Photos from the lectures

Photos from class are posted in a Google album. Anyone can add photos.

23 January 2017: Introduction to Comp 105

There are PDF slides for 1/23/2017.

Handout: 105 Tip Sheet

Handout: 105 Syllabus

Handout: Experiences of successful 105 students

Handout: Homework


Why so many languages?

Topic of study: the stuff software is made of

Conclusion: make it easier to write programs that really work

Your language can make you or break you.

What this course isn’t

Reusable Principles

What if the course were called “Houses”?

The same thing for programming languages:

What Programming Languages is, technically

What can you get out of Comp 105?

Students who get the most out of 105 report

Induction and recursion: linguistic structure


Show some example numerals

Q: How many numerals are there?

In-class Exercise: Inductive definitions of numerals
“Calculator expressions”


    1  2  3  +
    4  5  6  *
    7  8  9  -
    (  0  )  /

In-class Exercise: Inductive description of calculator expressions

define the set of calculator expressions

Concrete and abstract Syntax

Programming-languages people are wild about compositionality.

Example of compositionality: concrete syntax (grammar)

Programming languages more orderly than natural language.

Example of non-compositionality

Inductive descriptions of concrete syntax: Backus-Naur Form


Looky here:

9 - (6 / 13)    9 - 6 / 13     (9 - 6) / 13

Superficially different ways of writing one expression?

Introducing our common framework

Our common framework

Goal: eliminate superficial differences

No new language ideas.

Imperative programming with an IMPerative CORE:

Idea of LISP syntax

Parenthesized prefix syntax:


(+ 2 2)
(if (isbound? x rho) (lookup rho x) (error 99))

(For now, we use just the round brackets)

Impcore structure

Two syntactic categories: expressions, definitions

No statements!—expression-oriented

(if e1 e2 e3)
(while e1 e2)
(set x e)
(begin e1 ... en)
(f e1 ... en)

Evaluating e has value, may have side effects

Functions f named (e.g., + - * / = < > print)

The only type of data is “machine integer” (deliberate oversimplification)

Syntactic structure of Impcore

An Impcore program is a sequence of definitions

(define mod (m n) (- m (* n (/ m n))))


int mod (int m, int n) { 
  return m - n * (m / n); 

Impcore variable definition


(val n 99)


int n = 99;

Concrete syntax for Impcore

Definitions and expressions:

def ::= (define f (x1 ... xn) exp)  ;; "true" defs
     |  (val x exp)                
     |  exp
     |  (use filename)              ;; "extended" defs
     |  (check-expect exp1 exp2)
     |  (check-error exp)

exp ::= integer-literal
     |  variable-name
     |  (set x exp)
     |  (if exp1 exp2 exp3)
     |  (while exp1 exp2)
     |  (begin exp1 ... expn)
     |  (function-name exp1 ... expn)  

Recursion: a review

Case analysis depends on the inductive structure of the input data

Consuming natural numbers recursively

Q: How many natural numbers are there?

Q: What is a structure that works for natural numbers?

Ways a recursive function could decompose a natural number n.

  1. Peel back one (Peano numbers):

    n = 0
    n = m + 1,    m is also a natural number
  2. Split into two pieces:

    n = 0
    n = k + (n - k)    0 < k < n   (everything gets smaller)
  3. Sequence of decimal digits (see study problems on digits)

    n = d,               where 0 <= d < 10
    n = 10 * m + d,      where 0 <= d < 10 and m > 0

To do your homework problems, which I recommend starting today, you’ll need to invent at least one more.

Let’s try one now!

Recursive-function problem

Exercise: all-fours?

Write a function that takes a natural number n and returns true (1) if and only if all the digits in n’s numeral are 4’s.

Begin with unit tests (which also document):

(check-expect (all-fours?  4) 1)
(check-expect (all-fours?  5) 0)
(check-expect (all-fours? 44) 1)
(check-expect (all-fours? 14) 0)

Choose inductive structure for natural numbers:

Solution to ``all-fours?’’

(check-expect (all-fours?  4) 1)
(check-expect (all-fours?  5) 0)
(check-expect (all-fours? 44) 1)
(check-expect (all-fours? 14) 0)

;; induction: n is d,  where 0 < d < 10, or
;;            n is 10 * m + d,  where m > 0 ...

(define all-fours? (n)
   (if (< n 10)
       (= n 4)
       (and (= 4 (mod n 10))
            (all-fours? (/ n 10)))))

(Now we can talk a bit about the course.)

Course logistics: Books

Either available, or first part will go up on web right after class.

Course logistics: Recitation

25 January 2017: Abstract syntax and operational semantics

There are PDF slides for 1/25/2017.

Handout: 105 Impcore Semantics, Part 1

Opening sermon: What the course is like

Why get a university education?

The course is hard, but we are totally committed to your success

In practice, about half A’s—sometimes more, sometimes less.

Your approach

If you’re not sure how to work mindfully, start with frequent pauses (try every 15 minutes)

Course logistics and administration

You must get my book (Both Volumes!!!)

You won’t need the book on ML for about a month


Homework will be frequent and challenging:

Both individual and pair work:

Arc of the homework looks something like this:

And it’s more or less four-star homeworks from there on out.

Lesson: Don’t make plans based on the first couple of homeworks!

The role of lectures

In a 100-level course, you are responsible for your own learning


Questions and answers on Piazza

Other policies and procedures in the syllabus

Who am I? What am I called?

Call me “Norman,” “Professor Ramsey”, or “Mister Ramsey.” In a pinch, “Professor” will do. Do not call me “Ramsey”; where I come from, that form of address is insulting.

Who are you? What are you called?

Thinking about programming languages

Where have we been?

Short discussion: Two things you learned in the first class

This week: abstract syntax and operational semantics (next homework)

Programming-language semantics

“Semantics” means “meaning.”

We want a computational notion of meaning.

What problem are we trying to solve?

Know what’s supposed to happen when you run the code

Ways of knowing:

(For homework, you’ll prove that our specification is unambiguous.)

Q: Does anyone know the beginner exercise “make a peanut butter and jelly sandwich”? (Videos on YouTube)

Why bother with precise semantics?

(Needed to build implementation, tests)

Same reason as other forms of math:

The programming languages you encounter after 105 will certainly look different from what we study this term. But most of them will actually be the same. Studying semantics helps you identify that.

The idea: your new skills will apply

Behavior decomposes

What happens when we run (* y 3)?

We must know something about *, y, 3, and function application.

Knowledge is expressed inductively

Review: Concrete syntax for Impcore

Definitions and expressions:

def ::= (define f (x1 ... xn) exp)
     |  (val x exp)                
     |  exp
     |  (use filename)            
     |  (check-expect exp1 exp2)
     |  (check-error exp)

exp ::= integer-literal      ;; atomic forms
     |  variable-name
     |  (set x exp)          ;; compound forms
     |  (if exp1 exp2 exp3)
     |  (while exp1 exp2)
     |  (begin exp1 ... expn)
     |  (function-name exp1 ... expn)  

How to define behaviors inductively

Expressions only

Base cases (plural): numerals, names

Inductive steps: compound forms

First, simplify the task of definition

What’s different? What’s the same?

 x = 3;               (set x 3)

 while (i * i < n)    (while (< (* i i) n)
   i = i + 1;            (set i (+ i 1)))

Abstract away gratuitous differences

(See the bones beneath the flesh)

Abstract syntax

Same inductive structure as BNF

More uniform notation

Good representation in computer

Concrete syntax: sequence of symbols

Abstract syntax: ???

The abstraction is a tree

The abstract-syntax tree (AST):

Exp = LITERAL (Value)
    | VAR     (Name)
    | SET     (Name name, Exp exp)
    | IFX     (Exp cond, Exp true, Exp false)
    | WHILEX  (Exp cond, Exp exp)
    | BEGIN   (Explist)
    | APPLY   (Name name, Explist actuals)

One kind of “application” for both user-defined and primitive functions.


Question: What do we assign behavior to?

Answer: The Abstract Syntax Tree (AST) of the program.

Question: How can we represent all while loops?

while (i < n && a[i] < x) { i++ }


As a data structure:

In C, trees are a bit fiddly

typedef struct Exp *Exp;
typedef enum {
} Expalt;        /* which alternative is it? */

struct Exp {  // only two fields: 'alt' and 'u'!
    Expalt alt;
    union {
        Value literal;
        Name var;
        struct { Name name; Exp exp; } set;
        struct { Exp cond; Exp true; Exp false; } ifx;
        struct { Exp cond; Exp exp; } whilex;
        Explist begin;
        struct { Name name; Explist actuals; } apply;
    } u;

Let’s picture some trees

An expression:

  (f x (* y 3))

(Representation uses Explist)

A definition:

  (define abs (n)
    (if (< n 0) (- 0 n) n))

Behaviors of ASTs, part I: Atomic forms

Numeral: stands for a value

Name: stands for what?

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``Environment’’ is pointy-headed theory

You may also hear:

Influence of environment is “scope rules”

Find behavior using environment


  (* y 3)   ;; what does it mean?

Your thoughts?

Impcore uses three environments

Global variables ξ

Functions ϕ

Formal parameters ρ

There are no local variables

Function environment ϕ not shared with variables—just like Perl

Syntax {&} environments determine behavior

Behavior is called evaluation

Evaluation is

You know code. You will learn math.

Key ideas apply to any language




Rules written using operational semantics

Evaluation on an abstract machine

Idea: “mathematical interpreter”

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OK, let’s dive in

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30 January 2017: Syntactic Proofs, Metatheory

There are PDF slides for 1/30/2017.

Handout: Impcore expression rules

Last time: rules of operational semantics, how they correspond with code

Both math and code on homework

You’re good with code—lecture and recitation will focus on math

The big idea

Every terminating computation is described by a data structure—we’re going to turn computation into a data structure. Proofs about computations are hard (see: COMP 170), but proofs about data structures are lots easier (see: COMP 61).


Code example

  (define and (p q)
    (if p q 0))

  (define digit? (n)
    (and (<= 0 n) (< n 10)))

Suppose we evaluate (digit? 7)


  1. In the body of digit?, what expressions are evaluated in what order?

  2. As a function application, the body matches template (f e1 e2). In this example,

    • What is f?
    • What is e1?
    • What is e2?

Let’s develop the ApplyUser rule for the special case of two arguments: APPLY(f, e1, e2),ξ, ϕ, ρ⟩⇓?

What is the result of (digit? 7)?

How do we know it’s right?

From rules to proofs

What can a proof tell us?

Which of these judgments correctly describes what code does at run time?

To know for sure, we need a proof

Judgment speaks truth when ``derivable’’

Special kind of proof: derivation

A form of “syntactic proof”

Recursive evaluator travels inductive proof

Root of derivation at the bottom (surprise!)


First let’s see a movie

Example derivation (in handout)

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Building derivations

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Proofs about derivations: metatheory

Syntactic proofs empower meta reasoning

Proof $\mathcal D$ is a data structure

Got a fact about all proofs?

Prove facts by structural induction over derivations

Example: evaluating an expression doesn’t change the set of global variables

Metatheorems often help implementors

More example metatheorems:

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Metatheorems are proved by induction

Induction over structure (or height) of derivation trees $\mathcal D$

These are “math-class proofs” (not derivations)


Let’s try it!

