Env . model evaluator: Motivation

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• Saves repeated renaming and substitution: explicit substitution is replaced by variable

bindings using new data structures (frame, environment).

• Can be utilized to gain further improvements that cannot be achieved using the

substitution model (The analyzer/compiler)

Env. model evaluator: Motivation

Env. model evaluator: Structure and implementation

ASPASP

Derived expressions

Derived expressions

CoreCore

TestTest

Data structures

Data structures

Utils(a racket lib)

Utils(a racket lib)

Input text

Data structures:

• New data structures: frame, environment

• Additional procedures are added to handle the new ADTs

• A closure keeps the Env in which it was created.

• renaming and substitution are no longer needed.


ASPASP

Derived expressions

Derived expressions

TestTestUtils(a racket lib)

Utils(a racket lib)

Input text

CoreCoreData structures

Data structures

ASP: No changes here!

Core:

• Procedure application (apply procedure) and evaluation of

expressions are done with respect to a certain environment.

Tests:

• Updated accordingly.

4


Box, unbox:

• The global environment may change (E.g., by definitions).

• Values that may change should be boxed (similar to pointers).

#<primitive:+>+Binding: implemented as a Pair

• A correspondence between a variable and its value.

‘((x foo) (4 <procedure (x) x>))

• A substitution from variables to values. Each variable is bound to a single value.

Frame: Implemented as a substitution

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Environment: Implemented as a list of boxed frames

• A finite sequence of frames in which the last frames is the-global-environment.


…

…

E1

box-pointer diagram:

frame

t-g-e

‘((foo …) ( ))

‘((+ - …) (prim+ prim- …))

box box

proc params body

frame

E2

E3

7

Env. model evaluator: Evaluation examples

(1) Evaluating a definition expression:

(define eval-special-form (lambda (exp env) (cond ... ((definition? exp) (if (not (eq? env t-g-e)) (error 'eval "non global definition: ~s" exp) (eval-definition exp))) ...)))

(define eval-definition (lambda (exp) (add-binding! (make-binding (definition-variable exp)

(env-eval (definition-value exp) t-g-e))) 'ok))

(define env-eval (lambda (exp env) (cond ((atomic? exp) (eval-atomic exp env)) ((special-form? exp) (eval-special-form exp env)) ...)))

> (env-eval (derive '(define f (lambda (x) (lambda (y) (+ x y))))) t-g-e)

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Env. model evaluator: Evaluation example (1)

Evaluating a definition expression:

(define eval-special-form (lambda (exp env) (cond ... ((lambda? exp) (eval-lambda exp env)) ...)))


(define eval-lambda (lambda (exp env) (make-procedure (lambda-parameters exp) (lambda-body exp) env)))

(define make-procedure (lambda (parameters body env) (attach-tag (list parameters body env) 'procedure)))

(attach-tag (list '(x) '(lambda (y) (+ x y)) t-g-e)) 'procedure)

P: xB: (lambda (y) (+ x y))

t-g-e

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Evaluating a definition expression:



(define eval-special-form (lambda (exp env) (cond ... ((definition? exp) (if (not (eq? env t-g-e)) (error 'eval "non global definition: ~s" exp) (eval-definition exp))) ...)))

(define eval-definition (lambda (exp) (add-binding! (make-binding (definition-variable exp) (env-eval (definition-value exp) t-g-e))) 'ok))

Done!

(add-binding! (make-binding f (list 'procedure '(x) '(lambda (y) (+ x y)) t-g-e))

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Evaluating a user procedure:


> (derive-eval 'f t-g-e)

(define eval-atomic (lambda (exp env) (if (or (number? exp) (boolean? exp) (null? exp)) exp (lookup-variable-value exp env))))

> '(procedure (x) ((lambda (y) (+ x y))) t-g-e)

• We look up f in the given env.

• The result is our representation of a procedure.

x: 1

GE

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Evaluating an application of a user procedure:

> (derive-eval '(define g (f 1)) t-g-e)

(apply-procedure <procedure (x) ((lambda (y) (+ x y))) t-g-e> (1))

Create a new frame where x maps to 1(make-frame ‘(x) ‘(1))

Extend the environment(extend-env new-frame t-g-e)

Evaluate the body(eval-sequence body new-env)

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• Can be regarded as a nested let expression.

