Photos from the lectures
Photos from class are posted in a Google album. Anyone can add photos.
23 January 2017: Introduction to Comp 105
There are PDF slides for 1/23/2017.
Handout: Experiences of successful 105 students
Overview
Why so many languages?
Topic of study: the stuff software is made of
- Thousands
- Each one unique
- Why do you suppose so many?
Conclusion: make it easier to write programs that really work
- An invaluable skill for software practitioners
Your language can make you or break you.
What this course isn’t
- Java in January
- F# in February
- Modula-3 in March
- Ada in April
Reusable Principles
What if the course were called “Houses”?
You’d need to know something about houses work
- Want to cut a new door? How does wood-frame construction work?
- Digging in yard with backhoe. Then the house blows up. Why?
You’d want to know something about what makes a house livable:
Northern temperate zone: face south, roof with eaves
There are a few different kitchen layouts that work (galley, farmhouse); know what they are and how to exploit them.
Problem: keep water out—while letting water vapor escape
The same thing for programming languages:
How programming languages work: MATH
What makes programming languages usable: Great features for writing CODE
What Programming Languages is, technically
The marriage of math and code
Principal tools: Induction and recursion
What can you get out of Comp 105?
New ways of thinking about programming (found in many languages)
Double your productivity
Become a sophisticated consumer, aware of old wine in new bottles
Learn new languages quicklly
Bonus: preparation for advanced study
(Course serves everyone from recent 15/61 grads to grad students)
Students who get the most out of 105 report
- They enjoy programming (15)
- They also like math (61, sort of—induction and proofs)
- They work hard
Induction and recursion: linguistic structure
Numerals
Show some example numerals
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
“Calculator expressions”
Calculator
1 2 3 +
4 5 6 *
7 8 9 -
( 0 ) /
9 - (6 / 3)
is a calculator expression))7+*94
is not a calculator expression
In-class Exercise: Inductive description of calculator expressions
define the set of calculator expressions
Concrete and abstract Syntax
Programming-languages people are wild about compositionality.
- Build sensible things from sensible pieces using just a few construction principles.
Example of compositionality: concrete syntax (grammar)
(x + y)
is a grammatical expression(x - y)
is a grammatical expression- Any two expressions can be be multiplied
(x + y) * (x - y)
is an expressionfish
is a noun phrase;red
is an adjective;red fish
is a noun phrase
Programming languages more orderly than natural language.
Example of non-compositionality
fish
vsghoti
Both composed from letters, but no rules of composition
Inductive descriptions of concrete syntax: Backus-Naur Form
Board: BNF for numerals
Exercise: BNF for calculator expressions
Ambiguity
Looky here:
9 - (6 / 13) 9 - 6 / 13 (9 - 6) / 13
Superficially different ways of writing one expression?
Introducing our common framework
Recursion: a review
Case analysis depends on the inductive structure of the input data
Consuming natural numbers recursively
Q: How many natural numbers are there?
Q: What is a structure that works for natural numbers?
Ways a recursive function could decompose a natural number n.
Peel back one (Peano numbers):
n = 0 n = m + 1, m is also a natural number
Split into two pieces:
n = 0 n = k + (n - k) 0 < k < n (everything gets smaller)
Sequence of decimal digits (see study problems on digits)
n = d, where 0 <= d < 10 n = 10 * m + d, where 0 <= d < 10 and m > 0
To do your homework problems, which I recommend starting today, you’ll need to invent at least one more.
Let’s try one now!
Course logistics: Books
Either available, or first part will go up on web right after class.
Course logistics: Recitation
- Not all recitations filled
- Two will likely be cancelled. If this is yours, find another (SIS or Jared at end of class).
- Some students are waitlisted for recitation. Find one that’s not full.
25 January 2017: Abstract syntax and operational semantics
There are PDF slides for 1/25/2017.
Handout: 105 Impcore Semantics, Part 1
Opening sermon: What the course is like
Why get a university education?
- The opportunity to stretch yourself
The course is hard, but we are totally committed to your success
- There is no grading curve
- Everyone can get A’s
- But you have to work hard
In practice, about half A’s—sometimes more, sometimes less.
Your approach
- It’s nice to be smart
- It’s good to work hard
- What’s most important are mindful, thoughtful work habits
If you’re not sure how to work mindfully, start with frequent pauses (try every 15 minutes)
Course logistics and administration
You must get my book (Both Volumes!!!)