Cases to try:

For your homework, “Theory Impcore” leaves out While and Begin rules.

1 February 2017: Metatheory wrapup, functional programming

There are PDF slides for 2/1/2017.

Today: more induction and recursion

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Where are we going?

Recursion and composition:

Recursion comes from inductive structure of input

Structure of the input drives the structure of the code.

You’ll learn to use a three-step design process:

  1. Inductive structure
  2. Equations (“algebraic laws”)
  3. Code

To discover recursive functions, write algebraic laws:

sum 0 = 0
sum n = n + sum (n - 1)

Which direction gets smaller?


(define sum (n)
   (if (= n 0) 0 (+ n (sum (- n 1)))))

Another example:

exp x 0 = 1
exp x (n + 1) = x * (exp x n)

Can you find a direction in which something gets smaller?


(define exp (x m) 
  (if (= m 0) 
      (* x (exp x (- m 1)))))

For a new language, five powerful questions

As a lens for understanding, you can ask these questions about any language:

  1. What is the abstract syntax? What are the syntactic categories, and what are the terms in each category?

  2. What are the values? What do expressions/terms evaluate to?

  3. What environments are there? That is, what can names stand for?

  4. How are terms evaluated? What are the judgments? What are the evaluation rules?

  5. What’s in the initial basis? Primitives and otherwise, what is built in?

(Initial basis for μScheme on page 149)

Introduction to Scheme

Question 2: what are the values?

Two new kinds of data:

Picture of a cons cell: (cons 3 (cons 2 (cons 1 ’())))

Scheme Values

Values are S-expressions.

An S-expression is either

Many predefined functions work with a list of S-expressions

A list of S-expressions is either

S-Expression operators

Like any other abstract data type, S-Expresions have:

N.B. creators + producers = constructors


Subset of S-Expressions.

Can be defined via a recursion equation or by inference rules:

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Constructors: '(),cons

Observers: null?, pair?, car, cdr (also known as “first” and “rest”, “head” and “tail”, and many other names)

6 February 2017: More programming with lists and S-expressions

There are PDF slides for 2/6/2017.

Why are lists useful?

These “cheap and cheerful” representations are less efficient than balanced search trees, but are very easy to implement and work with—see the book.

The only thing new here is automatic memory management. Everything else you could do in C. (You can have automatic memory management in C as well.)

Immutable data structures

Key idea of functional programming. Instead of mutating, build a new one. Supports composition, backtracking, parallelism, shared state.

Review: Algebraic laws of lists

You fill in these right-hand sides:

(null? '()) == 
(null? (cons v vs)) == 
(car (cons v vs)) == 
(cdr (cons v vs)) == 

(length '()) ==
(length (cons v vs)) ==

Combine creators/producers with observers

Can use laws to prove properties of code and to write better code.

Recursive functions for recursive types

Any list is therefore constructed with '() or with cons applied to an atom and a smaller list.

Example: length

Algebraic Laws for length


;; you fill in this part

Algebraic laws to design list functions

Using informal math notation with .. for “followed by” and e for the empty sequence, we have these laws:

xs .. e         = xs
e .. ys         = ys
(z .. zs) .. ys = z .. (zs .. ys)
xs .. (y .. ys) = (xs .. y) .. ys

The underlying operations are append, cons, and snoc. Which ..’s are which?

You fill in these right-hand sides:

(append '()         ys) == 

(append (cons z zs) ys) == 

Equations and function for append

(append '()         ys) == ys

(append (cons z zs) ys) == (cons z (append zs ys))

(define append (xs ys)

  (if (null? xs)


      (cons (car xs) (append (cdr xs) ys))))

Why does it terminate?

Cost model

The major cost center is cons because it corresponds to allocation.

How many cons cells are allocated?

Let’s rigorously explore the cost of append.

Induction Principle for List(A)

Suppose I can prove two things:

  1. IH (’())

  2. Whenever a in A and also IH(as), then IH (cons a as)

then I can conclude

Forall as in List(A), IH(as)

Claim: Cost (append xs ys) = (length xs)

Proof: By induction on the structure of xs.

Base case: xs = ’()

Inductive case: xs = (cons z zs)

Conclusion: Cost of append is linear in length of first argument.

Costs of list reversal

Algebraic laws for list reversal:

reverse '() = '()
reverse (x .. xs) = reverse xs .. reverse '(x) = reverse xs .. '(x)

And the code?

Naive list reversal

(define reverse (xs)
   (if (null? xs)
       (append (reverse (cdr xs))
               (list1 (car xs)))))

The list1 function maps an atom x to the singleton list containing x.

How many cons cells are allocated? Let’s let n = |xs|.

8 February 2017: Accumulating parameters, let-bound names and anonymous functions

There are PDF slides for 2/8/2017.


What is an algebraic law?

Examples of algebraic laws from math class

x + 0     = x
(x + y)^2 = x^2 + 2ay + y^2
x * 1     = x
x * 0     = 0

Examples of algebraic laws from computer-science class

&a[i] == a + i                /* law of C code */
(member? x emptyset) == #f    ;; law of uScheme sets
(reverse (cons z zs)) == (append (reverse zs) (list1 z)) ;; list law

Your turn!

Homework asks you to discover a new law. Work with your neighbor to develop four laws involving some combination of append and reverse:

  1. (append (reverse '()) ys) =

  2. (append (reverse (cons z zs)) ys) =

  3. (append (reverse xs) '()) =

  4. One new law that you discover:

N.B. There is a deep trick behind left-hand sides 1 and 2

The method of accumulating parameters

Write laws for

(revapp xs ys) = (append (reverse xs) ys)

Who could write the code?

Reversal by accumulating parameters

(define revapp (xs ys)
   (if (null? xs)
       (revapp (cdr xs) 
               (cons (car xs) ys))))

(define reverse (xs) (revapp xs '()))

The cost of this version is linear in the length of the list being reversed.

Parameter ys is the accumulating parameter.
(A powerful, general technique.)

Association lists represent finite maps (code not to be covered in class)

Implementation: list of key-value pairs

'((k1 v1) (k2 v2) ... (kn vn))

Picture with spine of cons cells, car, cdar, caar, cadar.

A-list example

    -> (find 'Building 
             '((Course 105) (Building Barnum) 
               (Instructor Ramsey)))
    -> (val nr (bind 'Office 'Halligan-222
               (bind 'Courses '(105 150TW)
               (bind 'Email 'comp105-grades '()))))
    ((Email comp105-grades) 
     (Courses (105 150TW)) 
     (Office Halligan-222))
    -> (find 'Office nr) 
    -> (find 'Favorite-food nr)


Algebraic laws of association lists

Laws of association lists

(find k (bind k v l)) = v
(find k (bind k' v l)) = (find k l), provided k != k'
(find k '()) =  '() --- bogus!

μScheme’s new syntax

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An alternative to local variables: let binding

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Evaluate e1 through en, bind answers to x1, … xn

Also let* (one at a time) and letrec (local recursive functions)

Note that we would love to have definititions and it might be easier to read if McCarthy had actually used definition syntax, which you’ll see in ML, Haskell, and other functional languages:

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From Impcore to uScheme: Lambda

Things that should offend you about Impcore:

All these problems have one solution: lambda

Anonymous, first-class functions

From Church’s lambda-calculus:

(lambda (x) (+ x x))

“The function that maps x to x plus x”

At top level, like define. (Or more accurately, define is a synonym for lambda that also gives the lambda a name.)

In general, \x.E or (lambda (x) E)

The ability to “capture” free variables is what makes it interesting.

Functions become just like any other value.

First-class, nested functions

(lambda (x) (+ x y))  ; means what??

What matters is that y can be a parameter or a let-bound variable of an enclosing function.

First example: Finding roots. Given n and k, find an x such that x^n = k.

Step 1: Write a function that computes x^n - k.

Step 2: Write a function that finds a zero between lo and hi bounds.

Picture of zero-finding function. Algorithm uses binary search over integer interval between lo and hi. Finds point in that interval in which function is closest to zero.

Code that computes the function x^n - k given n and k:

Function escapes!

-> (define to-the-n-minus-k (n k)
        ([x-to-the-n-minus-k (lambda (x) 
                                (- (exp x n) k))])
-> (val x-cubed-minus-27 (to-the-n-minus-k 3 27))
-> (x-cubed-minus-27 2)

The function to-the-n-minus-k is a higher-order function because it returns another (escaping) function as a result.

No need to name the escaping function

-> (define to-the-n-minus-k (n k)
      (lambda (x) (- (exp x n) k)))

-> (val x-cubed-minus-27 (to-the-n-minus-k 3 27))
-> (x-cubed-minus-27 2)

General purpose zero-finder that works for any function f:

The zero-finder

(define findzero-between (f lo hi)
   ; binary search
   (if (>= (+ lo 1) hi)
       (let ([mid (/ (+ lo hi) 2)])
          (if (< (f mid) 0)
              (findzero-between f mid hi)
              (findzero-between f lo mid)))))
(define findzero (f) (findzero-between f 0 100))

findzero-between is also a higher-order function because it takes another function as an argument. But nothing escapes; you can do this in C.

Example uses:

Cube root of 27 and square root of 16

-> (findzero (to-the-n-minus-k 3 27))                                    
-> (findzero (to-the-n-minus-k 2 16))

Your turn!!

Lambda questions

(define combine (p? q?)
   (lambda (x) (if (p? x) (q? x) #f)))

(define divvy (p? q?)
   (lambda (x) (if (p? x) #t (q? x))))

(val c-p-e (combine prime? even?))
(val d-p-o (divvy   prime? odd?))

(c-p-e 9) == ?            (d-p-o 9) == ?
(c-p-e 8) == ?            (d-p-o 8) == ?
(c-p-e 7) == ?            (d-p-o 7) == ?

Lambda answers

(define combine (p? q?)
   (lambda (x) (if (p? x) (q? x) #f)))

(define divvy (p? q?)
   (lambda (x) (if (p? x) #t (q? x))))

(val c-p-e (combine prime? even?))
(val d-p-o (divvy   prime? odd?))

(c-p-e 9) == #f           (d-p-o 9) == #t
(c-p-e 8) == #f           (d-p-o 8) == #f
(c-p-e 7) == #f           (d-p-o 7) == #t

13 Feb 2017: Higher-order functions on lists; currying; tail calls

There are PDF slides for 2/13/2017.

Last time: conjunction and disjunction

sv - Conjunction, disjunction of (define conjoin (p? q?) (lambda (x) (if (p? x) (q? x) #f)))

(define disjoin (p? q?) (lambda (x) (if (p? x) #t (q? x)))) ev



Higher-order functions

Preview: in math, what is the following equal to?

(f o g)(x) == ???

Another algebraic law, another function:

(f o g) (x) = f(g(x))
(f o g) = \x. (f (g (x)))

Functions create new functions

-> (define o (f g) (lambda (x) (f (g x))))
-> (define even? (n) (= 0 (mod n 2)))
-> (val odd? (o not even?))
-> (odd? 3)
-> (odd? 4)

Another example: (o not null?)

