Checked complexity with typed simplicity

Checked complexity with
typed simplicity
Adelbert Chang
Box, Inc.
1 / 23

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When it comes to controlling the
complexity of developing and,
more importantly, maintaining a
large system, the only game in
town is modularity.
"Modules Matter Most" Robert Harper
2 / 23

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Code/concept reuse
3 / 23

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A type system is a syntactic method
for enforcing levels of abstraction
in programs.
"Types and Programming Languages" Benjamin C. Pierce
4 / 23

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These slides were compiled with Scala 2.11.7, Cats 0.4.1, and Kind-projector 0.7.1
using tut 0.4.2.
5 / 23

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Functions as values
d
e
f d
o
u
b
l
e
I
n
t
s
(
i
n
t
s
: L
i
s
t
[
I
n
t
]
)
: L
i
s
t
[
I
n
t
] = i
n
t
s m
a
t
c
h {
c
a
s
e N
i
l =
> N
i
l
c
a
s
e h :
: t =
> (
h * 2
) :
: d
o
u
b
l
e
I
n
t
s
(
t
)
}
6 / 23

View Slide

Functions as values
d
e
f d
o
u
b
l
e
I
n
t
s
(
i
n
t
s
: L
i
s
t
[
I
n
t
]
)
: L
i
s
t
[
I
n
t
] = i
n
t
s m
a
t
c
h {
c
a
s
e N
i
l =
> N
i
l
c
a
s
e h :
: t =
> (
h * 2
) :
: d
o
u
b
l
e
I
n
t
s
(
t
)
}
d
e
f a
d
d
5
0
(
i
n
t
s
: L
i
s
t
[
I
n
t
]
)
: L
i
s
t
[
I
n
t
] = i
n
t
s m
a
t
c
h {
c
a
s
e N
i
l =
> N
i
l
c
a
s
e h :
: t =
> (
h + 5
0
) :
: a
d
d
5
0
(
t
)
}
6 / 23

View Slide

Functions as values
d
e
f d
o
u
b
l
e
I
n
t
s
(
i
n
t
s
: L
i
s
t
[
I
n
t
]
)
: L
i
s
t
[
I
n
t
] = i
n
t
s m
a
t
c
h {
c
a
s
e N
i
l =
> N
i
l
c
a
s
e h :
: t =
> (
h * 2
) :
: d
o
u
b
l
e
I
n
t
s
(
t
)
}
d
e
f a
d
d
5
0
(
i
n
t
s
: L
i
s
t
[
I
n
t
]
)
: L
i
s
t
[
I
n
t
] = i
n
t
s m
a
t
c
h {
c
a
s
e N
i
l =
> N
i
l
c
a
s
e h :
: t =
> (
h + 5
0
) :
: a
d
d
5
0
(
t
)
}
d
e
f m
a
p
I
n
t
s
(
i
n
t
s
: L
i
s
t
[
I
n
t
]
)
(
f
: I
n
t =
> I
n
t
)
: L
i
s
t
[
I
n
t
] = i
n
t
s m
a
t
c
h {
c
a
s
e N
i
l =
> N
i
l
c
a
s
e h :
: t =
> f
(
h
) :
: m
a
p
I
n
t
s
(
t
)
(
f
)
}
6 / 23

View Slide

Type parameters
d
e
f l
i
s
t
S
t
r
i
n
g
C
o
n
c
a
t
(
l
e
f
t
: L
i
s
t
[
S
t
r
i
n
g
]
, r
i
g
h
t
: L
i
s
t
[
S
t
r
i
n
g
]
)
: L
i
s
t
[
S
t
r
i
n
g
] =
l
e
f
t m
a
t
c
h {
c
a
s
e N
i
l =
> r
i
g
h
t
c
a
s
e h :
: t =
> h :
: l
i
s
t
S
t
r
i
n
g
C
o
n
c
a
t
(
t
, r
i
g
h
t
)
}
7 / 23

View Slide

Type parameters
d
e
f l
i
s
t
I
n
t
C
o
n
c
a
t
(
l
e
f
t
: L
i
s
t
[
I
n
t
]
, r
i
g
h
t
: L
i
s
t
[
I
n
t
]
)
: L
i
s
t
[
I
n
t
] =
l
e
f
t m
a
t
c
h {
c
a
s
e N
i
l =
> r
i
g
h
t
c
a
s
e h :
: t =
> h :
: l
i
s
t
I
n
t
C
o
n
c
a
t
(
t
, r
i
g
h
t
)
}
d
e
f l
i
s
t
S
t
r
i
n
g
C
o
n
c
a
t
(
l
e
f
t
: L
i
s
t
[
S
t
r
i
n
g
]
, r
i
g
h
t
: L
i
s
t
[
S
t
r
i
n
g
]
)
: L
i
s
t
[
S
t
r
i
n
g
] =
l
e
f
t m
a
t
c
h {
c
a
s
e N
i
l =
> r
i
g
h
t
c
a
s
e h :
: t =
> h :
: l
i
s
t
S
t
r
i
n
g
C
o
n
c
a
t
(
t
, r
i
g
h
t
)
}
7 / 23

View Slide

Type parameters
d
e
f l
i
s
t
I
n
t
C
o
n
c
a
t
(
l
e
f
t
: L
i
s
t
[
I
n
t
]
, r
i
g
h
t
: L
i
s
t
[
I
n
t
]
)
: L
i
s
t
[
I
n
t
] =
l
e
f
t m
a
t
c
h {
c
a
s
e N
i
l =
> r
i
g
h
t
c
a
s
e h :
: t =
> h :
: l
i
s
t
I
n
t
C
o
n
c
a
t
(
t
, r
i
g
h
t
)
}
Abstract out irrelevant information
d
e
f l
i
s
t
S
t
r
i
n
g
C
o
n
c
a
t
(
l
e
f
t
: L
i
s
t
[
S
t
r
i
n
g
]
, r
i
g
h
t
: L
i
s
t
[
S
t
r
i
n
g
]
)
: L
i
s
t
[
S
t
r
i
n
g
] =
l
e
f
t m
a
t
c
h {
c
a
s
e N
i
l =
> r
i
g
h
t
c
a
s
e h :
: t =
> h :
: l
i
s
t
S
t
r
i
n
g
C
o
n
c
a
t
(
t
, r
i
g
h
t
)
}
7 / 23

