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# 5 September 2018: Introduction to Comp 105

There are PDF slides for 9/6/2018.

Handout: Experiences of successful 105 students

Handout: Programming with proof systems and algebraic laws

Handout: Recommended software process

There is a shared album with photos of lecture. Anybody can add photos.

### Introduction

My pronouns are he, him, his.

Please call me “Norman,” “Professor Ramsey,” or “Mr Ramsey”

## Why we are here

- Write code from scratch
- In a language you’ve never seen
- That sails past code review

What it involves:

- Programming practice (emphasize widely used features: functions, types, modules, objects)
- Mathematical description (how things work, see patterns)

(Example: see patterns of recursion in homework)

Today:

- Start with technical work
- Then a little about experience and expectations

## What is a programming language?

What is a PL? Think. Then pair. Then share.

- If you’ve taken 170, a caution: every programming language is a formal language, but not every formal language is a programming language.

This week: new ways of thinking about stuff you already know

### Syntactic structure: induction

#### Numerals

Show some example numerals

Numerals:

```
2018
73
1852
```

Not numerals:

```
3l1t3
2+2
-12
```

**Q:** What does a numeral stand for?

That is, if a numeral appears in the syntax, what sort of value flies around at run time?

**Q:** Does every numeral stand for one of these things?

**Q:** How many of them are there?

**Q:** How many numerals are there?

##### In-class Exercise: Inductive definitions of numerals

Write an inductive definition of numerals

There is more than one! When you finish one, look for another

### Value structure: induction again

In-class exercise: inductive definition of natural number

- Again, there is more than one
- What makes it easy to know meaning of numerals?

## Syntax in programming languages

### Concrete Syntax

Programming-languages people are wild about **compositionality**.

Build sensible things from sensible pieces using just a few construction principles.

Example: expressions,

*e*_{1}+*e*_{2}

### Syntactic structure of Impcore

Watch for syntactic categories, syntactic composition

### Live coding in Impcore

Try `square`

function. E.g., `(square (+ 2 2))`

## Proofs to values to code

Coding from scratch begins with data

- Forms of data (proof system or otherwise)
*Algebraic laws*(new)- Code

Example function: `all-fours?`

# 10 September 2018: Abstract syntax and operational semantics

There are PDF slides for 9/11/2018.

Handout: 105 Impcore Semantics, Part 1

## Review

**Short discussion**: *One thing you learned in the first class*

Language: syntax, rules/organization/grammar

*Programming* language: you can run it (values)

## Review: algebraic laws

## Approaching 105

100-level course

Responsibility for your own learning: lecture is the tip of the iceberg

Systematic programming process: we can’t observe process, but it will affect your time spent by a factor of 2, 3, or 4.

*Awareness* of your own learning: **Metacognition**

### Office hours

Students: go to office hours! An idle TA is an unhappy TA.

### Course logistics: Recitation

We have capacity for 120, with 112 registered. We can’t fill recitations first-come, first-served. Sorry we don’t seem to be able to communicate with the registrar.

You should have an email from Megan Monaghan, or if you registered after August 29, from me. Trust it, not SIS.

If you have no time or your time does not work, I can shift a few people. Post to Piazza and I will send email by Wednesday.

## Preparing for a language you’ve never seen before

You need a vocabulary. It will involve math.

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

### Concrete and abstract Syntax

Programming-languages people are wild about **compositionality**.

- Build sensible things from sensible pieces using just a few construction principles.

Example of compositionality: **concrete syntax** (grammar)

- How many different kinds of things can be composed:
*syntactic categories*

## 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:

*People*learn from examples- You can build intuition from words

(Book is full of examples and words) - To know
*exactly*,*unambiguously*, you need more precision

(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)

- You can watch and learn
- A computer can’t
- “Put the peanut butter on the bread”

### Why bother with precise semantics?

(Needed to build implementation, tests)

Same reason as other forms of math:

- Distill down your understanding and express it
- Prove properties people care about (e.g., private information doesn’t leak; device driver can’t bring kernel down)
- Most important for you:
*things that look different are actually the same*

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**

Atomic forms: Describe behavior directly (e.g., constants, variables)

Compound forms: Behavior specified by composing behaviors of parts

### ASTs

Question: What do we assign behavior to?

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

An AST is a

*data structure*that represents a program.A

*parser*converts program text into an AST.

Question: How can we represent all while loops?

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

Answer:

- Tag code as a while loop
- Identify the condition, which can be any expression
- Identify the body, which can be any expression

As a data structure:

- WHILEX(exp1, exp2), where
- exp1 is the representation of (i < n && a[i] < x), and
- exp2 is the representation of i++

# 12 September 2018: Rules, proofs, and metaproofs

There are PDF slides for 9/13/2018.

**Students: Take a blank white card**

Handout: Redacted Impcore rules

Today:

- How do we know what happens when we run the code?
**valid derivations** - What can we prove about it?
**metatheory**

### From ASTs to behaviors

## Rules and metatheory

Review: **What elements are needed to know run-time behavior?**

**OK, your turn:** what do you think is rule for evaluating literal, variable? (Base cases)

## Good and bad judgments

## Proofs: Putting rules together

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

**Exercise**:

In the body of

`digit?`

, what expressions are evaluated in what order?As a function application, the body matches template

`(`

*f**e*_{1}*e*_{2}`)`

. In this example,- What is
*f*? - What is
*e*_{1}? - What is
*e*_{2}?

- What is

Let’s develop the ApplyUser rule for the special case of two arguments: ⟨*A**P**P**L**Y*(*f*, *e*_{1}, *e*_{2}), *ξ*, *ϕ*, *ρ*⟩ ⇓ ?

What is the result of `(digit? 7)`

?

How do we know it’s right?

## From rules to proofs

### Building a derivation

### Building derivations

## Proofs about derivations: metatheory

Cases to try:

- Literal
- GlobalVar
- SetGlobal
- IfTrue
- ApplyUser2

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

# 17 Sep 2018: Metatheory; Scheme

There are PDF slides for 9/18/2018.

