# 3 September 2019: Introduction to Comp 105

There are PDF slides for 9/3/2019.

Handout: Lessons in program design

## 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.

First three classes: 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?

## Writing code with recursion and induction

Data-driven design process. Connections to design ideas from 1970s, 1980s, 1990s, plus test-driven development (2000s).

Design checklist:

Forms of data

Example inputs

Function’s name

Function’s contract

Example results

Algebraic laws

Code case analysis

Code right-hand sides

Revisit tests

## 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))`

# 9 September 2019: Abstract syntax and operational semantics

There are PDF slides for 9/9/2019.

## Live coding: Impcore

All fours, tests, big literals

`square`

, `negated`

, interactive `check-expect`

## 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.

## Review

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

Language: syntax, rules/organization/grammar

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

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

You need a vocabulary. It will involve math.

This week: abstract syntax and operational semantics

### 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*

```
C/C++ Impcore
Compute w/ existing names
- value and behavior exp exp
- just behavior stm
Add new names to program
- initialized def def
- uninitialized decl
```

Example rules of composition:

C/C++: expressions, definitions, statements

Impcore

## 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

### 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++

## Key question: names (review)

### From ASTs to behaviors

# 11 September 2019: Judgments and Rules

There are PDF slides for 9/11/2019.

**Students: Take a blank white card**

Handout: Redacted Impcore rules

Today:

How do we know what happens when we run the code?

- Judgment
- Rules
**Valid derivations**

What can we prove about it?

**Metatheory**

## Review: environments

## 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

# 16 September 2019: Derivations, metatheory, better semantics

There are PDF slides for 9/16/2019.

## Review: calls

## Review: Derivations

### 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.

# 18 September 2019: Scheme

There are PDF slides for 9/18/2019.

### Announcements

Note: course not graded on a curve

Scheme homework:

- ramp-up
- not frightfully hard concepts, but…
- new language (notable load)

## 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)

## For a new language, five powerful questions

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

**What is the abstract syntax?**What are the syntactic categories, and what are the forms in each category?**What are the values?**What do expressions/terms evaluate to?**What environments are there?**That is, what can names stand for?**How are terms evaluated?**What are the judgments? What are the evaluation rules?**What’s in the initial basis?**Primitives and otherwise, what is built in?

(Initial basis for *μ*Scheme on page 159)

## 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

Most 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: students pose problem on list of numbers

- Live coding: detect duplicate elements

*μ*Scheme’s new syntax

### Immutable data structures

Key idea of functional programming: **constructed data**. Instead of mutating, construct a new one. Supports composition, backtracking, parallelism, shared state.

### List-design shortcuts

Three forms of “followed by”

**Given:**Element .. list of values =`(cons x xs)`

**Define:**List of values .. list of values =`(append xs ys)`

**Ignore:**List of values .. element =`(snoc xs y)`

Or`(append xs (list1 y))`

Two lists? Four cases!

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?

But we have no

`snoc`

If we cross out the

`snoc`

law, we are left with three cases… but case analysis on the first argument is complete.So cross out the law

`xs .. e == xs`

.Which rules look useful for writing append?

You fill in these right-hand sides:

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

Why does it terminate?

### Cost model

The major cost center is `cons`

because it corresponds to **allocation**.

How many cons cells are allocated?

Let’s rigorously explore the cost of append.

### Induction Principle for List(A)

Suppose I can prove two things:

IH (’())

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

then I can conclude

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

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

*Proof*: By induction on the structure of xs.

Base case: xs = ’()

I am not allowed to make any assumptions.

`(append '() ys) = { because xs is null } ys`

Nothing has been allocated, cost is zero

`(length xs)`

is also zero.Therefore, cost =

`(length xs)`

.

Inductive case: xs = (cons z zs)

I am allowed to assume the inductive hypothesis for

`zs`

.Therefore, I may assume the number of cons cells allocated by

`(append zs ys)`

equals`(length zs)`

Now, the code:

`(append (cons z zs) ys) = { because first argument is not null } = { because (car xs) = z } = { because (cdr xs) = zs } (cons z (append zs ys))`

The number of cons cells allocated is 1 + the number of cells allocated by

`(append zs ys)`

.`cost of (append xs ys) = { reading the code } 1 + cost of (append zs ys) = { induction hypothesis } 1 + (length zs) = { algebraic law for length } (length (cons z zs)) = { definition of xs } (length xs)`

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

## Costs of list reversal

Algebraic laws for list reversal:

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

And the code?

The `list1`

function maps an atom `x`

to the singleton list containing `x`

.

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

.

- Q: How many calls to
`reverse`

? A:`n`

- Q: How many calls to
`append`

? A:`n`

- Q: How long a list is passed to
`reverse`

? A:`n-1`

,`n-2`

, … ,`0`

- Q: How long a list is passed as first argument to
`append`

? A:`n-1`

,`n-2`

, … ,`0`

- Q: How many
`cons`

cells are allocated by call to`list1`

? A: one per call to`reverse`

. - Conclusion:
*O*(*n*^{2}) cons cells allocated. (**We could prove it by induction**.)

## The method of accumulating parameters

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.)

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

Implementation: list of key-value pairs

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

Picture with spine of cons cells, `car`

, `cdar`

, `caar`

, `cadar`

.