View Slide

Type parameters
d
e
f l
i
s
t
I
n
t
C
o
n
c
a
t
(
l
e
f
t
: L
i
s
t
[
I
n
t
]
, r
i
g
h
t
: L
i
s
t
[
I
n
t
]
)
: L
i
s
t
[
I
n
t
] =
l
e
f
t m
a
t
c
h {
c
a
s
e N
i
l =
> r
i
g
h
t
c
a
s
e h :
: t =
> h :
: l
i
s
t
I
n
t
C
o
n
c
a
t
(
t
, r
i
g
h
t
)
}
d
e
f c
o
n
c
a
t
[
A
]
(
l
e
f
t
: L
i
s
t
[
A
]
, r
i
g
h
t
: L
i
s
t
[
A
]
)
: L
i
s
t
[
A
] =
l
e
f
t m
a
t
c
h {
c
a
s
e N
i
l =
> r
i
g
h
t
c
a
s
e h :
: t =
> h :
: c
o
n
c
a
t
(
t
, r
i
g
h
t
)
}
d
e
f l
i
s
t
S
t
r
i
n
g
C
o
n
c
a
t
(
l
e
f
t
: L
i
s
t
[
S
t
r
i
n
g
]
, r
i
g
h
t
: L
i
s
t
[
S
t
r
i
n
g
]
)
: L
i
s
t
[
S
t
r
i
n
g
] =
l
e
f
t m
a
t
c
h {
c
a
s
e N
i
l =
> r
i
g
h
t
c
a
s
e h :
: t =
> h :
: l
i
s
t
S
t
r
i
n
g
C
o
n
c
a
t
(
t
, r
i
g
h
t
)
}
7 / 23

View Slide

Type parameters
d
e
f l
i
s
t
I
n
t
C
o
n
c
a
t
(
l
e
f
t
: L
i
s
t
[
I
n
t
]
, r
i
g
h
t
: L
i
s
t
[
I
n
t
]
)
: L
i
s
t
[
I
n
t
] =
l
e
f
t m
a
t
c
h {
c
a
s
e N
i
l =
> r
i
g
h
t
c
a
s
e h :
: t =
> h :
: l
i
s
t
I
n
t
C
o
n
c
a
t
(
t
, r
i
g
h
t
)
}
d
e
f c
o
n
c
a
t
[
A
]
(
l
e
f
t
: L
i
s
t
[
A
]
, r
i
g
h
t
: L
i
s
t
[
A
]
)
: L
i
s
t
[
A
] =
l
e
f
t m
a
t
c
h {
c
a
s
e N
i
l =
> r
i
g
h
t
c
a
s
e h :
: t =
> h :
: c
o
n
c
a
t
(
t
, r
i
g
h
t
)
}
c
o
n
c
a
t
(
L
i
s
t
(
1
, 2
) , L
i
s
t
(
3
, 4
, 5
)
)
c
o
n
c
a
t
(
L
i
s
t
(
"
h
e
l
l
o
"
, " "
)
, L
i
s
t
(
"
w
o
r
l
d
"
)
)
c
o
n
c
a
t
(
L
i
s
t
(
'
a
'
, '
b
'
) , L
i
s
t
(
'
c
'
)
)
d
e
f l
i
s
t
S
t
r
i
n
g
C
o
n
c
a
t
(
l
e
f
t
: L
i
s
t
[
S
t
r
i
n
g
]
, r
i
g
h
t
: L
i
s
t
[
S
t
r
i
n
g
]
)
: L
i
s
t
[
S
t
r
i
n
g
] =
l
e
f
t m
a
t
c
h {
c
a
s
e N
i
l =
> r
i
g
h
t
c
a
s
e h :
: t =
> h :
: l
i
s
t
S
t
r
i
n
g
C
o
n
c
a
t
(
t
, r
i
g
h
t
)
}
7 / 23

View Slide

Functions as values + type parameters
d
e
f f
o
l
d
R
i
g
h
t
[
A
, B
]
(
a
s
: L
i
s
t
[
A
]
, b
: B
)
(
f
: (
A
, B
) =
> B
)
: B =
a
s m
a
t
c
h {
c
a
s
e N
i
l =
> b
c
a
s
e h :
: t =
> f
(
h
, f
o
l
d
R
i
g
h
t
(
t
, b
)
(
f
)
)
}
8 / 23

View Slide

Functions as values + type parameters
d
e
f f
o
l
d
R
i
g
h
t
[
A
, B
]
(
a
s
: L
i
s
t
[
A
]
, b
: B
)
(
f
: (
A
, B
) =
> B
)
: B =
a
s m
a
t
c
h {
c
a
s
e N
i
l =
> b
c
a
s
e h :
: t =
> f
(
h
, f
o
l
d
R
i
g
h
t
(
t
, b
)
(
f
)
)
}
d
e
f s
i
z
e
[
A
]
(
a
s
: L
i
s
t
[
A
]
)
: I
n
t =
f
o
l
d
R
i
g
h
t
(
a
s
, 0
)
(
(
_
, s
z
) =
> s
z + 1
)
d
e
f m
a
p
[
A
, B
]
(
a
s
: L
i
s
t
[
A
]
)
(
f
: A =
> B
)
: L
i
s
t
[
B
] =
f
o
l
d
R
i
g
h
t
(
a
s
, L
i
s
t
.
e
m
p
t
y
[
B
]
)
(
(
a
, b
s
) =
> f
(
a
) :
: b
s
)
d
e
f c
o
n
c
a
t
[
A
]
(
l
e
f
t
: L
i
s
t
[
A
]
, r
i
g
h
t
: L
i
s
t
[
A
]
)
: L
i
s
t
[
A
] =
f
o
l
d
R
i
g
h
t
(
l
e
f
t
, r
i
g
h
t
)
(
_ :
: _
)
d
e
f f
l
a
t
t
e
n
[
A
]
(
a
a
s
: L
i
s
t
[
L
i
s
t
[
A
]
]
)
: L
i
s
t
[
A
] =
a
a
s m
a
t
c
h {
c
a
s
e N
i
l =
> N
i
l
c
a
s
e a
s :
: t =
> c
o
n
c
a
t
(
a
s
, f
l
a
t
t
e
n
(
t
)
)
}
d
e
f f
l
a
t
M
a
p
[
A
, B
]
(
a
s
: L
i
s
t
[
A
]
)
(
f
: A =
> L
i
s
t
[
B
]
)
: L
i
s
t
[
B
] =
f
l
a
t
t
e
n
(
m
a
p
(
a
s
)
(
f
)
)
8 / 23