Higher-order list functions

-> (all? even? '(1 2 3 4))
-> (exists? even? '(1 2 3 4))
-> (map (lambda (n)
            (if (even? n) n 'odd))
        '(1 2 3 4))
(odd 2 odd 4)
-> (filter even? '(1 2 3 4))
(2 4)

Reasoning about functions

Truth about S-expressions and functions consuming functions

Q: Can you do case analysis on a function?

Reasoning principles

Recursive function consuming A is related to proof about A


Currying converts a binary function f(x,y) to a function f' that takes x and returns a function f'' that takes y and returns the value f(x,y).

What is the benefit? Functions like exists?, all?, map, and filter all expect a function of one argument. To get there, we use Currying and partial application.

To get one-argument functions: Curry

-> (val positive? (lambda (y) (< 0 y)))
-> (positive? 3)
-> (val <-c (lambda (x) (lambda (y) (< x y))))
-> (val positive? (<-c 0)) ; "partial application"
-> (positive? 0)

Curried functions take their arguments “one-at-a-time.”

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Higher-Order Functions on lists

Goal: Start with functions on elements, end up with functions on lists

Goal: Capture common patterns of computation or algorithms

Fold also called reduce, accum, a “catamorphism”

Your turn!!

Your turn!

-> (map     ((curry +) 3) '(1 2 3 4 5))
-> (exists? ((curry =) 3) '(1 2 3 4 5))
-> (filter  ((curry >) 3) '(1 2 3 4 5))
???                        ; tricky


-> (map     ((curry +) 3) '(1 2 3 4 5))
(4 5 6 7 8)
-> (exists? ((curry =) 3) '(1 2 3 4 5))
-> (filter  ((curry >) 3) '(1 2 3 4 5)) 
(1 2)

List search: exists?

Algorithm encapsulated: linear search

Example: Is there an even element in the list?

Algebraic laws:

(exists? p? '())          == ???
(exixts? p? '(cons a as)) == ???

(exists? p? '())          == #f
(exixts? p? '(cons a as)) == p? x or exists? p? xs

Defining exists?

-> (define exists? (p? xs)
      (if (null? xs)
          (if (p? (car xs)) 
              (exists? p? (cdr xs)))))
-> (exists? pair? '(1 2 3))
-> (exists? pair? '(1 2 (3)))
-> (exists? ((curry =) 0) '(1 2 3))
-> (exists? ((curry =) 0) '(0 1 2 3))

Slide 9 

List selection: filter

Algorithm encapsulated: Linear filtering

Example: Given a list of numbers, return only the even ones.

Algebraic laws:

(filter p? '())          == ???
(filter p? '(cons m ms)) == ???

(filter p? '())          == '()
(filter p? '(cons m ms)) == if (p? m)
                               (cons m (filter p? ms)) 
                               (filter p? ms)

Defining filter

-> (define filter (p? xs)
     (if (null? xs)
       (if (p? (car xs))
         (cons (car xs) (filter p? (cdr xs)))
         (filter p? (cdr xs)))))
-> (filter (lambda (n) (>  n 0)) '(1 2 -3 -4 5 6))
(1 2 5 6)
-> (filter (lambda (n) (<= n 0)) '(1 2 -3 -4 5 6))
(-3 -4)
-> (filter ((curry <)  0) '(1 2 -3 -4 5 6))
(1 2 5 6)
-> (filter ((curry >=) 0) '(1 2 -3 -4 5 6))
(-3 -4)

Composition Revisited: List Filtering

-> (val positive? ((curry <) 0))

-> (filter positive?         '(1 2 -3 -4 5 6))
(1 2 5 6)
-> (filter (o not positive?) '(1 2 -3 -4 5 6))
(-3 -4)

Slide 12 

“Lifting” functions to lists: map

Algorithm encapsulated: Transform every element

Example: Square every element of a list.

Algebraic laws:

(map f '())         ==  ???
(map f (cons n ns)) ==  ???

(map f '())         ==  '()
(map f (cons n ns)) ==  cons (f n) (map f ns)

Defining map

-> (define map (f xs)
     (if (null? xs)
       (cons (f (car xs)) (map f (cdr xs)))))
-> (map number? '(3 a b (5 6)))
(#t #f #f #f)
-> (map ((curry *) 100) '(5 6 7))
(500 600 700)
-> (val square* ((curry map) (lambda (n) (* n n))))

-> (square* '(1 2 3 4 5))
(1 4 9 16 25)

Slide 14 

The universal list function: fold

Slide 15 

foldr takes two arguments:

Example: foldr plus zero '(a b)

cons a (cons b '())
 |       |      |
 v       v      v
plus a (plus b zero)

Slide 16 

Slide 17 

In-class exercise

Slide 18 

Slide 19 

Slide 20 

Tail calls

Intuition: In a function, a call is in tail position if it is the last thing the function does.

A tail call is a call in tail position.

Important for optimizations: Can change complexity class.

What is tail position?

Tail position is defined inductively:

Idea: The last thing that happens

Anything in tail position is the last thing executed!

Key idea is tail-call optimization!

Slide 22 

Slide 23 

Slide 24 

Example: reverse '(1 2)

Question: How much stack space is used by the call?

Call stack:

reverse '() 
reverse '(2)
reverse '(1 2)

Answer: Linear in the length of the list

Slide 25 

Slide 26 

Example: revapp '(1 2) '()

Question: How much stack space is used by the call?

Call stack: (each line replaces previous one)

revapp ‘(1 2)’() –> revapp ‘(2)’(1) –> revapp ‘()’(2 1)

Answer: Constant

Slide 27 

Answer: a goto!!

Think of “tail call” as “goto with arguments”

15 Feb 2017: Continuations

There are PDF slides for 2/15/2017.

Remember tail calls? Suppose you call a parameter!

A continuation is code that represents “the rest of the computation.”

Different coding styles

Direct style: Last action of a function is to return a value. (This style is what you are used to.)

Continuation-passing style (CPS): Last action of a function is to “throw” value to a continuation.

Uses of continuations


Slide 1 

Motivating Example: From existence to witness

Slide 2 


Bad choices:

Good choice:

Slide 3 

Refine the laws

(witness-cps p? xs succ fail) = (succ x)
     ; where x is in xs and (p? x)
(witness-cps p? xs succ fail) = (fail)
     ; where (not (exists? p? xs))

(witness-cps p? '() succ fail) = ?

(witness-cps p? (cons z zs) succ fail) = ?
    ; when (p? z)

(witness-cps p? (cons z zs) succ fail) = ?
    ; when (not (p? z))

Coding with continuations

(define witness-cps (p? xs succ fail)
   (if (null? xs)
       (let ([x (car xs)])
         (if (p? x)
             (succ x)
             (witness-cps p? (cdr xs) succ fail)))))

Slide 6 

Question: How much stack space is used by the call?

Answer: Constant

Example Use: Instructor Lookup

-> (val 2016f '((Fisher 105)(Hescott 170)(Chow 116)))
-> (instructor-info 'Fisher 2016f)
(Fisher teaches 105)
-> (instructor-info 'Chow 2016f)
(Chow teaches 116)
-> (instructor-info 'Souvaine 2016f)
(Souvaine is-not-on-the-list)

Slide 8 

Slide 9 

Slide 10 

Slide 11 

Extended Example: A SAT Solver

Slide 12 

Slide 13 

Solving a Literal

start carries a partial truth assignment to variables current

Box describes how to extend current to make a variable, say x, true.

Case 1: current(x) = #t

Call success continuation with current

Pass fail as resume continuation (argument to success)

Case 2: current(x) = #f

Call fail continuation

Case 3: x not in current

Call success continuation with current{x -> #t}

Pass fail as resume continuation

Solving a literal

(define satisfy-literal-true (x current succ fail)
  (if (bound? x current)
      (if (find x current)
          (succ current fail)
      (succ (bind x #t current) fail)))

Solving a Negated Literal (Your turn)

start carries a partial truth assignment to variables current

Box describes how to extend current to make a negated variable, say not x, true.

Case 1: current(x) = #f

Call success continuation with current

Pass fail as resume continuation (argument to success)

Case 2: current(x) = #t

Call fail continuation

Case 3: x not in current

Call success cotinuation with current{x -> #f}

Pass fail as resume continuation

Solving A and B

  1. Solver enters A

  2. If A is solved, newly allocated success continuation starts B

  3. If B succeeds, we’re done! Use success continuation from context.

  4. If B fails, use resume continuation A passed to B as fail.

  5. If A fails, the whole thing fails. Use fail continuation from context.

Solving A or B

  1. Solver enters A

  2. If A is solved, we’re good! But what if context doesn’t like solution? It can resume A using the resume continuation passed out as fail.

  3. If A can’t be solved, don’t give up! Try a newly allocated failure continuation to start B.

  4. If ever B is started, we’ve given up on A entirely. So B’s success and failure continuations are exactly the ones in the context.

  5. If B succeeds, but the context doesn’t like the answer, the context can resume B.

  6. If B fails, abject failure all around; call the original fail continuation.

22 Feb 2017: Scheme Semantics

There are PDF slides for 2/22/2017.

Handout: Which let is which?

Last Time

Key cases: Make A ∧ B true, make A ∨ B true, make A ∨ B false, make A ∧ B false.


Scheme Semantics

New Syntax, New Values, New Environment, New Evaluation Rules

First four of five questions: Syntax, Values, Environments, Evaluation

Key changes from Impcore:

Slide 1 

Slide 2 

It’s not precisely true that rho never changes.
New variables are added when they come into scope.
Old variables are deleted when they go out of scope.
But the location associated with a variable never changes.

The book includes all rules for uScheme. Here we will discuss on key rules.


Slide 3 

Questions about Assign:


Slide 4 

Function Application

Slide 5 

Questions about ApplyClosure:

Slide 6 

Closure Optimizations


Which let is which and why?

Handout: Which let is which?


Lisp and Scheme Retrospective

Slide 7 

Common Lisp, Scheme


Down sides:

Bottom line: it’s all about lambda

Bonus content: Scheme as it really is

  1. Macros!
  2. Cond expressions (solve nesting problem)
  3. Mutation


Real Scheme: Macros

A Scheme program is just another S-expression

(See book sections 2.16, 2.17.4)

Conditional expressions

Real Scheme: Conditionals

(cond (c1 e1)    ; if c1 then e1
      (c2 e2)    ; else if c2 then e2
       ...            ...
      (cn en))   ; else if cn then en

; Syntactic sugar---'if' is a macro:
(if e1 e2 e3) == (cond (e1 e2)
                       (#t e3))


Real Scheme: Mutation

Not only variables can be mutated.

Mutate heap-allocated cons cell:

(set-car! '(a b c) 'd)  => (d b c)

Circular lists, sharing, avoids allocation

23 Feb 2017: Introduction to ML

There are PDF slides for 2/23/2017.

Ask the class: what are the pain points in your μScheme programming?