You won’t need the book on ML for about a month
Homework
Homework will be frequent and challenging:
- Many small programming problems
- Some theory problems, more like a math problem set
- The occasional larger project, like a type checker or a game solver
- Submit electronically
- Theory is best done by hand, then scanned or photographed
- The course is relentlessly cumulative.
Both individual and pair work:
- All problems should be discussed with others
(Essential to your success) - Discussions must be acknowleged
- Most problems must be completed individually
- Do not allow anyone else to see your code.
- For some problems larger in scope, you can work in pairs
- Be very careful to separate your pair work and your individual work. (A mistake could be major violation of academic integrity, with severe penalties.)
- Pair work is optional. Discussion is not.
Arc of the homework looks something like this:
- impcore one star
- opsem two stars
- scheme three stars
- hofs four stars
And it’s more or less four-star homeworks from there on out.
Lesson: Don’t make plans based on the first couple of homeworks!
The role of lectures
We don’t cover everything in lecture
Lecture is for just the hard parts
We’ll talk very little about the code (just the interesting bits)
In a 100-level course, you are responsible for your own learning
Like an internship: we point you in the right direction and identify traps and pitfalls, and we find good problems for you to work on.
If you’re expecting to see everything in lecture, you have a couple of choices: change your expectations, or take the course next year when you will have more experience and will be more prepared to manage your own learning.
Recitations
- Class goes very fast; recitations provide chance to dig in to key topics with classmates.
- Start this Thursday and Friday.
- Location information available in SIS.
- Designed to be interactive, so bring your thinking cap.
- An important part of your grade
Questions and answers on Piazza
- Don’t just ask questions; answer them too.
- Both activities count toward class participation.
- Don’t show your code. (This is an issue of academic integrity.) For help with code, email.
Other policies and procedures in the syllabus
- You are responsible!
- Treasure Hunt for class participation points
Who am I? What am I called?
Call me “Norman,” “Professor Ramsey”, or “Mister Ramsey.” In a pinch, “Professor” will do. Do not call me “Ramsey”; where I come from, that form of address is insulting.
Who are you? What are you called?
- First homework includes photo, preferred name, pronunciation
- Your photos will be graded!
Thinking about programming languages
Where have we been?
Short discussion: Two things you learned in the first class
This week: abstract syntax and operational semantics (next homework)
Programming-language semantics
“Semantics” means “meaning.”
We want a computational notion of meaning.
What problem are we trying to solve?
Know what’s supposed to happen when you run the code
Ways of knowing:
- 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++
OK, let’s dive in
30 January 2017: Syntactic Proofs, Metatheory
There are PDF slides for 1/30/2017.
Handout: Impcore expression rules
Last time: rules of operational semantics, how they correspond with code
Both math and code on homework
You’re good with code—lecture and recitation will focus on math
The big idea
Every terminating computation is described by a data structure—we’re going to turn computation into a data structure. Proofs about computations are hard (see: COMP 170), but proofs about data structures are lots easier (see: COMP 61).
Today:
- How we know what the code is supposed to do at run time: valid derivations
- What we know about valid derivations: metatheory
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 e1 e2)
. In this example,- What is f?
- What is e1?
- What is e2?
Let’s develop the ApplyUser rule for the special case of two arguments: ⟨APPLY(f, e1, e2),ξ, ϕ, ρ⟩⇓?
What is the result of (digit? 7)
?
How do we know it’s right?
From rules to proofs
What can a proof tell us?
Which of these judgments correctly describes what code does at run time?
⟨
(+ 2 2)
, ξ, ϕ, ρ⟩⇓⟨4, ξ, ϕ, ρ⟩⟨
(+ 2 2)
, ξ, ϕ, ρ⟩⇓⟨99, ξ, ϕ, ρ⟩⟨
(+ 2 2)
, ξ, ϕ, ρ⟩⇓⟨0, ξ, ϕ, ρ⟩⟨
(while 1 0)
, ξ, ϕ, ρ⟩⇓⟨77, ξ, ϕ, ρ⟩⟨
(begin (set n (+ n 1)) 17)
, ξ, ϕ, ρ⟩⇓⟨17, ξ, ϕ, ρ⟩
To know for sure, we need a proof
Example derivation (in handout)
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.
1 February 2017: Metatheory wrapup, functional programming
There are PDF slides for 2/1/2017.
