1. Describing Syntax and Semantics
Chapter 3
Describing Syntax and Semantics

2. Chapter 3 Topics Introduction The General Problem of Describing Syntax
Formal Methods of Describing Syntax
Attribute Grammars
Describing the Meanings of Programs: Dynamic Semantics

3. Introduction Language description includes two main components Syntax
The form
of expressions, statements, and program units
Semantics
The meaning
Syntax and semantics provide a language’s definition

4. Introduction Describing syntax is easier than describing semantics
Universally-accepted notations can be used to describe syntax.
No universally-accepted systems have been created for describing semantics.

5. The General Problem of Describing Syntax: Terminology
Definition: A sentence is an ordered string of characters over some alphabet
Definition: A language is a set of sentences
Definition: A lexeme is the lowest level syntactic unit of a language
(e.g., *, sum, begin)
Definition: A token is a category of lexemes
(e.g., identifier)

6. Formal Grammars and Formal Languages
A formal grammar G = (N,Σ,P,S) is a quad-tuple such that
N is a finite set of nonterminal symbols
Σ is a finite set of terminal symbols,
disjoint from N
P is a finite set of production rules of the form αNβ → γ
S Є N the start symbol
The language of a formal grammar G, denoted as L(G), is the set of all strings over Σ that can be generated by starting with the start symbol S and then applying the production rules in P until no nonterminal symbols are present.

7. Formal Methods of Describing Syntax
The most widely known methods for describing programming language syntax:
Backus-Naur Form (BNF)
Context-Free Grammars
Extended BNF (EBNF)
Improves readability and writability
Grammars and Recognizers

8. Context-Free Grammars
Developed by Noam Chomsky
mid-1950s
Language generators
meant to describe the syntax of natural languages
Defines a class of languages called context-free languages

9. Formal Definition of Languages
Recognizers
A device that reads input strings of the language
and decides whether the input strings
belong to the language
Generators
A device that generates sentences of a language
which are used to compare
with the syntax of a particular sentence

10. Backus-Naur Form (BNF)
Invented by John Backus and Peter Naur
Equivalent to context-free grammars
A metalanguage used to describe another language
Abstractions are used to represent classes of syntactic structures
act like syntactic variables
called nonterminal symbols

11. BNF Fundamentals Non-terminals: abstractions
Terminals: lexemes and tokens
Grammar: a collection of rules
Examples of BNF rules:
<id_list> -> ident | ident, <id_list>
<if_stmt> -> if <logic_expr> then <stmt>

12. BNF Rules A rule has a left-hand side (LHS) a right-hand side (RHS)
consists of terminal and nonterminal symbols
A grammar is a finite nonempty set of rules
An abstraction (or nonterminal symbol) can have more than one RHS
<stmt> -> <single_stmt>
| begin <stmt_list> end

13. Describing Lists Syntactic lists are described using recursion
<id_list> -> ident | ident, <id_list>
A derivation is
a repeated application of rules,
starting with the start symbol, and
ending with a sentence
all terminal symbols

14. An Example Grammar <program> -> <stmts>
<stmts> -> <stmt> | <stmt> ; <stmts>
<stmt> -> <var> = <expr>
<var> -> a | b | c | d
<expr> -> <term> + <term>
|<term> - <term>
<term> -> <var> | const

15. An Example Derivation <program> => <stmts>
=> <var> = <expr>
=> a = <expr>
=> a = <term> + <term>
=> a = <var> + <term>
=> a = b + <term>
=> a = b + const

16. Derivation
Every string of symbols in the derivation is a sentential form
A sentence is a sentential form that has only terminal symbols
A leftmost derivation is one in which the leftmost nonterminal in each sentential form is the one that is expanded
A derivation may be neither leftmost nor rightmost

17. A hierarchical representation of a derivation
Parse Tree
A hierarchical representation of a derivation
<program>
<stmts>
<stmt>
<var> =
<expr> a
<term> +
<term>
<var>
const b

18. Ambiguity in Grammars A grammar is ambiguous if and only if
it generates a sentential form
that has two or more distinct parse trees