Notes:

- attribute can be a list or any other value
- ‘nil’ stands for ‘not found’

### Algebraic laws of association lists

# 23 September 2019: First-class and higher-order functions

There are PDF slides for 9/23/2019.

## Skills refresh

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

Last time: new data forms (lists, S-expressions)

Today: New syntax, its rules and values

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

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

Already doing it: immutable data (`append`

)

- 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*?

# 25 Sep 2019: Vocabulary building: List HOFs, the function factory

There are PDF slides for 9/25/2019.

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

Today: both functions and lists

**Live coding interlude**

### 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
```

### List selection: filter

### “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?)`

# 30 Sep 2019: Tail calls and continuations

There are PDF slides for 9/30/2019.

Where have we been, where are we going: **functions**

Today: proofs, costs, control flow

## Proofs about functions

## 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”

But `goto`

is limited to labels named in the code.

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

# 9 Oct 2019: Lambda Calculus

There are PDF slides for 10/9/2019.

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

# 15 Oct 2019: Lambda-calculus semantics; encoding recursion

There are PDF slides for 10/15/2019.

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.

# 21 Oct 2019: Introduction to ML

There are PDF slides for 10/21/2019.

Handout: Learning Standard ML

**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

# 23 Oct 2019: Programming with constructed data and types

There are PDF slides for 10/23/2019.

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

## 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`

Every

`val`

*on this slide*is a**value constructor**`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

## 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.

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

## 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

# 30 October 2019: Hiding information with abstract data types

There are PDF slides for 10/30/2019.

Handout: Mastering Multiprecision Arithmetic

Announcements

- Hiring TAs for fall
- https://www.cs.tufts.edu/~molay/compta
- Also Send an email to me or to Jeff Foster (next spring, next fall)

- Midterm course evals:
- Experimental Wednesday night recitation continues this week
- Photo is not deleted from server, but access is locked down. (Same policy as SIS.)
- Review of homework solutions: when??
- Exercises deemed redundant or repetitive; busy work: I may ask on Piazza (Note varied skills on recursion.)
- Questions in lecture: I’ll try to keep on point
- Moving images (soccer games and others): back row only

Today’s learning objectives:

- Ideas found in language designs
- Terminology
- Understanding interfaces in Standard ML

## 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?

- Immersion in pile of interpreter code: not fun

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

## 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

# 13 Nov 2019: Types

There are PDF slides for 11/13/2019.

Notes on slide:

- Guide: What have I got? What holes am I trying to fill?

Where type systems are found:

- C#, Swift, Go
- Java, Scala, Rust (polymorphism)

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

## Type systems

What kind of thing is it?

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.

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 a word and a flag:

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

Can’t compare a word and a flag

```
(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 : word |- 5 : word
-------------------------
|- 3 + 5 : word
```

Rules out:

```
|- 'a' : char |- 5 : word
---------------------------
|- 'a' + 5 : ???
```

General form:

```
|- e1 : word |- e2 : word
-----------------------------
|- ARITH ( _ , e1, e2) : word
```

### Rule for comparisons

Informal example:

```
|- 7 : word |- 10 : word
-----------------------------
|- 7 < 10 : flag
```

General form:

```
|- e1 : word |- e2 : word
-----------------------------
|- CMP ( _ , e1, e2) : flag
```

### Rule for literals

Informal example:

`|- 14 : word`

General form:

```
--------------------
|- LIT (n) : word
```

### Rule for conditionals:

Informal example:

```
|- true : flag
|- 3 : word
|- 42 : word
--------------------------
|- IF (true, 3, 42) : word
```

General form:

```
|- e : flag
|- 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?

Weak: 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 (WORDTY, WORDTY) => WORDTY
| _ => raise IllTyped
| typeof (CMP (_, e1, e2)) =
case (typeof e1, typeof e2)
of (WORDTY, WORDTY) => FLAGTY
| _ => raise IllTyped
| typeof (LIT _) = WORDTY
| typeof (IF (e,e1,e2)) =
case (typeof e, typeof e1, typeof e2)
of (FLAGTY, 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`

.

## 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”.

# 18 Nov 2019: Type checking with type constructors

There are PDF slides for 11/18/2019.

### Type soundness

### Approaching types as programmers

## 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.

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

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

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

### 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

# 20 Nov 2019: Polymorphic Type Checking; Kinds classify types

There are PDF slides for 11/20/2019.

### Today

In class:

- 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”)

### Typing Rule Exercise

## 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

# 25 Nov 2019: Introduction to type inference

There are PDF slides for 11/25/2019.

## 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

### Extra office hours

- Tomorrow
- Maybe Friday
- Week after break

(And you’ll get an email)

## Today

How does the compiler know?

```
fun append (x::xs) ys = x :: append xs ys
| append [] ys = ys
```

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

After break, 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))`

# 2 Dec 2019: Making type inference precise

There are PDF slides for 12/2/2019.

## Review

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

### Bonus examples

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

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.

# 4 Dec 2019: Building and using a constraint solver

There are PDF slides for 12/4/2019.

Today:

- Solving constraints
- Final inference rules:
- 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 rule:

## 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*`val theta = solve C`

C-prime from

`map`

,`conjoinConstraints`

,`dom`

,`inter`

,`freetyvarsGamma`

`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 |