Today: more induction and recursion
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)
Recursion comes from inductive structure of input
Structure of the input drives the structure of the code.
You’ll learn to use a three-step design process:
- Inductive structure
- Equations (“algebraic laws”)
- Code
To discover recursive functions, write algebraic laws:
sum 0 = 0
sum n = n + sum (n - 1)
Which direction gets smaller?
Code:
(define sum (n)
(if (= n 0) 0 (+ n (sum (- n 1)))))
Another example:
exp x 0 = 1
exp x (n + 1) = x * (exp x n)
Can you find a direction in which something gets smaller?
Code:
(define exp (x m)
(if (= m 0)
1
(* x (exp x (- m 1)))))
For a new language, five powerful questions
As a lens for understanding, you can ask these questions about any language:
What is the abstract syntax? What are the syntactic categories, and what are the terms 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 μScheme on page 149)
Introduction to Scheme
Question 2: what are the values?
Two new kinds of data:
The function closure: the key to “first-class” functions
Pointer to automatically managed cons cell (mother of civilization)
Picture of a cons cell: (cons 3 (cons 2 (cons 1 ’())))
Scheme Values
Values are S-expressions.
An S-expression is either
a symbol
'Halligan
'tufts
a literal integer
0
77
a literal Boolean
#t
#f
(cons
v1 v2`), where v1 and v2 are S-expressions
Many predefined functions work with a list of S-expressions
A list of S-expressions is either
the empty list
'()
(cons
v1 v2`), where v1 is an S-expression and v2 is a list of S-expressionsWe say “an S-expression followed by a list of S-expressions”
S-Expression operators
Like any other abstract data type, S-Expresions have:
creators that create new values of the type
'()
producers that make new values from existing values
(cons s s')
mutators that change values of the type (not in uScheme)
observers that examine values of the type
number?
symbol?
boolean?
null?
pair?
car
cdr
N.B. creators + producers = constructors
Lists
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)
6 February 2017: More programming with lists and S-expressions
There are PDF slides for 2/6/2017.
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.)
Immutable data structures
Key idea of functional programming. Instead of mutating, build a new one. Supports composition, backtracking, parallelism, shared state.
Review: Algebraic laws of lists
You fill in these right-hand sides:
(null? '()) ==
(null? (cons v vs)) ==
(car (cons v vs)) ==
(cdr (cons v vs)) ==
(length '()) ==
(length (cons v vs)) ==
Combine creators/producers with observers
Can use laws to prove properties of code and to write better code.
Recursive functions for recursive types
Any list is therefore constructed with '()
or with cons
applied to an atom and a smaller list.
- How can you tell the difference between these types of lists?
- What, therefore, is the structure of a function that consumes a list?
Example: length
Algebraic Laws for length
Code:
;; you fill in this part
Algebraic laws to design list functions
Using informal math notation with .. for “followed by” and e for the empty sequence, we have these laws:
xs .. e = xs
e .. ys = ys
(z .. zs) .. ys = z .. (zs .. ys)
xs .. (y .. ys) = (xs .. y) .. ys
The underlying operations are append
, cons
, and snoc.
Which ..’s are which?
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 not 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 tolist1
? A: one per call toreverse
. - Conclusion: O(n2) cons cells allocated. (We could prove it by induction.)
8 February 2017: Accumulating parameters, let-bound names and anonymous functions
There are PDF slides for 2/8/2017.
Announcements:
- Thursday office hours via Webex (instructions to Piazza)
What is an algebraic law?
A law is an equation.
At least one side, usually both, contains at least one variable, say x.
For any value of x in the proper semantic domain, the equation holds. And similarly for all other variables.
(A semantic domain is a set of values like NUM or LIST(BOOL).)
The law justifies substitution of one side for another, which can be used to simplify a formula (math class) or a program (computer-science class)
Examples of algebraic laws from math class
x + 0 = x
(x + y)^2 = x^2 + 2ay + y^2
x * 1 = x
x * 0 = 0
Examples of algebraic laws from computer-science class
&a[i] == a + i /* law of C code */
(member? x emptyset) == #f ;; law of uScheme sets
(reverse (cons z zs)) == (append (reverse zs) (list1 z)) ;; list law
Your turn!
