EBNF: A Notation for Describing Syntax

1
EBNFA Notation for Describing Syntax

• n Languages and Syntax
• n EBNF Descriptions and Rules
• n More Examples of EBNF
• n Syntax and Semantics
• n EBNF Description of Sets
• n Advanced EBNF (recursion)

2
Quote of the Day

• When teaching a rapidly changing technology,
  perspective is more important than content.

3
Why Study EBNF

• EBNF is a notation for formally describing
  syntax how to write symbols in a language. We
  will use EBNF to describe the syntax of Java. But
  there is a more compelling reason to begin our
  study of programming with EBNF it is a microcosm
  of programming. There is a strong similarity
  between the control forms of EBNF and the control
  structures of Java sequence, decision,
  repetition, recursion, and the ability to name
  descriptions. There is also a strong similarity
  between the process of writing EBNF descriptions
  and writing Java programs. Finally studying EBNF
  introduces a level of formality that will
  continue throughout the semester.

4
Languages and Syntax

• EBNF Extended Backus-Naur Form
• John Backus (IBM) invented a notation called BNF
• He used it to describe FORTRANs syntax (1956)
• Peter Naur popularized BNF
• He used it to describe ALGOL's syntax (1958)
• Niklaus Wirth used and Extended form of BNF
  (called EBNF) to describe the syntax of his
  Pascal programming language (1976)
• Noam Chomsky (MIT linguist and philospher)
• Invented a Hierarchy of Notations for Natural
  Languages
• 4 levels 0-3 with 0 being the most powerful
• BNF is at level 2 programming languages are at
  level 0
• Formal Languages and Computability
• is the study of different families of notations
  and their power

5
EBNF Descriptions and Rules

• Each Description is a list of Rules
• Rule Form LHS Ü RHS (read Ü as is defined as)
• Rule Names (LHS) are italicized, hyphenated words
• Control Forms in RHS
• Sequence Items appear left to right order is
  important
• Choice Alternatives separated by (stroke)
  exactly one item is chosen from the alternatives
• Option Optional item enclosed between and it
  can be included or discarded
• Repetition Repeatable item enclosed between
  and it can be repeated 0 or more times

6
An EBNF Description of Integers

• A symbol (sequence of characters) is classified
  legal by an EBNF rule if we can process all the
  characters in the symbol when we reach the end of
  the right hand side of the EBNF rule.
• digit Ü 0123456789
• integer Ü -digitdigit
• digit is defined as any of the alternatives 0
  through 9
• integer is defined as a sequence of three items
  (1) an optional sign (if it is included, it must
  be the alternative or -), followed by (2) any
  digit, followed by (3) a repetition of zero or
  more digits.
• The integer RHS combines and illustrates all EBNF
• control forms sequence, option, alternative,
  repetition.

7
Proofs In English

• Is the symbol 7 an integer? Yes, the proof
• In the integer EBNF rule, start with the optional
  sign discard the option. Next in the sequence is
  a digit choose the 7 alternative. Next in the
  sequence is a repetition choose 0 repetitions.
  End of symbol integer reached.
• Is the symbol 127 an integer? Yes, the proof.
• In the integer EBNF rule, start with the optional
  sign include the option choose the
  alternative. Next in the sequence is a digit
  choose the 1 alternative. Next in the sequence is
  a repetition choose 2 repetitions choose the 2
  alternative for the first choose the 7
  alternative for the second. End of symbol
  integer reached.
• Are the symbols 1,024 A5 15- 12 an
  integer?

8
Tabular Proof

• Tabular Proof Replacement Rules
• (1) Replace a name (LHS) by its definition
  (RHS)
• (2) Choose an alternative
• (3) Include or Discard an Option
• (4) Choose the number of repetitions
• Status Reason
• integer Given
• -digitdigit Replace LHS by RHS (1)
• digitdigit Chose alternative (2)
• digitdigit Include option (3)
• 1digit Replace digit by 1 alternative (12)
• 1digit digit Choose two repetitions (4)
• 12digit Replace digit by 2 alternative (12)
• 127 Replace digit by 7 alternative (12)

9
Graphical Proof
integer
-
digit
digit

1
digit
digit

2
7
A graphical proof replaces multiple (equivalent)
tabular proofs, since the order of rule
application (which is unimportant) is often
absent in graphical proofs.
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Identical vs Equivalent Descriptions

• sign Ü -
• digit Ü 0123456789
• integer Ü signdigitdigit
• x Ü -
• y Ü 0123456789
• z Ü xyy
• These two descriptions are not identical but they
  are equivalent Although they use different EBNF
  rule names (consistently), asking whether a
  symbol is an integer is the same as asking
  whether the symbol is a z.

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Two Problematical Descriptions

• A simplified but equivalent definition of
  integer?
• sign Ü -
• digit Ü 0123456789
• integer Ü signdigit
• A good definition of integers with commas
  (1,024)?
• sign Ü -
• comma-digit Ü 0123456789,
• comma-integer Ü signcomma-digitcomma-digit
• Both definitions classify non-obvious symbols
  as legal integer or comma-integer. Find such
  symbols.