View Slide

Type parameters, universal quantification
9 / 23

View Slide

For all (∀) A. Given a term of type A, I can produce a term of type A
t
r
a
i
t F
o
r
A
l
l {
d
e
f a
p
p
l
y
[
A
]
(
a
: A
)
: A = a
}
9 / 23

View Slide

For all (∀) A. Given a term of type A, I can produce a term of type A
t
r
a
i
t F
o
r
A
l
l {
d
e
f a
p
p
l
y
[
A
]
(
a
: A
)
: A = a
}
You may choose any A
d
e
f f
o
r
A
l
l
I
n
t
(
f
: F
o
r
A
l
l
)
: I
n
t =
> I
n
t =
i =
> f
(
i
)
9 / 23

View Slide

Type members, existential quantification
10 / 23

View Slide

There exists (∃) an A such that given a term of type A, I can produce a term
of type A
t
r
a
i
t T
h
e
r
e
E
x
i
s
t
s {
t
y
p
e A
d
e
f a
p
p
l
y
(
a
: A
)
: A
}
10 / 23

View Slide

There exists (∃) an A such that given a term of type A, I can produce a term
of type A
t
r
a
i
t T
h
e
r
e
E
x
i
s
t
s {
t
y
p
e A
d
e
f a
p
p
l
y
(
a
: A
)
: A
}
I choose an A
d
e
f t
h
e
r
e
E
x
i
s
t
s
I
n
t
(
e
: T
h
e
r
e
E
x
i
s
t
s
)
: I
n
t =
> I
n
t =
i =
> e
(
i
)
/
/ <
c
o
n
s
o
l
e
>
:
1
3
: e
r
r
o
r
: t
y
p
e m
i
s
m
a
t
c
h
;
/
/ f
o
u
n
d : i
.
t
y
p
e (
w
i
t
h u
n
d
e
r
l
y
i
n
g t
y
p
e I
n
t
)
/
/ r
e
q
u
i
r
e
d
: e
.
A
/
/ i =
> e
(
i
)
/
/ ^
10 / 23

View Slide

Type Members vs. Type Parameters (Hi Jon!)
Type parameters = universal quantification = abstraction
(Abstract) Type members = existential quantification = information hiding
11 / 23

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Type Members vs. Type Parameters: Example
/
/ U
n
i
v
e
r
s
a
l
l
y q
u
a
n
t
i
f
i
e
d A - k
n
o
w
n t
o y
o
u
t
r
a
i
t S
t
a
c
k
[
A
] {
/
/ E
x
i
s
t
e
n
t
i
a
l
l
y q
u
a
n
t
i
f
i
e
d R
e
p
r - k
n
o
w
n t
o m
e
t
y
p
e R
e
p
r
d
e
f e
m
p
t
y
: R
e
p
r
d
e
f p
u
s
h
(
a
: A
, r
: R
e
p
r
)
: R
e
p
r
d
e
f p
o
p
(
r
: R
e
p
r
)
: O
p
t
i
o
n
[
(
A
, R
e
p
r
)
]
}
12 / 23

View Slide

/
/ U
n
i
v
e
r
s
a
l
l
y q
u
a
n
t
i
f
i
e
d A - k
n
o
w
n t
o y
o
u
t
r
a
i
t S
t
a
c
k
[
A
] {
/
/ E
x
i
s
t
e
n
t
i
a
l
l
y q
u
a
n
t
i
f
i
e
d R
e
p
r - k
n
o
w
n t
o m
e
t
y
p
e R
e
p
r
d
e
f e
m
p
t
y
: R
e
p
r
d
e
f p
u
s
h
(
a
: A
, r
: R
e
p
r
)
: R
e
p
r
d
e
f p
o
p
(
r
: R
e
p
r
)
: O
p
t
i
o
n
[
(
A
, R
e
p
r
)
]
}
d
e
f p
u
s
h
1
2
3
(
s
t
a
c
k
: S
t
a
c
k
[
I
n
t
]
)
: s
t
a
c
k
.
R
e
p
r = {
i
m
p
o
r
t s
t
a
c
k
.
_
p
u
s
h
(
3
, p
u
s
h
(
2
, p
u
s
h
(
1
, e
m
p
t
y
)
)
)
}
12 / 23