Handout: Impcore semantics (page 81), proof template from book (page 61)

Handout: Programming with Scheme Values and Algebraic Laws

## Announcements

Homework grades:

- Apologize for grades without feedback
- Do resubmit photos
`submit105-impcore`

Office hours:

- TAs cover the whole floor
- When open, will have photos up
- When closed, will have “next help at” signs up

Students are coming to have code debugged without having algebraic laws

- We’ll ask for your laws
- We’ll help you debug laws before debugging code

## Operational-semantics review

Review questions (not easy):

What is the evaluation judgment?

How do you say it?

What’s up with

*ξ*and*ξ*′?How do we capture the time element of a computation?

## Proofs about derivations: metatheory

Cases to try:

- Literal
- GlobalVar
- SetGlobal
- IfTrue
- ApplyUser2

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

## Better semantics

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.

# 19 September 2018: Scheme

There are PDF slides for 9/20/2018.

### Announcements

Handout: Q&A results

- Opsem as a
*vocabulary* - Hint of optimizing compiler (if performance matters)

Travel to Hopper? Tapia? Other professional?

- Make contact with me before you leave, figure out how to manage your workload

## Where are we going?

Recursion and composition:

Recursive functions in depth

Two recursive data structures: the list and the S-expression

More powerful ways of putting functions together (compositionality again, and it leads to reuse)

Today: **programming with lists and S-expressions** (around laws)

Difficulty with homework:

## Introduction to Scheme

Two new kinds of data:

The function closure: the key to “first-class” functions

Pointer to automatically managed cons cell (mother of civilization)

### Scheme Values

Values are *S-expressions*.

An *fully general S-expression* is one of

a

**symbol**`'Halligan`

`'tufts`

a

**literal integer**`0`

`77`

a

**literal Boolean**`#t`

`#f`

`(cons`

*v*_{1}*v*_{2}`)`

, where*v*_{1}and*v*_{2}are S-expressions

Many predefined functions work with a **list of S-expressions**

A list of S-expressions is either

the empty list

`'()`

`(cons`

*v**v**s*`)`

, where*v*is an S-expression and*v**s*is a list of S-expressionsWe say “an S-expression

*followed by*a list of S-expressions”

## Programming with lists and S-expressions

### Lists: A subset of S-Expressions.

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

**Constructors**: `'(),`

cons

**Observers**: `null?`

, `pair?`

, `car`

, `cdr`

(also known as “first” and “rest”, “head” and “tail”, and many other names)

**Any** list is therefore *constructed* with `'()`

or with `cons`

applied to an atom and a smaller list.

- How can you tell the difference between these types of lists?
- What, therefore, is the structure of a function that consumes a list?

### Why are lists useful?

Sequences a frequently used abstraction

Can easily approximate a set

Can implement finite maps with

**association lists**(aka dictionaries)You don’t have to manage memory

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.)

### Programming example: lists of numbers

Problem-solving: Dealer’s choice

*μ*Scheme’s new syntax

## An alternative to local variables: `let`

binding

Evaluate `e1`

through `en`

, bind answers to `x1`

, … `xn`

- Name intermediate results (simpler code, less error prone)
Creates new environment for local use only:

`rho {x1 |-> v1, ..., xn |-> vn}`

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:

# 24 Sep 2018: First-class and higher-order functions

There are PDF slides for 9/25/2018.

## Announcements

Office hours by appointment this week—send three times

## Where we’ve been and where we’re going: Functional programming

Techniques and features we’re learning fit under **functional programming**.

- Idea: reuse more code because of “better glue”

Already doing it: immutable data (`2-digit-elements`

)

- Always safe to share data (I can’t mess up things for you)
- Perfect for parallel/distributed (think Erlang)
- Perfect for web (JSON, XML)

Next up: better ways of gluing functions together

*μ*Scheme’s semantics

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.

### New syntax exploits semantics

Evaluate `e1`

through `en`

, bind answers to `x1`

, … `xn`

- Name intermediate results (simpler code, less error prone)
Creates new environment for local use only:

`rho {x1 |-> v1, ..., xn |-> vn}`

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:

So let’s see that semantics!

Key idea: **don’t worry about memory**

## From Impcore to uScheme: 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)`

`x`

is**bound**in`E`

- other variables are
**free**in`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.

- Can tell at compile time what is captured.
- To understand why anyone cares, you’ll need examples

**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`

:

The function `to-the-n-minus-k`

is a *higher-order function* because it **returns another (escaping) function as a result**.

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

:

`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:

### Your turn!!

## How `lambda`

works

### Rule for `lambda`

Key idea: ``every name I can see—remember where it is stored.’’

### Rule for function Application

Questions about ApplyClosure:

What if we used

*σ*instead of*σ*_{0}in evaluation of*e*_{1}?What if we used

*σ*instead of*σ*_{0}in evaluation of arguments?What if we used

*ρ*_{c}instead of*ρ*in evaluation of arguments?What if we did not require ℓ

_{1}, ℓ_{2}∉ dom(*σ*)?What is the relationship between

*ρ*and*σ*?

# 26 Sep 2018: Vocabulary building: List HOFs, the function factory

There are PDF slides for 9/27/2018.

Today: Using lambda to enlarge your vocabulary

- List computations
- Cheap functions from other functions

Similar: Haskell, ML, Python, JavaScript

Bonus: proving facts about functions

## Higher-Order Functions on lists

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

- Generalizes to sets, arrays, search trees, hash tables, …

Goal: Capture common **patterns of computation** or **algorithms**

`exists?`

(Ex: Is there a number?)`all?`

(Ex: Is everything a number?)`filter`

(Ex: Take only the numbers)`map`

(Ex: Add 1 to every element)**foldr**(General: can do all of the above.)

Fold also called `reduce`

, `accum`

, a “catamorphism”

### 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
```

### “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)) == ???
```

### The universal list function: `fold`

`foldr`

takes two arguments:

`zero`

: What to do with the empty list.`plus`

: How to combine next element with running results.

Example: `foldr plus zero '(a b)`

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

### In-class exercise

### Language design — why?

## One-argument functions: Curry and compose

Build one-argument functions from two-argument functions

*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**.

