Question

a) Give a Backus-Naur Form (BNF) grammar for Java integer literals (decimal, hexa, octal).

b) Show a BNF grammar that is ambiguous.

c) Is a BNF able to express “the maximum of a byte literal is 127” ? Why or why not?

Answer 1

a)

<package name> :::くidentifier <package name>·</dentifier <type name> ::-«identiner> 1 <package name>·</dentiner> simple type neme sidentifier expression name> : identifier»-ambiguous name>·</aentifier method name> :-<identifiers I <ambiguous name>-<!dentifies cembiguous name> identie cambiguous names, cldentfier </iterai> :- integer literal> \ <ficating point literal> hoolean literab «character literal> enteger stera/> :ー«decimal integer iiterap-hex integer iiterap-octa, integer iiteral c decimal integer literal> ::-decimal numere> 디nteger type sumx? shex integer literalchex numeralb sinteger type suffixe ? octal integer literal>ta numeralb sinteger type suffix? integer type sumx> ::-1 I L secima, numeral> :::0 <non zero digit>くdigits digitssdigits I "digits- *digit <digit> :::0 l non zero digit> non zero digit11213141516171819 <hex numerax <hex digitt | 0 X <hex digitz | <hex numeral shex digit> «nex digit> :: 0111213141516171819|albloldl01↑IAIBICIDIELE coctal numeral0 octal digit | coctal numerals coctai dligit octal digit01 2 31415 617 floating-point litera/> ::-<digits-.<digits? <exponent part? <float type surno? digits cexponent par? float type suffixp? exponent partexponent indicator signed integer? <string literal>-null literal>

cexponent indicetoo e|E sign> «noet type sumx> ::zfIFIdID boolean itea true false «character literal> ::-·«single character>- escape sequence> singie character input character except and «string litera» :-" <string characters?? string charactersstring character string characiers <string character? string character?input character exceptand Iescape character null teranul1

This is the BNF Grammar for Java Literals. Although the question has asked for integer literals only, other literals (floating-point, boolean, character, string etc.) have been added for completeness and clarity.  

b)

where x,y,z are terminals.

This grammar is ambiguous because the string yyxzx can have either of these two parse trees:

yAz A I1 yAz A y A x

c)

A BNF Grammar will not be able to express "the maximum of a byte literal is 127" because it cannot count. A BNF Grammar can only tell if a particular string is part of the grammar or not. There is nothing limiting the recurrence at 127 or for that matter any number. This may not cause a problem during compilation, however the code will be buggy because of this error during execution.