Abstract Syntax
Different Levels of Syntax
• Lexical syntax
– Basic symbols (names, values, operators, etc.)
• Concrete syntax
– Rules for writing expressions, statements,
programs
– Input to compilers/interpreters
• Abstract syntax
– “Internal” representation
– Captures semantics
CS784(PM)
2
Overview
Parse-expression
Concrete
Syntax
Abstract
Syntax
Unparse-expression
Interpreter
Results
CS784(PM)
3
Concrete vs. Abstract Syntax
• Expressions with common meaning (should)
have the same abstract syntax.
– C: a+b*c
• Assumes certain operator precedence (why?)
– Forth: bc*a+ (reverse Polish)
This expression tree represents the
meaning of expression
Not the same as parse tree (why?)
Does the value depend on traversal order?
abc*+ (or is this it?)
CS784(PM)
4
Parse tree vs. AST
expr
+
expr
expr
expr
expr
1
+
expr
+
(
CS784(PM)
2
expr
)
+
(
3
)
1
2
3
5
Abstract Syntax Tree
• More useful representation of syntax tree
– Less clutter
– Actual level of detail depends on your design
• Basis for semantic analysis
• Later annotated with various information
– Type information
– Computed values
CS784(PM)
6
Compilation in a Nutshell
Source code
(character stream)
if (b == 0) a = b;
Lexical analysis
Token stream if ( b == 0 ) a = b ;
Parsing
if
==
Abstract syntax tree
b
0
(AST)
if
boolean
==
Decorated AST
int
CS784(PM)
b
a
int
;
=
b
Semantic Analysis
int
=
0 int a
lvalue
;
int b
7
λ-expressions
1. <exp> ::=
2. <identifier>
3.
| (lambda (<identifier> ) <exp>)
4.
| (<exp> <exp>)
• Compare with Scheme S-expressions.
• EOPL3 p52: Lc-exp
CS784(PM)
8
Lc-exp ::=
• Identifier
– var-exp (var)
• (lambda (Identifier) Lc-exp)
– lambda-exp (bound-var body)
• (Lc-exp Lc-exp)
– app-exp (rator rand)
• Abstract syntax
CS784(PM)
9
EOPL3 Scheme: define-datatype
• Syntax definition:
(define-datatype type-name type-predicate-name
{(variant-name {(field-name predicate)}* )} + )
• An Example:
(define-datatype environment environment?
(empty-env-record)
(extended-env-record
(syms (list-of symbol?))
(vals (list-of scheme-value?))
(env environment?)))
• Data types built by define-datatype may be mutually recursive.
CS784(PM)
10
Syntax Driven Representation
(define-datatype expression expression?
(var-exp
(id symbol?))
(lambda-exp
(id symbol?)
(body expression?))
(app-exp
(rator expression?)
(rand expression?)))
CS784(PM)
11
Figure 2.2: (lambda (x) (f (f x)))
CS784(PM)
12
Concrete to Abstract Syntax
(define parse-expression
(lambda (datum)
(cond
((symbol? datum) (var-exp datum))
((pair? datum)
(if (eqv? (car datum) 'lambda)
(lambda-exp (caadr datum)
(parse-expression (caddr datum)))
(app-exp
(parse-expression (car datum))
(parse-expression (cadr datum))))
)
(else (eopl:error
'parse-expression
"Invalid concrete syntax ~s" datum))
CS784(PM)
)))
13
DrRacket
CS784(PM)
14
Unparse: Abstract to Concrete Syntax
(define unparse-expression
(lambda (exp)
(cases expression exp
(var-exp (id) id)
(lambda-exp (id body)
(list 'lambda (list id)
(unparse-expression body)) )
(app-exp (rator rand)
(list (unparse-expression rator)
(unparse-expression rand)) ))))
CS784(PM)
15
Role of Induction and Recursion
• Define data structures (infinite values) by
induction.
– Seed elements.
– Closure operations.
• Define functions (operations) by recursion.
– Boundary/Basis case.
– Composite/Recursive case.
• Prove properties using structural induction.
– Basis case.
– Inductive step.