View Slide

/
/ U
n
i
v
e
r
s
a
l
l
y q
u
a
n
t
i
f
i
e
d A - k
n
o
w
n t
o y
o
u
t
r
a
i
t S
t
a
c
k
[
A
] {
/
/ E
x
i
s
t
e
n
t
i
a
l
l
y q
u
a
n
t
i
f
i
e
d R
e
p
r - k
n
o
w
n t
o m
e
t
y
p
e R
e
p
r
d
e
f e
m
p
t
y
: R
e
p
r
d
e
f p
u
s
h
(
a
: A
, r
: R
e
p
r
)
: R
e
p
r
d
e
f p
o
p
(
r
: R
e
p
r
)
: O
p
t
i
o
n
[
(
A
, R
e
p
r
)
]
}
d
e
f p
u
s
h
1
2
3
(
s
t
a
c
k
: S
t
a
c
k
[
I
n
t
]
)
: s
t
a
c
k
.
R
e
p
r = {
i
m
p
o
r
t s
t
a
c
k
.
_
p
u
s
h
(
3
, p
u
s
h
(
2
, p
u
s
h
(
1
, e
m
p
t
y
)
)
)
}
/
/ I b
e
t t
h
e S
t
a
c
k i
s a L
i
s
t
d
e
f p
u
s
h
1
2
3
L
i
s
t
(
s
t
a
c
k
: S
t
a
c
k
[
I
n
t
]
) =
s
t
a
c
k
.
p
u
s
h
(
1
, L
i
s
t
(
2
, 3
, 4
)
)
/
/ <
c
o
n
s
o
l
e
>
:
1
4
: e
r
r
o
r
: t
y
p
e m
i
s
m
a
t
c
h
;
/
/ f
o
u
n
d : L
i
s
t
[
I
n
t
]
/
/ r
e
q
u
i
r
e
d
: s
t
a
c
k
.
R
e
p
r
/
/ s
t
a
c
k
.
p
u
s
h
(
1
, L
i
s
t
(
2
, 3
, 4
)
)
/
/ ^
12 / 23

View Slide

Kinds
The "type" of types
13 / 23

View Slide

Kinds
The "type" of types
Types are to Values as Kinds are to Types
13 / 23

View Slide

Kinds
The "type" of types
*
Int
Char
List[Int]
Either[String, Int]
13 / 23

View Slide

Kinds
The "type" of types
*
Int
Char
List[Int]
Either[String, Int]
* -
> *
List
Vector
Future
13 / 23

View Slide

Kinds
The "type" of types
*
Int
Char
List[Int]
Either[String, Int]
* -
> *
List
Vector
Future
(
*
, *
) -
> *
Either
Tuple2 13 / 23

View Slide

Higher-kinded types
Types that abstract over type constructors
"Generics of a Higher Kind" Adrian Moors, Frank Piessens, Martin Odersky
14 / 23

View Slide

Higher-kinded types
/
/ (
* -
> *
) -
> *
t
r
a
i
t F
u
n
c
t
o
r
[
F
[
_
]
] {
d
e
f m
a
p
[
A
, B
]
(
f
a
: F
[
A
]
)
(
f
: A =
> B
)
: F
[
B
]
}
t
r
a
i
t A
p
p
l
i
c
a
t
i
v
e
[
F
[
_
]
] e
x
t
e
n
d
s F
u
n
c
t
o
r
[
F
] {
d
e
f m
a
p
2
[
A
, B
, C
]
(
f
a
: F
[
A
]
, f
b
: F
[
B
]
)
(
f
: (
A
, B
) =
> C
)
: F
[
C
]
d
e
f p
u
r
e
[
A
]
(
a
: A
)
: F
[
A
]
d
e
f m
a
p
[
A
, B
]
(
f
a
: F
[
A
]
)
(
f
: A =
> B
)
: F
[
B
] =
m
a
p
2
(
f
a
, p
u
r
e
(
(
)
)
)
(
(
a
, _
) =
> f
(
a
)
)
}
14 / 23

View Slide

Higher-kinded types
/
/ (
* -
> *
) -
> *
t
r
a
i
t F
u
n
c
t
o
r
[
F
[
_
]
] {
d
e
f m
a
p
[
A
, B
]
(
f
a
: F
[
A
]
)
(
f
: A =
> B
)
: F
[
B
]
}
t
r
a
i
t A
p
p
l
i
c
a
t
i
v
e
[
F
[
_
]
] e
x
t
e
n
d
s F
u
n
c
t
o
r
[
F
] {
d
e
f m
a
p
2
[
A
, B
, C
]
(
f
a
: F
[
A
]
, f
b
: F
[
B
]
)
(
f
: (
A
, B
) =
> C
)
: F
[
C
]
d
e
f p
u
r
e
[
A
]
(
a
: A
)
: F
[
A
]
d
e
f m
a
p
[
A
, B
]
(
f
a
: F
[
A
]
)
(
f
: A =
> B
)
: F
[
B
] =
m
a
p
2
(
f
a
, p
u
r
e
(
(
)
)
)
(
(
a
, _
) =
> f
(
a
)
)
}
v
a
l l
i
s
t
F
u
n
c
t
o
r
: F
u
n
c
t
o
r
[
L
i
s
t
] = n
e
w F
u
n
c
t
o
r
[
L
i
s
t
] {
d
e
f m
a
p
[
A
, B
]
(
f
a
: L
i
s
t
[
A
]
)
(
f
: A =
> B
)
: L
i
s
t
[
B
] = f
a m
a
t
c
h {
c
a
s
e N
i
l =
> N
i
l
c
a
s
e h :
: t =
> f
(
h
) :
: m
a
p
(
t
)
(
f
)
}
}
14 / 23

View Slide

i
m
p
o
r
t c
a
t
s
.
A
p
p
l
i
c
a
t
i
v
e
d
e
f t
r
a
v
e
r
s
e
L
i
s
t
[
G
[
_
]
, A
, B
]
(
a
s
: L
i
s
t
[
A
]
)
(
f
: A =
> G
[
B
]
)
(
i
m
p
l
i
c
i
t G
: A
p
p
l
i
c
a
t
i
v
e
[
G
]
)
: G
[
L
i
s
t
[
B
]
] =
a
s
.
f
o
l
d
R
i
g
h
t
(
G
.
p
u
r
e
(
L
i
s
t
.
e
m
p
t
y
[
B
]
)
) { (
a
, b
s
) =
>
G
.
m
a
p
2
(
f
(
a
)
, b
s
)
(
(
h
, t
) =
> h :
: t
)
}
15 / 23

