Docsity
Docsity
Prepare for your exams
Prepare for your exams

Study with the several resources on Docsity

Earn points to download
Earn points to download

Earn points by helping other students or get them with a premium plan

Guidelines and tips
Guidelines and tips
Sell on Docsity
Sell on Docsity
Docsity AI
Docsity AI
Log inSign up

Prepare for your exams

Study with the several resources on Docsity

Find documents

Prepare for your exams with the study notes shared by other students like you on Docsity

Search for your university

Find the specific documents for your university's exams

Docsity AINEW

Summarize your documents, ask them questions, convert them into quizzes and concept maps

Explore questions

Clear up your doubts by reading the answers to questions asked by your fellow students

Earn points to download

Earn points by helping other students or get them with a premium plan

Share documents
download point icon20 Points

For each uploaded document

Answer questions
download point icon5 Points

For each given answer (max 1 per day)

All the ways to get free points
Get points immediately

Choose a premium plan with all the points you need

Study Opportunities

Choose your next study program

Get in touch with the best universities in the world. Search through thousands of universities and official partners

Community

Ask the community

Ask the community for help and clear up your study doubts

Free resources

Our save-the-student-ebooks!

Download our free guides on studying techniques, anxiety management strategies, and thesis advice from Docsity tutors

Lambda Calculus and Types in Programming Languages - Prof. Atif M. Memon, Study notes of Programming Languages

University of MarylandProgramming Languages
Prof. Atif M. Memon

Material Type: Notes; Professor: Memon; Class: ORGNZTN PROGM LANG; Subject: Computer Science; University: University of Maryland; Term: Unknown 1989;

Typology: Study notes

Pre 2010

Uploaded on 02/13/2009

koofers-user-4jo
koofers-user-4jo 🇺🇸

10 documents

1 / 5

Toggle sidebar

This page cannot be seen from the preview

Don't miss anything!

bg1
1
CMSC 330: Organization of
Programming Languages
Lambda Calculus and Types
CMSC 330 2
Lambda Calculus
• A lambda calculus expression is defined as
e ::= x variable
| x.e function
| e e function application
•x.e is like (fun x -> e) in OCaml
• That’s it! Only higher-order functions
CMSC 330 3
Beta-Reduction, Again
• Whenever we do a step of beta reduction...
–(x.e1) e2 e1[x/e2]
– ...alpha-conv ert variables as necessary
•Examples:
–(x.x (x.x)) z = ( x.x (y.y)) z z (y. y)
–(x.y.x y) y = (x.z.x z) y z.y z
CMSC 330 4
Encodings
• It turns out that this language is Turing complete
• That means we can encode any computation
we want in it
– ...if we’re suff iciently clever...
CMSC 330 5
Booleans
The lambda calculus was creat ed by logician Alonzo
Church in the 1930's to formulate a m athematical logical
system
true = x.y.x
false = x.y.y
if a then b else c is defined to be the expression: a b c
•Examples:
– if true then b else c (x.y.x) b c (y.b) c b
– if fal se then b else c (x.y.y) b c (y.y ) c c
CMSC 330 6
Booleans (continued)
Other Boolean operations:
• not = x.((x false) true)
• not true x.((x false) true) true
((true false) true) false
• and = x.y.((xy) false)
•or = x.y.((x true) y)
•Show not, and and or have the desired properties, …
• Given these operatio ns, can build up a logical inference
system
pf3
pf4
pf5

Partial preview of the text

Download Lambda Calculus and Types in Programming Languages - Prof. Atif M. Memon and more Study notes Programming Languages in PDF only on Docsity!

CMSC 330: Organization of

Programming Languages

Lambda Calculus and Types

CMSC 330 2

Lambda Calculus

  • A lambda calculus expression is defined as

e ::= x variable

| x.e function

| e e function application

  • x.e is like (fun x -> e) in OCaml
  • That’s it! Only higher-order functions

CMSC 330 3

Beta-Reduction, Again

  • Whenever we do a step of beta reduction...
    • (x.e1) e2  e1[x/e2]
    • ...alpha-convert variables as necessary
  • Examples:
    • (x.x (x.x)) z = (x.x (y.y)) z  z (y.y)
    • (x.y.x y) y = (x.z.x z) y  z.y z

CMSC 330 4

Encodings

  • It turns out that this language is Turing complete
  • That means we can encode any computation

we want in it

  • ...if we’re sufficiently clever...