Homework asks you to discover a new law. Work with your neighbor to develop four laws involving some combination of append
and reverse
:
(append (reverse '()) ys)
=(append (reverse (cons z zs)) ys)
=(append (reverse xs) '())
=One new law that you discover:
N.B. There is a deep trick behind left-hand sides 1 and 2
The method of accumulating parameters
Write laws for
(revapp xs ys) = (append (reverse xs) ys)
Who could write the code?
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
μScheme’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:
From Impcore to uScheme: Lambda
Things that should offend you about Impcore:
Look up function vs look up variable requires different interfaces!
To get a variable, must check 2 or 3 environments.
- Can’t create a function without giving it a name:
- High cognitive overhead
- A sign of second-class citizenship
All these problems have one solution: lambda
Anonymous, first-class functions
From Church’s lambda-calculus:
(lambda (x) (+ x x))
“The function that maps x to x plus x”
At top level, like define
. (Or more accurately, define
is a synonym for lambda
that also gives the lambda
a name.)
In general, \x.E
or (lambda (x) E)
x
is bound inE
- 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!!
13 Feb 2017: Higher-order functions on lists; currying; tail calls
There are PDF slides for 2/13/2017.
Last time: conjunction and disjunction
sv - Conjunction, disjunction of (define conjoin (p? q?) (lambda (x) (if (p? x) (q? x) #f)))
(define disjoin (p? q?) (lambda (x) (if (p? x) #t (q? x)))) ev
Today
Plan:
Higher-order functions that consume lists, in three parts:
Manipulating one-argument functions
Using one-argument functions to interrogate and transform lists
Turning two-argument functions into one-argument functions
Tail calls
Higher-order functions
Preview: in math, what is the following equal to?
(f o g)(x) == ???
Another algebraic law, another function:
(f o g) (x) = f(g(x))
(f o g) = \x. (f (g (x)))
Another example: (o not null?)
Reasoning about functions
Truth about S-expressions and functions consuming functions
Q: Can you do case analysis on a function?
Reasoning principles
Recursive function consuming A is related to proof about A
Q: How to prove two lists are
equal?
A: Prove they are both
'()
or that they are bothcons
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.
Currying
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.”
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”
Your turn!!
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
Algorithm encapsulated: Linear filtering
Example: Given a list of numbers, return only the even ones.
Algebraic laws:
(filter p? '()) == ???
(filter p? '(cons m ms)) == ???
(filter p? '()) == '()
(filter p? '(cons m ms)) == if (p? m)
(cons m (filter p? ms))
(filter p? ms)
“Lifting” functions to lists: map
Algorithm encapsulated: Transform every element
Example: Square every element of a list.
Algebraic laws:
(map f '()) == ???
(map f (cons n ns)) == ???
(map f '()) == '()
(map f (cons n ns)) == cons (f n) (map f ns)
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
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
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”
15 Feb 2017: Continuations
There are PDF slides for 2/15/2017.
Remember tail calls? Suppose you call a parameter!
A continuation is code that represents “the rest of the computation.”
- Not a normal function call because continuations never return
- Think “goto with arguments”
Different coding styles
Direct style: Last action of a function is to return a value. (This style is what you are used to.)
Continuation-passing style (CPS): Last action of a function is to “throw” value to a continuation.
Uses of continuations
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 tok
.A style of coding that can mimic exceptions
Call-backs in GUI frameworks
Implementation
First-class continuations require compiler support.
We’re going to simulate continuations with function calls in tail position.
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
Extended Example: A SAT Solver
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
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 asfail
.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.
22 Feb 2017: Scheme Semantics
There are PDF slides for 2/22/2017.
Last Time
- Continuations
- “gotos with arguments”
- Example: Handling missing values (association list)
- Example: Structuring a search (SAT solver)
Key cases: Make A ∧ B true, make A ∨ B true, make A ∨ B false, make A ∧ B false.
Today
Scheme Semantics
Closures
Stores
Semantics of let, let*, and letrec
New Syntax, New Values, New Environment, New Evaluation Rules
First four of five questions: Syntax, Values, Environments, Evaluation
Key changes from Impcore:
New constructs: let, lambda, application (not just names)
New values:
cons
cells and functions (closures)A single environment
Environments get copied (in closures).
Environment maps names to mutable locations, not values.
A store maps locations to values.
It’s not precisely true that rho never changes.
New variables are added when they come into scope.
Old variables are deleted when they go out of scope.
But the location associated with a variable never changes.
The book includes all rules for uScheme. Here we will discuss on key rules.
