1. Announcements Quiz 6 HW7 due Tuesday, October 30
Questions?
We’ll have grades for HWs 4 over the weekend
Fall 18 CSCI 4430, A Milanova/BG Ryder

2. Announcements Exam 2 on Friday, November 2nd Topics
You are allowed 2 “cheat” pages only
Practice tests under Course Materials
Topics
Binding and scoping
Attribute grammars
Scheme
Lists, recursion, higher-order functions (especially map and fold), tail recursion, let expressions, scoping
Lambda calculus and reduction strategies (as known as evaluation order)

3. Last class Scheme Tail recursion Binding with let expressions
Scoping in Scheme, closures
Scoping, revisited
Fall 18 CSCI 4430, A Milanova/BG Ryder

4. Today’s Lecture Outline
Lambda calculus
Introduction
Syntax and semantics
Free and bound variables
Substitution, formally
Rules of lambda calculus
Normal forms
Reduction strategies
Fall 18 CSCI 4430, A Milanova/BG Ryder

5. Lambda Calculus
Reading: Scott, Ch. 11 on CD

6. Lambda Calculus A theory of functions
Theory behind functional programming
Turing-complete: any computable function can be expressed and evaluated using the calculus
“Lingua franca” of programming language research
Lambda () calculus expresses function definition and function application
f(x)=x*x becomes x. x*x
g(x)=x+1 becomes x. x+1
f(5) becomes (x. x*x) 5  5*5  25
Fall 18 CSCI 4430, A Milanova

7. Syntax of Pure Lambda Calculus
Convention:
notation f, x, y, z for variables;
E, M, N, P, Q for expressions
E ::= x | ( x. E1 ) | ( E1 E2 )
A -expression is one of
Variable: x
Abstraction (i.e., function definition): x. E1
Application: E1 E2
-calculus formulae (e.g., ( x. (x y) )) are called expressions or terms
( x. (x y) ) corresponds to (lambda (x) (x y)) in Scheme!
I use notation f, x, y, z for variables; E, M, N, P, Q for expressions
Fall 18 CSCI 4430, A Milanova/BG Ryder

8. Syntactic Conventions
Parentheses may be dropped from ( E1 E2 ) or ( x.E )
E.g., ( f x ) may be written as f x
Function application groups from left-to-right (i.e., it is left-associative)
E.g., x y z abbreviates ( ( x y ) z )
E.g., E1 E2 E3 E4 abbreviates ( ( ( E1 E2 ) E3 ) E4 )
Parentheses in x (y z) are necessary! Why?
Fall 18 CSCI 4430, A Milanova/BG Ryder

9. Syntactic Conventions
Application has higher precedence than abstraction
Another way to say this is that the scope of the dot extends as far to the right as possible
E.g., x. x z = x. ( x z ) = ( x. ( x z ) ) = ( x. x z ) ≠ ( ( x. x ) z )
WARNING: This is the most common syntactic convention (e.g., Pierce 2002). However some books give abstraction higher precedence; you might have seen that different convention

10. Terminology Parameter (also, formal parameter)
E.g., x is the parameter in x. x z
Argument (also, actual argument)
E.g., expression z. z is the argument in
(x. x) (z. z)
Can you guess what this evaluates to?
Fall 18 CSCI 4430, A Milanova/BG Ryder

11. Currying In lambda calculus, all functions have one parameter
How do we express n-ary functions?
Currying expresses an n-ary function in terms of n unary functions
f(x,y) = x+y, becomes (x.y. x + y)
(x.y. x + y) 2 3  (y. 2 + y) 3  = 5
Fall 18 CSCI 4430, A Milanova/BG Ryder

12. Currying in Scheme (define curried-plus
(lambda (a) (lambda (b) (+ a b))))
(curried-plus 3) returns what?
Returns the plus-3 function (or more precisely, it returns a closure)
((curried-plus 3) 2) returns what? 5
Scheme departs from the lamda calculus and allows us to define n-ary functions.
Of course, we can still defined curried functions in Scheme. Here is an example.
Fall 18 CSCI 4430, A Milanova/BG Ryder

13. Currying f(x1, x2,…,xn) = g x1 x2 … xn
Function g is said to be the curried form of f.
g1 x2
g2 x3 …
Fall 18 CSCI 4430, A Milanova/BG Ryder

14. Semantics of Pure Lambda Calculus
An expression has as its meaning the value that results after evaluation is carried out
Somewhat informally, evaluation is the process of reducing expressions
E.g., (x.y. x + y) 3 2  (y. 3 + y) 2  = 5
(Note: this example is just an informal illustration. There is no + in the pure lambda calculus!)
x.y. x is assigned the meaning of TRUE
x.y. y is assigned the meaning of FALSE

15. Free and Bound Variables
Reducing expressions
Consider expression ( x.y. x y ) (y w)
Try 1:
Reducing this expression results in the following
( y. x y ) [(y w)/x] = ( y. (y w) y )
The above notation means: we substitute argument (y w) for every occurrence of parameter x in body ( y. x y ).
But what is wrong here?
( x.y. x y ) (y w): different y’s! If we substitute (y w) for x, the “free” y will become “bound”!