Apply your new knowledge in Standard ML:

Meta: Not your typical introduction to a new language

ML Overview

Designed for programs, logic, symbolic data

Theme: Precise ways to describe data

ML = uScheme + pattern matching + static types + exceptions

μScheme to ML Rosetta stone

uScheme                    SML

 (cons x xs)             x :: xs

 '()                     []
 '()                     nil

 (lambda (x) e)          fn x => e

 (lambda (x y z) e)      fn (x, y, z) => e

 ||  &&                  andalso    orelse

 (let* ([x e1]) e2)      let val x = e1 in e2 end

 (let* ([x1 e1]          let val x1 = e1
        [x2 e2]              val x2 = e2
        [x3 e3]) e)          val x3 = e3
                         in  e

Three new ideas

  1. Pattern matching is big and important. You will like it.
  2. Exceptions are easy
  3. Static types get two to three weeks in their own right.

Pattern matching makes code look more like algebraic laws: one pattern for each case

Exceptions solve the problem “I can’t return anything sensible!”

Static types tell us at compile time what the cases are.

And lots of new concrete syntax!


The length function.


fun length []      = 0
  | length (x::xs) = 1 + length xs

val res = length [1,2,3]


fun map f []      = []
  | map f (x::xs) = (f x) :: (map f xs)

val res1 =
  map length [[], [1], [1,2], [1,2,3]]

Map, without redundant parentheses

fun map f []      = []
  | map f (x::xs) =  f x  ::  map f xs

val res1 =
  map length [[], [1], [1,2], [1,2,3]]


fun filter pred []      = [] 
  | filter pred (x::xs) =  (* no 'pred?' *)
      let val rest = filter pred xs
      in  if pred x then
            (x :: rest) 

val res2 =
  filter (fn x => (x mod 2) = 0) [1,2,3,4]

Filter, without redundant parentheses

fun filter pred []      = [] 
  | filter pred (x::xs) =  (* no 'pred?' *)
      let val rest = filter pred xs
      in  if pred x then
             x :: rest

val res2 =
  filter (fn x => (x mod 2) = 0) [1,2,3,4]


fun exists pred []      = false
  | exists pred (x::xs) = 
      (pred x) orelse (exists pred xs)

val res3 =
  exists (fn x => (x mod 2) = 1) [1,2,3,4]
(* Note: fn x => e is syntax for lambda *)

Exists, without redundant parentheses

fun exists pred []      = false
  | exists pred (x::xs) = 
       pred x  orelse  exists pred xs

val res3 =
  exists (fn x => (x mod 2) = 1) [1,2,3,4]
(* Note: fn x => e is syntax for lambda *)


fun all pred []      = true
  | all pred (x::xs) =
      (pred x) andalso (all pred xs)

val res4 = all (fn x => (x >= 0)) [1,2,3,4]

All, without redundant parentheses

fun all pred []      = true
  | all pred (x::xs) =
       pred x  andalso  all pred xs

val res4 = all (fn x => (x >= 0)) [1,2,3,4]


exception TooShort
fun take 0 _       = []  (* wildcard! *)
  | take n []      = raise TooShort
  | take n (x::xs) = x :: (take (n-1) xs)

val res5 = take 2 [1,2,3,4]
val res6 = take 3 [1]
           handle TooShort =>
             (print "List too short!"; [])

(* Note use of exceptions. *)

Take, without redundant parentheses

exception TooShort
fun take 0 _       = []  (* wildcard! *)
  | take n []      = raise TooShort
  | take n (x::xs) = x ::  take (n-1) xs

val res5 = take 2 [1,2,3,4]
val res6 = take 3 [1]
           handle TooShort =>
             (print "List too short!"; [])

(* Note use of exceptions. *)


fun drop 0 zs      = zs
  | drop n []      = raise TooShort
  | drop n (x::xs) = drop (n-1) xs

val res7 = drop 2 [1,2,3,4]
val res8 = drop 3 [1]
           handle TooShort =>
             (print "List too short!"; [])


fun takewhile p [] = []
  | takewhile p (x::xs) = 
      if p x then (x :: (takewhile p xs))
      else []

fun even x = (x mod 2 = 0)
val res8 = takewhile even [2,4,5,7]
val res9 = takewhile even [3,4,6,8]

Takewhile, without redundant parentheses

fun takewhile p [] = []
  | takewhile p (x::xs) = 
      if p x then  x ::  takewhile p xs
      else []

fun even x = (x mod 2 = 0)
val res8 = takewhile even [2,4,5,7]
val res9 = takewhile even [3,4,6,8]


fun dropwhile p []              = []
  | dropwhile p (zs as (x::xs)) = 
      if p x then (dropwhile p xs) else zs
val res10 = dropwhile even [2,4,5,7]
val res11 = dropwhile even [3,4,6,8]

(* fancy pattern form: zs as (x::xs) *)

Dropwhile, without redundant parentheses

fun dropwhile p []              = []
  | dropwhile p (zs as (x::xs)) = 
      if p x then  dropwhile p xs  else a
val res10 = dropwhile even [2,4,5,7]
val res11 = dropwhile even [3,4,6,8]

(* fancy pattern form: zs as (x::xs) *)


fun foldr p zero []      = zero
  | foldr p zero (x::xs) = p (x, (foldr p zero xs))

fun foldl p zero []      = zero
  | foldl p zero (x::xs) = foldl p (p (x, zero)) xs

val res12 = foldr (op +)  0 [1,2,3,4] 
val res13 = foldl (op * ) 1 [1,2,3,4] 

(* Note 'op' to use infix operator as a value *)

Folds, without redundant parentheses

fun foldr p zero []      = zero
  | foldr p zero (x::xs) = p (x,  foldr p zero xs )

fun foldl p zero []      = zero
  | foldl p zero (x::xs) = foldl p (p (x, zero)) xs

val res12 = foldr (op +)  0 [1,2,3,4] 
val res13 = foldl (op * ) 1 [1,2,3,4] 

(* Note 'op' to use infix operator as a value *)

ML—The Five Questions

Syntax: definitions, expressions, patterns, types

Values: num/string/bool, record/tuple, algebraic data

Environments: names stand for values (and types)

Evaluation: uScheme + case and pattern matching

Initial Basis: medium size; emphasizes lists

(Question Six: type system—a coming attraction)

A note about books

Ullman is easy to digest

Ullman costs money but saves time

Ullman is clueless about good style


Details in course guide Learning Standard ML

27 Feb 2017: Programming with constructed data and types

There are PDF slides for 2/27/2017.


Foundation: Data

Syntax is always the presenting complaint, but data is what’s always important

“Distinguish one cons cell (or one record) from another”

Tuple types and arrow types

Background for datatype review (board):

This is all you need to know about the special built-in type constructors (cross and arrow).

Constructed data: Algebraic data types



Datatype declarations

datatype  suit    = HEARTS | DIAMONDS | CLUBS | SPADES

datatype 'a list  = nil           (* copy me NOT! *)
                  | op :: of 'a * 'a list

datatype 'a heap  = EHEAP
                  | HEAP of 'a * 'a heap * 'a heap

type suit          val HEARTS : suit, ...
type 'a list       val nil   : forall 'a . 'a list
                   val op :: : forall 'a . 
                               'a * 'a list -> 'a list
type 'a heap
val EHEAP: forall 'a.                          'a heap
val HEAP : forall 'a.'a * 'a heap * 'a heap -> 'a heap

Exegesis (on board):

Additional language support for algebraic types: case expressions

Eliminate values of algebraic types

New language construct case (an expression)

fun length xs =
  case xs
    of []      => 0
     | (x::xs) => 1 + length xs

Clausal definition is preferred
(sugar for val rec, fn, case)

works for any datatype

 fun toStr t = 
     case t 
       of EHEAP => "empty heap"
        | HEAP (v, left, right) =>
                   "nonempty heap"

But often a clausal definition is better style:

 fun toStr' EHEAP = "empty heap"
   | toStr' (HEAP (v,left,right)) =
                    "nonempty heap"

Making types work for you

The types survey:

Baffling         Noise I can ignore          Information I understand

Today, add to far right: type help me program

Talking type theory: Introduction and elimination constructs

Part of learning any new field: talk to people in their native vocabulary

It’s like knowing what to say when somebody sneezes.

Slide 4 

Types help me, part I: type-directed programming

Common idea in functional programming: “lifting:

val lift : forall 'a . ('a -> bool) -> ('a list -> bool)

Type-directed coding

Common idea in functional programming: “lifting”

val lift : forall 'a . ('a      -> bool) ->
                       ('a list -> bool)

What (sensible) functions have this type?


Type-directed coding (results)

val lift : ('a -> bool) -> ('a list -> bool)
fun lift p = (fn xs => (case xs
                          of [] => false
                           | z::zs => p z orelse
                                      lift p zs))

Merge top-level into

fun lift p xs = case xs of []    => false
                         | z::zs => p z orelse
                                    lift p zs

Merge top-level into

fun lift p []      = false
  | lift p (z::zs) = p z orelse lift p zs

I know this function!

fun exists p []      = false
  | exists p (z::zs) = p z orelse exists p zs

Bonus content: Even more algebraic datatypes

Algebraic datatype review:

Frequently overlooked

An algebraic data type is a collection of alternatives

Don’t forget:

The thing named is the value constructor

(Also called “datatype constructor”)

Enumerated types

Datatypes can define an enumerated type and associated values.

datatype suit = HEARTS | DIAMONDS | SPADES | CLUBS

Here suit is the name of a new type.

The value constructors HEARTS, DIAMONDS, SPADES, and CLUBS are the values of type suit.

Value constructors are separated by vertical bars.

Pattern matching

Datatypes are deconstructed using pattern matching.

fun toString HEARTS = "hearts"
  | toString DIAMONDS = "diamonds"
  | toString SPADES = "spades"
  | toString CLUBS = "clubs"

val suitName = toString HEARTS

But wait, there’s more: Value constructors can take arguments!

datatype int_tree = LEAF | NODE of int * int_tree * int_tree

int_tree is the name of a new type.

There are two data constructors: LEAF and NODE.

NODEs take a tuple of three arguments: a value at the node, and left and right subtrees.

The keyword of separates the name of the data constructor and the type of its argument.

When fully applied, data constructors have the type of the defining datatype (ie, int_tree).

Building values with constructors

We build values of type int_tree using the associated constructors: (Draw on board)

 val tempty = LEAF
 val t1 = NODE (1, tempty, tempty)
 val t2 = NODE (2, t1, t1)
 val t3 = NODE (3, t2, t2)

What is the in-order traversal of t3?


What is the pre-order traversal of t3?


Deconstruct values with pattern matching

(The @ symbol denotes append in ML)

fun inOrder LEAF = []
  | inOrder (NODE (v, left, right)) = 
       inOrder left @ [v] @ inOrder right

val il3 = inOrder t3

fun preOrder LEAF = []
  | preOrder (NODE (v, left, right)) = 
       v :: preOrder left @ preOrder right

val pl3 = inOrder t3

int_tree is monomorphic because it has a single type.

Note though that the inOrder and preOrder functions only cared about the structure of the tree, not the payload value at each node.

But wait, there’s still more: Polymorphic datatypes!

Polymorphic datatypes are written using type variables that can be instantiated with any type.

datatype 'a tree = CHILD | PARENT of 'a * 'a tree * 'a tree

tree is a type constructor (written in post-fix notation), which means it produces a type when applied to a type argument.


'a is a type variable: it can represent any type.

It is introduced on the left-hand of the = sign. References on the right-hand side are types.

CHILD and PARENT are value constructors.