Variables
Questions about Assign:
What changes are captured in σ′?
What changes are captured in σ′{ℓ↦v}?
What would happen if we used σ instead of σ′
What would happen if we used a fresh ℓ?
Some other ℓ in the range of ρ?
Lambdas
Function Application
Questions about ApplyClosure:
What if we used σ instead of σ0 in evaluation of e1?
What if we used σ instead of σ0 in evaluation of arguments?
What if we used ρc instead of ρ in evaluation of arguments?
What if we did not require ℓ1, …, ℓn ∉ dom(σ)?
What is the relationship between ρ and σ?
Closure Optimizations
- Major issue in making functional programs efficient
- Keep closures on the stack
- Share closures
- Eliminate closures (when functions don’t escape)
Lets
Which let
is which and why?
Recall:
let
puts the new bindings in scope only for the body expression.let*
adds each binding one at a time, so each binding is in scope for the later ones.letrec
considers all the bindings to be mutually recursive.
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
23 Feb 2017: Introduction to ML
There are PDF slides for 2/23/2017.
Ask the class: what are the pain points in your μScheme programming?
Apply your new knowledge in Standard ML:
- 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
μScheme to ML Rosetta stone
uScheme SML
(cons x xs) x :: xs
'() []
'() nil
(lambda (x) e) fn x => e
(lambda (x y z) e) fn (x, y, z) => e
|| && andalso orelse
(let* ([x e1]) e2) let val x = e1 in e2 end
(let* ([x1 e1] let val x1 = e1
[x2 e2] val x2 = e2
[x3 e3]) e) val x3 = e3
in e
end
Three new ideas
- Pattern matching is big and important. You will like it.
- 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
27 Feb 2017: Programming with constructed data and types
There are PDF slides for 2/27/2017.
THE MIDTERM IS COMING
- Need an accommodation? You have until Wednesday.
- 75 minutes
- Operational semantics, functional programming
- Available March 8
- I’ll be here (or upstairs) to answer questions
- But you don’t have to take it in class
- Just return it to CS office two hours after checkout
- There will be a short study guide
- Review the syllabus
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
“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!)
Board:
A “suit” is constructed using
HEARTS
,DIAMONDS
,CLUBS
, orSPADES
A “list of A” is constructed using
nil
ora :: as
, wherea
is an A andas
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
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 type
They are new and hide previous names
(Do not hide built-in names
nil
andlist
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 contextType 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 constructorsHEARTS
,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
Additional language support for algebraic types: case expressions
Making types work for you
The types survey:
Baffling Noise I can ignore Information I understand
Today, add to far right: type help me program
- (Possible bonus content: ML traps and pitfalls)
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.
Types help me, part I: type-directed programming
Common idea in functional programming: “lifting:
val lift : forall 'a . ('a -> bool) -> ('a list -> bool)
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 integersbool tree
is a tree of booleanschar tree
is a tree of charactersint 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 declarations introduce names into:
the type environment:
suit
,int_tree
,tree
the value environment:
HEART
,LEAF
,PARENT
Inductive
Datatype declarations are 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:
- Declaration:
exception EmptyQueue
- Introduction:
raise e
wheree : 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
1 March 2017: Type systems
There are PDF slides for 3/1/2017.
Today:
- Revisiting “code from types” idea
- Type system with two types
- Type checking
- Unbounded number of types! (Formation, introduction, elimination)
Why do we write a type checker? (probably for a later lecture)
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.
Code from types, revisited
Your turn: What possible types?
Our turn: write the code
Type systems
What kind of thing is it?
Slogan: “Types classify terms”
n + 1 : int
"hello" ^ "world" : string
(fn n => n * (n - 1)) : int -> int
if p then 1 else 0 : int, provided p : bool
Questions type systems can answer:
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 generally cannot answer:
Can it divide by zero?
Can it access an array out of bounds?
Can it take the
car
of `’()?Will it terminate?
Type System and Checker for a Simple Language
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:
typeWritten
|- 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 is a type?
OK: a set of values
Better: a conservative prediction about values
Best: 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 (tc e1, typeof e2)
of (INTTY, INTTY) => INTTY
| _ => raise IllTyped
| typeof (CMP (_, e1, e2)) =
case (tc e1, typeof e2)
of (INTTY, INTTY) => BOOLTY
| _ => raise IllTyped
| typeof (LIT _) = INTTY
| typeof (IF (e,e1,e2)) =
case (tc e, typeof e1, typeof e2)
of (BOOLTY, tau1, tau2) =>
if eqType (tau1, tau2)
then tau1 else raise IllTyped
| _ => raise IllTyped
An implementor’s trick: If you see identical types in a rule,
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
.
