CSI 3125, Grammars, page 1

1
Language description methods

• Major topics in this part of the course
• Syntax and semantics
• Grammars
• Axiomatic semantics (next handout)

2
Syntax and semantics

• Points to discuss
• The form and meaning of programming languages
• Types of processing
• Types of languages

3
Syntax

• The syntax of a language determines how programs
  are built from elementary units (keywords,
  identifiers, numbers, brackets, and so on).
• A syntactically correct program may still not be
  acceptable, or it may work in a way that we do
  not want (or do not expect).
• Formal syntax is a system for describing the
  structure of programs exactly.
• Such systems include grammars, BNF, syntactic
  diagrams (syntax graphs).

4
Grammars

• There are infinitely many different programs, but
  every program is finite and must be recognized in
  finite time.
• A grammar should allow a finite description of a
  usually infinite language.

5
Semantics

• Semantics of a language determines the meaning of
  elementary units and their combinations
• how does the meaning of a program derive from the
  meaning of its fragments?
• The effect of a compound statement (such as a
  loop, an "if") should depend only on the effect
  of the elementary statements (such as an
  assignment).

6
Methods of semantic description

• Operational semantics
• Simple lower-level operations explain how
  higher-level statements are performed.
• Denotational semantics
• A program computes a function, a mapping data ?
  results
• Axiomatic semantics
• A program establishes a relation data ? results

7
Lexical analysis

• Lexical analysis pre-processes a file that
  contains a source program
• recognize units larger than single characters
  (keywords, predefined names, identifiers,
  numbers, brackets, operators, and so on).
• remove white space.
• This helps make translators simpler, by keeping
  low-level details out.

8
Syntactic analysis

• Syntactic analysis, based on grammars, can mean
  two things
• Recognition (the program is / is not correct)
• Parsing (a representation of the syntactic
  structure is built for correct programs).
• Syntactic analysis is the essential part of any
  implementation of a programming language.
• By the way, syntactic generation, also based on
  grammars, is the flip side of analysis it runs
  from a syntactic structure to a source. Important
  in language technology, not much in programming
  languages.

9
What is a language?

• A language is a set of sentences.
• A sentence is a sequence of elementary pieces,
  built according to certain rules (usually grammar
  rules).
• In a natural language, sentences have the usual
  meaning.
• In programming languages, various syntactic unit
  can be considered as "sentences".
• For example, in a set of all expressions, each
  valid expression is a sentence.
• In a set of all programs, each valid complete
  program is a sentence. And so on.

10
A hierarchy of formal languages

• Formal languages are classified on their
  complexity.
• A four-level hierarchy, from the simplest to the
  most complicated
• regular lt
• context-free lt
• context-sensitive lt
• recursively enumerable.
• Grammars too are classified in this way.

11
A hierarchy of formal languages (2)

• Programming languages usually have
• context-free syntax,
• context-sensitive semantics.
• Context-freeness (important in syntactic
  analysis) means that a fragment we are analyzing
  does not depend on any other fragments of the
  program.
• for example, an occurrence of a variable is not
  related to its declaration
• a message to a method is analyzed separately of
  the definition of this method.

12
Formal grammars

• Points to discuss
• Concepts of formal grammars
• A sample grammar in BNF
• Derivations and parse trees
• Ambiguity in grammars

13
A formal grammar has four components

• Terminal symbols language elements (for
  example, variable names in Java, or English
  words).
• Non-terminal symbols auxiliary symbols,
  denoting classes of constructions (for example
  loop_statement, Boolean_expression).
• The goal (start) symbol denotes any sentence.
• Productions rewriting rules ("this structure
  has such and such components") used to recognize
  or generate sentences.

14
Two ways of rewriting

• From the start symbol, produce more and more
  specific approximations of a sentence, replacing
  non-terminals with their definitions
• Reduce the sentence into more and more general
  forms, replacing definitions with non-terminals,
  and reach the goal symbol (the same!).
• Productions are what makes a grammar regular,
  context-free or context-sensitive.