CHILD takes no arguments, and so has type 'a tree

When given a value of type 'a and two 'a trees, PARENT produces a 'a tree

Constructors build tree values

val empty = CHILD
val tint1 = PARENT (1, empty, empty)
val tint2 = PARENT (2, tint1, tint1)
val tint3 = PARENT (3, tint2, tint2)

val tstr1 = PARENT ("a", empty, empty)
val tstr2 = PARENT ("b", tstr1, tstr1)
val tstr3 = PARENT ("c", tstr2, tstr2)

Pattern matching deconstructs tree values

fun inOrder CHILD = []
  | inOrder (PARENT (v, left, right)) = 
       (inOrder left) @ [v] @ (inOrder right)

fun preOrder CHILD = []
  | preOrder (Parent (v, left, right)) = 
       v :: (preOrder left) @ (preOrder right)

Functions inOrder and preOrder are polymorphic: they work on any value of type 'a tree. 'a is a type variable and can be replaced with any type.


Datatype declarations introduce names into:

  1. the type environment: suit, int_tree, tree

  2. the value environment: HEART, LEAF, PARENT


Datatype declarations are inherently inductive:

Datatype Exercise

Slide 12 

Wait for it …

Exercise answers

datatype sx1 = ATOM1 of atom
             | LIST1 of sx1 list

datatype sx2 = ATOM2 of atom
             | PAIR2 of sx2 * sx2

Bonus content: Exceptions — Handling unusual circumstances


Informal Semantics:

Exception handling in action

loop (evaldef (reader (), rho, echo))
handle EOF            => finish ()
  | Div               => continue "Division by zero"
  | Overflow          => continue "Arith overflow"
  | RuntimeError msg  => continue ("error: " ^ msg)
  | IO.Io {name, ...} => continue ("I/O error: " ^
  | SyntaxError msg   => continue ("error: " ^ msg)
  | NotFound n        => continue (n ^ "not found")

Bonus Content: ML traps and pitfalls

Slide 16 

Order of clauses matters

fun take n (x::xs) = x :: take (n-1) xs
  | take 0 xs      = []
  | take n []      = []

(* what goes wrong? *)

Gotcha — overloading

- fun plus x y = x + y;
> val plus = fn : int -> int -> int
- fun plus x y = x + y : real;
> val plus = fn : real -> real -> real

Slide 19 

Gotcha — parentheses

Put parentheses around anything with |

case, handle, fn

Function application has higher precedence than any infix operator

Bonus content (seen in examples)

Syntactic sugar for lists

Syntactic sugar for lists

- 1 :: 2 :: 3 :: 4 :: nil; (* :: associates to the right *)
> val it = [1, 2, 3, 4] : int list

- "the" :: "ML" :: "follies" :: [];
> val it = ["the", "ML", "follies"] : string list

> concat it;
val it = "theMLfollies" : string

Bonus content: ML from 10,000 feet

Slide 22 


The value environment

Names bound to immutable values

Immutable ref and array values point to mutable locations

ML has no binding-changing assignment

Definitions add new bindings (hide old ones):

val pattern = exp
val rec pattern = exp
fun ident patterns = exp
datatype … = …

Nesting environments

At top level, definitions

Definitions contain expressions:

def ::= val pattern = exp

Expressions contain definitions:

exp ::= let defs in exp end

Sequence of defs has let-star semantics


What is a pattern?

pattern ::= variable
          | wildcard
          | value-constructor [pattern]
          | tuple-pattern
          | record-pattern
          | integer-literal
          | list-pattern

Design bug: no lexical distinction between

Workaround: programming convention


Function pecularities: 1 argument

Each function takes 1 argument, returns 1 result

For “multiple arguments,” use tuples!

 fun factorial n =
   let fun f (i, prod) = 
         if i > n then prod else f (i+1, i*prod)
   in  f (1, 1)

 fun factorial n =  (* you can also Curry *)
   let fun f i prod = 
         if i > n then prod else f (i+1) (i*prod)
   in  f 1 1

Tuples are “usual and customary.”

Slide 27 


Slide 28 

Slide 29 

Slide 30 

Slide 31 

1 March 2017: Type systems

There are PDF slides for 3/1/2017.


Why do we write a type checker? (probably for a later lecture)

Code from types, revisited

Slide 1 

Your turn: What possible types?

Our turn: write the code

Type systems

What kind of thing is it?

Slogan: “Types classify terms”

 n + 1  : int

 "hello" ^ "world"  : string

 (fn n => n * (n - 1))  : int -> int

 if p then 1 else 0  : int, provided  p : bool

Questions type systems can answer:

Questions type systems generally cannot answer:

Type System and Checker for a Simple Language

Define an AST for expressions with:

Language of expressions

Numbers and Booleans:

datatype exp = ARITH of arithop * exp * exp
             | CMP   of relop   * exp * exp
             | LIT   of int
             | IF    of exp     * exp * exp
and      arithop = PLUS | MINUS | TIMES | ...
and      relop   = EQ | NE | LT | LE | GT | GE

datatype ty = INTTY | BOOLTY

Examples to rule out

Can’t add an integer and a boolean:

3 + (3 < 99)


Can’t compare an integer and a boolean

(3 < (4 = 24))

CMP (LT, LIT 3, CMP(EQ (LIT 4, LIT 24)))

Inference rules to define a type system

Rule for arithmetic operators

Informal example:

|- 3 : int    |- 5 : int 
|- 3 + 5 : int

Rules out:

|- 'a' : char    |- 5 : int
|- 'a' + 5 : ???

General form:

|- e1 : int    |- e2 : int
|- ARITH ( _ , e1, e2) : int

Rule for comparisons

Informal example:

|- 7 : int    |- 10 : int
|- 7 < 10 : bool

General form:

|- e1 : int    |- e2 : int
|- CMP ( _ , e1, e2) : bool

Rule for literals

Informal example:

|- 14 : int

General form:

|- LIT (n) : int

Rule for conditionals:

Informal example:

|- true : bool    
|- 3    : int
|- 42   : int      
|- IF (true, 3, 42) : int

General form:

|- e : bool    
|- e1 : tau1   
|- e2 : tau2      tau1 equiv tau2
|- IF ( e, e1, e2) : tau1

Typing rules let us read off what a type checker needs to do.

What is a type?

Source of new language ideas for next 20 years

Needed if you want to understand advanced designs (or create your own)

Type checker in ML

val typeof : exp -> ty
exception IllTyped
fun typeof (ARITH (_, e1, e2)) = 
      case (tc e1, typeof e2) 
        of (INTTY, INTTY) => INTTY
         | _              => raise IllTyped
  | typeof (CMP (_, e1, e2)) = 
      case (tc e1, typeof e2) 
        of (INTTY, INTTY) => BOOLTY
         | _              => raise IllTyped
  | typeof (LIT _) = INTTY
  | typeof (IF (e,e1,e2)) = 
      case (tc e, typeof e1, typeof e2) 
        of (BOOLTY, tau1, tau2) => 
           if eqType (tau1, tau2) 
           then tau1 else raise IllTyped
         | _                    => raise IllTyped

An implementor’s trick: If you see identical types in a rule,

6 March 2017: Type checking with type constructors

There are PDF slides for 3/6/2017.

Announcement: NR extra office hours:

Review: typing rules for machine expressions

things that could go wrong:

(8 < 10) + 4

(8 == 8) < 9

x + (x :: xs)

let val y = 10 in length y end

Extended language of expressions

    datatype exp = ARITH of arithop * exp * exp
                 | CMP   of relop   * exp * exp
                 | LIT   of int
                 | IF    of exp     * exp * exp
                 | VAR   of name
                 | LET   of name    * exp * exp
    and      arithop = PLUS | MINUS | TIMES | ...
    and      relop   = EQ | NE | LT | LE | GT | GE

    datatype ty = INTTY | BOOLTY

Typing Rules: Contexts and Term Variables

Your turn:

Things to think about:

Q: What context do we need to evaluate an expression?

Q: Do we need all the same context to decide on a type?

Q: What do we need then?

Rule for var

x in dom Gamma        tau = Gamma(x) 
Gamma |- VAR x : tau

Rule for let

Gamma         |- e  : tau
Gamma{x->tau} |- e' : tau'   
Gamma |- LET x = e in e' : tau'

What is the information flow?

Type Checker

Type checker needs Gamma – gives type of each “term variable”.

val typeof : exp * ty env -> ty
fun typeof (ARITH ..., Gamma ) =  <as before>
  | typeof (VAR x, Gamma)      =
      (case maybeFind (x, Gamma)
         of SOME tau => tau
          | NONE     => raise IllTyped)
  | typeof (LET (x, e1, e2), Gamma) = 
        let tau1 = typeof (e1, Gamma)
        in  typeof (e2, extend Gamma x tau1)



Gamma{x->tau1} |- e : tau2   
Gamma |- (lambda ([x : tau1]) e)  : tau1 -> tau2


Gamma |- e  : tau1 -> tau2   
Gamma |- e1 : tau1
Gamma |- (e e1) : tau2

What’s coming

On the new homework,

This is a big chunk of what language designers do.

Type Checking with Type Constructors

Where we’ve been and where we’re going

New watershed in the homework

What’s next is much more sophisticated type systems, with an infinite number of types. We’ll focus on two questions about type systems:

We’ll look at these questions in two contexts: monomorphic and polymorphic languages.

Design and implementation of monomorphic languages


Language designer’s agenda:

Words “introduce” and “eliminate” are the native vocabulary of type-theoretic language design—it’s like knowing what to say when somebody sneezes.

Question for the class: If I add lists to a language, how many new types am I introducing?

Managing the set of types: Type formation

Examples: Well-formed types

These are types:

Examples: Not yet types, or not types at all

These “types in waiting” don’t classify any terms

These are utter nonsense

Type-formation rules

We need a way to classify type expressions into:

Type constructors

Technical name for “types in waiting”

Given zero or more arguments, produce a type:

More complex type constructors:

Slide 6 

Type judgments for monomorphic system

Two judgments:

Monomorphic type rules

Slide 8 

Slide 9 

Notice: one rule for if!!

Classic types for data structures

Slide 10 

(At run time, identical to cons, car, cdr)

Slide 11 

Slide 12 

Typical syntactic support for types

Explicit types on lambda and define:

Abstract syntax:

datatype exp = ...
 | LAMBDA of (name * tyex) list * exp
datatype def = ...
 | DEFINE of name * tyex * ((name * tyex) list * exp)

Slide 14 

Slide 15 

Typing Rule Exercise

Slide 16 

Wait for it …

Slide 18 

Coding the arrow-introduction rule

Slide 19 

Type-checking LAMBDA

datatype exp = LAMBDA of (name * tyex) list * exp 
fun ty (Gamma, LAMBDA (formals, body)) = 
  let val Gamma' = (* body gets new env *)
        foldl (fn ((x, ty), g) => bind (x, ty, g))
              Gamma formals
      val bodytype = ty (Gamma', body)
      val formaltypes = 
        map (fn (x, ty) => ty) formals
  in  FUNTY (formaltypes, bodytype)

13 March 2017: Polymorphic Type Checking; Kinds classify types

There are PDF slides for 3/13/2017.


