Transcript Chapter 1

Chapter 3
Part II
Describing Syntax
and Semantics
Chapter 3 Topics
• Introduction
• The General Problem of Describing Syntax
• Formal Methods of Describing Syntax
Copyright © 2012 Addison-Wesley. All rights reserved.
1-2
Static Semantics
• Context-free grammars (CFGs) cannot be
used to fully describe all legal forms for a
programming language.
– In cases where it is inefficient
– In cases where it is impossible
Copyright © 2012 Addison-Wesley. All rights reserved.
1-3
Attribute Grammars
• Attribute grammars are
– a formal approach to describing and checking
the correctness of the static semantics rules
of a programming language.
– context-free grammars to which have been
added attributes, computation functions, and
predicate functions.
Copyright © 2012 Addison-Wesley. All rights reserved.
1-4
Attribute Grammars : Definition
•
An attribute grammar is a context-free grammar 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.
•
More formally,
–
–
–
Let X0  X1 ... Xn be a rule
Functions of the form S(X0) = f(A(X1), ... , A(Xn))
define synthesized attributes
Functions of the form I(Xj) = f(A(X0), ... , A(Xn)), for i <= j <= n, define inherited attributes
–
Initially, there are intrinsic attributes on the leaves
Copyright © 2012 Addison-Wesley. All rights reserved.
1-5
Attribute Grammars: Definition
• Inherited attributes pass semantic
information down and across the tree.
• Synthesized attributes pass semantic
information up a parse tree.
• Initially, there are intrinsic attributes on the
leaves whose values are determined outside
the parse tree.
Copyright © 2012 Addison-Wesley. All rights reserved.
1-6
Attribute Grammars
Example 1: An attributed grammar for the syntax and static
semantics for the Ada rule which states that the name on
the end of an Ada procedure must match the procedure’s
name.
Syntax Rule:
<proc_def> -> procedure <proc_name>[1]
<proc_body>
end
<proc_name>[2];
Predicate:
<proc_name>[1].string == <proc_name>[2].string
Copyright © 2012 Addison-Wesley. All rights reserved.
1-7
Attribute Grammars
Example 2: An attributed grammar for the syntax and static semantics of an
assignment statement.
The syntax of the grammar is shown by the grammar:
<assign> -> <var> = <expr>
<expr> -> <var> + <var> | <var>
<var> -> A | B | C
The only variable names are A, B, and C.
The right side of the assignment can be either a variable or an expression in the form
of a variable added to another variable.
The static semantics of the grammar are:
• The variables can be of two types: int or real.
• When there are two variables on the right side of an assignment, they need not be
•
•
•
the same type
The type of the expression when the two types are not the same is real.
When they are the same the type of the expression is the type of the two variables.
The type of the lhs must match the type of the rhs.
Copyright © 2012 Addison-Wesley. All rights reserved.
1-8
Copyright © 2007 Addison-Wesley. All rights reserved.
1-9
Copyright © 2007 Addison-Wesley. All rights reserved.
1-10
Attribute Grammars : continued
Decorating the parse tree - computing the values of the attributes
1. <var>.actual_type  lookup (A)(R4)
2. <expr>.expected_type  <var.actual_type (R1)
3. <var>[2].actual_type  lookup (A) (R4)
4. <var>[3].actual_type  lookup (B) (R4)
5. <expr>.actual_type  either int or real (R2)
6. <expr>.expected_type == <expr>.actual_type
7.
True or False (R2)
Copyright © 2007 Addison-Wesley. All rights reserved.
1-11
Attribute Grammars : continued
Fully attributed parse tree where A is type real and B is type int
Predicate: <expr>.expected_type == <expr>.actual_type
TRUE
Copyright © 2007 Addison-Wesley. All rights reserved.
1-12
Attribute Grammars
• Adding the attributes, their functions and predicates to the
grammar of to describe the static semantics:
<assign> -> <var> = <expr>
<expr> -> <var> + <var> | <var>
<var> -> A | B | C
Attributes for nonterminals
• actual_type: a synthesized attribute for <var> and <expr>
used to store the type of int or real.
– <var> - intrinsically determined
– <expr> - determined from the actual types of the child(ren)
• expected_type: an inherited attribute for <expr> used to
store the expected type of int or real as determined by the
type of the variable on the LHS of the assignment statement
Copyright © 2012 Addison-Wesley. All rights reserved.