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Syntax and Semantics

• Syntax Form
• Semantics Meaning
• Key Questions
• Can two different symbols have the same meaning?
• Can a symbol have many meanings (depending on
  context)?
• Do the following symbols have the same meaning?
• 1 and 1, 000193 and 193
• 9.000 and 9.0
• Rich and rich
• EBNF specifies syntax, not semantics
• Semantics is supplied informally English,
  examples, ...
• Formal semantics is a research area in CS, AI,
  Linguistics, ...

13
Structured Integers

• Allow non-adjacent embedded underscores to add a
  special structure to a number
• 2_10_54
• 1_800_555_1212
• 1_000_000 (compared to 1000000 figure each
  value fast)
• Define structured-integer
• digit Ü 0123456789
• structured-integer Ü signdigit_digit
• Semantically, the underscore is ignored
• 1_2 has the same meaning as 12
• How can we fix the date problem 12_5_1987 and
  1_25_1987

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Syntax Charts
Sequence
Choice
A B C D
ABCD A B C D
ABCD
Option
Repetition
A A
A A
15
Syntax Charts for integer and digit
0 1 2 3 4 5 6 7 8 9
digit

digit
-
digit
integer
16
A Syntax Chart with no other names
0 1 2 3 4 5 6 7 8 9
0 1 2 3 4 5 6 7 8 9

-
integer
Which Syntax chart for integer is simpler? The
previous one (because it is smaller) or this one
(because it it doesnt need another name for
digit)?
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Interesting Rules Their Charts
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Description of Sets

• Set syntax
• Sets start with ( and end with )
• Sets contain 0 or more integers
• A comma appears between every pair of integers
• integer-list Ü integer,integer
• integer-set Ü (integer-list)
• Set semantics
• Order is unimportant
• (1,3,5) is equivalent to (5,1,3) and any other
  permutation
• Duplicate elements are unimportant
• (1,3,5,1,3,3,5) is equivalent to (1,3,5)

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Proof (5,-2,11) is an integer-set

• Status Reason
• integer-set Given
• (integer-list) Replace integer-set by its RHS
• (integer-list) Include option
• (integer,integer) Replace integer-list by its
  RHS
• (5,integer) Lemma 5 is an integer
• (5,integer,integer) Choose two repetitions
• (5,-2,integer) Lemma -2 is an integer
• (5,-2,11) Lemma 11 is an integer

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Description of Sets with Ranges

• Ranges syntax
• A ranges is a single integer or a pair separated
  by ..
• integer-range Ü integer..integer
• integer-list Ü integer-range,integer-range
• integer-set Ü (integer-list)
• Range semantics X..Y
• XY all integers from X up to Y (inclusive)
• 1..5 is equivalent to 1,2,3,4,5 5..5 is
  equivalent just to 5
• XgtY a null range it contains no values
• (1..4,10,5..4,11..13) is equivalent to
  (1,2,3,4,10,11,12,13)

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Recursive Descriptions

• A directly recursive EBNF rule has its LHS in its
  RHS
• r1 Ü Ar1
• We read this as r1 is defined as the choice of
  nothing or an A followed by an r1. The symbols
  recognized as an r1 are of the form An, n³ 0.
  Proof that AAA is an r1
• r1 Given
• Ar1 Replace r1 by the second alternative in
  its RHS
• AAr1 Replace r1 by the second alternative in
  its RHS
• AAAr1 Replace r1 by the second alternative in
  its RHS
• AAA Replace r1 by the first (empty)
  alternative in its RHS
• This rule is equivalent to r1 Ü A

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The Power of Recursion

• To recognize symbols of the form form An Bn , n³
  0 we cannot write r1 Ü AB, because nothing
  constrains us choosing different repetitions of A
  and B AAB
• The recursive rule r1 Ü Ar1B works, because
  each choice of the second alternative uses
  exactly one A and one B. Proof that AAABBB is an
  r1
• r1 Given
• Ar1B Replace r1 by the second alternative in
  its RHS
• AAr1BB Replace r1 by the second alternative in
  its RHS
• AAAr1BBB Replace r1 by the second alternative
  in its RHS
• AAABBB Replace r1 by the first (empty)
  alternative in its RHS
• Symbols of the form form An Bn , n³ 0

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Problems

• Read the EBNF Handout (all but Section 2.7)
• Study and Understand the Review Questions
• 2 (page 10), 23 (page 12), 1 (page 16), 2
  (page 18)
• Be prepared to discuss in class solutions to the
  following Exercises (starting on page 23)
• 1, 2, 4, and expecially 8
• See next slide for more problems

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Problems (continued)

• Translate the following RHS of an EBNF rule into
  its equivalent syntax chart. Then, classify each
  of the examples below as legal or illegal
  according to this rule (or its equivalent chart).
• ABAZ
• AZ BZ ABZ ABAZ
• ABABZ ABA AAAZ ABABBZ
• ABCZ
• BZ ABC ABBBZ ACCZ
• ABCZ ABCBCZ ABBCBBZ ABCZBCZ
• ABCZ
• AB ABC ABBBZ BBZ
• ABBCCZ ACCBBZ ACBBCZ ABCZBCZ