CMSC 330 5

Booleans

The lambda calculus was created by logician Alonzo Church in the 1930's to formulate a mathematical logical system true = x.y.x false = x.y.y

if a then b else c is defined to be the  expression: a b c

  • Examples:
    • if true then b else c  (x.y.x) b c  (y.b) c  b
    • if false then b else c  (x.y.y) b c  (y.y) c  c

CMSC 330 6

Booleans (continued)

Other Boolean operations:

  • not = x.((x false) true)
  • not true  x.((x false) true) true  ((true false) true)  false
  • and = x.y.((xy) false)
  • or = x.y.((x true) y)
  • Show not, and and or have the desired properties, …
  • Given these operations, can build up a logical inference system

CMSC 330 7

Pairs

(a,b) = x.if x then a else b

fst = f.f true

snd = f.f false

  • Examples:
    • fst (a,b) = (f.f true) (x.if x then a else b)  (x.if x then a else b) true  if true then a else b  a
    • snd (a,b) = (f.f false) (x.if x then a else b)  (x.if x then a else b) false  if false then a else b  b CMSC 330 8

Natural Numbers (Church*)

0 = f.y.y

1 = f.y.f y

2 = f.y.f (f y)

3 = f.y.f (f (f y))

i.e., n = f.y.

succ = z.f.y.f (z f y)

iszero = g.g (y.false) true

  • Recall that this is equivalent to g.((g (y.false)) true)

*(Named after Alonzo Church, developer of lambda calculus)

CMSC 330 9

Natural Numbers (cont’d)

  • Examples: succ 0 = (z.f.y.f (z f y)) (f.y.y)  f.y.f ((f.y.y) f y)  f.y.f y = 1

iszero 0 = (z.z (y.false) true) (f.y.y)  (f.y.y) (y.false) true  (y.y) true  true CMSC 330 10

Arithmetic defined

  • Addition, if M and N are integers (as  expressions): M + N = x.y.(M x)((N x) y) Equivalently: + = M.N.x.y.(M x)((N x) y)
  • Multiplication: M * N = x.(M (N x))
  • Prove 1+1 = 2. 1+1 = x.y.(1 x)((1 x) y)  x.y.((x.y.x y) x)(((x.y.x y) x) y)  x.y.(y.x y)(((x.y.x y) x) y)  x.y.(y.x y)((y.x y) y)  x.y.x ((y.x y) y)  x.y.x (x y) = 2
  • With these definitions, can build a theory of integer arithmetic.

CMSC 330 11

Looping

  • Define D = x.x x
  • Then
    • D D = (x.x x) (x.x x)  (x.x x) (x.x x) = D D
  • So D D is an infinite loop
    • In general, self application is how we get looping

CMSC 330 12

The “Paradoxical” Combinator

Y = f.(x.f (x x)) (x.f (x x))

  • Then Y F = (f.(x.f (x x)) (x.f (x x))) F  (x.F (x x)) (x.F (x x))  F ((x.F (x x)) (x.F (x x))) = F (Y F)
  • Thus Y F = F (Y F) = F (F (Y F)) = ...

CMSC 330 19

Type Judgments

  • We will construct a type system that proves

judgments of the form

A  e : t

  • “In type environment A, expression e has type t”
  • If for a program e we can prove that it has some

type, then the program type checks

  • Otherwise the program has a type error, and we’ll reject the program as bad CMSC 330 20

Type Environments

  • A type environment is a map from variables

names to their types

  • Just like in our operational semantics for Scheme
  • • is the empty type environment
  • A, x:t is just like A, except x now has type t
  • When we see a variable in the program, we’ll

look up its type in the environment

CMSC 330 21

Type Rules

A  n : int

e ::= n | x | x:t.e | e e

A  x : A(x)

x  A

A  x:t.e : t  t'

A, x : t  e : t' A  e e' : t'

A  e : t  t' A  e' : t

CMSC 330 22

Example

A  (x:int.+ x 3) 4 : int

A = + : int  int  int

A  (x:int.+ x 3) : int  int A  4 : int

B  + x 3 : int

B  3 : int

B  + : iii B  + x : int  int

B  x : int

B = A, x : int

CMSC 330 23

Discussion

  • The type rules are a kind of logic for reasoning

about types of programs

  • The tree of judgments we just saw is a kind of proof in this logic that the program has a valid type
  • So the type checking problem is like solving a

jigsaw puzzle

  • Can we apply the rules to a program in such a way as to produce a typing proof?
  • It turns out we can easily decide whether or not we can do this. CMSC 330 24

An Algorithm for Type Checking

(Write this in OCaml!) TypeCheck : type env ™ expression  type

TypeCheck(A, n) = int TypeCheck(A, x) = if x in A then A(x) else fail TypeCheck(A, x:t.e) = let t' = TypeCheck((A, x:t), e) in t  t' TypeCheck(A, e1 e2) = let t1 = TypeCheck(A, e1) in let t2 = TypeCheck(A, e2) in if dom(t1) = t2 then range(t1) else fail

CMSC 330 25

Type Inference

  • We could extend the rules to show how a

language could figure out, even if types aren't

specified, what the types of everything are in a

program

  • Can you believe there are languages which can actually do this?
  • We could do these things, but we actually won't.

CMSC 330 26

Summary

  • Lambda calculus shows all the issues with

scoping and higher-order functions

  • It's useful for understanding how languages

work

CMSC 330 27

Practice

  • Reduce the following:
    • (x.y.x y y) (a.a) b
    • (or true) (and true false)
    • (* 1 2) (* m n = M.N.x.(M (N x)) )
  • Derive and prove the type of:
    • (f:int->int.n:int.f n) (x:int. 3 + x) 6
    • x:int->int->int. y:int->int. z:int.x z (y z)