1-13
Attribute Grammars
: Evaluated
• Every compiler must be written to check the
static semantic rules of a programming
language whether formally done or not.
Copyright © 2012 Pearson Education. All rights reserved.
1-14
Dynamic Semantics
• There is no single widely acceptable notation or formalism for
describing dynamic semantics.
• Three example methods for describing dynamic semantics.
– Operational
– Denotational
– Axiomatic
• How can these be useful?
– By programmers
– By compiler writers
– By programming language designers
– When correctness proofs are wanted
– For compiler generators
Copyright © 2012 Addison-Wesley. All rights reserved.
1-15
Operational Semantics
• Operational Semantics
– Describe the meaning of a program by executing its statements on a
machine, either simulated or actual.
• Uses of operational semantics:
- Language manuals and textbooks
- Teaching programming languages
• Two levels of uses: Natural & Structural
• Evaluation
- Good if used informally
- Can be extremely complex if used formally
Copyright © 2012 Addison-Wesley. All rights reserved.
1-16
Operational Semantics
•
Examples of the form of a few statements
–
–
•
Ident = var bin_op var
Ident = un_op var
Example of usage for the description of the semantics of the C for statement:
for (expre1; expre2; expre3) {
…
}
Meaning
expr1;
Loop: if expre2 == 0 goto Out
…
expre3;
goto Loop
Out:
Copyright © 2012 Pearson Education. All rights reserved.
1-17
Denotational Semantics
•
The process of building a denotational specification for a language:
-Define a mathematical object for each language entity
-Define a function that maps instances of the language entities onto instances
of the corresponding mathematical objects
The meaning of language constructs are defined by only the values of the program's
variables
<dec_num> 
'0' | '1' | '2' | '3' | '4' | '5' |
'6' | '7' | '8' | '9' |
<dec_num> ('0' | '1' | '2' | '3' |
'4' | '5' | '6' | '7' |
'8' | '9')
Mdec('0') = 0,
Mdec (<dec_num>
Mdec (<dec_num>
…
Mdec (<dec_num>
Mdec ('1') = 1, …, Mdec ('9') = 9
'0') = 10 * Mdec (<dec_num>)
'1’) = 10 * Mdec (<dec_num>) + 1
'9') = 10 * Mdec (<dec_num>) + 9
Copyright © 2012 Addison-Wesley. All rights reserved.
1-18
Evaluation of Denotational Semantics
• Can be used to prove the correctness of
programs
• Provides a rigorous way to think about
programs
• Can be an aid to language design
• Has been used in compiler generation
systems
• Because of its complexity, it are of little use
to language users
Copyright © 2012 Addison-Wesley. All rights reserved.
1-19
Axiomatic Semantics
• Based on formal logic (predicate calculus)
• Original purpose: formal program
verification
• Axioms or inference rules are defined for
each statement type in the language (to
allow transformations of logic expressions
into more formal logic expressions)
• The logic expressions are called assertions
Copyright © 2012 Addison-Wesley. All rights reserved.
1-20
Axiomatic Semantics (continued)
• An assertion before a statement (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
• A weakest precondition is the least
restrictive precondition that will guarantee
the postcondition
Copyright © 2012 Addison-Wesley. All rights reserved.
1-21
Axiomatic Semantics Form
• Pre-, post form: {P} statement {Q}
• An example
– a = b + 1 {a > 1}
– One possible precondition: {b > 10}
– Weakest precondition:
{b > 0}
• An example
– sum = 2 * x + 1 {sum > 1}
– One possible precondition: {x > 10}
– Weakest precondition:
{x > 0}
Copyright © 2012 Addison-Wesley. All rights reserved.
1-22
Program Proof Process
• The postcondition for the entire program is
the desired result
– Work back through the program to the first
statement. If the precondition on the first
statement is the same as the program
specification, the program is correct.
{P1} statement1 {Q1}
{P2} statement2 {Q2}
{P3} statement3 {Q3}
Copyright © 2012 Addison-Wesley. All rights reserved.
1-23
Axiomatic Semantics: Assignment
•
Typical notation for the axiomatic semantics of a given statement form is:
{P} S {Q}
•
An axiom for assignment statements: (x = E): {Qx->E} x = E {Q}
•
•
•
•
•
•
Example: a = b / 2 – 1 { a < 10 }
•
•
•
•
•
Example: x = 2 * y – 3 {x > 25}
2 * y – 3 > 25
2 * y > 28
y > 14
P is {y > 14}
a < 10 , where a = b / 2 - 1
b / 2 – 1 < 10
b / 2 < 11
b < 22
• P is {b < 22}
Copyright © 2012 Addison-Wesley. All rights reserved.
