6 September 2017: Introduction to Comp 105

There are PDF slides for 9/7/2017.

Handout: 105 Tip Sheet

Handout: 105 Syllabus

Handout: Experiences of successful 105 students

Handout: Homework

Video: The Big Picture



Why so many languages?

Question: What languages have you programmed in?

There are thousands of programming languages, each unique.

Question: Why do you suppose there are so many?

The right language for the job makes it easier to write programs that really work

The Blub paradox

What this course isn’t

Why not?

What this course is:


What are reusable principles?

What if the course were called “Cooking”?

The same division for programming languages:

What Programming Languages are, technically

What can you get out of Comp 105?

Students who get the most out of 105 report

Great languages begin with great features

In Comp 105,

How will we study language features?

Common Framework

Course logistics and administration



Homework will be frequent and challenging:

Both individual and pair work:

Arc of the homework looks something like this:

Assignment Difficulty
impcore one star
opsem two stars
scheme three stars
hofs four stars

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

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

Just as intellectually challenging as COMP 40, but in an entirely different direction.

Everyone who takes this class has the ability to master the material; Succeeding just requires digging in.

We provide lots of resources to help:

We encourage you to form study groups so you have thought partners.

Two two bad habits to avoid:

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 on the web

What am I called?

Call me “Kathleen,” “Professor Fisher”, or “Profesor.”

ImpCore: The first language in our common framework

Exercise: all-fours?

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


Impcore variable definition


(val n 99)


int n = 99;

Also, expressions at top level (definition of it)

Impcore expressions

No statements means expression-oriented:

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

Each one has a value and may have side effects!

Functions are primitive (+ - * / = < > print)
or defined with (define f ...).

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

Scoping rules for Impcore

Scopes also called “name spaces”; we will call them “environments” because that’s the pointy-headed theory term—and if you want to read some of the exciting papers, pointy-headed theory has to be second nature.

Names known in ``environments"

Ways to talk about meanings of names:

Impcore vars in 2 environments: globals, formals

There are no local variables

Functions live in their own environment (not shared with variables)

Environmental abuse

Abuse of separate name spaces:

-> (val f 33)
-> (define f (x) (+ x x))
-> (f f)

Recursion: a review

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.

11 September 2017: Introduction to Semantics

There are PDF slides for 9/12/2017.

Handout: 105 Impcore Semantics, Part 1

Today: Abstract Syntax and Operational Semantics

Discussion: Two things you learned last class.

Programming-language semantics

Semantics means meaning.

What problem are we trying to solve?

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

Ways of knowing:

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

Why bother with precise semantics?

Same reason as other forms of math:

Plus, needed to build language implementation and tests

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: The skills you learn in this class will apply

Behavior decomposes

We want a computational notion of meaning.

What happens when we run (* y 3)?

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

Knowledge is expressed inductively

(Non)-Example of compositionality: Spelling/pronunciation in English

By design, programming languages more orderly than natural language.

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 and 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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With that as background, we can now dive in to the semantics for Impcore!

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13 September 2017: Semantics, Syntactic Proofs, Metatheory

There are PDF slides for 9/14/2017.

Handout: Impcore expression rules



Last Time

Both math and code on homework

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

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Using Operational Semantics

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).

Valid derivations, or “How do I know what this program should evaluate to?”

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?

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What is the result of (digit? 7)?

How do we know it’s right?

From rules to proofs

What can a proof tell us?

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Judgment is valid 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 (rules in handout)

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Building derivations

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At this point, we’ve now covered derivations and how a single derivation corresponds to evaluating a particular program.

Proofs about all derivations: Metatheory

Derivations (aka syntactic proofs) enable meta-reasoning

Derivation D is a data structure

Got a fact about all derivations?

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.

18 September 2017: Metatheory wrapup. Intro to functional programming

There are PDF slides for 9/19/2017.



Last Time

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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 157)

Introduction to Scheme

Question 2: What are the values?

Two new kinds of data:


Picture of two cons cells

(cons 3 (cons 2 ’()))

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

Examples of S-Expression operators

 (cons 'a '())         also written '(a)

 (cons 'b '(a))        equals '(b a)

 (cons 'c '(b a))      equals '(c b a)

 (null? '(c b a))      equals #f

 (cdr '(c b a)         equals '(b a)

 (car '(c b a)         equals 'c

The symbol ’ is pronounced “tick.”
It indicates that what follows is a literal.

Picture of (cons c (cons b (cons a '())))

Your turn!