19. An Ambiguous Expression Grammar
<expr>  <expr> <op> <expr> | const
<op>  / | -
<expr>
<expr>
<expr>
<op>
<expr>
<expr>
<op>
<op>
<expr>
<expr>
<op>
<expr>
<expr>
<op>
<expr>
const -
const /
const
const -
const /
const

20. An Unambiguous Expression Grammar
If we use the parse tree to indicate precedence levels of the operators, we cannot have ambiguity
<expr>  <expr> - <term> | <term>
<term>  <term> / const | const
<expr>
<expr> -
<term>
<term>
<term> /
const
const
const

21. Associativity of Operators
Operator associativity can also be indicated by a grammar.
ambiguous:
<expr> -> <expr> + <expr> | const
unambiguous:
<expr> -> <expr> + const | const
<expr>
<expr>
<expr> +
const
<expr> +
const
const

22. Extended BNF Optional parts are placed in brackets []
<proc_call> -> ident [(<expr_list>)]
Alternative parts of RHSs are placed inside parentheses and separated via vertical bars
<term> → <term> (+|-) const
Repetitions (0 or more) are placed inside braces {}
<ident> → letter {letter|digit}

23. BNF and EBNF BNF <expr> -> <expr> + <term>
<term> -> <term> * <factor>
| <term> / <factor>
| <factor>
EBNF
<expr> -> <term> {(+|-) <term> }
<term> -> <factor> {(*|/) <factor> }

24. The Chomsky Hierarchy Grammar Languages Automaton Production Rules
Type-0
unrestricted
Recursively enumerable
Turing machine
No restrictions
Type-1
context-sensitive
Context-sensitive
Linear-bounded non-deterministic Turing machine
αAβ -> αγβ
Type-2
context-free
Context-free
Non-deterministic pushdown automaton
A -> γ
Type-3
regular
Regular
Finite state automaton
A -> aB A -> a

25. Semantics The meaning, not the form
Need a language to describe the semantics of languages
Assorted mathematical formalisms
Static Semantics
Attribute Grammars
Dynamic Semantics
Operational Semantics
Axiomatic Semantics
Denotational Semantics

26. Static Semantics
Static semantics of a language are those non-syntactic rules which
1) define legality of a program
2) can be analyzed at compile time
Consider a rule which states “a variable must be declared before it is referenced”
Cannot be specified in a context-free grammar
Can be tested at compile time
Some rules can be specified in a grammar, but unnecessarily complicate the grammar

27. Attribute Grammars
Describes more than can be described with a context-free grammar (Knuth, 1968)
Uses three additional elements to CFGs to carry some semantic information along parse trees
Attributes
value holders which describe language symbols
Attribute computation functions
describe how attribute values are determined
also called semantic functions
Predicate functions
describe language rules, associated with grammar rules

28. Attribute Grammars Most attribute grammar work uses the parse trees:
Inherited attributes pass information down a parse tree
Synthesized attributes pass information up a parse tree
Intrinsic attributes are synthesized attributes for parse tree leaves

29. Attribute Grammars
Definition: An attribute grammar is a context-free grammar G = (N,Σ,P,S) with the following additions:
For each grammar symbol x there is a set A(x) of attribute values
Each rule has a set of functions that define certain attributes of the nonterminals in the rule
Each rule has a (possibly empty) set of predicates to check for attribute consistency

30. Attribute Grammars Definitions
Let X0  X1 ... Xn be a rule
Functions of the form
S(X0) = f(A(X1), ... , A(Xn))
define synthesized attributes
I(Xj) = f(A(X0), ... , A(Xn)),
for i <= j <= n,
define inherited attributes
Initially, there are intrinsic attributes on the leaves

31. Attribute Grammar Example
Syntax
<expr> -> <var> + <var> | <var>
<var> -> A|B|C
Attributes
actual_type
synthesized for <var> and <expr>
expected_type
inherited for <expr>