6 March 2017: Type checking with type constructors
There are PDF slides for 3/6/2017.
Announcement: NR extra office hours:
- 3pm today
- 11am tomorrow
Review: typing rules for machine expressions
I gave you syntax for simple language
You came up with typing rules
I showed you how to implement the type checker.
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:
Complete typing rules from last time
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
What’s coming
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.
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
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.
Question for the class: If I add lists to a language, how many new types am I introducing?
Managing the set of types: Type formation
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
13 March 2017: Polymorphic Type Checking; Kinds classify types
There are PDF slides for 3/13/2017.
Announcements
Midterms are graded, but half physical, half virtual
- Will have collated things for you to pick up by tomorrow, in Halligan
Midterm course evaluations:
44 responses
Enhanced learning:
- 38 homework (8+3 CQ’s, related reading)
- 28 lectures
- 21 recitations
- 14 readings or book
Changes desired
- 16 office hours (10 daytime, more hours)
- 15 reduce homework load (8, fewer small problems)
- 23 other homework
- 19 other lecture
Your part
- 8 Organizing study groups
- 8 Starting earlier
- 7 Piazza
- 6 Ask more questions, especially in class
Today
Last week: Typed Impcore, but in μScheme syntax
Today: Typed μScheme
Recitation this week: a chance to practice coding in Typed μScheme. You could choose to hold off on problem TD.
Limitations of monomorphic type systems
Notes:
- Implementing arrays on homework
- Writing rules for lists on homework
Quantified types
Type formation via kinds
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
Bonus content: a definition manipulates three environments
15 March 2017: Type inference
There are PDF slides for 3/15/2017.
Midterm grades look good:
Points Grade
72+ Excellent
51.5-71 Very Good
41-51 Good
30-40.5 Fair
under 30 Poor
Questions: where do explicit types appear in C?
Where do they appear in Typed μScheme?
Get rid of all that:
- Guess a type for each formal parameter
- Guess a return type
- Guess a type for each instantiation
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 ′a1x
has type ′a2+
has typeint * int -> int
(+ x x)
is an application, what does it require?- ′a2 =
int
and ′a2 =int
- ′a2 =
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 ′a3,1
has typeint
,0
has typeint
Requires what constraints? (
int
=int
, ′a3 = bool`)
Example:
(if z z (- 0 z))
z
has type ′a3,0
has typeint
,-
has typeint * int -> int
Requires what constraints? (′a3 =
bool
/\
int
=int
/\
′a3 =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))
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
27 March 2017: Making type inference precise
There are PDF slides for 3/27/2017.
Announcements
Study-group tool: 12 people, will wait til afternoon
- Will finalize at 4:00
Grades delayed by suspicions of cheating
Review
Board:
- Name bound in environment may have a polymorphic type
- Name bound by
lambda
always has monomorphic type - Every use of a polymorphic name gets a fresh instance (
car
) - For each unknown type, a fresh type variable
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
What you know and can do now
Writing the constraint solver
Two questions: what’s substitution, and when is a constraint satisfied?
29 March 2017: Building and using a constraint solver
There are PDF slides for 3/29/2017.
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
What you can know and do now
Write type inference for everything except VAL, VALREC, and LETX
Write the solver
Instantiate and generalize
Moving from type scheme to types (Instantiation)
Moving from types to type scheme (Generalization)
From Type Scheme to Types
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 θΓ = Γ: Γ is “input”: it can’t be changed.
The condition ensures that θ doen’t conflict with Γ.
We can’t generalize over free type variables in Γ.
Their presence indicates they can be used somewhere else, and hence they aren’t free to be instantiated with any type.
Type Inference for Lets and Recursive Definitions
Let with constraints, operationally:
typesof
: returns τ1, …, τn and CC-prime from
map
,conjoinConstraints
,dom
,inter
,freetyvarsGamma
val theta = solve C'
freetyvarsGamma
,union
,freetyvarsConstraint
Map anonymous lambda using
generalize
, get all the σiExtend 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 |
3 April 2017: Lambda Calculus
There are PDF slides for 4/3/2017.