1-24
Axiomatic Semantics: Assignment
Example 3: x = x + y -3 {x > 10}
Calculate P
x + y – 3 > 10
x + y > 13
y > 13 – x
P {y > 13 – x}
Copyright © 2012 Addison-Wesley. All rights reserved.
1-25
Axiomatic Semantics: Sequences
• An inference rule for sequences of the form
S1; S2
{P1} S1 {P2}
{P2} S2 {P3}
{P 1}S1 {P 2},{P 2}S2 {P 3}
{P 1}S1; S2 {P 3}
Copyright © 2012 Addison-Wesley. All rights reserved.
1-26
Axiomatic Semantics
Sequences Example:
y = 3 * x + 1;
x = y + 3;
{ x < 10 }
Start with:
{p}
x = y + 3;
{ x < 10 }
To find P:
y + 3 < 10
y<7
Result:
{y < 7}
x = y + 3;
{ x < 10 }
Next find P for first Final Result is
statement:
{ X < 2}
{P}
y = 3 * x + 1;
y = 3 * x + 1;
{y < 7}
{y < 7}
x = y + 3;
{ x < 10 }
To find P:
y = 3 * x + 1;
3*x+1<7
3*x<6
x<2
{P 1}S1 {P 2},{P 2}S2 {P 3}
Result:
{P 1}S1; S2 {P 3}
{x < 2}
y = 3 * x + 1;
{ y < 7}
Copyright © 2012 Addison-Wesley. All rights reserved.
1-27
Axiomatic Semantics: Selection
• An inference rules for selection
- if B then S1 else S2
{B and P} S1 {Q}, {(not B) and P} S2 {Q}
{P} if B then S1 else S2 {Q}
Copyright © 2012 Addison-Wesley. All rights reserved.
1-28
Axiomatic Semantics: Selection
{B and P} S1 {Q}, {(not B) and P} S2 {Q}
{P} if B then S1 else S2 {Q}
Example: if x > 0 then
y = y -1
else
y=y+1
{y > 0 }
Apply to then clause: y – 1 > 0 yields P = {y>1}
Apply to else clause: y + 1 > 0 yields P = {y>-1}
Because y>1 => y>-1, we use {y > 1} as the
precondition for the entire statement.
Copyright © 2012 Addison-Wesley. All rights reserved.
1-29
Axiomatic Semantics: Loops
• An inference rule for logical pretest loops
{P} while B do S end {Q}
(I and B) S {I}
{I} while B do S {I and (not B)}
where I is the loop invariant (the inductive
hypothesis)
Copyright © 2012 Addison-Wesley. All rights reserved.
1-30
Axiomatic Semantics: Axioms
• Characteristics of the loop invariant: I must
meet the following conditions:
–
–
–
–
–
P => I -- the loop invariant must be true initially
{I} B {I} -- evaluation of the Boolean must not change the validity of I
{I and B} S {I} -- I is not changed by executing the body of the loop
(I and (not B)) => Q
-- if I is true and B is false, Q is implied
The loop terminates -- can be difficult to prove
Copyright © 2012 Addison-Wesley. All rights reserved.
1-31
Evaluation of Axiomatic Semantics
• Developing axioms or inference rules for all
of the statements in a language is difficult
• It is a good tool for correctness proofs, and
an excellent framework for reasoning about
programs, but it is not as useful for
language users and compiler writers
• Its usefulness in describing the meaning of
a programming language is limited for
language users or compiler writers
Copyright © 2012 Addison-Wesley. All rights reserved.
1-32
Denotation Semantics vs Operational
Semantics
• In operational semantics, the state changes
are defined by coded algorithms
• In denotational semantics, the state
changes are defined by rigorous
mathematical functions
Copyright © 2012 Addison-Wesley. All rights reserved.
1-33
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 semantics
description
– Operation, axiomatic, denotational
Copyright © 2012 Addison-Wesley. All rights reserved.
1-34
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 semantics
description
– Operation, axiomatic, denotational
Copyright © 2012 Addison-Wesley. All rights reserved.
1-35