32. Attribute Grammar Example
1. Syntax rule:
<expr>  <var>[1] + <var>[2]
Semantic rules:
<expr>.actual_type  <var>[1].actual_type
Predicate:
<var>[1].actual_type == <var>[2].actual_type
<expr>.expected_type == <expr>.actual_type
2. Syntax rule:
<var>  id
Semantic rule:
<var>.actual_type  lookup (<var>.string)

33. How are attribute values computed?
Attribute Grammars
How are attribute values computed?
If all attributes were inherited, the tree could be decorated in top-down order.
If all attributes were synthesized, the tree could be decorated in bottom-up order.
In many cases, both kinds of attributes are used, and it is some combination of top-down and bottom-up that must be used.

34. Attribute Grammars <expr>.expected_type  inherited from parent
<var>[1].actual_type  lookup (A)
<var>[2].actual_type  lookup (B)
<var>[1].actual_type =? <var>[2].actual_type
<expr>.actual_type  <var>[1].actual_type
<expr>.actual_type =? <expr>.expected_type

35. Attribute Grammars Primary value of AGs:
1. Static semantics specification
2. Static semantics checking

36. Dynamic Semantics
There is no single widely acceptable notation or formalism for describing semantics
Operational Semantics
Describe the meaning of a program by executing its statements on a machine, either simulated or actual.
The change in the state of the machine (memory, registers, etc.) defines the meaning of each statement.

37. Dynamic Semantics
The dynamic semantics of a program is the meaning of its expressions, statements, and program units
Accurately describing semantics is essential
Semantics are often described in English:
thus, ambiguity!
Three different methods are commonly used to describe semantics formally

38. Operational Semantics Denotational Semantics
Dynamic Semantics
Operational Semantics
Axiomatic Semantics
Denotational Semantics

39. Operational Semantics
Describe the meaning of a program by executing its statements on a machine, either simulated or actual.
The change in the state of the machine (memory, registers, etc.) defines the meaning of the statement

40. Operational Semantics
To use operational semantics for a high-level language, a virtual machine is needed
A hardware pure interpreter would be too expensive
A software pure interpreter also has problems

41. Operational Semantics
A better alternative:
A complete computer simulation
The process:
Build a translator (translates source code to the machine code of an idealized computer)
Build a simulator for the idealized computer

42. Uses of Operational Semantics
compiler writing
rigorously proving properties of the program
program modeling for model checking
the “gold standard” for validating other semantics

43. Axiomatic Semantics Based on formal logic (predicate calculus)
Original purpose: formal program verification
Define axioms or inference rules for each statement type in the language
allowa transformations of expressions to other expressions
The expressions are called assertions

44. Axiomatic Semantics An assertion before a statement is a precondition
states the relationships and constraints among variables that are true at that point in execution
An assertion following a statement is a postcondition
The weakest precondition
least restrictive precondition that guarantees the postcondition
Pre-post form: {P} statement {Q}

45. Axiomatic Semantics Program proof process
The postcondition for the whole program is the desired results
Work back through the program to the first statement
If the precondition on the first statement is the same as the program spec, then the program is correct

46. Axiomatic Semantics An axiom for assignment statements (x = E):
{Qx->E} x = E {Q}
The Rule of Consequence:
{P} S {Q}, P' => P, Q => Q'
{P’} S {Q’}

47. Axiomatic Semantics An inference rule for statement sequences:
For a sequence S1;S2:
{P1} S1 {P2}
{P2} S2 {P3}
the inference rule is:
{P1} S1 {P2}, {P2} S2 {P3}
{P1} S1; S2 {P3}

48. Axiomatic Semantics An inference rule for logical pretest loops:
For the loop construct
{P} while B do S end {Q}
the inference rule is
{I and B} S {I}
{I} while B do S {I and (not B)}
where I is the loop invariant

49. Characteristics of the the following conditions:
Loop Invariant
It must meet
the following conditions:
1. P => I
2. {I} B {I}
3. {I and B} S {I}
4. (I and (not B)) => Q
5. The loop terminates