• Similar to let, but defined variable can be used in the following definitions.

> (let ((a 1) (c (* a 2))) (+ a c))

> (define a 10)

Env. model evaluator: Supporting let*

Reminder: What is a let* expression?

Example:

> (let* ((a 1) (c (* a 2))) (+ a c))

> (let ((a 1)) (let ((c (* a 2))) (+ a c)))

> ((λ (a) ((λ (c) (+ a c)) (* a 2)) 1)

> 3

= =

> 21

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An expression (let* ((v1 e1) (v2 e2) … (vn en)) b1 b2 … bm),

with respect to environment E0, is defined as follows:

E1 E0 * make-frame [(v1),(env-eval e1 E0)]E2 E1 * make-frame [(v2),(env-eval e2 E1)]...En En-1* make-frame [(vn),(env-eval en En-1)]

env-eval b1 En) ... (env-eval bm-1 En)Return (env-eval bm En)

Q: How would the evaluation rule for let look like?

A: e1…en will be evaluated with respect to E0. A new frame mapping (v1…vn) to (e1…en)

will extend E0, creating a new environment, E1.

Evaluation rule for a let* expression:

Env. model evaluator: Supporting let*

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(define (let*? exp) (tagged-list? exp 'let*))

(define let*-bindings cadr)

(define let*-body cddr)

(define (let*-variables exp) (map car (let*-bindings exp)))

(define (let*-initial-values exp) (map cadr (let*-bindings exp)))

(define (let*-first-variable exp) (caar (let*-bindings exp)))

(define (let*-first-initial-value exp) (cadar (let*-bindings exp)))

(define (make-let*-bindings vars vals) (map list vars vals))

(define (make-let* bindings body) (append (list 'let* bindings) body))

Env. model evaluator: Supporting let* - as a derived expression

1. Add the required ADT procedures to the ASP

(let* ((v1 e1) (v2 e2) … (vn en)) b1 b2 … bm)

ASPASP

Derived expressions

Derived expressions


Data structures

15


2. Modify the procedure derived?(define (derived? exp) (or (cond? exp) (function-definition? exp) (let? exp) (let*? exp)))

(define (shallow-derive exp) (cond ((cond? exp) (cond->if exp)) ... ((let*? exp) (let*->nested-let exp)) (else "Unhandled " exp)))

3. Modify the procedure shallow-derive


ASPASP

Derived expressions

Derived expressions


Data structures

16


4. Add a translation procedure let*->nested-let(define (let*->nested-let exp) (let ((bindings (let*-bindings exp)) (body (let*-body exp))) (if (null? (cdr bindings)) (make-let bindings body)) (make-let

(make-let-bindings (list (let*-first-variable exp)) (list (let*-first-initial-value exp)))(list (make-let* (cdr bindings) body))))))


ASPASP

Derived expressions

Derived expressions


Data structures

No changes to the core!

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(define (let*->nested-let exp) (let ((bindings (let*-bindings exp)) (body (let*-body exp))) (if (null? (cdr bindings)) (make-let bindings body)) (make-let

(make-let-bindings (list (let*-first-variable exp)) (list (let*-first-initial-value exp)))(list (make-let* (cdr bindings) body))))))

Q: The result is a let exp. Shouldn’t it be further derived?

A: Nah.. It will be taken care of by derive which calls itself until the result exp does not change.

Q: What about the let* expression created by make-let*?

A: Nope. derive is recursively called for all sub expressions.

Q: When the if-condition does not hold, we wrap the body parameter in a list. Why?

A: Since the body of a let expression may consist of several expressions.