Where have we been and where are we going?
Long tour of expressive power at the level of a function or a group of functions. Type systems considered a significant aid to programmers.
A week of foundations: the test bench for new language features
To finish the term, language features designed for larger systems: modules and objects
What is a calculus?
Demonstration of calculus: reduce
d/dx $(x^2 + y^2)$
What is a calculus? Manipulation of syntax.
What corresponds to evaluation? “Reduction to normal form”
Calculus examples:
Concurrency | CCS (Robin Milner) |
Security | Ambient calculus (Cardelli and Gordon) |
Distributed computing | pi calculus (Milner) |
Biological networks | stochastic pi calculus (Regev) |
Computation | lambda calculus (Church) |
Why study lambda calculus?
Theoretical underpinnings for most programming languages (all in this class).
A metalanguage for describing other languages
(Church-Turing Thesis: Any computable operator can be encoded in lambda calculus)
Test bench for new language features
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
Alert to the reading: Wikipedia is reasonably good on this topic
- EXCEPT for the way they encode lists, which is bogus (violates abstraction)
Everything is continuation-passing style
Q: Who remembers the boolean equation solver?
Q: What classes of results could it produce?
Q: How were the results delivered?
Q: How shall we do Booleans?
Coding Booleans
A Boolean takes two continuations:
true = \x.\y.x
false = \x.\y.y
if M then N else P = ???
Coding pairs
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?
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
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 numbers: Church Numerals
5 April 2017: Lambda-calculus semantics; encoding recursion
There are PDF slides for 4/5/2017.
Review: Church encodings
Question: What’s missing from this picture?
Answer: We’re missing recursive functions.
Astonishing fact: we don’t need letrec
or val-rec
(to com)
Operational semantics of lambda calculus
New kind of semantics: small-step
New judgment form
M --> N ("M reduces to N in one step")
No context!! No turnstile!!
Just pushing terms around == calculus
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 it
Model 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!
Fixed points, recursion
Suppose g F = F
. Proof that F
is factorial.
For all n
, g F n
= n!
, by induction:
F 0 = g F 0 = 1
F n
= { by assumption }
g F n
= { definition of g }
if n = 0 then 1 else n * F (n-1)
= { assumption, n > 0 }
n * F (n-1)
= { induction hypothesis }
n * (n-1)!
= { definitiion of factorial }
n!
Now you do it
Fixed-point operators
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 languages
A 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.
10 April 2017: Hiding information with abstract data types
There are PDF slides for 4/10/2017.
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
/ \
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
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
12 April 2017: more ML modules
There are PDF slides for 4/12/2017.
Abstract data types
Data abstraction for reuse
Functors and an Extended SML Example
19 April 2017: Object-orientation
There are PDF slides for 4/19/2017.
Demo: circle, square, triangle, with these methods:
position:
cardinal-pointset-position:to:
cardinal-point coordinatedraw
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
Private instance variables
- Only object knows its instance variables and can see them
- C++ calls these “members”
- Like the coordinate of the geometric figure
Code attached to objects and classes
- Code needed to draw the object is associated with the object
Dynamic dispatch
- 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”
24 April 2017: Inheritance. Numbers and magnitudes
There are PDF slides for 4/24/2017.
History and overview of objects
History of objects
We know that mixing code and data can create powerful abstractions (function closures)
Objects are another way to mix code and data
Pioneers were Nygaard and Dahl, who added objects to Algol 60, producing SIMULA-67, the first object-oriented language
Bjarne Stroustrup liked Simula but wanted complete control of costs, so he created C++
James Gosling wanted something a little cleaner and a little more like Simula, created Java
Microsoft funded C#
Objects are everywhere
What’s an object?
Agglutination containing
Some mutable state (instance variables)
Code that can respond to messages (code is called methods)
A lot like a closure
What are objects good at?
Not really useful for building small things
If you build a big, full-featured abstraction, you can easily use inheritance to build another, similar abstraction. Very good at adding new kinds of things that behave similarly to existing things.
Programs that are evolving
A particular kind of evolution: operations stay the same, but we add new kinds of things
Example: GUIs (operations are
paint
andrespond-to-mouse-click
)Example: numbers
For your homework, you’ll take a Smalltalk system that has three kinds of numbers, and you’ll add a fourth kind of number.
What’s hard about objects?
