DEL
UNIT(✓)*
*UNIT preserves the result of DEL
  if START had been called before.

When choosing the order in which the above transformations are to be applied, it has to be considered that some transformations may destroy the result achieved by other ones. For example, START will re-introduce a unit rule if it is applied after UNIT. The table shows which orderings are admitted.

Moreover, the worst-case bloat in grammar size[note 5] depends on the transformation order. Using |G| to denote the size of the original grammar G, the size blow-up in the worst case may range from |G|2 to 22 |G|, depending on the transformation algorithm used.[6]:7 The blow-up in grammar size depends on the order between DEL and BIN. It may be exponential when DEL is done first, but is linear otherwise. UNIT can incur a quadratic blow-up in the size of the grammar.[6]:5 The orderings START,TERM,BIN,DEL,UNIT and START,BIN,DEL,UNIT,TERM lead to the least (i.e. quadratic) blow-up.

Example[edit]

The following grammar, with start symbol Expr, describes a simplified version of the set of all syntactical valid arithmetic expressions in programming languages like C or Algol60. Both number and variable are considered terminal symbols here for simplicity, since in a compiler front-end their internal structure is usually not considered by the parser. The terminal symbol "^" denoted exponentiation in Algol60.

ExprTerm| ExprAddOpTerm| AddOpTerm
TermFactor| TermMulOpFactor
FactorPrimary| Factor ^ Primary
Primarynumber| variable| ( Expr )
AddOp→ +| −
MulOp→ *| /

In step "START" of the above conversion algorithm, just a rule S0Expr is added to the grammar. After step "TERM", the grammar looks like this:

S0Expr
ExprTerm| ExprAddOpTerm| AddOpTerm
TermFactor| TermMulOpFactor
FactorPrimary| FactorPowOpPrimary
Primarynumber| variable| OpenExprClose
AddOp→ +| −
MulOp→ *| /
PowOp→ ^
Open→ (
Close→ )

After step "BIN", the following grammar is obtained:

S0Expr
ExprTerm| ExprAddOp_Term| AddOpTerm
TermFactor| TermMulOp_Factor
FactorPrimary| FactorPowOp_Primary
Primarynumber| variable| OpenExpr_Close
AddOp→ +| −
MulOp→ *| /
PowOp→ ^
Open→ (
Close→ )
AddOp_TermAddOpTerm
MulOp_FactorMulOpFactor
PowOp_PrimaryPowOpPrimary
Expr_CloseExprClose

Since there are no ε-rules, step "DEL" doesn't change the grammar. After step "UNIT", the following grammar is obtained, which is in Chomsky normal form:

S0number| variable| OpenExpr_Close| FactorPowOp_Primary| TermMulOp_Factor| ExprAddOp_Term| AddOpTerm
Exprnumber| variable| OpenExpr_Close| FactorPowOp_Primary| TermMulOp_Factor| ExprAddOp_Term| AddOpTerm
Termnumber| variable| OpenExpr_Close| FactorPowOp_Primary| TermMulOp_Factor
Factornumber| variable| OpenExpr_Close| FactorPowOp_Primary
Primarynumber| variable| OpenExpr_Close
AddOp→ +| −
MulOp→ *| /
PowOp→ ^
Open→ (
Close→ )
AddOp_TermAddOpTerm
MulOp_FactorMulOpFactor
PowOp_PrimaryPowOpPrimary
Expr_CloseExprClose

The Na introduced in step "TERM" are PowOp, Open, and Close. The Ai introduced in step "BIN" are AddOp_Term, MulOp_Factor, PowOp_Primary, and Expr_Close.

Alternative definition[edit]

Chomsky reduced form[edit]

Another way[2]:92[7] to define the Chomsky normal form is:

A formal grammar is in Chomsky reduced form if all of its production rules are of the form:

or
,

where , and are nonterminal symbols, and is a terminal symbol. When using this definition, or may be the start symbol. Only those context-free grammars which do not generate the empty string can be transformed into Chomsky reduced form.

