CS 251 Fall 20192019 Principles of of Programming Programming Languages Languages Ben Wood Topics λ Ben Wood • Function definitions Defining Racket: • Function application Functions • Functions are first-class values. https://cs.wellesley.edu/~cs251/ Functions 1 Functions 2 Anonymous function definition expressions Function application (call) Syntax: (lambda (x1 … xn) e) Syntax: (e0 e1 ... en) – parameters: x1 through xn are identifiers – body: e is any expression Evaluation: 1. Under the current dynamic environment, E, Evaluation: evaluate e0 through en to values v0, …, vn. 1. The result is a function closure, ⟨E, (lambda (x1 … xn) e)⟩, 2. If v0 is a function closure of n arguments, holding the current environment, E, and the function. ⟨E', (lambda (x1 … xn) e)⟩ then [closure] The result is the result of evaluating the closure E ⊢ (lambda (x1 … xn) e) ↓ ⟨E, (lambda (x1 … xn) e)⟩ body, e, under the closure environment, E', Note: extended with argument bindings: – An anonymous function definition is an expression. x1 ⟼ v1, …, xn ⟼ vn. – A function closure is a new kind of value. Closures are not expressions. Otherwise, there is a type error. – This is a definition, not a call. The body, e, is not evaluated now. – lambda from the λ-calculus. Functions 3 Functions 4 Function application (call) Function application derivation example Syntax: (e0 e1 … en) Assume initial environment is empty. ((lambda (x) (* x x)) (- 12 8)) Evaluation: E ⊢ e0 ↓ ⟨E', (lambda (x1 … xn) e)⟩ E ⊢ e1 ↓ v1 … E ⊢ en ↓ vn x1 ⟼ v1, …, xn ⟼ vn, E' ⊢ e ↓ v [apply] E ⊢ (e0 e1 … en) ↓ v Functions 5 Functions 6 Function bindings and recursion PL design/implementation: layers A function is an expression, so define works: (define square (lambda (x) (* x x))) kernel define also adds self-binding to function's environment*, primitive values, supporting recursion. data types (define pow (lambda (base exp) syntactic sugar (if (< exp 1) 1 (* base (pow base (- exp 1)))))) standard libraries During an application of this function, pow will be bound to this function. * Magic for now. We will be precise later. user libraries Functions 7 Functions 8 syntax syntax syntax syntax syntax syntax syntax syntax syntax syntax syntax Syntactic sugar for function bindings syntax More syntactic sugar (define (pow base exp) (if (< exp 1) What else looks like a function application? 1 (* base (pow base (- exp 1))))) Syntactic sugar: simpler syntax for common idiom. What looks like a function application but – Static textual translation to existing features. really is not? – i.e., not a new feature. Desugar (define (x0 x1 … xn) e) to (define x0 (lambda (x1 … xn) e)) Functions 9 Functions 10 So sweet Racket so far Expressions like (+ e1 e2), (< e1 e2), and (not e)are Racket declaration bindings: really just function calls! (define x e) Racket expressions (most of the kernel language!) Initial top-level environment binds built-in functions: literal values (numbers, booleans, strings): 251, 3.141, #t, "PL" + ⟼ addition function, variable references: x, fact, positive?, fib@n-1 - ⟼ subtraction function, conditionals: (if e1 e2 e3) * ⟼ multiplication function, functions: (lambda (x1 … xn) e) < ⟼ less-than function, function application: (e0 e1 … en) not ⟼ boolean negation function, What about: – Assignment? Unnecessary! Thread state through. … – Loops? Unnecessary! Use recursion. (where some built-in functions do primitive things) – Data structures? lambda is all we need, but other options soon. Functions 11 Functions 12 Racket kernel syntax so far Meta-syntax so far Bindings – Syntax of our evaluation model. b ::= (define x e) – Not part of the Racket syntax. "*" is grammar meta-syntax – Cannot write in source programs. Expressions that means "zero or more of e ::= v | x | (if e e e) the preceding symbol." | (lambda (x*) e) | (e e*) Values (+closures) v ::= … | ⟨E, (lambda (x*) e)⟩ Literal Values (booleans, numbers, strings) v ::= #f | #t | n | s Environments Identifiers (variable names) E ::= . | x ⟼ v, E x (see valid identifier explanation) Functions 13 Functions 14 Racket kernel dynamic semantics so far Binding evaluation: E ⊢ b ß E' [define] E ⊢ e ↓ v E ⊢ (define x e) ß x ⟼ v, E' Expression evaluation: E ⊢ e ↓ v [if nonfalse] [value] [var] E ⊢ e1 ↓ v1 E ⊢ v ↓ v E(x) = v E ⊢ e2 ↓ v2 E ⊢ x ↓ v v1 is not #f [apply] E ⊢ (if e1 e2 e3) ↓ v2 E ⊢ e0 ↓ ⟨E', (lambda (x1 … xn) e)⟩ [if false] E ⊢ e1 ↓ v1 … E ⊢ en ↓ vn E ⊢ e1 ↓ #f x1 ⟼ v1, …, xn ⟼ vn, E' ⊢ e ↓ v E ⊢ e3 ↓ v3 E ⊢ (e0 e1 … en) ↓ v E ⊢ (if e1 e2 e3) ↓ v3 Functions 15.