If you do anything at all interesting, your control flow becomes smeared out over half a dozen classes, and your algorithms are nearly impossible to undrstand.
Smalltalk objects
Why Smalltalk?
- Another Turing Award
- Small, simple, pure objects
- Almost the complete language can be done in a relatively small interpreter
Alive and well today
- At the core of Ruby
- As an extension to C (“Objective C”) for Apple products
The five questions:
Values are objects (even
true
, 3,"hello"
)Even classes are objects!
There are no functions—only methods on objects
Syntax:
mutable variables
message send
sequential composition of mutations and message sends (side effects)
“blocks” (really closures, objects and closures in one, used as continuations)
No
if
orwhile
. These are implemented by passing continuations to Boolean objects.
(Smalltalk programmers have been indoctrinated and don’t even notice)
Environments
Name stands for a mutable cell containing an object:
- Global variables
- “Instance variables” (new idea, not yet defined)
Types
There is no compile-time type system.
At run time, Smalltalk uses behavioral subtyping, known to Rubyists as “duck typing”
Dynamic semantics
- Main rule is method dispatch (complicated)
- The rest is familiar
The initial basis is enormous
- Why? To demonstrate the benefits of reuse, you need something big enough to reuse.
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)
N.B. BLOCK
and LITERAL
are special objects.
Magnitudes and numbers
Key problems on homework
Natural
is aMagnitude
“Large integer” is a
Number
26 April 2017: Double dispatch, collections
There are PDF slides for 4/26/2017.
COURSE EVALUATIONS: email me your response
Two topics for today
Information hidden and revealed; three layers
(Focus on extending open systems)
Collections: more OO design, plus useful for homework
Making open system extensible
Bonus content 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?
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
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
!
1 May 2017: Programming languages past, present, and future
There were no PDF slides on 5/1/2017.
The timeline
Threads:
- Syntax: (FORTRAN Expressions; Algol while/if/begin/case)
- Higher-order programming (APL; Backus FP, Scheme; ML; Haskell)
- Pattern matching (HOPE, Prolog; …, Erlang)
- Types & polymorphism (Reynolds, CLU, ML)
- Modules (Cedar, Modula-2; Ada, Standard~ML, Modula-3; Units)
- Objects (Simula-67; Smalltalk; Objective C, C++, Self; Java, Ruby; JavaScript)
- Message-passing concurrency (CSP; Pegasus; CML; Erlang)
Your questions
Prelude and disclaimer
My job is to know programming-language research. Most questions are (appropriately) about programming-language design.
- The difference between research and design
Best language for the job
(side board)
Next steps
Key next step is to adapt your skills in context
That said,
If you liked Scheme, Racket system—exploit the fact that every Scheme program is an S-expression. Factory for building new languages. See “Creating languages in Racket” by Matthew Flatt
If you liked types and functional programming: Haskell
If you liked Smalltalk: Ruby
If you liked math: Coq, Software Foundations, OTT
Also, go deeper:
- Real experience with modules
- Real experience with objects
And well beyond:
Scripting (and string processing) — Lua, ksh
Types for systems programming: Rust
Message-passing concurrency, parallelism (Erlang, Concurrent ML, Go)
Code inference, operator overloading: Haskell type classes
Major paradigms not covered in class
Not new, but
- Concurrent
- Distributed
- Dependent types (hoping to become major)
- Logic programming (minor)
- Parallel programming (minor)
My go-to languages
- Quick and dirty? Strings? Lua
- Develop a new idea? Haskell
- Big program has to work? Standard ML
- Systems programming? C
Note: each of these has replaced some other language (Awk, Icon, Modula-3, …).
Efficiency
IT DOESN’T MATTER
Programming-language problems
Problems begging for new languages:
- Teaching
- Programming the GPU
- Control the bits, safely
- Arduino
Of personal interest:
- Build run-time systems and operating systems
Note: the person who thinks of the problem is a super genius. I am not that person.
Today’s PL problems:
Recognize Sturgeon’s law: 90% of everything is crap. (PL research is not exempt.)
Heavy bet on types
Modest bet on probability
Not being tackled: usability (a tragic omission)
Future PL needs: I have no idea, but
- Usability
- The always-on compiler
- Dirt-cheap IDEs
What’s exciting right now:
- Universal deployment through the web browser
- Types
- Haskell, Rust, Elm
Other questions
Best first language
The biggest tragedy in PL
What is it?
My favorite question
Tie.