  1. What is the representation of

'((a b) (c d))

which can be alternatively written

cons( (cons a (cons b '()))

 `(cons (cons c (cons d '())) '()))`
  1. What is the representation of

cons('a 'b)

Contrast this representation with the one for

cons('a '())

Both of these expressions are S-expressions, but only cons('a '()) is a list.

Picture of '((a b) (c d))

Picture of cons('a 'b)

20 September 2017: More Scheme

There are PDF slides for 9/21/2017.



Last Time


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)

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 to create laws.

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?

Example: Append

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(Z)

Suppose I can prove two things:

  1. IH (’())

  2. Whenever z in Z and also IH(zs), then IH (cons z zs)

then I can conclude

Forall zs in List(Z), IH(zs)

Example: The cost of append

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.

Example: 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|.

The method of accumulating parameters

The function revapp takes two list arguments xs and ys.
It reverses xs and appends the result to ys:

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

Write algebraic laws for revapp involving different possible forms for xs.

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.)

Linear reverse, graphically

We call reverse on the list '(1 2 3):

Function reverse calls the helper function revapp with '() as the ys argument:

The xs parameter isn’t '(), so we recursively call revapp with the cdr of xs and the result of consing the car of xs onto ys:

The xs parameter still isn’t '(), so we again call revapp recursively:

Still not '(), so we recurse again:

This time xs is '(), so now we just return ys, which now contains the original list, reversed!

PDF slides of revapp

25 September 2017: Let and Lambda

There are PDF slides for 9/26/2017.



Last Time


Association lists represent finite maps [Not covered in class]

Implementation: List of key-value pairs

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

Picture with spine of cons cells

Functions car, cdar, caar, cadar can help navigate.

Recall that the left box in a cons cell is the address and the right box is the data. Read the a as “address” and the d as “data” from right to left.

In association lists, these operations correspond to

A-list example

    -> (find 'Building 
             '((Course 105) (Building Robinson) 
               (Instructor Fisher)))
    -> (val ksf (bind 'Office 'Halligan-242
                (bind 'Courses '(105)
                (bind 'Email 'comp105-staff '()))))
    ((Email comp105-staff) 
     (Courses (105)) 
     (Office Halligan-242))
    -> (find 'Office ksf) 
    -> (find 'Favorite-food ksf)


Algebraic laws of association lists

Laws of assocation lists

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

Handy new feature of Scheme: let binding

Introduce local names into environment

    (let ((x1 e1)
          (xn en))

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

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

Note that we have definititions in the language 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:

Syntax McCarthy should have used

(let ((val x1 e1)
      (val xn en))

From Impcore to uScheme

Things that should offend you about Impcore:

All these problems have one solution: Lambda! (λ)

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

Escaping functions

“Escape” means “outlive the function in which the lambda was evaluated.”

We have already seen an example:

An ``escaping’’ function

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

Where are n and k stored???

Picture of activation record for to-the-n-minus-k with n and k being popped.

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An example:

What’s the closure for conjunction?

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

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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?)


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).

As we study higher-order functions in more detail, you will see why currying is useful.

Classic functional technique: Currying

-> (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.”

What’s the algebraic law for curry?

     ...   (curry f) ...    =  ... f ...

     Keep in mind: 
     All you can do with a function is apply it!

     (((curry f) x) y) = f (x, y)

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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)

Bonus content: Lambda as an abstraction barrier

Bonus content: vulnerable variables?

-> (val seed 1)
-> (val rand (lambda ()
      (set seed (mod (+ (* seed 9) 5) 1024)))))
-> (rand)
-> (rand)
-> (set seed 1)
-> (rand)

Q: What’s the problem with this approach?

A: The seed is exposed to the end user, who can break the abstraction of the rand function.

Bonus: Lambda as abstraction barrier!

-> (val mk-rand (lambda (seed)
     (lambda ()
       (set seed (mod (+ (* seed 9) 5) 1024))))))
-> (val rand (mk-rand 1))
-> (rand)
-> (rand)
-> (set seed 1)
error: set unbound variable seed
-> (rand)

27 September 2017: Higher-order functions

There are PDF slides for 9/28/2017.



Last Time


-> (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)


Reasoning about code

Reasoning principle for lists

Recursive function that consumes A has the same structure as a proof about A

Reasoning principle for functions

Q: Can you do case analysis on a function?

A: No!

Q: So what can you do then?

A: Apply it!

Higher-Order Functions

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, or a “catamorphism”

List search: exists?