View Slide

i
m
p
o
r
t c
a
t
s
.
A
p
p
l
i
c
a
t
i
v
e
d
e
f t
r
a
v
e
r
s
e
L
i
s
t
[
G
[
_
]
, A
, B
]
(
a
s
: L
i
s
t
[
A
]
)
(
f
: A =
> G
[
B
]
)
(
i
m
p
l
i
c
i
t G
: A
p
p
l
i
c
a
t
i
v
e
[
G
]
)
: G
[
L
i
s
t
[
B
]
] =
a
s
.
f
o
l
d
R
i
g
h
t
(
G
.
p
u
r
e
(
L
i
s
t
.
e
m
p
t
y
[
B
]
)
) { (
a
, b
s
) =
>
G
.
m
a
p
2
(
f
(
a
)
, b
s
)
(
(
h
, t
) =
> h :
: t
)
}
i
m
p
o
r
t c
a
t
s
.
i
m
p
l
i
c
i
t
s
.
_
i
m
p
o
r
t s
c
a
l
a
.
c
o
n
c
u
r
r
e
n
t
.
F
u
t
u
r
e
i
m
p
o
r
t s
c
a
l
a
.
c
o
n
c
u
r
r
e
n
t
.
E
x
e
c
u
t
i
o
n
C
o
n
t
e
x
t
.
I
m
p
l
i
c
i
t
s
.
g
l
o
b
a
l
d
e
f p
a
r
s
e
I
n
t
O
p
t
i
o
n
(
s
: S
t
r
i
n
g
)
: O
p
t
i
o
n
[
I
n
t
] =
i
f (
s
.
m
a
t
c
h
e
s
(
"
-
?
[
0
-
9
]
+
"
)
) S
o
m
e
(
s
.
t
o
I
n
t
) e
l
s
e N
o
n
e
t
r
a
v
e
r
s
e
L
i
s
t
(
L
i
s
t
(
"
1
"
, "
2
"
, "
3
"
)
)
(
p
a
r
s
e
I
n
t
O
p
t
i
o
n
)
/
/ r
e
s
2
: O
p
t
i
o
n
[
L
i
s
t
[
I
n
t
]
] = S
o
m
e
(
L
i
s
t
(
1
, 2
, 3
)
)
t
r
a
v
e
r
s
e
L
i
s
t
(
L
i
s
t
(
"
h
e
l
l
o
"
, "
1
"
, "
w
o
r
l
d
"
)
)
(
p
a
r
s
e
I
n
t
O
p
t
i
o
n
)
/
/ r
e
s
3
: O
p
t
i
o
n
[
L
i
s
t
[
I
n
t
]
] = N
o
n
e
15 / 23

View Slide

i
m
p
o
r
t c
a
t
s
.
A
p
p
l
i
c
a
t
i
v
e
d
e
f t
r
a
v
e
r
s
e
L
i
s
t
[
G
[
_
]
, A
, B
]
(
a
s
: L
i
s
t
[
A
]
)
(
f
: A =
> G
[
B
]
)
(
i
m
p
l
i
c
i
t G
: A
p
p
l
i
c
a
t
i
v
e
[
G
]
)
: G
[
L
i
s
t
[
B
]
] =
a
s
.
f
o
l
d
R
i
g
h
t
(
G
.
p
u
r
e
(
L
i
s
t
.
e
m
p
t
y
[
B
]
)
) { (
a
, b
s
) =
>
G
.
m
a
p
2
(
f
(
a
)
, b
s
)
(
(
h
, t
) =
> h :
: t
)
}
i
m
p
o
r
t c
a
t
s
.
i
m
p
l
i
c
i
t
s
.
_
i
m
p
o
r
t s
c
a
l
a
.
c
o
n
c
u
r
r
e
n
t
.
F
u
t
u
r
e
i
m
p
o
r
t s
c
a
l
a
.
c
o
n
c
u
r
r
e
n
t
.
E
x
e
c
u
t
i
o
n
C
o
n
t
e
x
t
.
I
m
p
l
i
c
i
t
s
.
g
l
o
b
a
l
d
e
f p
a
r
s
e
I
n
t
O
p
t
i
o
n
(
s
: S
t
r
i
n
g
)
: O
p
t
i
o
n
[
I
n
t
] =
i
f (
s
.
m
a
t
c
h
e
s
(
"
-
?
[
0
-
9
]
+
"
)
) S
o
m
e
(
s
.
t
o
I
n
t
) e
l
s
e N
o
n
e
t
r
a
v
e
r
s
e
L
i
s
t
(
L
i
s
t
(
"
1
"
, "
2
"
, "
3
"
)
)
(
p
a
r
s
e
I
n
t
O
p
t
i
o
n
)
/
/ r
e
s
2
: O
p
t
i
o
n
[
L
i
s
t
[
I
n
t
]
] = S
o
m
e
(
L
i
s
t
(
1
, 2
, 3
)
)
t
r
a
v
e
r
s
e
L
i
s
t
(
L
i
s
t
(
"
h
e
l
l
o
"
, "
1
"
, "
w
o
r
l
d
"
)
)
(
p
a
r
s
e
I
n
t
O
p
t
i
o
n
)
/
/ r
e
s
3
: O
p
t
i
o
n
[
L
i
s
t
[
I
n
t
]
] = N
o
n
e
t
r
a
v
e
r
s
e
L
i
s
t
(
L
i
s
t
(
"
h
e
l
l
o
"
, "
w
o
r
l
d
"
)
)
(
s =
> F
u
t
u
r
e
(
s +
+ s
)
)
/
/ r
e
s
4
: s
c
a
l
a
.
c
o
n
c
u
r
r
e
n
t
.
F
u
t
u
r
e
[
L
i
s
t
[
S
t
r
i
n
g
]
] = s
c
a
l
a
.
c
o
n
c
u
r
r
e
n
t
.
i
m
p
l
.
P
r
o
m
i
s
e
$
D
e
f
a
u
l
t
P
r
o
m
15 / 23