Midterms are graded, but half physical, half virtual

Midterm course evaluations:


Last week: Typed Impcore, but in μScheme syntax

Today: Typed μScheme

Recitation this week: a chance to practice coding in Typed μScheme. You could choose to hold off on problem TD.

Typechecking review, part I

fun ty (IFX (e1, e2, e3)) =
      if eqType (ty e1, booltype) then
        let val (t2, t3) = (ty e2, ty e3)
        in  ...  YOU FILL IN 1 ... 
        ... YOU FILL IN 2 ...
  | ty (SET (x, e)) =
      let val tau_x = find (x, Gamma)
          val tau_e = ty e
      in  ... YOU FILL IN 3 ...

Typechecking review, part II

fun ty (APPLY (f, actuals)) = 
  let val atys = map ty actuals
  in  case ty f
        of FUNTY (formals, result) =>
             if eqTypes (atys, formals) then
                ... YOU FILL IN 4 ...
                ... YOU FILL IN 5 ...
         | _ => ... YOU FILL IN 6 ...

Limitations of monomorphic type systems

Monomorphic types are limiting

Each new type constructor requires

Slide 4 


Monomorphism hurts programmers too

Monomorphism leads to code duplication

User-defined functions are monomorphic:

(define int lengthI ([xs : (list int)])
   (if (null? xs) 0 (+ 1 (lengthI (cdr xs)))))
(define int lengthB ([xs : (list bool)])
   (if (null? xs) 0 (+ 1 (lengthB (cdr xs)))))
(define int lengthS ([xs : (list sym)])
   (if (null? xs) 0 (+ 1 (lengthS (cdr xs)))))

Quantified types

Slide 6 

Type formation via kinds


Back up here—what types do we have?

Type formation: Composing types

Typed Impcore:

Typed μScheme:

Standard ML:

Can’t add new syntactic forms and new type formation rules for every new type.

Slide 9 

Slide 10 

Slide 11 

Well-formed types

We still need to classify type expressions into:

Idea: kinds classify types

one-off type-formation rules

Δ tracks type constructors, vars

Polymorphic Type Checking

Quantified types

Slide 13 

Representing quantified types

Two new alternatives for tyex:

datatype tyex
  = TYCON  of name
  | CONAPP of tyex * tyex list 
  | FUNTY  of tyex list * tyex
  | TYVAR  of name
  | FORALL of name list * tyex

Slide 15 

Programming with quantified types

Substitute for quantified variables

-> length
<procedure> : (forall ('a) ((list 'a) -> int))
-> (@ length int)
<procedure> : ((list int) -> int)
-> (length '(1 2 3))
type error: function is polymorphic; instantiate before applying
-> ((@ length int) '(1 2 3))
3 : int

Substitute what you like

-> length
 : (forall ('a) ((list 'a) -> int))
-> (@ length bool)
 : ((list bool) -> int)
-> ((@ length bool) '(#t #f))
2 : int

More ``Instantiations’’

-> (val length-int (@ length int))
length-int : ((list int) -> int)
-> (val cons-bool (@ cons bool))
cons-bool : ((bool (list bool)) ->
                                (list bool))
-> (val cdr-sym (@ cdr sym))
cdr-sym : ((list sym) -> (list sym))
-> (val empty-int (@ '() int))
() : (list int)

Bonus instantiation:

-> map
<procedure> : 
  (forall ('a 'b) 
    (('a -> 'b) (list 'a) -> (list 'b)))
-> (@ map int bool)
<procedure> : 
  ((int -> bool) (list int) -> (list bool))

Create your own!

Abstract over unknown type using type-lambda

  -> (val id (type-lambda ['a]
                (lambda ([x : 'a]) x )))
  id : (forall ('a) ('a -> 'a))

'a is type parameter (an unknown type)

This feature is parametric polymorphism

Two forms of abstraction:

Power comes at notational cost

Function composition

-> (val o (type-lambda ['a 'b 'c]         
    (lambda ([f : ('b -> 'c)] 
             [g : ('a -> 'b)])
     (lambda ([x : 'a]) (f (g x))))))    

o : (forall ('a 'b 'c) 
       (('b -> 'c) ('a -> 'b) -> ('a -> 'c)))

Aka o :

Type rules for polymorphism

Slide 21 

Slide 22 

A phase distinction embodied in code

-> (val x 3)
3 : int
-> (val y (+ x x))
6 : int

fun processDef (d, (delta, gamma, rho)) =
  let val (gamma', tystring)  = elabdef (d, gamma, delta)
      val (rho',   valstring) = evaldef (d, rho)
      val _ = print (valstring ^ " : " ^ tystring)
  in  (delta, gamma', rho')

Slide 24 

Type formation through kinds

Slide 25 

Slide 26 

Slide 27 

Slide 28 

Slide 29 

Bonus content: a definition manipulates three environments

Slide 30 

Slide 31 

Slide 32 

15 March 2017: Type inference

There are PDF slides for 3/15/2017.

Midterm grades look good:

  Points       Grade
    72+        Excellent
    51.5-71    Very Good
    41-51      Good
    30-40.5    Fair
    under 30   Poor

Questions: where do explicit types appear in C?

Where do they appear in Typed μScheme?

Get rid of all that:

What type inference accomplishes

-> (define     double (x)       (+ x x))
double                         ;; uScheme
-> (define int double ([x : int]) (+ x x))
double : (int -> int)          ;; Typed uSch.
-> (define     double (x)       (+ x x))
double : int -> int            ;; nML

What else type inference accomplishes

-> ((@ cons bool) #t ((@ cons bool) #f (@ '() bool)))
(#t #f) : (list bool)    ;; typed uScheme
-> (   cons       #t (   cons       #f    '()      ))
(#t #f) : bool list      ;; nML

How it works

  1. For each unknown type, a fresh type variable

  2. Every typing rule adds equality constraints

  3. Instantiate every variable automatically

  4. Introduce polymorphism at let/val bindings

Slide 4 

Let’s do an example on the board

N.B. Book is “constraints first;” lecture will be “type system first.” Use whatever way works for you

(val-rec double (lambda (x) (+ x x)))

What do we know?

Key idea: Record the constraint in a typing judgement.

'a2 = int /\ 'a2 = int, { double : 'a1, x : 'a2 } |- (+ x x) : int

Example: if


Inferring polymorphic types

(val app2 (lambda (f x y)
                (f x)
                (f y))))

Assume f : ’a_f

Assume x : ’a_x

Assume y : ’a_y

f x implies ’a_f ~ (’a_x -> ’a)

f y implies ‘a_f ~ (’a_y -> ’a’)

Together, these constraints imply ‘a_x = ’a_y and ’a = ’a’

begin implies result of function is ’a


app2 : (('a_x -> 'a) 'a_x 'a_x -> 'a)

’a_x and ’a aren’t mentioned anywhere else in program, so

we can generalize to:

(forall ('a_x 'a) (('a_x -> 'a) 'a_x 'a_x -> 'a))

which is the same thing as:

app2 : (forall ('a 'b) (('a -> 'b) 'a 'a -> 'b))

Your turn! The type of cc

-> (val cc (lambda (nss) (car (car nss))))

Assume nss : ’b

We know car : forall ’a . ’a list -> ’a

=> car_1 : ’a1 list -> ’a1

=> car_2 : ’a2 list -> ’a2

(car_1 nss) => ’b = ’a1 list

(car_2 (car_1 nss)) => ’a1 = ’a2 list

(car_2 (car_1 nss)) : ’a2

nss : ’b

: 'a1 list                    

: ('a2 list) list

So, cc : (’a2 list) list -> ’a2

Because ’a2 is unconstrained, we can generalize:

cc : forall ’a . (’a2 list) list -> ’a

Your turn! The type of cc

-> (val cc (lambda (nss) (car (car nss))))
cc : (forall ('a) ((list (list 'a)) -> 'a))

27 March 2017: Making type inference precise

There are PDF slides for 3/27/2017.


Study-group tool: 12 people, will wait til afternoon

Grades delayed by suspicions of cheating



Last time

-> (val cc (lambda (nss) (car (car nss))))
cc : (forall ('a) ((list (list 'a)) -> 'a))

Refresh your skills!

-> (val second (lambda (xs) (car (cdr xs))))
second : ...
-> (val two    (lambda (f) (lambda (x) (f (f x)))))
two : ...

Skills refreshed

-> (val second (lambda (xs) (car (cdr xs))))
second : (forall ('a) ((list 'a) -> 'a))
-> (val two    (lambda (f) (lambda (x) (f (f x)))))
two :    (forall ('a) (('a -> 'a) -> ('a -> 'a)))

Infer the type of function two:

Precise inference with Hindley-Milner types

Making Type Inference Precise

Sad news:



Slide 5 

Representing Hindley-Milner types

datatype ty
  = TYCON  of name        
  | CONAPP of ty * ty list
  | TYVAR  of name        

datatype type_scheme
  = FORALL of name list * ty

Slide 7 

Slide 8 

To code the type-inference algorithm, replace eqType with constraint generation!

All the pieces

  1. Hindley-Milner types
  2. Bound names : σ, expressions : τ
  3. Type inference yields type-equality constraint
  4. Constraint solving produces substitution
  5. Substitution refines types
  6. Call solver, introduce polytypes at val
  7. Call solver, introduce polytypes at let

The inference algorithm, formally

Slide 10 

Slide 11 

Slide 12 

Slide 13 

What you know and can do now

Your skills so far

You can complete typeof

(Except for let forms.)

Next up: solving constraints!

Writing the constraint solver

Representing Constraints

datatype con = ~   of ty  * ty
             | /\  of con * con
             | TRIVIAL
infix 4 ~
infix 3 /\

Slide 16 

Two questions: what’s substitution, and when is a constraint satisfied?

Slide 17 

Slide 18 


Which have solutions?

 1. int           ~ bool
 2. (list int)    ~ (list bool)
 3. 'a            ~ int
 4. 'a            ~ (list int)
 5. 'a            ~ ((args int) -> int)
 6. 'a            ~ 'a
 7. (args 'a int) ~ (args bool 'b)
 8. (args 'a int) ~ ((args bool) -> 'b)
 9. 'a            ~ (pair 'a int)
10. 'a            ~ tau    // arbitrary tau

29 March 2017: Building and using a constraint solver

There are PDF slides for 3/29/2017.


Which have solutions?

   1. int           ~ bool                // no
   2. (list int)    ~ (list bool)         // no
   3. 'a            ~ int
   4. 'a            ~ (list int)
   5. 'a            ~ ((args int) -> int)
   6. 'a            ~ 'a
   7. (args 'a int) ~ (args bool 'b)
   8. (args 'a int) ~ ((args bool) -> 'b) // no
   9. 'a            ~ (pair 'a int)       // no
  10. 'a            ~ tau   // it's complicated

Solving simple type equalities

Slide 2 

Question: in solving tau1 ~ tau2, how many potential cases are there to consider?

Question: how are you going to handle each case?

Solving conjunctions

Slide 3 

What you can know and do now

Write type inference for everything except VAL, VALREC, and LETX

Write the solver

Instantiate and generalize

Moving from type scheme to types (Instantiation)

Moving from types to type scheme (Generalization)

From Type Scheme to Types

Slide 4 

Slide 5 

From Types to Type Scheme

Slide 6 

Slide 7 

The set A above will be useful when some variables in τ are mentioned in the environment.