Algorithm encapsulated: linear search

Example: Is there a 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)
          (or (p? (car xs)) 
              (exists? p? (cdr xs)))))
-> (exists? even? '(1 3))
-> (exists? even? '(1 2 3))
-> (exists? ((curry =) 0) '(1 2 3))
-> (exists? ((curry =) 0) '(0 1 2 3))

Slide 4 

Your turn: Does everything match: all?

Example: Is every element in a list even?

Algebraic laws:

(all? p? '())          == ???
(all? p? '(cons a as)) == ???

(all? p? '())          == #t
(all? p? '(cons a as)) == p? x and all? p? xs

Defining all?

-> (define all? (p? xs)
      (if (null? xs)
          (and (p? (car xs)) 
               (all? p? (cdr xs)))))

-> (all? even? '(1 3)) 
-> (all? even? '(2)) 
-> (all? ((curry =) 0) '(1 2 3))
-> (all? ((curry =) 0) '(0 0 0))

Slide 6 

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 9 

“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 11 

The universal list function: fold

Slide 12 

foldr takes two arguments:

Example: foldr plus zero '(a b)

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

Slide 13 

Slide 14 

In-class exercise: Folding combine?

Slide 15 

Slide 16 

Tail calls

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

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 18 

Slide 19 

Slide 20 

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 21 

Slide 22 

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

Question: Why can’t we do this same optimization on reverse?

Answer: reverse has to do further computation with results of recursive calls, so can’t eliminate the stack frame until later.

Slide 23 

Answer: a goto!!

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

2 October 2017: Continuations

There are PDF slides for 10/3/2017.

Last Time




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” a value to a continuation.

Uses of continuations


Slide 1 

Motivating Example: From existence to witness

Slide 2 


Bad choices:

Good choice:

Slide 3 

Your turn: 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))

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) = (fail)

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

(witness-cps p? (cons z zs) succ fail) = 
     (witness-cps p? 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 7 

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

Answer: Constant

Example Use: Instructor Lookup

-> (val 2017f '((Fisher 105)(Cowen 170)(Chow 116)))
-> (instructor-info 'Fisher 2017f)
(Fisher teaches 105)
-> (instructor-info 'Chow 2017f)
(Chow teaches 116)
-> (instructor-info 'Souvaine 2017f)
(Souvaine is-not-on-the-list)

Slide 9 

Slide 10 

Slide 11 

Slide 12 

Extended Example: A SAT Solver

Exercise: Find a satisfying assignment if one exists

(val f1 '(and x y z w p q (not x)))

(val f2 '(not (or x y)))

(val f3 '(not (and x y z)))

(val f4 '(and (or x y z) 
              (or (not x) (not y) (not z))))

Satisfying assignments

(val f1 '(and x y z w p q (not x))) ; NONE

(val f2 '(not (or x y))) 
                  ; { x |-> #f, y |-> #f }

(val f3 '(not (and x y z))) 
                  ; { x |-> #f, ... }
(val f4 '(and (or x y z) 
              (or (not x) (not y) (not z))))
              ; { x |-> #f, y |-> #t, ... }

Slide 15 

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 cotinuation with current{x -> #t}

Pass fail as resume continuation

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

These diagrams (and the corresponding code) compose!

Solving A and B

Picture of 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

Picture of 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.

4 October 2017: Scheme Semantics

There are PDF slides for 10/5/2017.



Scheme Semantics

Last Time

New Syntax, Values, Environments, and 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 

Board: Picture of environment pointing to store.

Questions about Assign:


Slide 4 

Function Application


  (val even (lambda (x) (= 0 (mod x 2)))) 

  (val f    (lambda (y) (if (even y) 5 15)))

  (val even 3)

  (f 10)

Question: Which even is referenced when f is called?
Answer: With static scoping, it’s the predicate. With dynamic scoping it’s the one bound to 3.

Slide 6 

Questions about ApplyClosure:

Slide 7 

Picture of environment and store that results from executing above program.

Closure Optimizations

11 October 2017: Scheme Wrap-up; ML Intro

There are PDF slides for 10/12/2017.

Handout: Which let is which?



Last Time


Which let is which and why?

Three versions of let:

Handout: Which let is which?

Lisp and Scheme Retrospective

Slide 1 

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

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

Introduction to ML

Apply your new knowledge in Standard ML:

Lectures on ML:

  1. Algebraic types and pattern matching
  2. Exceptions
  3. An introduction to types

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 + exceptions + static types

uScheme -> 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.