CS784(PM)
16
The Environment
• Environment:
– table of variable-value pairs
– Chronologically later var-value pair overrides
• Constructors
– empty-env
– extend-env
• Observer
– apply-env
CS784(PM)
17
The Environment Spec
• (empty-env) = ∅
• (apply-env f var) = f (var)
• (extend-env var v  f] ) = g,
– where g(var1 )
= v, if var1 = var
= f (var1). otherwise
• from EOPL3 p36
CS784(PM)
18
An Example Env e
• (define e
(extend-env ’d 6
(extend-env ’y 8
(extend-env ’x 7
(extend-env ’y 14
(empty-env))))))
• e(d)=6, e(x)=7, e(y)=8
CS784(PM)
19
Representing Environment
•
•
•
•
Many representations are possible
Speedy access
Memory frugal
Change in the interface: Syms x Vals
(define extend-env
(lambda (syms vals env) … ))
CS784(PM)
20
Alt-1: env is a List of Ribs
•
•
•
left rib: list of variables
right rib: corresponding list of values.
Exercise 2.11 EOPL3 (Ribcage)
CS784(PM)
21
Alt-1: List of Ribs (Ribcage)
(define empty-env
(lambda () '()))
(define extend-env
(lambda (syms vals env)
(cons (list syms vals) env) ))
(define apply-env
(lambda (env sym)
(if (null? env)
(eopl:error 'apply-env "No binding for ~s" sym)
(let ((syms (car (car env)))
(vals (cadr (car env)))
(env (cdr env)))
(let ((pos (rib-find-position sym syms)))
(if (number? pos)
(list-ref vals pos)
(apply-env env sym)))))))
CS784(PM)
22
Alt-2: env is a Unary Function
(define empty-env
(lambda ()
(lambda (sym)
(eopl:error 'apply-env "No binding for ~s" sym)) ))
(define extend-env
(lambda (syms vals env)
(lambda (sym)
(let ((pos (list-find-position sym syms)))
(if (number? pos)
(list-ref vals pos)
(apply-env env sym)))) ))
(define apply-env
(lambda (env sym)
(env sym) ))
CS784(PM)
23
Alt-3: Tagged Records
(define-datatype environment environment?
(empty-env-record)
(extended-env-record
(syms (list-of symbol?))
(vals (list-of scheme-value?))
(env environment?)))
(define scheme-value? (lambda (v) #t))
(define empty-env
(lambda ()
(empty-env-record) ))
(define extend-env
(lambda (syms vals env)
(extended-env-record syms vals env)))
CS784(PM)
24
Alt-3: Tagged records
(define apply-env
(lambda (env sym)
(cases environment env
(empty-env-record ()
(eopl:error 'apply-env "No binding for ~s" sym))
(extended-env-record (syms vals env)
(let ((pos (list-find-position sym syms)))
(if (number? pos)
(list-ref vals pos)
(apply-env env sym)))))))
CS784(PM)
25
Queue
(define
(define
(define
(define
(define
reset
empty?
enqueue
dequeue
Q
(lambda
(lambda
(lambda
(lambda
(q)
(q)
(q)
(q)
(vector-ref
(vector-ref
(vector-ref
(vector-ref
q
q
q
q
0)))
1)))
2)))
3)))
(create-queue))
((enqueue Q) 55)
((empty? Q))
((dequeue Q))
((empty? Q))
((reset Q))
((dequeue Q))
CS784(PM)
26
1. (define create-queue
2.
(lambda ()
3.
(let ((q-in '()) (q-out '()))
4.
(letrec
5.
((reset-queue
6.
(lambda ()
7.
(set! q-in '()) (set! q-out '())) )
8.
(empty-queue?
9.
(lambda ()
10.
(and (null? q-in) (null? q-out))) )
11.
(enqueue
12.
(lambda (x)
13.
(set! q-in (cons x q-in))) )
14.
(dequeue
15.
(lambda ()
16.
(if (empty-queue?)
17.
(eopl:error 'dequeue "Not on an empty queue")
18.
(begin
19.
(if (null? q-out)
20.
(begin
21.
(set! q-out (reverse q-in)) (set! q-in '())))
22.
(let ((ans (car q-out)))
23.
(set! q-out (cdr q-out))
24.
ans))))) )
25.
(vector reset-queue empty-queue? enqueue dequeue))
26. )))
CS784(PM)
27