(let*->nested-let

(derive

4. Add a translation procedure let*->nested-let

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Env. model evaluator: Supporting let* - as a special form



ASPASP

Derived expressions

Derived expressions


Data structures

DONE

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(define (special-form? exp) (or (quoted? exp) (lambda? exp) (definition? exp) (if? exp) (begin? exp) (let*? exp)))

(define (eval-special-form exp env) (cond ((quoted? exp) (text-of-quotation exp)) ((lambda? exp) (eval-lambda exp env)) … ((let*? exp) (eval-let* exp env))))


ASPASP

Derived expressions

Derived expressions


Data structures


2. Modify the procedure special-form?

3. Modify eval-special-form

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(define (eval-let* exp env) (let ((vars (let*-variables exp)) (vals (let*-initial-values exp)) (body (let*-body exp))) (letrec ((helper

(lambda (vars-lst vals-lst env) (if (null? vars-lst) (eval-sequence body env) (helper (cdr vars-lst) (cdr vals-lst) (extend-env

(make-frame (list (car vars-lst)) (list (env-eval (car vals-lst) env)))

env)))))) (helper vars vals env))))


4. Add the procedure eval-let*

• Extend the environment with a new frame per variable.

• When no variables remain, evaluate the body.

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Env. model Compiler: Motivation

(define length (lambda (lst) (if (null? lst) 0 (* 1 (length (cdr lst))))))

Special form?

What is ? *

What’s fact?

Each time fact is called, the interpreter:

• Uses the ASP to identify the body as an if exp,

• Extracts the required components for evaluation,

• Looks-up variables, etc.

To avoid repeated analysis, the compiler:

• Analyses a given expression in static (compilation) time.

• Returns a procedure that awaits an environment argument.

• Once applied, it evaluates the analyzed expression with respect to the given environment.

• No further Analysis is performed!

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• Using the compiler, we distinguish analysis from evaluation.

• The compiler returns a procedure ready for execution.

• Given an env, the procedure will evaluate the analyzed code with respect to that env.

• No analysis is performed during evaluation.

(define env-eval (lambda (exp env) <body>))

(define analyze (lambda (exp) (lambda (env) <analyzed -body>)))

Env. model Compiler: Introduction

env-eval:

[Exp*Env->Scheme-

Type]

compiler:

[Exp->[Env->Scheme-

Type]

Interpreter vs Compiler:

(define (analyze exp) (cond ((atomic? exp) (analyze-atomic exp)) ((special-form? exp) (analyze-special-form exp)) ((application? exp) (analyze-application exp)) (else

(error "Unknown expression type -- EVAL“ exp))))

Env. model Compiler: Comparing with env-eval

The analyze procedure:

Handling an if expression:(define eval-if (lambda (exp env) (if (true? (env-eval (if-predicate exp) env)) (env-eval (if-consequent exp) env) (env-eval (if-alternative exp) env))))

(define analyze-if (lambda (exp) (let ((pred (analyze (if-predicate exp))) (consequent (analyze (if-consequent exp))) (alternative (analyze (if-alternative exp)))) (lambda (env) (if (true? (pred env)) (consequent env) (alternative env))))))

Evaluator

Analyzer

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(define (analyze-special-form exp) (cond ((quoted? exp) (analyze-quoted exp)) ((lambda? exp) (analyze-lambda exp)) … ((let*? exp) (analyze-let* exp))))

As a derived expression? Already done in the ASP!


2. identify a let* expression as a special form

3. Modify the procedure analyze-special-form:

Env. model Compiler: Supporting let*

As a special form?

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Env. model Compiler: Supporting let*

(define (analyze-let* exp) (let ((vars (let*-variables exp)) (vals (map analyze (let*-initial-values exp))) (body (analyze-sequence (let*-body exp)))) (letrec ((helper (lambda (vars-lst vals-lst env) (if (null? vars-lst) (body env) (helper (cdr vars-lst) (cdr vals-lst) (extend-env (make-frame (list (car vars-lst)) (list ((car vals-lst) env))) env)))))) (lambda (env) (helper vars vals env)))))

4. Add the procedure analyze-let*:

“Translate” eval-let*, to an analyzer procedure: • Curry the env parameter. • Inductively analyze all sub expressions.

Env . model evaluator: Motivation

Documents

exp evallambda exp env

evallambda lambda exp

evaldefinition lambda

exp evalspecial

form lambda exp env

s exp evaldefinition

form exp env application

values operands exp