EBNF: A Notation for Describing Syntax

EBNF:A Notation for Describing Syntax nLanguages and Syntax nEBNF Descriptions and Rules nMore Examples of EBNF nSyntax and Semantics nEBNF Description of Sets nAdvanced EBNF (recursion)

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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.

Languages and Syntax • EBNF: Extended Backus-Naur Form • John Backus (IBM) invented a notation called BNF • He used it to describe FORTRAN’s 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

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

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. digitÜ 0|1|2|3|4|5|6|7|8|9 integerÜ [+|-]digit{digit} digit is defined as any of the alternatives0 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.

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 1,024A515-1+2 an integer?

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 integerGiven [+|-]digit{digit} Replace LHS by RHS (1) [+]digit{digit} Chose + alternative (2) +digit{digit} Include option (3) +1{digit} Replace digit by 1 alternative (1&2) +1digit digitChoose two repetitions (4) +12digitReplace digit by 2 alternative (1&2) +127 Replace digit by 7 alternative (1&2)

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.

Identical vs Equivalent Descriptions signÜ +|- digitÜ 0|1|2|3|4|5|6|7|8|9 integerÜ [sign]digit{digit} xÜ +|- yÜ 0|1|2|3|4|5|6|7|8|9 zÜ [x]y{y} 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.

Two Problematical Descriptions A “simplified but equivalent” definition of integer? signÜ +|- digitÜ 0|1|2|3|4|5|6|7|8|9 integerÜ [sign]{digit} A “good” definition of integers with commas (1,024)? signÜ +|- comma-digitÜ 0|1|2|3|4|5|6|7|8|9|, comma-integerÜ [sign]comma-digit{comma-digit} Both definitions classify “non-obvious” symbols as legal integer or comma-integer. Find such symbols.

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, ...

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 digitÜ 0|1|2|3|4|5|6|7|8|9 structured-integerÜ [sign]digit{[_]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

Syntax Charts Sequence Choice A B C D A|B|C|D ABCD A B C D Option Repetition A [A] A {A}

Syntax Charts for integer and digit 0 1 2 3 4 5 6 7 8 9 digit + digit - digit integer

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 doesn’t need another name for digit)?

A A B {A|B} B {A}|{B} A B C AB|C Interesting Rules & Their Charts

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-listÜinteger{,integer} integer-setÜ ([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)

Proof: (5,-2,11) is an integer-set Status Reason integer-setGiven ([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

Description of Sets with Ranges • Ranges syntax • A ranges is a single integer or a pair separated by .. integer-rangeÜinteger[..integer] integer-listÜinteger-range{,integer-range} integer-setÜ ([integer-list]) • Range semantics X..Y • X£Y: 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 • X>Y: 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)

Recursive Descriptions A directly recursive EBNF rule has its LHS in its RHS r1Ü| 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 r1Given Ar1Replace r1 by the second alternative in its RHS AAr1Replace r1 by the second alternative in its RHS AAAr1Replace r1 by the second alternative in its RHS AAA Replace r1 by the first (empty) alternative in its RHS This rule is equivalent to r1Ü{A}

The Power of Recursion To recognize symbols of the form form An Bn , n³ 0 we cannot write r1Ü{A}{B}, because nothing constrains us choosing different repetitions of A and B: AAB The recursive rule r1Ü| Ar1B works, because each choice of the second alternative uses exactly one A and one B. Proof that AAABBB is an r1 r1Given 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

Problems Read the EBNF Handout (all but Section 2.7) • Study and Understand the Review Questions • 2 (page 10), 2&3 (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

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). • A{BA}Z • AZ BZ ABZ ABAZ • ABABZ ABA AAAZ ABABBZ • A{B[C]}Z • BZ ABC ABBBZ ACCZ • ABCZ ABCBCZ ABBCBBZ ABCZBCZ • A{B|C}Z • AB ABC ABBBZ BBZ • ABBCCZ ACCBBZ ACBBCZ ABCZBCZ

EBNF: A Notation for Describing Syntax