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

### Currying and list HOFs

### One-argument functions compose

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)))
```

Another example: `(o not null?)`

# 1 Oct 2018: Tail calls and continuations

There are PDF slides for 10/2/2018.

## Reasoning principles

Recursive function consuming *A* is related to proof about *A*

Q: How to prove two lists are

`equal?`

A: Prove they are both

`'()`

or that they are both`cons`

cells cons-ing equal car’s to equal cdr’sQ: How to prove two

*functions*equal?A: Prove that when applied to equal arguments they produce equal results.

## 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.

Anything in tail position is the **last thing executed**!

Key idea is **tail-call optimization**!

Example: `reverse '(1 2)`

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

Call stack:

```
reverse '()
append
reverse '(2)
append
reverse '(1 2)
```

Answer: Linear in the length of the list

## Tail calls and the method of accumulating parameters

Trick: put answer in parameter Write laws for

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

**Who could write the code?**

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.)

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

Answer: a **goto**!!

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

Remember tail calls? Suppose you call a **parameter**!

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

- Not a normal function call because continuations never return
- Think “goto with arguments”

## Continuations

### 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.* For us, tail call to a parameter.

### Uses of continuations

Compiler representation: Compilers for functional languages often convert direct-style user code to CPS because CPS matches control-flow of assembly.

Some languages provide a construct for

*capturing*the*current continuation*and giving it a name`k`

. Control can be resumed at captured continuation by*throwing*to`k`

.A style of coding that can mimic

*exceptions*Callbacks in GUI frameworks

Event loops in game programming and other concurrent settings

Even web services!

### Implementation

We’re going to simulate continuations with function calls in tail position.

First-class continuations require compiler support: primitive function that materializes “current continuation” into a variable. (Missing chapter number 3.)

## Motivating Example: From existence to witness

Ideas?

Bad choices:

- nil
- special symbol
`'fail`

- run-time error

Good choice:

- exception (not in uScheme)

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

Answer: Constant

# 3 Oct 2018: Continuations for backtracking

There are PDF slides for 10/4/2018.

### Continuations for Search

### 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

**Board**: pictures of two solvers:

- Make
*either*A or B equal to`b`

(last time) [“or true”, “and false”] - Make
*both*A and B equal to`b`

[“and true”, “or false”]

### 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

Solver enters A

If A is solved, newly allocated success continuation starts B

If B succeeds, we’re done! Use

`success`

continuation from context.If B fails, use

`resume`

continuation A passed to B as`fail`

.If A fails, the whole thing fails. Use

`fail`

continuation from context.

### Solving A or B

Solver enters A

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`

.If A can’t be solved, don’t give up! Try a

**newly allocated failure continuation**to start B.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.

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

If B fails, abject failure all around; call the original

`fail`

continuation.

## Lisp and Scheme Retrospective

### Common Lisp, Scheme

Advantages:

- High-level data structures
- Cheap, easy recursion
- Automatic memory management (garbage collection!)
- Programs as data!
- Hygenic macros for extending the language
- Big environments, tiny interpreters, everything between
- Sophisticated Interactive Development Environments
- Used in AI applications; ITA; Paul Graham’s company Viaweb

Down sides:

- Hard to talk about data
- Hard to detect errors at compile time

Bottom line: it’s all about `lambda`

- Major win
- Real implementation cost (heap allocation)

## Bonus content: Scheme as it really is

- Macros!
- Cond expressions (solve nesting problem)
- Mutation
- …

### Macros!

### Conditional expressions

### Mutation

# 9 October 2018: Introduction to ML

There are PDF slides for 10/10/2018.

Handout: Learning Standard ML

Handout: Program Design with ML Types and Pattern Matching

**Ask the class:** what are the strengths you found in *μ*Scheme programming? What are the pain points?

Apply your new knowledge in Standard ML:

- You’ve already learned (most of) the ideas
- There will be a lot of new detail
- Good language for implementing language features
- Good language for studying type systems

**Meta**: Not your typical introduction to a new language

- Not definition before use, as in a manual
- Not tutorial, as in Ullman
- Instead, the
**most important**ideas that are**most connected**to your work up to now

### ML Overview

Designed for programs, logic, symbolic data

Theme: Precise ways to describe data

ML = uScheme + pattern matching + static types + exceptions

### ML forms of data

Live coding:

```
bool
string
int
list of int
list of bool
ill-typed list
pair
triple
function
```

More live coding:

```
sum of list of int (fold)
reverse (with fold)
```

Still more live coding? (New forms of data)

```
Ordinary S-expression
Binary tree of integers
```

Live coding memories:

```
(* list selection *)
fun nth ([], 0) = raise Subscript
| nth (y :: ys, 0) = y
| nth ([], n) = raise Subscript
| nth (y :: ys, n) = nth (ys, n - 1)
(* better version: *)
fun nth ([], _) = raise Subscript
| nth (y :: ys, 0) = y
| nth (y :: ys, n) = nth (ys, n - 1)
(* binary trees of integers *)
datatype itree
= LEAF
| NODE of itree * int * itree
(* val root : itree -> int option *)
(* if tree is not empty, returns value at root *)
fun root LEAF = NONE
| root (NODE (left, n, right)) = SOME n
(* testing code *)
(* val singleton : int -> itree
`singleton n` returns tree with `n` as its only node *)
fun singleton n = NODE (LEAF, n, LEAF)
val t35 = NODE (singleton 3, 5, LEAF)
val () =
Unit.checkExpectWith (Unit.optionString Int.toString)
"root of t35"
(fn () => root t35) (SOME 5)
```

*μ*Scheme 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
end
```

### Three new ideas

- Pattern matching is big and important. You will like it. It’s “coding with algebraic laws”
- Exceptions are easy
- 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!

### Examples

The `length`

function.

Algebraic laws:

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

The code:

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

Things to notice:

No brackets! (I hate the damn parentheses)

Function application by juxtaposition

Function application has

**higher precedence than any infix operator**Compiler checks all the cases (try in the interpreter)

Let’s try another! map, filter, exists, all, take, drop, takewhile, dropwhile

# 10 Oct 2018: Programming with constructed data and types

There are PDF slides for 10/11/2018.

Today’s lecture: lots of info in the notes, but won’t see in class. Because not everybody has a book.

## Review: Improving on Scheme

## Foundation: Data

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

- Base types:
`int`

,`real`

,`bool`

,`char`

,`string`

- Functions
**Constructed data**:- Tuples: pairs, triples, etc
- (Records with named fields)
- Lists and other algebraic types

**Deconstruct**using*pattern matching*

“Language support for forms of data”

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

### Tuple types and arrow types

**Background for datatype review (board):**

if A and B are sets, what is

`A * B`

?if A and B are sets, what is

`A -> B`

?if A, B, and C are sets, what is

`A * B * C`

?

