Posted: June 13th, 2022

;; Towards a Scheme Interpreter for the Lambda Calculus — Part 1: Syntax

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;; 5 points

;; , and pre-requisite for all subsequent parts of the project

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;; All programming is to be carried out using the pure functional sublanguage of R5RS Scheme.

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;; 1. The lambda calculus is a particularly simple programming language consisting only of

;; variable references, lambda expressions with a single formal parameter, and function

;; applications. A BNF definition of lambda calculus expressions is

;; <expr> ::= <variable> | (lambda ( <variable> ) <expr> ) | ( <expr> <expr> )

;; Design a data type for the lambda calculus, with constructors, selectors, and classifiers.

;; For concrete representation, use Scheme, as follows: an identifier should be represented as

;; a quoted Scheme variable, a lambda expression (lambda (x) E) as the quoted 3-element list

;; ‘(lambda (x) [list representing E]), and an application (E1 E2) as the quoted 2-element list

;; ‘([list representing E1] [list representing E2])

;; 2. In (lambda (<variable>) <expr>), we say that <variable> is a binder that

;; binds all occurrences of that variable in the body, <expr>, unless some intervening

;; binder of the same variable occurs. Thus in (lambda (x) (x (lambda (x) x))),

;; the first occurrence of x binds the second occurrence of x, but not

;; the fourth. The third occurrence of x binds the fourth occurrence of x.

;; A variable x occurs free in an expression E if there is some occurrence of x which is not

;; bound by any binder of x in E. A variable x occurs bound in an expression E if it is

;; not free in E. Thus x occurs free in (lambda (y) x), bound in (lambda (x) x), and both

;; free and bound in (lambda (y) (x (lambda (x) x))).

;; As a consequence of this definition, we can say that a variable x occurs free in a

;; lambda calculus expression E iff one of the following holds:

;; (i) E = x

;; (ii) E = (lambda (y) E’), where x is distinct from y and x occurs free in E’

;; (iii) E = (E’ E”) and x occurs free in E’ or x occurs free in E”

;; Observe that this is an inductive definition, exploiting the structure of lambda calculus

;; expressions.

;; Similarly, a variable x occurs bound in a lambda calculus expression E iff one of the

;; following holds:

;; (i) E = (lambda (x) E’) and x occurs free in E’

;; (ii) E = (lambda (y) E’), and x occurs bound in E’: here, y may be x, or distinct from x

;; (iii) E = (E1 E2) and x occurs bound in either E1 or E2

;; Develop and prove correct a procedure free-vars that inputs a list representing a lambda calculus

;; expression E and outputs a list without repetitions (that is, a set) of the variables occurring

;; free in E.

;; Develop and prove correct a procedure bound-vars that inputs a list representing a lambda calculus

;; expression E and outputs the set of variables which occur bound in E.

;; 3. Define a function all-ids which returns the set of all symbols — free or bound variables,

;; as well as the lambda identifiers for which there are no bound occurrences — which occur in

;; a lambda calculus expression E.

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