15
Example a grammar of expressions

• Seven terminal symbols
• - ( ) x y
• Four non-terminal symbols
• expr term factor var
• These names are what we choose writing a grammar
  is not different from writing a program. The
  names are meant to help us read the grammar.
• Start/goal symbol
• expr

16
Notation

• Angle brackets distinguish non-terminal from
  terminal symbols. (It's like distinguishing
  strings and keywords in Java "class" and class
  are different.)
• LHS ? RHS means"the Left-Hand Side consists of
  things on the Right-Hand Side".
• The bar separates alternative Right-Hand Sides
  with the same Left-Hand Side.

17
Productions of our grammar

• expr ? term expr term
  expr - term
• term ? factor term factor
• factor ? var ( expr )
• var ? x y

18
Top-down and bottom-up

• For any sentence ?, productions can be applied in
  two directions.
• Top-down
• Derive ? from the start symbol.
• ? will then be an example of an expression.
• Bottom-up
• Fold ? into the initial symbol.

19
Derivations

• Let us take the sequence of terminal symbols
• ( x - y ) x y
• It is an expression, but we must first show that
  it is.
• Consider two derivations (the next two pages).
• On each line, the highlighted part is involved in
  rewriting into the next line, according to some
  grammar production.

20
A top-down derivation

• expr ?
• expr term ?
• term term ?
• term factor term ?
• factor factor term ?
• ( expr ) factor term ?
• ( expr - term ) factor term ?
• ( term - term ) factor term ?
• ( factor - term ) factor term ?
• ( var - term ) factor term ?
• ( x - term ) factor term ?
• ( x - factor ) factor term ?
• ( x - var ) factor term ?
• ( x - y ) factor term ?
• ( x - y ) var term ?
• ( x - y ) x term ?
• ( x - y ) x factor ?
• ( x - y ) x var ?
• ( x - y ) x y

21
A bottom-up derivation

• ( x - y ) x y ?
• ( var - y ) x y ?
• ( factor - y ) x y ?
• ( term - y ) x y ?
• ( expr - y ) x y ?
• ( expr - var ) x y ?
• ( expr - factor ) x y ?
• ( expr - term ) x y ?
• ( expr ) x y ?
• factor x y ?
• term x y ?
• term var y ?
• term factor y ?
• term y ?
• expr y ?
• expr var ?
• expr factor ?
• expr term ?
• expr

22
And both side by side

• ( x - y ) x y ?
• ( var - y ) x y ?
• ( factor - y ) x y ?
• ( term - y ) x y ?
• ( expr - y ) x y ?
• ( expr - var ) x y ?
• ( expr - factor ) x y ?
• ( expr - term ) x y ?
• ( expr ) x y ?
• factor x y ?
• term x y ?
• term var y ?
• term factor y ?
• term y ?
• expr y ?
• expr var ?
• expr factor ?
• expr term ?
• expr

expr ? expr term ? term term
? term factor term ? factor
factor term ? ( expr ) factor
term ? ( expr - term ) factor term
? ( term - term ) factor term ? (
factor - term ) factor term ? ( var
- term ) factor term ? ( x - term )
factor term ? ( x - factor ) factor
term ? ( x - var ) factor term ? ( x
- y ) factor term ? ( x - y ) var
term ? ( x - y ) x term ? ( x - y ) x
factor ? ( x - y ) x var ? ( x - y ) x
y
23
Is it really so easy?

• In both derivations, guessing is required which
  production should we choose to apply next?
• Strategies of choice are at the heart of parsing
  algorithms. Ideally, we would always guess
  correctly. Less ideally, we may have to try a
  production, fail, and return to try another.
• Both processes recognize the given sequence of
  symbols
• ( x - y ) x y
• as an expression that is well-formed according
  to our grammar.

