[SOLVED] CS Lambda Calculus algorithm cse130

cse130
https://ucsd-cse130.github.io/wi21/lectures/01-lambda.html
win Appeal
Conditionals Loops
as
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Lambda Calculus
Have youheardof 1
Your Favorite Language
Probably has lots of features:
15hr5
Assignment (x = x + 1) O
E8
Booleans, integers, characters, strings,
return, break, continue
Functions
ROI
Recursion
References / pointers Objects and classes Inheritance

Which ones can we do without?
What is the smallest universal language?
YET

cse130 https://ucsd-cse130.github.io/wi21/lectures/01-lambda.html
What is computable? Before 1930s
Turing Mach 105 Informal notion of an e!ectively calculable function:
t
O
can be computed by a human with pen and paper, following an algorithm
1936: Formalization
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cse130
https://ucsd-cse130.github.io/wi21/lectures/01-lambda.html
What is the smallest universal language?
turns
105
Alan Turing
19301
Alonzo Chardy
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cse130 https://ucsd-cse130.github.io/wi21/lectures/01-lambda.html
Alonzo Church
The Next 700 Languages
Peter Landin
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cse130 https://ucsd-cse130.github.io/wi21/lectures/01-lambda.html
Peter Landin, 1966
The Lambda Calculus
Has one feature: Functions
Function Calls
Whatever the next 700 languages turn out to be, they will surely be variants of lambda calculus.
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cse130 https://ucsd-cse130.github.io/wi21/lectures/01-lambda.html
No, really
Assignment (x = x + 1)
Booleans, integers, characters, strings, Conditionals
Loops
return, break, continue Functions
Recursion
References / pointers Objects and classes Inheritance
Reflection
More precisely, only thing you can do is:
Define a function Call a function
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cse130
https://ucsd-cse130.github.io/wi21/lectures/01-lambda.html
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PW
Describing a Programming Language
Syntax: what do programs look like? Semantics: what do programs mean?
Operational semantics: how do programs execute step-by-step?
Syntax: What Programs Look Like

cse130 Express variable
e ::= x E applehorse
https://ucsd-cse130.github.io/wi21/lectures/01-lambda.html
notexpr arsine ae
Programs are expressions e (also called -terms) of one of three kinds:
Variable
x, y, z
Abstraction (aka nameless function definition)
x->e
x is the formal parameter, e is the body for any x compute e
Application (aka function call)
e1 e2
e1 is the function, e2 is the argument in your favorite language: e1(e2)
(Here each of e , e1 , e2 can itself be a variable, abstraction, or application)
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| x -> e funchfiniti dn
| e1 e2
return e
IX
valid Ax
Gx lx
fencer
net
y fencer
ret
y
s
y

cse130
https://ucsd-cse130.github.io/wi21/lectures/01-lambda.html
Examples
x -> x
function y return y id The identity function (id)
(for any x compute x)
oI
x -> (y -> y)
f -> (f (x -> x)) A function that applies its argument to id
function a x
a
ya
c
QUIZ
x
CxcyDC
g
A function that returns (id)
return
t
funcaonly
return y 33
Which of the following terms are syntactically incorrect?
t
A. (x->x)->y
B. x -> x x
C. x -> x (y x)
D. A and C
E Alloftheabove 9 of 68
At functionCx returnseen
B funonin se funches ret x
return hisses return a
yes
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cse130
https://ucsd-cse130.github.io/wi21/lectures/01-lambda.html
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If
5
235
ME
so
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E. all of the above
lx Ix
Examples
x -> x
x -> (y -> y)
on
f -> f (x -> x)
xx
x
valid
valid
x
The identity function
(for any x compute x)
A function that returns the identity functi
A function that applies its argument
to the identity function
How do I define a function with two arguments?
e.g. a function that takes x and y and returns y ?
wit
21
3
5
e
v

cse130
i Cx x
asf x -> (y -> y)
https://ucsd-cse130.github.io/wi21/lectures/01-lambda.html
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I
y
funcC4E.e3iixIxI
on
sxJdognLna.I
anonymous
Kk
functions
must be a variable e NOT an expr
A function that returns the identity functi
OR: a function that takes two arguments
and returns the second one!
EDxusax
sxCd4x
9 func Arg
t
How do I apply a function to two arguments?
e.g. apply x -> (y -> y) to apple and banana ?
dog