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

### Constructed data: Algebraic data types

Tidbits:

The most important idea in ML!

Originated with Hope (Burstall, MacQueen, Sannella), in the same lab as ML, at the same time (Edinburgh!)

Notes:

A “suit” is

*constructed*using`HEARTS`

,`DIAMONDS`

,`CLUBS`

, or`SPADES`

A “list of A” is constructed using

`nil`

or`a :: as`

, where`a`

is an A and`as`

is a “list of A”A “heap of A” is either empty or it’s an A and two child heaps

Exegesis (on board):

Notation

`'a`

is a**type variable**- On left-hand side, it is a
**formal type parameter** - On right-hand side it is an ordinary type
- In both cases it represents a single
**unknown type**

- On left-hand side, it is a
Name before

`=`

introduces a new**type constructor**into the**type environment**. Type constructors may be**nullary**.Alternatives separated by bars are

**value constructors**of the typeThey are new and

**hide previous names**(Do not hide built-in names

`nil`

and`list`

from the initial basis!)Value constructors build

**constructed data**Value constructors participate in

**pattern matching****Complete**by themselves:`HEARTS`

,`SPADES`

,`nil`

**Expect parameters**to make a value*or pattern*:`::`

,`HEAP`

`op`

enables an**infix operator**to appear in a**nonfix context****Type application is***postfix*- A list of integer lists is written:
`int list list`

- A list of integer lists is written:
New names into

**two**environments:`suit`

,`list`

,`heap`

stand for new**type constructors**`HEARTS`

,`CLUBS`

,`nil`

,`::`

,`EHEAP`

,`HEAP`

stand for new**value constructors**

Algebraic datatypes are inherently inductive (

`list`

appears in its own definition)—to you, that means**finite trees**`'a * 'a list`

is a**pair type**—*infix operators are always applied to pairs*

### Your turn: S-expressions

mds - Structure of algebraic types An algebraic data type is a **collection of alternatives**

- Each alternative
**must have a name**

The thing named is the **value constructor**

(Also called “datatype constructor”) es

## The other form of constructed data: tuples

## Additional language support for algebraic types: case expressions

## Making types work for you

### Types help me, part I: type-directed programming

Common idea in functional programming: "lifting:

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

Potential bonus content: inorder traversal of binary tree.

## Bonus content: Even more algebraic datatypes

Algebraic datatype review:

### 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`

.

`NODE`

s 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`

?

` [1,2,1,3,1,2,1]`

What is the *pre-order* traversal of `t3`

?

` [3,2,1,1,2,1,1]`

### 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.

Examples:

`int tree`

is a tree of integers`bool tree`

is a tree of booleans`char tree`

is a tree of characters`int list tree`

is a tree of a list of integers.

`'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 tree`

s, `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.

### Environments

Datatype definitions introduce names into:

the type environment:

`suit`

,`int_tree`

,`tree`

the value environment:

`HEART`

,`LEAF`

,`PARENT`

### Inductive

Datatype definitions inherently **inductive**:

the type

`int_tree`

appears in its own definitionthe type

`tree`

appears in its own definition

### Datatype Exercise

## Bonus content: Exceptions — Handling unusual circumstances

### Syntax:

- Definition:
`exception EmptyQueue`

- Introduction:
`raise e`

where`e : exn`

- Elimination:
`e1 handle pat => e2`

### Informal Semantics:

- alternative to normal termination
- can happen to any expression
- tied to function call
- if evaluation of body raises exn, call raises exn

Handler uses pattern matching

e handle pat1 => e1 | pat2 => e2

## Bonus Content: ML traps and pitfalls

## Bonus content (seen in examples)

### Syntactic sugar for lists

## Bonus content: ML from 10,000 feet

### Environments

### Patterns

### Functions

Tuples are “usual and customary.”

### Types

# 15 October 2018: Types

There are PDF slides for 10/16/2018.

Announcement:

- Requirement for instructor’s office hours is cancelled

Type systems:

- C#, Swift, Go
- Java, Scala, Rust (polymorphism)
- Most effective example of
*static analysis*

(Without running the code, what can we know about run time)

Trajectory:

- Formalize familiar,
*monomorphic*type systems (like C) - Learn
*polymorphic*type systems - Eventually,
*infer*polymorphic types

Monomorphic types systems are the past.

Inference and polymorphism are the present and future.

(Visible trend just like `lambda`

.)

Today:

- Type system with two types
- Type checking
- Unbounded number of types! (Formation, introduction, elimination)
- Revisiting “code from types” idea

### Types help me: type-directed programming

The types survey

- Baffling
- Noise I can ignore
- Information I understand
- Help me program

## Talking type theory: Introduction and elimination constructs

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

**Introduce**means “produce”, “create”, “make”, “define”**Eliminate**means “consume”, “examine”, “observe”, “use”, “mutate”

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

## 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:

What kind of value does it evaluate to (if it terminates)?

What is the contract of the function?

Is each function called with the right number of arguments?

(And similar errors)Who has the rights to look at it/change it?

Is the number miles or millimeters?

Questions type systems typically do not answer:

Can it divide by zero?

Can it access an array out of bounds?

Can it take the

`car`

of`'()`

?Will it terminate?

Type systems **designed to solve a problem**

- Confirm behavior
- Help the compiler

## Type System and Checker for a Simple Language

**Why do we write a type checker?**

To be educated about programming languages, you must be able to realize inference rules in code. Eventually you should learn to “speak” inference rules like a native. Implementing a type system is a valuable way to build these competencies. And to get you there, a handout!

If (when!) you get to do your own language designs, type systems are an area where you are most likely to be able to innovate. The ideas behind type systems apply any time you need to validate user input, for example. This is the highest level of cognitive task: creation of new things.

Also your introduction to static analysis. Used in code improvement, security.