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) =   (* pred? not legal *)
      let val rest = filter pred xs 
      in if pred x then
         else rest

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

(* Note fn x => e is syntax for lambda in SML *)

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]

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]


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

val res5 = take 2 [1,2,3,4]
val res6 = take 3 [1] 
           handle ListTooShort => 
             (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     l   = l
  | drop n    []   = raise ListTooShort
  | drop n (x::xs) = (drop (n-1) xs)

val res7 = drop 2 [1,2,3,4]
val res8 = drop 3 [1] 
           handle ListTooShort => 
              (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]

Drop while

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 zs
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 an 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)

16 Oct 2017: Programming with constructed data and types

There are PDF slides for 10/17/2017.



Last Time

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

Foundation: Data

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

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

Algebraic Datatypes

Enumerated types

Datatypes can define an enumerated type and associated values.

datatype suit = heart | diamond | spade | club

Here suit is the name of a new type.

The data constructors heart, dimaond, spade, and club are the values of type suit.

Data constructors are separated by vertical bars.

Pattern matching

Datatypes are deconstructed using pattern matching.

fun toString heart = "heart"
  | toString diamond = "diamond"
  | toString spade = "spade"
  | toString club = "club"

val suitName = toString heart

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

datatype IntTree = Leaf | Node of int * IntTree * IntTree

IntTree 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, IntTree).

Building values with constructors

We build values of type IntTree 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 = preOrder t3

IntTree 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 data 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.

Things to notice about datatypes


Datatype declarations introduce names into:

  1. the type environment: suit, IntTree, tree

  2. the value environment: heart, Leaf, Parent


Datatype declarations are inherently inductive:

Datatype Exercise

Slide 2 

Exercise answers

datatype sx1 = ATOM1 of atom
             | LIST1 of sx1 list

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

Case expressions: How we use datatypes

Eliminate values of algebraic types

New language construct case (an expression)

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

At top level, fun better than case

When possible, write

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

case works for any datatype

 fun toStr t = 
     case t 
       of Leaf => "Leaf"
        | Node(v,left,right) => "Node"

But often pattern matching is better style:

 fun toStr' Leaf = "Leaf"
   | toStr' (Node (v,left,right)) = "Node"

Bonus: 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 7 

Tuple Pattern Matching

val (x,y) = (1,2)

val (left, pivot, right) = split xs

val (n,xs) = (3, [1,2,3])

val (x::xs) = [1,2,3]

val (_::xs) = [1,2,3]

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 9 

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 12 

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 15 


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 20 


Slide 21 

Slide 22 

Slide 23 

Slide 24 

18 October 2017: Types

There are PDF slides for 10/19/2017.



Type systems

What kind of value do we have?

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 that p : bool

Questions type systems can answer:

Questions type systems generally cannot answer:

Decidability and Type Checking

Suppose L is a “Turing-Complete” Language.

TP is the set of programs in L that terminate.

Wish: a type system to statically classify terminating programs:

Expression e in L has type T (e : T) iff e terminates.

But: Undecideable!

We can prove no such type system exists.


Static vs. Dynamic Type Checking

Most languages use a combination of static and dynamic checks

Static: “for all inputs”

Dynamic: “for some inputs”

Type System and Checker for a Simple Language

Define an AST for expressions with:

Language of expressions

    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

Experience shows it is better to test two types for equivalence than to write rule with same type appearing twice.

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

What is a type?

Type checker in ML

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

Typing Rules: Contexts and Term Variables

Add variables and let binding to our language, what happens?

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

What could go wrong with a variable?

Key idea: Type environment (Gamma) tracks the types of variables.

Rule for var

x in domain 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'

Type Checker

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

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


This is a big chunk of what language designers do.

23 October 2017: Type Checking with Type Constructors

There are PDF slides for 10/24/2017.


Last Time




Gamma{x->tau1} |- e : tau2   
Gamma |- fn x : tau1 => e  : tau1 -> tau2


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

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.

Monomorphic vs Polymorphic Types

Monomorphic types have “one shape.”

Polymorphic types have “many shapes.”

Design and implementation of monomorphic languages


Language designer’s process when adding new kinds of types:

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

Question: 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 5 

Type judgments for monomorphic system

Two judgments:

Monomorphic type rules

Slide 7 

Slide 8 

Notice: one rule for if!!

Classic types for data structures

Slide 9 

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

Slide 10 

Slide 11 

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 13 

Slide 14 

Typing Rule Exercise

Slide 15 

Slide 16 

Coding the arrow-introduction rule

Slide 17 

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  funtype (formaltypes, bodytype)