View Slide

i
m
p
o
r
t c
a
t
s
.
A
p
p
l
i
c
a
t
i
v
e
d
e
f t
r
a
v
e
r
s
e
L
i
s
t
[
G
[
_
]
, A
, B
]
(
a
s
: L
i
s
t
[
A
]
)
(
f
: A =
> G
[
B
]
)
(
i
m
p
l
i
c
i
t G
: A
p
p
l
i
c
a
t
i
v
e
[
G
]
)
: G
[
L
i
s
t
[
B
]
] =
a
s
.
f
o
l
d
R
i
g
h
t
(
G
.
p
u
r
e
(
L
i
s
t
.
e
m
p
t
y
[
B
]
)
) { (
a
, b
s
) =
>
G
.
m
a
p
2
(
f
(
a
)
, b
s
)
(
(
h
, t
) =
> h :
: t
)
}
i
m
p
o
r
t c
a
t
s
.
i
m
p
l
i
c
i
t
s
.
_
i
m
p
o
r
t s
c
a
l
a
.
c
o
n
c
u
r
r
e
n
t
.
F
u
t
u
r
e
i
m
p
o
r
t s
c
a
l
a
.
c
o
n
c
u
r
r
e
n
t
.
E
x
e
c
u
t
i
o
n
C
o
n
t
e
x
t
.
I
m
p
l
i
c
i
t
s
.
g
l
o
b
a
l
d
e
f p
a
r
s
e
I
n
t
O
p
t
i
o
n
(
s
: S
t
r
i
n
g
)
: O
p
t
i
o
n
[
I
n
t
] =
i
f (
s
.
m
a
t
c
h
e
s
(
"
-
?
[
0
-
9
]
+
"
)
) S
o
m
e
(
s
.
t
o
I
n
t
) e
l
s
e N
o
n
e
t
r
a
v
e
r
s
e
L
i
s
t
(
L
i
s
t
(
"
1
"
, "
2
"
, "
3
"
)
)
(
p
a
r
s
e
I
n
t
O
p
t
i
o
n
)
/
/ r
e
s
2
: O
p
t
i
o
n
[
L
i
s
t
[
I
n
t
]
] = S
o
m
e
(
L
i
s
t
(
1
, 2
, 3
)
)
t
r
a
v
e
r
s
e
L
i
s
t
(
L
i
s
t
(
"
h
e
l
l
o
"
, "
1
"
, "
w
o
r
l
d
"
)
)
(
p
a
r
s
e
I
n
t
O
p
t
i
o
n
)
/
/ r
e
s
3
: O
p
t
i
o
n
[
L
i
s
t
[
I
n
t
]
] = N
o
n
e
t
r
a
v
e
r
s
e
L
i
s
t
(
L
i
s
t
(
"
h
e
l
l
o
"
, "
w
o
r
l
d
"
)
)
(
s =
> F
u
t
u
r
e
(
s +
+ s
)
)
/
/ r
e
s
4
: s
c
a
l
a
.
c
o
n
c
u
r
r
e
n
t
.
F
u
t
u
r
e
[
L
i
s
t
[
S
t
r
i
n
g
]
] = s
c
a
l
a
.
c
o
n
c
u
r
r
e
n
t
.
i
m
p
l
.
P
r
o
m
i
s
e
$
D
e
f
a
u
l
t
P
r
o
m
t
r
a
v
e
r
s
e
L
i
s
t
(
L
i
s
t
(
'
a
'
, '
b
'
, '
c
'
)
)
(
c =
> L
i
s
t
(
c
, c
, c
)
)
/
/ r
e
s
5
: L
i
s
t
[
L
i
s
t
[
C
h
a
r
]
] = L
i
s
t
(
L
i
s
t
(
a
, b
, c
)
, L
i
s
t
(
a
, b
, c
)
, L
i
s
t
(
a
, b
, c
)
, L
i
s
t
(
a
, b
, c
15 / 23

View Slide

Embedded DSLs
16 / 23

View Slide

Embedded DSLs
Build an AST of your DSL
17 / 23

View Slide

Embedded DSLs
Separate structure from interpretation
17 / 23

View Slide

Embedded DSLs
A program inside your program
17 / 23

View Slide

Embedded DSLs
Pass AST around in your program, avoiding interpretation until the end
17 / 23

View Slide

Embedded DSLs
Pass AST around in your program, avoiding interpretation until the end
Free monad, free applicative, finally tagless
17 / 23

View Slide

i
m
p
o
r
t c
a
t
s
.
M
o
n
a
d
/
*
* T
h
e i
n
t
e
r
p
r
e
t
e
r *
/
a
b
s
t
r
a
c
t c
l
a
s
s R
W
I
n
t
e
r
p
[
F
[
_
]
] {
d
e
f F
: M
o
n
a
d
[
F
]
d
e
f r
e
a
d
: F
[
S
t
r
i
n
g
]
d
e
f w
r
i
t
e
(
s
: S
t
r
i
n
g
)
: F
[
U
n
i
t
]
}
18 / 23

View Slide

i
m
p
o
r
t c
a
t
s
.
M
o
n
a
d
/
*
* T
h
e i
n
t
e
r
p
r
e
t
e
r *
/
a
b
s
t
r
a
c
t c
l
a
s
s R
W
I
n
t
e
r
p
[
F
[
_
]
] {
d
e
f F
: M
o
n
a
d
[
F
]
d
e
f r
e
a
d
: F
[
S
t
r
i
n
g
]
d
e
f w
r
i
t
e
(
s
: S
t
r
i
n
g
)
: F
[
U
n
i
t
]
}
/
*
* T
h
e s
t
r
u
c
t
u
r
e
/
A
S
T *
/
s
e
a
l
e
d a
b
s
t
r
a
c
t c
l
a
s
s R
W
[
A
] {
d
e
f r
u
n
[
F
[
_
]
]
(
i
n
t
e
r
p
: R
W
I
n
t
e
r
p
[
F
]
)
: F
[
A
]
}
i
m
p
l
i
c
i
t v
a
l c
o
n
s
o
l
e
M
o
n
a
d
: M
o
n
a
d
[
R
W
] = n
e
w M
o
n
a
d
[
R
W
] {
d
e
f f
l
a
t
M
a
p
[
A
, B
]
(
f
a
: R
W
[
A
]
)
(
f
: A =
> R
W
[
B
]
)
: R
W
[
B
] =
n
e
w R
W
[
B
] {
d
e
f r
u
n
[
F
[
_
]
]
(
i
n
t
e
r
p
: R
W
I
n
t
e
r
p
[
F
]
)
: F
[
B
] =
i
n
t
e
r
p
.
F
.
f
l
a
t
M
a
p
(
f
a
.
r
u
n
(
i
n
t
e
r
p
)
)
(
a =
> f
(
a
)
.
r
u
n
(
i
n
t
e
r
p
)
)
}
d
e
f p
u
r
e
[
A
]
(
a
: A
)
: R
W
[
A
] =
n
e
w R
W
[
A
] {
d
e
f r
u
n
[
F
[
_
]
]
(
i
n
t
e
r
p
: R
W
I
n
t
e
r
p
[
F
]
)
: F
[
A
] =
i
n
t
e
r
p
.
F
.
p
u
r
e
(
a
)
}
}
18 / 23