Define an AST for expressions with:

- Simple integer arithmetic operations
- Numeric comparisons
- Conditional
- Numeric literal

### Examples to rule out

Can’t add an integer and a boolean:

```
3 + (3 < 99)
(ARITH(PLUS, LIT 3, CMP (LT, LIT 3, LIT 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

Form of judgment

*Context*`|-`

*term*`:`

*type*Written

`|- e : tau`

**Contexts vary**between type systems(Right now, the empty context)

Judgment is proved by

**derivation**Derivation made using

*inference rules*Inference rules determine how to code

`val typeof : exp -> ty`

:Given

*e*, find*tau*such that`|- e : tau`

What inference rules do you recommend for this language?

### 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.

input to checker: e

output from checker: tau

### What types can rule out

things that could go wrong:

```
(8 < 10) + 4
(8 == 8) < 9
x + (x :: xs)
let val y = 10 in length y end
```

### What is a type?

OK: a

*set of values*Better: a

*conservative prediction*about valuesBest: the precise definition:

**classifier for terms**!!The relationship to values becomes a

**proof obligation**.Note: a computation can have a type even if it doesn’t terminate! (Or doesn’t produce a value)

**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 (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,

Give each type a distinct subscript

Introduce equality constraints

Remember to

**be careful using primitive equality**to check types—you are better off with`eqType`

.

# 17 October 2018: Type checking with type constructors

There are PDF slides for 10/18/2018.

Review:

- Types classify terms
- Types serve a purpose (guide compiler, prevent bugs)
- Type system with two types
- Types relate to syntax (Introduction, Elimination)
- Typing rules
- Handout (clarify ugly code)

Today:

- Add variables to a language
- Unbounded number of types
- Design of type systems

Upcoming (on the new homework)

- You will design new syntax and typing rules for lists
- You will extend an existing type checker
- You will implement a full type checker from scratch

First two parts: today is good

Third part: Monday’s lecture

This is a big chunk of **what language designers do.**

Review: typing rules for machine expressions

I gave you syntax for simple language

I showed typing rules

things that could go wrong:

```
(8 < 10) + 4
(8 == 8) < 9
x + (x :: xs)
let val y = 10 in length y end
```

## Typing Rules: Contexts and Term Variables

**Your turn**:

- What you need for VAR and LET

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)
end
```

### Functions

Introduction:

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

Elimination:

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

## Type Checking with Type Constructors

Type checking with type constructors

Formation, Introduction, and Elimination

## Where we’ve been and where we’re going

New watershed in the homework

You’ve been developing and polishing programming skills:

**recursion**,**higher-order functions**,**using types to your advantage**. But the problems have been mostly simple problems around simple data structures, mostly lists.We’re now going to shift and spend the next several weeks doing real programming-languages stuff, starting with type systems.

You’ve already seen everything you need to know to implement a basic type checker, and you are almost fully equipped to add array operations and types to Typed Impcore.

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

What is and is not a good type, that is, a classifier for terms?

How shall we represent types?

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

## Design and implementation of monomorphic languages

Mechanisms:

Every new variety of type requires

**special syntax**(examples: structs, pointers, arrays)Implementation is a straightforward application of what you already know.

Language designer’s agenda:

What new types do I have (

**formation rules**)?What new syntax do I have to

*create new values with that type*(**introduction rules**)?For

**introduce**think “produce”, “create”, “make”, “define”What new syntax do I have to

*observe terms of a type*(**elimination rules**)?For

**eliminate**think “consume”, “examine”, “interrogate”, “look inside”, or “take apart”, “observe”, “use”, “mutate”

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

**Your turn**: If I “add lists” to a language, how many new *types* am I introducing?

### Managing the set of types: Type formation

### Monomorphic type rules

Notice: **one rule for if**!!

## Classic types for data structures

(At run time, identical to `cons`

, `car`

, `cdr`

)

### Typing Rule Exercise

### Coding the arrow-introduction rule

# 22 October 2018: Polymorphic Type Checking; Kinds classify types

There are PDF slides for 10/23/2018.

## Announcements

### Recitation this week

Designed for final stages of homework. Your choice:

- Regression testing your type checker
- Coding in Typed uScheme (You could choose to hold off on problem TD.)
- Open office hours or AMA with your recitation leaders

**If** you have submitted your homework, OK to skip—but notify your recitation leader.

### Today

In class:

- Type soundness
- Type-checking review
- Burdens of monomorphism
- On designers and implementors
- On programmers

- Lift burden with quantified types
- Extensible type formation

Language targets: Java, Scala, C# (with “generics”)

### Type soundness

## Typechecking review

Last week: Typed Impcore, but in *μ*Scheme syntax

Today: Typed *μ*Scheme

## Limitations of monomorphic type systems

Notes:

**Implementing arrays**on homework**Writing rules for lists**on homework

## Quantified types

## Polymorphic Type Checking

### Quantified types

Bonus instantiation:

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

Two forms of abstraction:

### Type rules for polymorphism

## Type formation through kinds

## Opening the closed world

## Bonus content: a definition manipulates three environments

# 24 October 2018: Type inference

There are PDF slides for 10/25/2018.

**Note:** Type-system questions and answers are online.

Questions: where do *explicit types* appear in C?

Where do they appear in Typed *μ*Scheme?

Get rid of all that:

- Guess a type for each formal parameter
- Guess a return type
- Guess a type for each instantiation

**Plan of study**:

Today, I’ll walk through several examples, showing how to

*generate*constraintsAt end of class, you’ll do an example

Next time, we’ll

*solve*constraintsAnd for the next homework, you’ll implement both algorithms

#### 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?

`double`

has type ′*a*_{1}`x`

has type ′*a*_{2}`+`

has type`int * int -> int`

`(+ x x)`

is an application, what does it require?- ′
*a*2 =`int`

and ′*a*2 =`int`

- ′
Is this possible?

Key idea: *Record the constraint in a typing judgement.*

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

Example: `if`

`(if y 1 0)`

`y`

has type ′*a*3,`1`

has type`int`

,`0`

has type`int`

Requires what constraints? (

`int`

=`int`

, ′*a*3 = bool`)

Example:

`(if z z (- 0 z))`

`z`

has type ′*a*3,`0`

has type`int`

,`-`

has type`int * int -> int`

Requires what constraints? (′

*a*3 =`bool`

`/\`

`int`

=`int`

`/\`

′*a*3 =`int`

)Is this possible?

Why not?

