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Abstract Syntax

Learn about the abstract syntax of the L6AST language for parsing lambda expressions. Understand concrete vs. abstract syntax, parsing techniques, interpreting results, and representing syntax with records. Discover examples with the Scheme language. Explore the role of induction, recursion, defining data structures, representing environments, and using alternative approaches.

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Abstract Syntax

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  1. Abstract Syntax L6AST

  2. Language of l-expressions <exp> ::= <identifier> | (lambda( <identifier>)<exp>) | (<exp> <exp>) E.g.,concrete syntax Scheme S-expressions ( lambda (x) ( f ( f x ) ) ) L6AST

  3. Abstract Syntax (vs Concrete Syntax) lambda-exp id body app-exp rand rator var-exp app-exp rator rand id var-exp var-exp id id L6AST

  4. Overview Parse-expression Concrete Syntax Abstract Syntax Unparse-expression Interpreter Results L6AST

  5. Representing Abstract Syntax with Records (define-datatype expression expression? (var-exp (id symbol?)) (lambda-exp (id symbol?) (body expression?)) (app-exp (rator expression?) (rand expression?))) L6AST

  6. Parse: 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)) ))) L6AST

  7. Example (Petite Scheme) > (current-directory “I:\\tkprasad\\cs784\\EOPL-CODE\\interps") > (load "chez-init.scm") > (load "2-2-2.scm") > (parse-expression 'x) (var-exp x) > (parse-expression '(lambda (x) (f x))) (lambda-exp x (app-exp (var-exp f) (var-exp x))) > (parse-expression 45) Error reported by parse-expression: Invalid concrete syntax 45 debug>e >(unparse-expression '(lambda-exp x (app-exp (var-exp f) (var-exp x)))) (lambda (x) (f x)) L6AST

  8. Example (PLT Scheme) L6AST

  9. 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)) ) ))) L6AST

  10. 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. L6AST

  11. Representing Environment L6AST

  12. Alternative 1 (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))))) )) L6AST

  13. Alternative 2 (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) ) ) L6AST

  14. Alternative 3 (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)) L6AST

  15. (cont’d) (define empty-env (lambda () (empty-env-record) )) (define extend-env (lambda (syms vals env) (extended-env-record syms vals env))) (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)))) ) )) L6AST

  16. Queue (define reset (lambda (q) (vector-ref q 0))) (define empty? (lambda (q) (vector-ref q 1))) (define enqueue (lambda (q) (vector-ref q 2))) (define dequeue (lambda (q) (vector-ref q 3))) (define Q (create-queue)) ((enqueue Q) 55) ((empty? Q)) ((dequeue Q)) ((empty? Q)) ((reset Q)) ((dequeue Q)) L6AST

  17. (define create-queue (lambda () (let ((q-in '()) (q-out '())) (letrec ((reset-queue (lambda () (set! q-in '()) (set! q-out '())) ) (empty-queue? (lambda () (and (null? q-in) (null? q-out))) ) (enqueue (lambda (x) (set! q-in (cons x q-in))) ) (dequeue (lambda () (if (empty-queue?) (eopl:error 'dequeue "Not on an empty queue") (begin (if (null? q-out) (begin (set! q-out (reverse q-in)) (set! q-in '()))) (let ((ans (car q-out))) (set! q-out (cdr q-out)) ans))))) ) (vector reset-queue empty-queue? enqueue dequeue)) ))) L6AST

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