16. Free and Bound Variables
Try 2:
Rename “bound” y in y. x y to z: z. x z
(x.y. x y) (y w) => (x.z. x z) (y w)
E.g., in C, int id(int p) { return p; } is exactly the same as int id(int q) { return q; }
Applying the reduction rule results in
( z. x z ) [(y w)/x] => ( z. (y w) z )
Fall 18 CSCI 4430, A Milanova/BG Ryder

17. Free and Bound Variables
Abstraction ( x. E ) is also referred as binding
Variable x is said to be bound in x. E
The set of free variables of E is the set of variables that are unbound in E
Defined by cases on E
Var x:
App E1 E2:
Abs x.E:
free(x) = {x}
free(E1 E2) = free(E1) U free(E2)
free(x.E) = free(E) - {x}
Fall 18 CSCI 4430, A Milanova

18. Free and Bound Variables
A variable x is bound if it is in the scope of a lambda abstraction: as in x. E
Variable is free otherwise
1. (x. x) y
2. (z. z z) (x. x)
3. x.y.z. x z (y (u. u))
y is free.
No free variables
Fall 18 CSCI 4430, A Milanova

19. Free and Bound Variables
We must take free and bound variables into account when reducing expressions
E.g., (x.y. x y) (y w)
First, rename bound y in y. x y to z: z. x z (more precisely, we have to rename to a variable that is NOT free in (y w))
(x.y. x y) (y w)  (x.z. x z) (y w)
Second, apply the reduction rule that substitutes (y w) for x in the body ( z. x z )
( z. x z ) [(y w)/x]  ( z. (y w) z ) = z. y w z

20. Substitution, formally
(x.E) M  E[M/x] replaces all free occurrences of x in E by M
E[M/x] is defined by cases on E:
Var: y[M/x] =
y[M/x] =
App: (E1 E2)[M/x] =
Abs: (y.E1)[M/x] =
(y.E1)[M/x] =
M if x = y
y otherwise
(E1[M/x] E2[M/x])
y.E1 if x = y
z.((E1[z/y])[M/x]) otherwise,
where z NOT in free(E1) U free(M) U {x}
Fall 18 CSCI 4430, A Milanova

21. Substitution, formally
(x.y. x y) (y w)
(y. x y)[(y w)/x]
1_. ( ((x y)[1_/y])[(y w)/x] )
1_. ( (x 1_)[(y w)/x] )
1_. ( (y w) 1_ )
1_. y w 1_
You will have to implement this substitution algorithm in Haskell
Fall 18 CSCI 4430, A Milanova/BG Ryder

22. Rules (Axioms) of Lambda Calculus
 rule (-conversion): renaming of parameter (choice of parameter name does not matter)
x. E  z. E[z/x] provided that z is not free in E
e.g., x. x x is the same as z. z z
 rule (-reduction): function application (substitutes argument for parameter)
(x.E) M  E[M/x]
Note: E[M/x] as defined on previous slide!
e.g., (x. x) z  z
Fall 18 CSCI 4430, A Milanova

23. Rules of Lambda Calculus: Exercises
Use -conversion and/or β-reduction:
(x. x) y  ?
(x. x) (y. y)  ?
(x.y.z. x z (y z)) (u. u) (v. v) 
Notation:  denotes that expression on the left reduces
to the expression on the right, through a sequence -conversions
and β-reductions.
Fall 18 CSCI 4430, A Milanova

24. Reductions
An expression ( x.E ) M is called a redex (for reducible expression)
An expression is in normal form if it cannot be β-reduced
The normal form is the meaning of the term, the “answer”
Fall 18 CSCI 4430, A Milanova

25. Questions Is z. z z in normal form?
Answer: yes, it cannot be beta-reduced
Is (z. z z) (x. x) in normal form?
Answer: no, it can be beta-reduced
Fall 18 CSCI 4430, A Milanova/BG Ryder

26. Definitions of Normal Form
Normal form (NF): a term without redexes
Head normal form (HNF)
x is in HNF
(x. E) is in HNF if E is in HNF
(x E1 E2 … En) is in HNF
Weak head normal form (WHNF)
x is in WHNF
(x. E) is in WHNF
(x E1 E2 … En) is in WHNF
Fall 18 CSCI 4430, A Milanova (from MIT’s 2015 Program Analysis OCW)

27. Questions z. z z is in NF, HNF, or WHNF? (z. z z) (x. x) is in?
x.y.z. x z (y (u. u)) is in?
(x.y. x) z ((x. z x) (x. z x)) is in?
z ((x. z x) (x. z x)) is in?
z.(x.y. x) z ((x. z x) (x. z x)) is in?
(We will be reducing to NF, mostly)

28. More Reduction Exercises
C = x.y.f. f x y
H = f. f (x.y. x) T = f. f (x.y. y)
What is H (C a b)?
(f. f (x.y. x)) (C a b)
(C a b) (x.y. x)
((x.y.f. f x y) a b) (x.y. x)
(f. f a b) (x.y. x)
(x.y. x) a b a
Fall 18 CSCI 4430, A Milanova (from MIT 2015 Program Analysis OCW)

29. Exercise S = x.y.z. x z (y z) I = x. x What is S I I I?
An expression with no free
variables is called combinator.
S, I, C, H, T are combinators.
S = x.y.z. x z (y z)
I = x. x
What is S I I I?
( x.y.z. x z (y z) ) I I I
(y.z. I z (y z) ) I I
(z. I z (I z) ) I
I I (I I) = (x. x) I (I I)
I (I I) = (x. x) (I I)
I I = (x. x) I  I
Reducible expression is underlined
at each step.
Fall 18 CSCI 4430, A Milanova

30. Today’s Lecture Outline
Lambda calculus
Introduction
Syntax and semantics
Free and bound variables
Substitution, formally
Rules of the lambda calculus
Normal forms
Reduction strategies
Fall 18 CSCI 4430, A Milanova/BG Ryder