View Slide

v
a
l r
e
a
d
: R
W
[
S
t
r
i
n
g
] =
n
e
w R
W
[
S
t
r
i
n
g
] {
d
e
f r
u
n
[
F
[
_
]
]
(
i
n
t
e
r
p
: R
W
I
n
t
e
r
p
[
F
]
)
: F
[
S
t
r
i
n
g
] =
i
n
t
e
r
p
.
r
e
a
d
}
d
e
f w
r
i
t
e
(
s
: S
t
r
i
n
g
)
: R
W
[
U
n
i
t
] =
n
e
w R
W
[
U
n
i
t
] {
d
e
f r
u
n
[
F
[
_
]
]
(
i
n
t
e
r
p
: R
W
I
n
t
e
r
p
[
F
]
)
: F
[
U
n
i
t
] =
i
n
t
e
r
p
.
w
r
i
t
e
(
s
)
}
19 / 23

View Slide

v
a
l r
e
a
d
: R
W
[
S
t
r
i
n
g
] =
n
e
w R
W
[
S
t
r
i
n
g
] {
d
e
f r
u
n
[
F
[
_
]
]
(
i
n
t
e
r
p
: R
W
I
n
t
e
r
p
[
F
]
)
: F
[
S
t
r
i
n
g
] =
i
n
t
e
r
p
.
r
e
a
d
}
d
e
f w
r
i
t
e
(
s
: S
t
r
i
n
g
)
: R
W
[
U
n
i
t
] =
n
e
w R
W
[
U
n
i
t
] {
d
e
f r
u
n
[
F
[
_
]
]
(
i
n
t
e
r
p
: R
W
I
n
t
e
r
p
[
F
]
)
: F
[
U
n
i
t
] =
i
n
t
e
r
p
.
w
r
i
t
e
(
s
)
}
v
a
l p
r
o
g =
f
o
r {
x <
- r
e
a
d
y <
- r
e
a
d
_ <
- w
r
i
t
e
(
x +
+ y
)
} y
i
e
l
d (
)
/
/ p
r
o
g
: R
W
[
U
n
i
t
] = $
a
n
o
n
$
1
$
$
a
n
o
n
$
2
@
e
b
4
9
8
c
b
19 / 23

View Slide

i
m
p
o
r
t c
a
t
s
.
I
d
i
m
p
o
r
t s
c
a
l
a
.
i
o
.
S
t
d
I
n
/
*
* A
n i
n
t
e
r
p
r
e
t
e
r t
h
a
t u
s
e
s c
o
n
s
o
l
e I
O *
/
v
a
l s
t
d
i
o
I
n
t
e
r
p = n
e
w R
W
I
n
t
e
r
p
[
I
d
] {
v
a
l F = i
m
p
l
i
c
i
t
l
y
[
M
o
n
a
d
[
I
d
]
]
d
e
f r
e
a
d
: S
t
r
i
n
g = S
t
d
I
n
.
r
e
a
d
L
i
n
e
(
)
d
e
f w
r
i
t
e
(
s
: S
t
r
i
n
g
)
: U
n
i
t = p
r
i
n
t
l
n
(
s
)
}
/
/ s
t
d
i
o
I
n
t
e
r
p
: R
W
I
n
t
e
r
p
[
c
a
t
s
.
I
d
]
{
v
a
l F
: c
a
t
s
.
M
o
n
a
d
[
c
a
t
s
.
I
d
]
} = $
a
n
o
n
$
1
@
3
4
e
2
0
d
2
f
d
e
f g
o
(
)
: U
n
i
t = p
r
o
g
.
r
u
n
(
s
t
d
i
o
I
n
t
e
r
p
)
/
/ g
o
: (
)
U
n
i
t
20 / 23

View Slide

i
m
p
o
r
t c
a
t
s
.
d
a
t
a
.
S
t
a
t
e
c
a
s
e c
l
a
s
s R
W
S
t
a
t
e
(
i
n
: L
i
s
t
[
S
t
r
i
n
g
]
, o
u
t
: L
i
s
t
[
S
t
r
i
n
g
]
) {
d
e
f r
e
a
d
: (
R
W
S
t
a
t
e
, S
t
r
i
n
g
) = (
c
o
p
y
(
i
n = i
n
.
t
a
i
l
)
, i
n
.
h
e
a
d
) /
/ Y
O
L
O
d
e
f w
r
i
t
e
(
s
: S
t
r
i
n
g
)
: (
R
W
S
t
a
t
e
, U
n
i
t
) = (
c
o
p
y
(
o
u
t = s :
: o
u
t
)
, (
)
)
}
/
/ d
e
f
i
n
e
d c
l
a
s
s R
W
S
t
a
t
e
/
*
* A
n i
n
t
e
r
p
r
e
t
e
r t
h
a
t u
s
e
s t
h
e S
t
a
t
e m
o
n
a
d *
/
v
a
l s
t
a
t
e
I
n
t
e
r
p = n
e
w R
W
I
n
t
e
r
p
[
S
t
a
t
e
[
R
W
S
t
a
t
e
, ?
]
] {
v
a
l F = i
m
p
l
i
c
i
t
l
y
[
M
o
n
a
d
[
S
t
a
t
e
[
R
W
S
t
a
t
e
, ?
]
]
]
v
a
l r
e
a
d
: S
t
a
t
e
[
R
W
S
t
a
t
e
, S
t
r
i
n
g
] = S
t
a
t
e
(
_
.
r
e
a
d
)
d
e
f w
r
i
t
e
(
s
: S
t
r
i
n
g
)
: S
t
a
t
e
[
R
W
S
t
a
t
e
, U
n
i
t
] =
S
t
a
t
e
(
c
s =
> c
s
.
w
r
i
t
e
(
s
)
)
}
/
/ s
t
a
t
e
I
n
t
e
r
p
: R
W
I
n
t
e
r
p
[
[
β
]
c
a
t
s
.
d
a
t
a
.
S
t
a
t
e
T
[
c
a
t
s
.
E
v
a
l
,
R
W
S
t
a
t
e
,
β
]
]
{
v
a
l F
: c
a
t
s
.
M
o
n
a
d
[
[
β
]
c
a
21 / 23