24
Parse trees
The results of both derivations can be summarized
in the same parse tree (or abstract syntax tree).
Note that we do not show in this tree the order
in which productions have been applied during
derivations.
25
Ambiguity

• A grammar is ambiguous when an expression defined
  by this grammar has more than one structurally
  different parse tree.
• For example, here is a grammar of arithmetic
  expressions
• E ? E E E E N
• where N denotes any unsigned integer.
• The expression 6 17 23 has two different
  derivation trees.

26
Two different parse trees...
E ? E E E E N
6 17 23
27
... and their meaning...

• These trees represent two different ways of
  computing the value of the expression!
• Ambiguity should be avoided.

28
... and what to do with ambiguity

• In our previous example, we should have written
  the usual two-level definition instead of a
  definition with and at the same level.
• An expressions E is a sum of terms T.
• A term is a product of numbers N.
• E ? T ltEgt ltTgt
• T ? N ltTgt ltNgt

29
A long phrase...
Examples

• the dog
• the dog that chased the cat
• the dog that chased the cat that caught the mouse
• the dog that chased the cat that caught the
  mouse that chewed the shoe
• the dog that chased the cat that caught the
  mouse that chewed the shoe that squashed the
  fruit
• the dog that chased the cat that caught the
  mouse that chewed the shoe that squashed the
  fruit that stained the chair
• and so on...

30
... a grammar of long phrases
Examples

• ltlong phrasegt ?
• the ltnoungt
• the ltnoungt that ltverbgt
• ltlong phrasegt
• ltnoungt ?
• cat chair dog
• fruit mouse shoe ...
• ltverbgt ?
• caught chased chewed
• squashed stained ...

31
A clause...
Examples

• the dog that chased the catthat caught the mouse
  that chewed the shoethat squashed the fruit that
  stained the chairgrabbed the sausage that
  tempted the wolfthat fought the fox that scared
  the squirrelthat bit the twig that cracked the
  nutthat hit the boy that lifted the hat

32
... a grammar of clauses
Examples

• ltclausegt ?
• ltlongPhrasegt ltverbgt ltlongPhrasegt
• ltlongPhrasegt ?
• the ltnoungt
• the ltnoungt that ltverbgt
• ltlongPhrasegt
• ltnoungt ? boy cat ...
• ltverbgt ? bit caught ...
• (Add maybe 1500 rules and you will havea
  reasonable grammar of English. ?)

33
Simple lists in Scheme...
Examples

• A list is either empty
• ()
• or it is a sequence of elements separated by
  blank spaces, all enclosed in parentheses
• ( element ... element )
• Each element is either a list, or an atom. An
  atom is an identifier made of small letters. We
  assume that a scanner converts a text on input
  into a sequence of tokensatoms and parentheses.
• Example ( ab ( xyz br ) () ( no ) yes )

34
... a grammar of lists
Examples

• ltlistgt ? () ( ltelementsgt )
• ltelementsgt ? ltelementgt
• ltelementgt ltelementsgt
• ltelementgt ? ltatomgt ltlistgt
• ltatomgt ? ltlettergt
• ltlettergt ltatomgt
• ltlettergt ???a b c ... z

35
A flower garden...
Examples

• We have four kinds of things in our garden
• ? a wall
• ? a large flower
• ? a small flower
• ? a house
• Starting from the left, a garden has a wall, then
  at least one large flower, another wall, some
  small flowers (more than we have large ones) and
  finally a house.

36
... and a few examples ...
Examples

• ??????????????? a garden?
• ??????????????? a garden?
• ??????????????? a garden?
• ????????????? a garden?

37
... a grammar of gardens
Examples

• ltgardengt ?
• ? ltlargeWallSmallgt ltmoreSmallgt ?
• ltlargeWallSmallgt ?
• ? ? ?
• ? ltlargeWallSmallgt ?
• ltmoreSmallgt ?
• ? ? ltmoreSmallgt