Floyd normal form[edit]

In a paper where he proposed a term Backus–Naur form (BNF), Donald E. Knuth implied a BNF "syntax in which all definitions have such a form may be said to be in 'Floyd Normal Form'",

or
or
,

where , and are nonterminal symbols, and is a terminal symbol, because Robert W. Floyd found any BNF syntax can be converted to the above one in 1961.[8] But he withdrew this term, "since doubtless many people have independently used this simple fact in their own work, and the point is only incidental to the main considerations of Floyd's note."[8]

Application[edit]

Besides its theoretical significance, CNF conversion is used in some algorithms as a preprocessing step, e.g., the CYK algorithm, a bottom-up parsing for context-free grammars, and its variant probabilistic CKY.[9]

See also[edit]

Notes[edit]

References[edit]

  1. ^Chomsky, Noam (1959). "On Certain Formal Properties of Grammars"(PDF). Information and Control. 2: 137–167. doi:10.1016/S0019-9958(59)90362-6. 
  2. ^ abcdefHopcroft, John E.; Ullman, Jeffrey D. (1979). Introduction to Automata Theory, Languages and Computation. Reading, Massachusetts: Addison-Wesley Publishing. ISBN 0-201-02988-X. 
  3. ^Hopcroft, John E.; Motwani, Rajeev; Ullman, Jeffrey D. (2006). Introduction to Automata Theory, Languages, and Computation (3rd ed.). Addison-Wesley. ISBN 0-321-45536-3.  Section 7.1.5, p.272
  4. ^Rich, Elaine (2007). Automata, Computability, and Complexity: Theory and Applications (1st ed.). Prentice-Hall. ISBN 978-0132288064. [page needed]
  5. ^Wegener, Ingo (1993). Theoretische Informatik - Eine algorithmenorientierte Einführung. Leitfäden und Mongraphien der Informatik (in German). Stuttgart: B. G. Teubner. ISBN 978-3-519-02123-0.  Section 6.2 "Die Chomsky-Normalform für kontextfreie Grammatiken", p. 149–152
  6. ^ abcLange, Martin; Leiß, Hans (2009). "To CNF or not to CNF? An Efficient Yet Presentable Version of the CYK Algorithm"(PDF). Informatica Didactica. 8. 
  7. ^Hopcroft et al. (2006)[page needed]
  8. ^ abKnuth, Donald E. (December 1964). "Backus Normal Form vs. Backus Naur Form". Communications of the ACM. 7 (12): 735–736. doi:10.1145/355588.365140. 
  9. ^Jurafsky, Daniel; Martin, James H. (2008). Speech and Language Processing (2nd ed.). Pearson Prentice Hall. p. 465. ISBN 978-0-13-187321-6. 

Further reading[edit]

  • Cole, Richard. Converting CFGs to CNF (Chomsky Normal Form), October 17, 2007. (pdf) — uses the order TERM, BIN, START, DEL, UNIT.
  • John Martin (2003). Introduction to Languages and the Theory of Computation. McGraw Hill. ISBN 0-07-232200-4. (Pages 237–240 of section 6.6: simplified forms and normal forms.)
  • Michael Sipser (1997). Introduction to the Theory of Computation. PWS Publishing. ISBN 0-534-94728-X. (Pages 98–101 of section 2.1: context-free grammars. Page 156.)
  • Sipser, Michael. Introduction to the Theory of Computation, 2nd edition.
  1. ^that is, one that produces the same language
  2. ^For example, Hopcroft, Ullman (1979) merged TERM and BIN into a single transformation.
  3. ^indicating a kept and omitted nonterminal N by N and , respectively
  4. ^If the grammar had a rule S0 → ε, it could not be "inlined", since it had no "call sites". Therefore it couldn't be deleted in the next step.
  5. ^i.e. written length, measured in symbols

- Вычитайте, да побыстрее. Джабба схватил калькулятор и начал нажимать кнопки. - А что это за звездочка? - спросила Сьюзан.

0 comments

Leave a Reply

Your email address will not be published. Required fields are marked *