;; Towards a Scheme Interpreter for the Lambda Calculus — Part 1: Syntax ;; 5 points ;; , and pre-requisite for all subsequent parts of the project ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; ;; All programming is to be carried out using the pure functional sublanguage of R5RS Scheme. ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; ;; You might want to have a look at http://www.cs.unc.edu/~stotts/723/Lambda/overview.html ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; ;; 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 ;; ::= | (lambda ( ) )  |  ( ) ;; 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 () ), we say that is a binder that ;; binds all occurrences of that variable in the body, , 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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