View Slide

i
m
p
o
r
t c
a
t
s
.
d
a
t
a
.
S
t
a
t
e
c
a
s
e c
l
a
s
s R
W
S
t
a
t
e
(
i
n
: L
i
s
t
[
S
t
r
i
n
g
]
, o
u
t
: L
i
s
t
[
S
t
r
i
n
g
]
) {
d
e
f r
e
a
d
: (
R
W
S
t
a
t
e
, S
t
r
i
n
g
) = (
c
o
p
y
(
i
n = i
n
.
t
a
i
l
)
, i
n
.
h
e
a
d
) /
/ Y
O
L
O
d
e
f w
r
i
t
e
(
s
: S
t
r
i
n
g
)
: (
R
W
S
t
a
t
e
, U
n
i
t
) = (
c
o
p
y
(
o
u
t = s :
: o
u
t
)
, (
)
)
}
/
/ d
e
f
i
n
e
d c
l
a
s
s R
W
S
t
a
t
e
/
*
* A
n i
n
t
e
r
p
r
e
t
e
r t
h
a
t u
s
e
s t
h
e S
t
a
t
e m
o
n
a
d *
/
v
a
l s
t
a
t
e
I
n
t
e
r
p = n
e
w R
W
I
n
t
e
r
p
[
S
t
a
t
e
[
R
W
S
t
a
t
e
, ?
]
] {
v
a
l F = i
m
p
l
i
c
i
t
l
y
[
M
o
n
a
d
[
S
t
a
t
e
[
R
W
S
t
a
t
e
, ?
]
]
]
v
a
l r
e
a
d
: S
t
a
t
e
[
R
W
S
t
a
t
e
, S
t
r
i
n
g
] = S
t
a
t
e
(
_
.
r
e
a
d
)
d
e
f w
r
i
t
e
(
s
: S
t
r
i
n
g
)
: S
t
a
t
e
[
R
W
S
t
a
t
e
, U
n
i
t
] =
S
t
a
t
e
(
c
s =
> c
s
.
w
r
i
t
e
(
s
)
)
}
/
/ s
t
a
t
e
I
n
t
e
r
p
: R
W
I
n
t
e
r
p
[
[
β
]
c
a
t
s
.
d
a
t
a
.
S
t
a
t
e
T
[
c
a
t
s
.
E
v
a
l
,
R
W
S
t
a
t
e
,
β
]
]
{
v
a
l F
: c
a
t
s
.
M
o
n
a
d
[
[
β
]
c
a
v
a
l c
s = R
W
S
t
a
t
e
(
L
i
s
t
(
"
h
e
l
l
o "
, "
n
e
s
c
a
l
a
"
, "
n
o
t r
e
a
d
"
)
, L
i
s
t
.
e
m
p
t
y
)
/
/ c
s
: R
W
S
t
a
t
e = R
W
S
t
a
t
e
(
L
i
s
t
(
h
e
l
l
o , n
e
s
c
a
l
a
, n
o
t r
e
a
d
)
,
L
i
s
t
(
)
)
v
a
l s
t
a
t
e = p
r
o
g
.
r
u
n
[
S
t
a
t
e
[
R
W
S
t
a
t
e
, ?
]
]
(
s
t
a
t
e
I
n
t
e
r
p
)
/
/ s
t
a
t
e
: c
a
t
s
.
d
a
t
a
.
S
t
a
t
e
T
[
c
a
t
s
.
E
v
a
l
,
R
W
S
t
a
t
e
,
U
n
i
t
] = c
a
t
s
.
d
a
t
a
.
S
t
a
t
e
T
@
5
2
e
f
f
0
e
6
v
a
l r
u
n
S
t
a
t
e = s
t
a
t
e
.
r
u
n
S
(
c
s
)
.
v
a
l
u
e
/
/ r
u
n
S
t
a
t
e
: R
W
S
t
a
t
e = R
W
S
t
a
t
e
(
L
i
s
t
(
n
o
t r
e
a
d
)
,
L
i
s
t
(
h
e
l
l
o n
e
s
c
a
l
a
)
)
21 / 23

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Summary
Functions allow reuse of computation
22 / 23

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Summary
Functions as values allow for highly generic combinators
22 / 23

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Summary
Type parameters abstract away unnecessary details
22 / 23

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Summary
Abstract type members enforce hiding of irrelevant detail
22 / 23

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Summary
Higher-kinds allow for higher order abstraction
22 / 23

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Summary
EDSLs separate structure from interpretation, allowing the same structure
to be run against multiple interpreters
22 / 23

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Summary
EDSLs separate structure from interpretation, allowing the same structure
to be run against multiple interpreters
Learn more:
"On understanding types, data abstraction, and polymorphism" Luca
Cardelli, Peter Wenger
"On understanding data abstraction, revisited" William R. Cook
"Type Classes vs. the World" Edward Kmett
22 / 23

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EOF
23 / 23

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Checked complexity with typed simplicity

Checked complexity with typed simplicity

Adelbert Chang

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