## Inferring polymorphic types

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

So,

`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))`

# 29 October 2018: Making type inference precise

There are PDF slides for 10/30/2018.

### Review

**Board**

The basics:

- For each unknown type, a fresh type variable
- Instead of
`eqType`

, type-equality constraint

The secret sauce:

- Every polymorphic value is instantiated with fresh type variables—a “fresh instance” (e.g.,
`car`

)

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

Infer the type of function `two`

:

### Precise inference with Hindley-Milner types

**To code the type-inference algorithm**, replace `eqType`

with constraint generation!

### The inference algorithm, formally

Apply rule: *e*’s type could just be a type variable—we need to force it into the arrow form we need.

### What you know and can do now

# 31 October 2018: Building and using a constraint solver

There are PDF slides for 11/1/2018.

(Guest lecturer Kathleen Fisher)

Today:

- Solving constraints
- Final inference rules:
- Review IF, see APPLY
- Instantiate variables
- Generalize at VAL, LET, and LETREC

## Solving constraints

What’s going on with substitutions? We have the informal idea—we just formalize it. It’s a function that behaves a certain way.

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

Constraint satisfaction: equal types mean **equal constructors** applied to **equal arguments**—same is in `eqType`

.

### Solving simple type equalities

**Question**: in solving `tau1 ~ tau2`

, how many potential cases are there to consider?

**Question**: how are you going to handle each case?

### Solving conjunctions

## Final pieces: completing the algorithm

Revisit a few rules:

Apply rule: *e*’s type could just be a type variable—we need to force it into the arrow form we need. (In Typed *μ*Scheme, we did this by pattern matching. Here, we form a function type and kick out a constraint.)

## What you can know and do now

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

Write function `solve`

, which, given a constraint C, has one of three outcomes:

Returns the identity substitution in the case where C is trivially satisfied

Returns a non-trivial substitution if C is satisfiable otherwise.

Calls

`unsatisfiableEquality`

in when C cannot be satisfied

## Instantiate and generalize

Moving from type scheme to types (Instantiation)

Moving from types to type scheme (Generalization)

## From Type Scheme to Types

## From Types to Type Scheme

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

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

On the condition *θ**Γ* = *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

Let with constraints, operationally:

`typesof`

: returns*τ*_{1}, …,*τ*_{n}and*C*C-prime from

`map`

,`conjoinConstraints`

,`dom`

,`inter`

,`freetyvarsGamma`

`val theta = solve C'`

`freetyvarsGamma`

,`union`

,`freetyvarsConstraint`

Map anonymous lambda using

`generalize`

, get all the*σ*_{i}Extend the typing environment Gamma (pairfoldr)

Recursive call to type checker, gets

`C_b`

,`\tau`

Return

`(tau, C' /\ C_b)`

Forall things

`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 |

# 5 November 2018: Hiding information with abstract data types

There are PDF slides for 11/6/2018.

Handout: Mastering Multiprecision Arithmetic

Handout: Program Design with Abstract Data Types

## Announcements

November schedule:

- Cheating: just because you’ve been accused doesn’t mean I think you cheated
- Take-home midterm:
- Dress rehearsal for final
- Operational semantics, functional programming, ML, type systems
- 75 minutes (longer for those with accommodations)
- Friday 11/16, Monday 11/19, or Monday 11/20
- You choose the time that is best for you

(Do it Friday, and you have Thanksgiving week off.) - If you can find no other time, this time and room are available on Monday 11/10. There is no class, but someone will be here to answer questions.
- Balance of theory, practice, basics
- When exam is finished, I will post a summary of the questions
- Calculational proof of a property about lists
- Translating imperative code into functional code
- Continuation-passing style
- More to come

- Choppy holidays: this week full, next unit spread over 2 weeks
- I like to do data-abstraction topics together, but they can’t survive the schedule. So: modules now, objects at end of term

## Data abstraction

Where have we been?

- Programming in the small
- Expressive power
Success stories:

- First-class functions
- Algebraic data types and pattern matching
- Polymorphic type systems

What about *big* programs?

An area of agreement and a great divide:

```
INFORMATION HIDING
(especially: mutable data)
/ \
/ \
modular reasoning / \ code reuse
/ \
internal access / \ interoperability
to rep / \ between reps
/ \
MODULES OBJECTS
ABSTRACT TYPES
```

### Why modules?

Unlocking the final door for building large software systems

You have all gotten good at first-class functions, algebraic data types, and polymorphic types

Modules are the last tool you need to build big systems

### Modules overview

Functions of a true module system:

**Hide***representations*,*implementations*, private**names**“Firewall” separately compiled units (promote independent compilation)

Possibly reuse units

Real modules include *separately compilable* **interfaces** and **implementations**

Designers almost always choose static type checking, which should be “modular” (i.e., independent)

C and C++ are cheap imitations

- C doesn’t provide namespaces
- C++ doesn’t provide modular type checking for templates

### Interfaces

Collect *declarations*

- Unlike
*definition*, provides*partial*information about a name (usually*environment*and*type*)

Things typically declared:

Variables or constants (values, mutable or immutable)

Types

Procedures (if different from values)

Exceptions

Key idea: **a declared type can be abstract**

- Just like a primitive type

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

Roles of interfaces in programming:

The unit of

*sharing*and*reuse*Explainer of libraries

Underlie component technology

The best-proven technology for structuring large systems.

### Ways of thinking about interfaces

Means of hiding information (ask “what secret does it hide?”)

A way to

**limit what we have to understand**about a program- Estimated force multiplier x10

A contract between programmers

- Essential for large systems
- Parties might be you and your future self

Interface is the

*specification*or*contract*that a*module*implements- Includes contracts for all declared functions

### Module Implementations

Holds all dynamically executed code (expressions/statements)

May include private names

May depend only on interfaces, or on interfaces and implementations both (max cognitive benefits when all dependency is mediated by interfaces)

Dependencies may be implicit or explicit (

`import`

,`require`

,`use`

)

## Standard ML Modules

The Perl of module languages?

Has all known features

Supports all known styles

**Rejoice**at the expressive power**Weep**at the terminology, the redundancy, the bad design decisions

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

Modules are a **separate language** layered on top.

### Signature basics

### ML Modules examples, part I

# 7 November 2018: more ML modules

There are PDF slides for 11/8/2018.