We can’t generalize over those variables.

Applying idea to the type inferred for the function fst:

generalize(’a * ’b -> ’a, emptyset) = forall ’a, ’b. ’a * ’b -> ’a

Slide 8 

Note the new judgement form above for type checking a declaration.

Slide 9 

Slide 10 

Slide 11 

On the condition θΓ = Γ: Γ is “input”: it can’t be changed.
The condition ensures that θ doen’t conflict with Γ.

We can’t generalize over free type variables in Γ.

Their presence indicates they can be used somewhere else, and hence they aren’t free to be instantiated with any type.

Type Inference for Lets and Recursive Definitions

Slide 12 

Let Examples

(lambda (ys)  ; OK
   (let ([s (lambda (x) (cons x '()))])
      (pair (s 1) (s #t))))

(lambda (ys)  ; Oops!
   (let ([extend (lambda (x) (cons x ys))])
      (pair (extend 1) (extend #t))))

(lambda (ys)  ; OK
    (let ([extend (lambda (x) (cons x ys))])
       (extend 1)))

Slide 14 

Let with constraints, operationally:

  1. typesof: returns τ1, …, τn and C

  2. C-prime from map, conjoinConstraints, dom, inter, freetyvarsGamma

  3. val theta = solve C'

  4. freetyvarsGamma, union, freetyvarsConstraint

  5. Map anonymous lambda using generalize, get all the σi

  6. Extend the typing environment Gamma (pairfoldr)

  7. Recursive call to type checker, gets C_b, \tau

  8. Return (tau, C' /\ C_b)

Slide 15 

Slide 16 

Slide 17 

Forall things

Managing Quantified types
val and val-rec let, letrec, … lambda
FORALL contains all variables (because none are free in the context) FORALL contains variables not free in the context FORALL is empty
Generalize over all variables (because none are free in the context) Generalize over variables not free in the context Never generalize

3 April 2017: Lambda Calculus

There are PDF slides for 4/3/2017.

Where have we been and where are we going?

What is a calculus?

Demonstration of calculus: reduce

d/dx $(x^2 + y^2)$

What is a calculus? Manipulation of syntax.

What corresponds to evaluation? “Reduction to normal form”

Calculus examples:

A selection of calculi
Concurrency CCS (Robin Milner)
Security Ambient calculus (Cardelli and Gordon)
Distributed computing pi calculus (Milner)
Biological networks stochastic pi calculus (Regev)
Computation lambda calculus (Church)

Why study lambda calculus?

The world’s simplest reasonable programming language

Only three syntactic forms:

M ::= x | \x.M | M M'

Everything is programming with functions

First example:

(\x.\y.x) M N --> (\y.M) N --> M

Crucial: argument N is never evaluated (could have an infinite loop)

Programming in Lambda Calculus

Alert to the reading: Wikipedia is reasonably good on this topic

Everything is continuation-passing style

Q: Who remembers the boolean equation solver?

Q: What classes of results could it produce?

Q: How were the results delivered?

Q: How shall we do Booleans?

Coding Booleans

A Boolean takes two continuations:

true  = \x.\y.x
false = \x.\y.y

if M then N else P = ???

Coding pairs

Coding lists

Coding numbers: Church Numerals

Slide 1 

Slide 2 

Church Numerals to machine integers

; uscheme or possibly uhaskell
-> (define to-int (n)
          ((n ((curry +) 1)) 0))
-> (to-int three)
-> (to-int ((times three) four))

Church Numerals in λ

zero  = \f.\x.x;
succ  = \n.\f.\x.f (n f x);
plus  = \n.\m.n succ m;
times = \n.\m.n (plus m) zero;
-> four;
\f.\x.f (f (f (f x)))
-> three;
\f.\x.f (f (f x))
-> times four three;
\f.\x.f (f (f (f (f (f (f (f (f (f (f (f x)))))))))))

5 April 2017: Lambda-calculus semantics; encoding recursion

There are PDF slides for 4/5/2017.

Review: Church encodings

Review: Church Encodings

true  = \x.\y.x;          // Booleans
false = \x.\y.y;

pair  = \x.\y.\f.f x y;   // pairs
fst   = \p.p (\x.\y.x);
snd   = \p.p (\x.\y.y);

noreduce bot   = (\x.x x)(\x.x x);  // divergence

                          // S-expressions
         nil   =        \n.\c.n; 
         cons  = \y.\ys.\n.\c.c y ys;
         null? = \xs.xs true (\y.\ys.false);
noreduce car   = \xs.xs bot  (\y.\ys.y);
noreduce cdr   = \xs.xs bot  (\y.\ys.ys);

Review: Church Numerals

zero  = \f.\x.x;
succ  = \n.\f.\x.f (n f x);
plus  = \n.\m.n succ m;
times = \n.\m.n (plus m) zero;
-> four;
\f.\x.f (f (f (f x)))
-> three;
\f.\x.f (f (f x))
-> times four three;
\f.\x.f (f (f (f (f (f (f (f (f (f (f (f x)))))))))))

Question: What’s missing from this picture?

Answer: We’re missing recursive functions.

Astonishing fact: we don’t need letrec or val-rec (to com)

Operational semantics of lambda calculus

New kind of semantics: small-step

New judgment form

M --> N   ("M reduces to N in one step")

No context!! No turnstile!!

Just pushing terms around == calculus

Slide 3 

Board examples:

  1. Are these functions the same?

  2. Are these functions the same?


Slide 4 

Examples of free variables:

\x . + x y

\x. \y. x

Your turn! Free Variables

What are the free variables in each expression?

  \x.\y. y z
  \x.x (\y.x)
  \x.\y.\x.x y
  \x.\y.x (\z.y w)
  y (\x.z)
  (\x.\y.x y) y        

Your turn! Free Variables

What are the free variables in each expression?

  \x.\y. y z           - z
  \x.x (\y.x)          - nothing
  \x.\y.\x.x y         - nothing
  \x.\y.x (\z.y w)     - w
  y (\x.z)             - y z
  (\x.\y.x y) y        - y


The substitution in the beta rule is the heart of the lambda calculus

Slide 7 


(\yes.\no.yes)(\ ->

and never

\no.\    // WRONG!!!!!!

Really wrong!

(\yes.\no.yes) (\ tuesday
   ->   WRONG!!!
(\no.\ tuesday

Must rename the bound variable:

(\yes.\no.yes) (\ tuesday
(\yes.\z.yes)  (\ tuesday
(\z.\  tuesday

Slide 8 

Slide 9 

Nondeterminism of conversion:

              / \
             V   V
            B     C

Now what??

Slide 10 

Normal forms

Idea: normal form

A term is a normal form if

It cannot be reduced

What do you suppose it means to say

Idea: normal form

A term is a normal form if

It cannot be reduced

A term has a normal form if

There exists a sequence of reductions that terminates (in a normal form)

A term has no normal form if

It always reduces forever
(This term diverges)

Slide 13 

Reduction strategies (your homework, part 2)

Applicative-order reduction

Given a beta-redex

(\x.M) N

do the beta-reduction only if N is in normal form

Q: Does applicative order ever prevent you from making progress?

A: No. We can prove it by induction on the number of lambdas in N

Slide 14 

Normal-order reduction

Always choose leftmost, outermost redex

“Normal-order” stands for produces a normal form, not for “the normal way of doing things”

Slide 15 

Not your typical call-by-value semantics!

Fixed points, recursion

Slide 16 

Slide 17 

Your turn: Recursion equations

Is there a solution? Is it unique? If so, what is it?

f1 = \n.\m.(eq? n m) n 
              (plus n (f1 (succ n) m));

f2 = \n.f2 (isZero? n 100 (pred n));

f3 = \xs.xs nil (\z.\zs.cons 0 (f3 zs));

f4 = \xs.\ys.f4 ys xs;

Slide 19 

Your turn: Recursion equations

f1 = \n.\m.(eq? n m) n 
              (plus n (f1 (succ n) m));
    ; sigma (sum from n to m)

f2 = \n.f2 (isZero? n 100 (pred n));
    ; no unique solution (any constant f2)

f3 = \xs.xs nil (\z.\zs.cons 0 (f3 zs));
    ; map (const 0)

f4 = \xs.\ys. f4 xs ys;
    ; not unique: constant functions, commutative ops

Slide 21 

Suppose g F = F. Proof that F is factorial.

For all n, g F n = n!, by induction:

F 0 = g F 0 = 1
F n
  = { by assumption }
g F n 
  = { definition of g }
if n = 0 then 1 else n * F (n-1)
  = { assumption, n > 0 }
n * F (n-1)
  = { induction hypothesis }
n * (n-1)!
  = { definitiion of factorial }

Now you do it

Conversion to fixed point

length = \xs.null? xs 0 (+ 1 (length (cdr xs)))

lg = \lf.\xs.null? xs 0 (+ 1 (lf (cdr xs)))

One startling idea

You can define a fixed-point operator

Fixed-point operators

A simple algebraic law


    fix g = g (fix g)

then fix g can define recursive functions!

The only recursion equation you’ll ever need

Slide 25 

Lambda calculus in context

What’s its role in the world of theory?

Operational semantics          Type theory     Denotational      Lambda 
(Natural deducation style)                       semantics      calculus
--------------------------     -----------     ------------     --------

Interpreters like Python       type checkers   compilers        *models*

Why is it “calculus”?

What’s the role of calculi in computer science:

Why so many calculi? They have simple metatheory and proof technique.

10 April 2017: Hiding information with abstract data types

There are PDF slides for 4/10/2017.

Where have we been?

What about big programs?

An area of agreement and a great divide:

                 INFORMATION HIDING
                     /         \
 modular reasoning  /           \  code reuse
                   /             \ 
internal access   /               \  interoperability 
to rep           /                 \  between reps
                /                   \
            MODULES               OBJECTS           

Why modules?

Unlocking the final door for building large software systems

Modules overview

Functions of a true module system:

Real modules include separately compilable interfaces and implementations


Collect declarations

Things typically declared:

Key idea: a declared type can be abstract

Terminology: a module is a client of the interfaces it depends on

Roles of interfaces in programming:

The best-proven technology for structuring large systems.

Ways of thinking about interfaces

Two approaches to writing interfaces

Interface “projected” from implementation:

Full interfaces:

Module Implementations

Standard ML Modules

The Perl of module languages?

What we’ve been using so far is the core language

Modules are a separate language layered on top.

ML module terminology

Interface is a signature

Implementation is a structure

Generic module is a functor

Structures and functors match signature

Analogy: Signatures are the ``types’’ of structures.

Signature basics

Signature says what’s in a structure

Specify types (w/kind), values (w/type), exceptions.

Ordinary type examples:

    type t        // abstract type, kind *
    eqtype t
    type t = ...  // 'manifest' type
    datatype t = ...

Type constructors work too

    type 'a t     // abstract, kind * => *   
    eqtype 'a t
    type 'a t = ...
    datatype 'a t = ...