50. Loop Invariant The loop invariant I is a weakened version
of the loop postcondition,
and it is also a precondition.
I must be weak enough
to be satisfied
prior to the beginning of the loop,
I must be strong enough
(combined with the loop exit condition)
to force the truth of the postcondition

51. Evaluation of Axiomatic Semantics
1. DIFFICULT:
developing axioms or inference rules for all of the statements in a language
2. A GOOD TOOL:
for correctness proofs, and an excellent framework for reasoning about programs
3. NOT VERY USEFUL:
for language users and compiler writers

52. Denotational Semantics
Based on recursive function theory
The most abstract semantics description method
The most rigorous formal technique which is widely known
Its complexity makes it of little use to language users

53. Denotational Semantics
The process of building a denotational spec for a language:
1. Define a mathematical object for each language entity
2. Define a function that maps instances of the language entities onto instances of the corresponding mathematical objects

54. Denotational Semantics
The difference between
denotational and operational
semantics:
In operational semantics,
state changes are defined by
coded algorithms
In denotational semantics,
rigorous mathematical functions

55. Denotational Semantics
The state of a program is the values of all its current variables
s = {<i1, v1>, <i2, v2>, …, <in, vn>}
where ik is a variable name and vk is the value of the variable.
Define
VARMAP(ij, s) = vj
a function mapping values vj to names ij in a state s

56. Denotational Semantics
Decimal Numbers
<digit> => 0|1|2|3|4|5|6|7|8|9
<dec_num> => <digit>|<dec_num><digit>
Mdec('0') = 0, Mdec('1') = 1, …, Mdec('9') = 9
Mdec(<dec_num> '0') = 10*Mdec(<dec_num>)
Mdec(<dec_num> '1') = 10*Mdec(<dec_num>) + 1 …
Mdec(<dec_num> '9') = 10*Mdec(<dec_num>) + 9

57. Denotational Semantics Expressions: Map expressions onto Z  {ERROR}
assume <b> is <binary_expr>, <l> is <left_expr>, <r> is <right_expr>
Me(<expr>,s) = case <expr> of
<dec_num> => Mdec(<dec_num>,s)
<var> => if VARMAP(<var>,s) == UNDEF
then ERROR
else VARMAP(<var>,s)
<b> => if ( Me(<b>.<l>,s) == UNDEF
or Me(<b>.<r>,s) == UNDEF)
then ERROR
elseif <b>.<operator> == ‘+’
then Me(<b>.<l>,s) + Me(<b>.<r>,s)
else Me(<b>.<l>,s) * Me(<b>.<r>,s)

58. Denotational Semantics
Assignment Statements: Maps state sets to state sets
Ma(x := E,s) =
if Me(E,s) == ERROR
then ERROR
else s’={<i1’,v1’>,<i2’,v2’>,...,<in’,vn’>},
where for j = 1, 2, ..., n,
if ij <> x, vj’ = VARMAP(ij,s)
if ij == x, vj’ = Me(E,s)

59. Denotational Semantics
Logical Pretest Loops: Maps state sets to state sets
Ml(while B do L, s) =
if Mb(B,s) == UNDEF then ERROR
elseif Mb(B,s) == false then s
elseif Msl(L,s) == ERROR then ERROR
else Ml(while B do L, Msl(L, s))

60. The Meaning of a Loop The meaning of the loop is:
the value of the program variables after the statements in the loop have been executed the prescribed number of times, assuming there have been no errors
In essence, the loop has been converted from iteration to recursion, where the recursive control is mathematically defined by other recursive state mapping functions
Recursion, when compared to iteration, is easier to describe with mathematical rigor

61. Evaluation of Denotational Semantics
USEFUL:
for proving the correctness of programs
can be an aid to language design
RIGOROUS:
provides a way to think about programs
has been used in compiler generation systems
COMPLEX:
of little use to language users

62. Summary BNF and context-free grammars are equivalent meta-languages
Well-suited for describing the syntax of programming languages
An attribute grammar is a descriptive formalism that can describe both the syntax and the semantics of a language
Three primary methods of dynamic semantics description
Operation, axiomatic, denotational