## Abstract data types

## Data abstraction for reuse

## Functors and an Extended SML Example

## Extended example: Error-tracking Interpreter

### Why this example?

Lots of interfaces using ML signatures

Idea of how to compose large systems

Some ambitious, very abstract abstractions—it’s toward the end of term, and you should see something ambitious.

Practice implementing functors

Error modules: *Three common implementations*

Collect

*all*errorsKeep just the

*first*errorKeep the

*most severe*error

Your obligations: two types, three functions, algebraic laws

### Computations Abstraction

Ambitious! (will make your head hurt a bit)

Computations either:

return a value

produce an error

Errors must be threaded through everything :-(

That’s really painful!

We can extend the computation abstraction with sequencing operations to help.

### Example:

```
eval e1 + eval e2
(op +) (eval e1, eval e2)
curry (op +) (eval e1) (eval e2)
curry (op +) <$> eval e1 <*> eval e2
```

The first three versions are not well typed. Why?

The last version will thread errors through the compuation behind the scenes.

Note:

```
eval e1, eval e2 : int comp
curry (op +) : int -> (int -> int)
<$> : (int -> (int -> int)) * (int comp) -> (int -> int) comp
<*> : (int -> int) comp * int comp -> int comp
curry (op +) <$> eval e1 : (int -> int) comp
let pa = curry (op +) <$> eval e1
note by above, pa : (int -> int) comp
pa <*> eval e2 : int comp
```

## ML Module summary

# 14 November 2018: Lambda Calculus

There are PDF slides for 11/15/2018.

Today: Living with just application, abstraction, variables

- Living without
`let`

,`while`

- Living without
`if`

- Living without recursive
`define`

- Coding data structures
- Coding natural numbers

### Why study lambda calculus?

A

*metalanguage*for describing other languages, known to all educated people(

*Church-Turing Thesis*: Any computable operator can be encoded in lambda calculus)*Test bench*for new language featuresTheoretical underpinnings for most programming languages (all in this class).

### The world’s simplest reasonable programming language

Only three syntactic forms:

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

Everything is **programming with functions**

Everything is Curried

Application associates to the left

**Arguments are not evaluated**

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

Absolute minimum of code forms: no `set`

, `while`

, `begin`

, but also no `if`

and no `define`

!

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

**EXCEPT**for the way they encode lists, which is bogus (violates abstraction)

Systematic approach to **constructed data**:

Everything is continuation-passing style

Q: Who remembers the boolean-formula solver?

Q: What classes of results could it produce?

Q: How were the results delivered?

Q: How shall we do Booleans?

### Coding Booleans and if expressions

A Boolean takes **two continuations**.

Laws:

```
true s f = s
false s f = f
```

Code:

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

Coding the if expression, **laws**:

```
if true then N else P = N
if false then N else P = P
```

Therefore, code is:

`if M then N else P = ???`

Your turn: implement `not`

Laws for

`not`

: what are the forms of the input data?Code for

`not`

## Coding data structures

### Coding pairs

If you have a pair containing a name and a type, how many

*alternatives*are there?How many continuations?

What information does each expect?

What are the algebraic laws?

`fst (pair x y) = x snd (pair x y) = y`

Code

`pair`

,`fst`

,`snd`

`pair x y k = k x y fst p = p (\x.\y.x) snd p = p (\x.\y.y) pair = \x.\y.\f.f x y fst = \p.p (\x.\y.x) snd = \p.p (\x.\y.y)`

### Coding lists

List laws

`null? nil = true null? (cons x xs) = false car (cons x xs) = x cdr (cons x xs) = xs`

How many ways can lists be created?

How many continuations?

What does each continuation expect?

For each creator, what are the laws regarding its continuations?

`cons y ys n c = c y ys nil n c = n car xs = xs error (\y.\ys.y) cdr xs = xs error (\y.\ys.ys) null? xs = xs true (\y.\ys.false)`

What are the definitions?

`cons = \y.\ys.\n.\c.c y ys nil = \n.\c.n car = \xs.xs error (\y.\ys.y) cdr = \xs.xs error (\y.\ys.ys) null? = \xs.xs true (\y.\ys.false)`

What do those second continuations look like? (This is the source of Wikipedia’s terrible hack)

## Coding recursion

### Using fixed points

**Now you do it**

## Coding numbers: Church Numerals

# 26 November 2018: Lambda-calculus semantics; encoding recursion

There are PDF slides for 11/27/2018.

Calculus examples:

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) |

Substitution examples:

- bash
- tcl/tk
- TeX/LaTeX

### What is a calculus?

Demonstration of differential calculus: reduce

d/dx (

x^{2}+y^{2})

Rules:

```
d/dx k = 0
d/dx x = 1
d/dx y = 0 where y is different from x
d/dx (u + v) = d/dx u + d/dx v
d/dx (u * v) = u * d/dx v + v * d/dx u
d/dx (e^n) = n * e^(n-1) * d/dx e
```

So

d/dx (

x+y)^{2}2 ⋅ (

x+y) ⋅d/dx(x+y)2 ⋅ (

x+y) ⋅ (d/dxx+d/dxy)2 ⋅ (

x+y) ⋅ (1 +d/dxy)2 ⋅ (

x+y) ⋅ (1 + 0)2 ⋅ (

x+y) ⋅ 12 ⋅ (

x+y)

What is a calculus? Manipulation of syntax.

What corresponds to evaluation? “Reduction to normal form”

### Today

Lambda:

- Operational semantics
- Beta reduction and capture-avoiding substitution
- Reduction strategies (making deterministic)

## Review: Live coding

Algebraic laws for

- Booleans
- Pairs
- Lists

Review: Church encodings

**Your turn:** write `and`

## 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

Board examples:

Are these functions the same?

`\x.\y.x \w.\z.w`

Are these functions the same?

`\x.\y.z \w.\z.z`

Examples of free variables:

```
\x . + x y
\x. \y. x
```

## Beta-reduction

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

- It’s hard to get right
- It’s a stupid design for real programming (shell, tex, tcl)
- It’s even hard for theorists!
- But it’s the simplest known thing

Example:

```
(\yes.\no.yes)(\time.no) ->
\z.\time.no
```

and never

`\no.\time.no // WRONG!!!!!!`

Really wrong!