ML Modules examples, part I

Signature example: Ordering

signature ORDERED = sig
  type t
  val lt : t * t -> bool
  val eq : t * t -> bool

Signature example: Integers

signature INTEGER = sig
  eqtype int             (* <-- ABSTRACT type *)
  val ~   : int -> int
  val +   : int * int -> int
  val -   : int * int -> int
  val *   : int * int -> int
  val div : int * int -> int
  val mod : int * int -> int
  val >   : int * int -> bool
  val >=  : int * int -> bool
  val <   : int * int -> bool
  val <=  : int * int -> bool
  val compare : int * int -> order
  val toString   : int    -> string
  val fromString : string -> int option

Implementations of integers

A selection…

structure Int    :> INTEGER
structure Int31  :> INTEGER  (* optional *)
structure Int32  :> INTEGER  (* optional *)
structure Int64  :> INTEGER  (* optional *)
structure IntInf :> INTEGER  (* optional *)

Homework: natural numbers

signature NATURAL = sig
   type nat   (* abstract, NOT 'eqtype' *)
   exception Negative
   exception BadDivisor

   val of_int   : int -> nat 
   val /+/      : nat * nat -> nat
   val /-/      : nat * nat -> nat
   val /*/      : nat * nat -> nat
   val sdiv     : nat * int -> 
                  { quotient : nat, remainder : int }
   val compare  : nat * nat -> order
   val decimals : nat -> int list

12 April 2017: more ML modules

There are PDF slides for 4/12/2017.

Signature review: collect

signature QUEUE = sig
  type 'a queue    (* another abstract type *)
  exception Empty

  val empty : 'a queue
  val put : 'a * 'a queue -> 'a queue
  val get : 'a queue -> 'a * 'a queue   (* raises Empty *)

  (* LAWS:  get(put(a, empty))     ==  (a, empty)

Structure: collect definitions

structure Queue :> QUEUE = struct   (* opaque seal *)
  type 'a queue = 'a list
  exception Empty

  val empty = []
  fun put (x,q) = q @ [x]
  fun get [] = raise Empty
    | get (x :: xs) = (x, xs)

  (* LAWS:  get(put(a, empty))     ==  (a, empty)

Dot notation to access components

fun single x = Queue.put (Queue.empty, x)
val _ = single : 'a -> 'a Queue.queue

What interface with what implementation?

Maybe mixed together, extracted by compiler!

Maybe matched by name:

Best: any interface with any implementation:

But: not “any”—only some matches are OK

Signature Matching


 structure Queue :> QUEUE = QueueImpl

if principal signature of QueueImpl matches ascribed signature QUEUE:

Signature Ascription

Ascription attaches signature to structure

Slogan: “use the beak”

Opaque Ascription



 structure Queue :> QUEUE = struct
   type 'a queue = 'a list
   exception Empty

   val empty = []
   fun put (x, q) = q @ [x]
   fun get [] = raise Empty
    | get (x :: xs) = (x, xs)

Not exposed: 'a Queue.queue = 'a list

Abstract data types

How opaque ascription works

Outside module, no access to representation

Inside module, complete access to representation

Abstract data types and your homework

Natural numbers

Data abstraction for reuse

Abstract data types and your homework

Two-player games:

Problems abstraction must solve:

Result: a very wide interface

Abstraction design: Computer player

Computer player should work with any game, provided

Brute force: exhaustive search

Your turn! What does computer player need?

Our computer player: AGS

Any game has two key types:

  type config
  structure Move : sig
     type move
     ...  (* string conversion, etc *)

Key functions use both types:

  val possmoves : config -> Move.move list
  val makemove  : config -> Move.move -> config

Multiple games with different config, move?

Yes! Using key feature of ML: functor

Functors and an Extended SML Example

Game interoperability with functors

functor AgsFun (structure Game : GAME) :> sig
  structure Game : GAME
  val bestmove : Game.config -> Game.Move.move option
  val forecast : Game.config -> Player.outcome
   where type Game.Move.move = Game.Move.move
   and   type Game.config    = Game.config
= struct
    structure Game = Game
    ... definitions of `bestmove`, `forecast` ...

Functors: baby steps

A functor abstracts over a module

Formal parameters are declarations:

functor AddSingle(structure Q:QUEUE) = 
     structure Queue = Q
     fun single x = Q.put (Q.empty, x)

Combines familiar ideas:

Using Functors

Functor applications are evaluated at compile time.

functor AddSingle(structure Q:QUEUE) = 
     structure Queue = Q
     fun single x = Q.put (Q.empty, x)

Actual parameters are definitions

structure QueueS  = AddSingle(structure Q = Queue)
structure EQueueS = AddSingle(structure Q = EQueue)

where EQueue is a more efficient implementation

Slide 16 

Slide 17 

Functors on your homework

Separate compilation:

Code reuse with type abstraction

Trick: Functor instead of function

AGS expects game with fixed initial configuration.

What about family of games? 3 sticks? 14 sticks? 1000 sticks?

Functor to rescue:

functor SticksFun (val N : int) :> GAME =
  struct ... end

structure S14 = SticksFun(val N = 14)

ML module summary

New syntactic category: declaration

Signature groups declarations: interface

Structure groups definitions: implementation

Functor enables reuse:

Opaque ascription hides information

19 April 2017: Object-orientation

There are PDF slides for 4/19/2017.

Demo: circle, square, triangle, with these methods:

Instructions to student volunteers


  1. Object 1, adjust your coordinate to place your South control point at (0, 0).

  2. Object 1, what is the coordinate position of your North control point?

  3. Object 2, adjust your coordinate to place your South control point at (0, 2).

  4. Object 2, what is the coordinate position of your North control point?

  5. Object 3, adjust your coordinate to place your Southwest control point at (0, 4).

  6. Object 1, draw yourself on the board

  7. Object 2, draw yourself on the board

  8. Object 3, draw yourself on the board

Key concepts of object-orientation

Key mechanisms

Private instance variables

Code attached to objects and classes

Dynamic dispatch

Key idea

Protocol determines behavioral subtyping

Class-based object-orientation

Dynamic dispatch determined by class definition

Code reuse by sending messages around like crazy

Example: list filter

Example: list filter

-> (val ns (new List))
List( )
-> (addAll: ns #(1 2 3 4 5 6))
( 1 2 3 4 5 6 )
-> ns
List( 1 2 3 4 5 6 )
-> (select: ns [block (n) (= 0 (mod: n 2))])
List( 2 4 6 )

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Blocks and Booleans

Blocks are objects

Block Examples

-> (val twice [block (n) (+ n n)])

-> (value twice 3)
-> (val delayed {(println #hello) 42})

-> delayed

-> (value delayed)

Booleans use continuation-passing style

Boolean example: minimum

-> (val x 10)
-> (val y 20)
-> (ifTrue:ifFalse: (<= x y) {x} {y})

Slide 8 

Booleans implemented with two classes True and False

Slide 9 

Method dispatch in the Booleans

Slide 10 

Board - Method dispatch

To answer a message:

  1. Consider the class of the receiver

  2. Is the method with that name defined?

  3. If so, use it

  4. If not, repeat with the superclass

Run out of superclasses?

“Message not understood”

Slide 11 

Dispatching to

(class True Boolean ()
  (method ifTrue:ifFalse: (trueBlock falseBlock) 
      (value trueBlock))
  ; all other methods are inherited

Slide 13 

Slide 14 

24 April 2017: Inheritance. Numbers and magnitudes

There are PDF slides for 4/24/2017.

Slide 1 

Slide 2 

Your turn: Short-circuit

(class Boolean Object
  (method not ()          
    (ifTrue:ifFalse: self {false} {true}))
  (method and: (aBlock)

Your turn: Short-circuit

(class Boolean Object
  (method not ()          
    (ifTrue:ifFalse: self {false} {true}))
  (method and: (aBlock)
    (ifTrue:ifFalse: self aBlock {self})))

History and overview of objects

History of objects

We know that mixing code and data can create powerful abstractions (function closures)

Objects are another way to mix code and data

Pioneers were Nygaard and Dahl, who added objects to Algol 60, producing SIMULA-67, the first object-oriented language

What’s an object?

Agglutination containing

A lot like a closure

What are objects good at?

Not really useful for building small things

If you build a big, full-featured abstraction, you can easily use inheritance to build another, similar abstraction. Very good at adding new kinds of things that behave similarly to existing things.

For your homework, you’ll take a Smalltalk system that has three kinds of numbers, and you’ll add a fourth kind of number.

What’s hard about objects?

If you do anything at all interesting, your control flow becomes smeared out over half a dozen classes, and your algorithms are nearly impossible to undrstand.

Smalltalk objects

Why Smalltalk?

The five questions:

Slide 5 

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Slide 7 

Message passing

Look at SEND

N.B. BLOCK and LITERAL are special objects.

Magnitudes and numbers

Key problems on homework

Slide 8 

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Implementation of

(class Magnitude Object 
  () ; abstract class
  (method =  (x) (subclassResponsibility self)) 
                    ; may not inherit = from Object
  (method <  (x) (subclassResponsibility self))
  (method >  (y) (< y self))
  (method <= (x) (not (> self x)))
  (method >= (x) (not (< self x)))
  (method min: (aMagnitude)
     (if (< self aMagnitude) {self} {aMagnitude}))
  (method max: (aMagnitude)
     (if (> self aMagnitude) {self} {aMagnitude}))

Slide 13 

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Slide 15 

Example class : initialization

(class Fraction Number
    (num den) ;; representation (concrete!)
    (class-method num:den: (a b)
        (initNum:den: (new self) a b))
    (method initNum:den: (a b) ; private
        (setNum:den: self a b)
        (signReduce self)
        (divReduce self))
    (method setNum:den: (a b)
        (set num a) (set den b) self) ; private
    .. other methods of class Fraction ...

Slide 17 

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Slide 19 

26 April 2017: Double dispatch, collections

There are PDF slides for 4/26/2017.

COURSE EVALUATIONS: email me your response

Two topics for today

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Making open system extensible

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Bonus content not covered in class: Collections

Why collections?

Goal of objects is reuse

Key to successful reuse is a well-designed class hierarchy

Slide 16 

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Implementing Collections

Slide 20 

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Question: what’s the most efficient way to find the size of a list?

Question: what’s the most efficient way to find the size of an array?

Slide 22 


Key strategy for reuse in object-oriented languages: subtype polymorphism

Slide 23 

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Example collection - Sets

Slide 26 

Most subclass methods work by delegating all or part of work to list members

N.B. Set is a client of List, not a subclass!

Next example highlight: class method and super!

Slide 27 

1 May 2017: Programming languages past, present, and future

There were no PDF slides on 5/1/2017.

The timeline


Your questions

Prelude and disclaimer

My job is to know programming-language research. Most questions are (appropriately) about programming-language design.

Best language for the job

(side board)

Next steps

Key next step is to adapt your skills in context

That said,

Also, go deeper:

And well beyond:

Major paradigms not covered in class

Not new, but

My go-to languages

Note: each of these has replaced some other language (Awk, Icon, Modula-3, …).



Programming-language problems

Problems begging for new languages:

Of personal interest:

Note: the person who thinks of the problem is a super genius. I am not that person.

Today’s PL problems:

Future PL needs: I have no idea, but

What’s exciting right now:

Other questions

Best first language

The biggest tragedy in PL

What is it?

My favorite question