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

Must **rename the bound variable**:

```
(\yes.\no.yes) (\time.no) tuesday
->
(\yes.\z.yes) (\time.no) tuesday
->
(\z.\time.no) tuesday
->
\time.no
```

Nondeterminism of conversion:

```
A
/ \
V V
B C
```

Now what??

## Normal forms

## 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**

- Good model for ML and Scheme, so-called “call by value” languages
- Think “arguments before bodies”

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

### Normal-order reduction

Always choose **leftmost, outermost redex**

**Normalization theorem:**if a normal form exists, this will find itModel for Haskell, Clean

You can try ‘uhaskell’, but if it does anything useful, we’re all surprised and pleased

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

**Not** your typical call-by-value semantics!

## 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”?

Differential calculus example: d/dx x^n equals what?

What’s going on here?

Answer:

**pure formal manipulation**No understanding of functions required; you could write a program to do it (and many people have)

What’s the role of calculi in computer science:

Lambda calculus:

A

*metalanguage*for describing other languagesA

*starter kit*for experimenting with new features

Process calculus:

Concurrent and parallel programming

Biological processes!

Pi calculus:

- Mobile computing and mobile agents

Ambient calculus:

- Security and protection domains

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

# 28 November 2018: Object-orientation

There are PDF slides for 11/29/2018.

Announcement: Recitation is “choose your own adventure”

What it’s about:

- Encapsulation
- Higher-order programming
- Dynamic dispatch
- (Inheritance)

In common with modules, lambda calculus: you can’t touch things directly.

### Object-oriented demo

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

`position:`

*cardinal-point*`set-position:to:`

*cardinal-point**coordinate*`draw`

Instructions to student volunteers

- You have one instance variable, which is a coordinate position.

Messages:

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

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

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

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

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

Object 1, draw yourself on the board

Object 2, draw yourself on the board

Object 3, draw yourself on the board

## Key concepts of object-orientation

### Key mechanisms

**Encapsulate: Private instance variables**

- Only object knows its instance variables and can see them
- C++ calls these “members”
- Like the coordinate of the geometric figure
- (This is the information hiding)

**Higher-order: Code attached to objects and classes**

- Code needed to draw the object is associated with the object

(A species of higher-order programming)

**Dynamic dispatch (NEW)**

- We don’t know what function will be called
- In fact, there is no function; code is a ``method’’

### 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

## Blocks and Booleans

`[block (formals) expressions]`

For parameterless blocks (normally continuations),

`{expressions}`

Blocks are **objects**

- You don’t “apply” a block; you “send it the
`value`

message”

Booleans use continuation-passing style

- Blocks delay evaluation

Booleans *implemented* with two classes `True`

and `False`

- one value apiece

### Method dispatch in the Booleans

**Board** - Method dispatch

To answer a message:

Consider the

**class**of the**receiver**Is the method with that name defined?

If so, use it

If not, repeat with the

**superclass**

Run out of superclasses?

“Message not understood”

# 3 December 2018: Inheritance. Dispatch. Numbers and magnitudes

There are PDF slides for 12/4/2018.

### Announcements

Exam results:

```
75 and up: Excellent
51 to 75: Very Good
31 to 51: Good
22 to 31: Fair
under 22: Poor
```

White cards: AMA (emphasize programming languages past, present, and future)

Final course evaluations: screen shot to `nr@cs.tufts.edu`

.

Hoping to evaluate recitation leaders as well

WICS is a Tufts organization focused on representing and empowering women and non-binary students in Tufts Computer Science. They are having elections for leadership positions, and they are looking for candidates. More info on Piazza (and on board).

### Plan for the week

#### Object-oriented mechanisms

Key mechanisms

- Dynamic dispatch
- Private instance variables

#### Object-oriented design

Key ideas (today, context for understanding, start h/w):

- Big protocols on small foundations
- Two roles of classes

Key ideas (next time, build on today):

- Power of dynamic dispatch
- Behavioral subtyping (“duck typing”, protocols)

Case studies related to the homework.

## Method dispatch in the Booleans

Booleans *implemented* with two classes `True`

and `False`

- one value apiece

**Board** - Method dispatch

To answer a message:

Consider the

**class**of the**receiver**Is the method with that name defined?

If so, use it

If not, repeat with the

**superclass**

Run out of superclasses?

“Message not understood”

## Smalltalk syntax and values

**Values first:**

Every value is an object.

Every class is an object!

Message passing:

Look at `SEND`

- Message identified
**by name**(messages are not values) - Always sent to a
**receiver** - Optional arguments must match
**arity**of message name

(no other static checking)

## Case study: Magnitudes and numbers

Key problems on homework

`Natural`

is a`Magnitude`

“Large integer” is a

`Number`

## Bonus case study not covered in class: Collections

### Why collections?

Goal of objects is **reuse**

Key to successful reuse is a well-designed **class hierarchy**

Killer app: toolkits for building user interfaces

Smalltalk blue book is 90 pages on language, 300 pages on library

Lots of abstract classes

- Define protocols
- Build reusable stuff, just like
`Boolean`

,`Magnitude`

,`Number`

### Implementing Collections

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?

# 5 December 2018: Double dispatch, collections

There are PDF slides for 12/6/2018.

More AMA’s

**COURSE EVALUATIONS**: email me a screen shot

### Two topics for today

Initialization and invariants

Information hidden and revealed; three layers

(Focus on extending open systems)

## Initialization and invariants

### Making open system extensible

## Bonus: Subtyping

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

A value of the

*subtype*can be used wherever a value of the*supertype*is expected.Board:

**SUBTYPE != SUBCLASS****SUPERTYPE != SUPERCLASS**

Only crippled languages like C++ identify subtype with subclass

Only the ignorant and uneducated don’t know the difference

## Bonus case study not covered in class: Collections

### Why collections?

Goal of objects is **reuse**

Key to successful reuse is a well-designed **class hierarchy**

Killer app: toolkits for building user interfaces

Smalltalk blue book is 90 pages on language, 300 pages on library

Lots of abstract classes

- Define protocols
- Build reusable stuff, just like
`Boolean`

,`Magnitude`

,`Number`

### Implementing Collections

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?

### Example collection - Sets

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`

!