. While several substructural type systems have been proposed and - - PowerPoint PPT Presentation
. . Substructural logics provide a framework for designing resource-aware type systems. . While several substructural type systems have been proposed and implemented, these either have been developed for a special purpose or have been too
.
.
Substructural logics provide a framework for designing resource-aware type systems. While several substructural type systems have been proposed and implemented, these either have been developed for a special purpose or have been too unwieldy for practical use.
Practical Affine Types
Jesse A. Tov and Riccardo Pucella
Northeastern University
January 27, 2010 .
. .
3
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3
Example: OpenGL on Android
.. .
All you have to do to initialize a GLSurfaceView is call setRenderer(). However, if desired, you can modify the default behavior of GLSurfaceView by calling one or more of these methods before setRenderer:
.
(Android 2.2 API Reference)
.
All you have to do to initialize a GLSurfaceView is call setRenderer(). However, if desired, you can modify the default behavior of GLSurfaceView by calling one or more of these methods before setRenderer:
setDebug() setChooser() setWrapper()
.
You can optionally modify the behavior of GLSurfaceView by calling one or more of the debugging methods setDe-
bug(), and setWrapper(). These methods may be called be-
fore and/or after setRenderer, …
.
You can optionally modify the behavior of GLSurfaceView by calling one or more of the debugging methods setDe-
bug(), and setWrapper(). These methods may be called be-
fore and/or after setRenderer, …
.
You can optionally modify the behavior of GLSurfaceView by calling one or more of the debugging methods setDe-
bug(), and setWrapper(). These methods may be called be-
fore and/or after setRenderer, …
.
Once the renderer is set, you can control whether the renderer draws continuously or on-demand by calling setRenderMode().
. .
.
. raw
.
. ready .
new GLSurfaceView
.
setChooser setDebug setWrapper
.
setRenderer
.
rendering operations
.
setRenderMode setDebug setWrapper
. . .
4
Example: OpenGL on Android
.. .
All you have to do to initialize a GLSurfaceView is call setRenderer(). However, if desired, you can modify the default behavior of GLSurfaceView by calling one or more of these methods before setRenderer:
.
(Android 2.2 API Reference)
.
All you have to do to initialize a GLSurfaceView is call setRenderer(). However, if desired, you can modify the default behavior of GLSurfaceView by calling one or more of these methods before setRenderer:
setDebug() setChooser() setWrapper()
.
You can optionally modify the behavior of GLSurfaceView by calling one or more of the debugging methods setDe-
bug(), and setWrapper(). These methods may be called be-
fore and/or after setRenderer, …
.
You can optionally modify the behavior of GLSurfaceView by calling one or more of the debugging methods setDe-
bug(), and setWrapper(). These methods may be called be-
fore and/or after setRenderer, …
.
You can optionally modify the behavior of GLSurfaceView by calling one or more of the debugging methods setDe-
bug(), and setWrapper(). These methods may be called be-
fore and/or after setRenderer, …
.
Once the renderer is set, you can control whether the renderer draws continuously or on-demand by calling setRenderMode().
. .
.
. raw
.
. ready .
new GLSurfaceView
.
setChooser setDebug setWrapper
.
setRenderer
.
rendering operations
.
setRenderMode setDebug setWrapper
. . .
4
Example: OpenGL on Android
.. .
All you have to do to initialize a GLSurfaceView is call setRenderer(). However, if desired, you can modify the default behavior of GLSurfaceView by calling one or more of these methods before setRenderer:
.
(Android 2.2 API Reference)
.
All you have to do to initialize a GLSurfaceView is call setRenderer(). However, if desired, you can modify the default behavior of GLSurfaceView by calling one or more of these methods before setRenderer:
setDebug() setChooser() setWrapper()
.
You can optionally modify the behavior of GLSurfaceView by calling one or more of the debugging methods setDe-
bug(), and setWrapper(). These methods may be called be-
fore and/or after setRenderer, …
.
You can optionally modify the behavior of GLSurfaceView by calling one or more of the debugging methods setDe-
bug(), and setWrapper(). These methods may be called be-
fore and/or after setRenderer, …
.
You can optionally modify the behavior of GLSurfaceView by calling one or more of the debugging methods setDe-
bug(), and setWrapper(). These methods may be called be-
fore and/or after setRenderer, …
.
Once the renderer is set, you can control whether the renderer draws continuously or on-demand by calling setRenderMode().
.
.
. raw
.
. ready .
new GLSurfaceView
.
setChooser, setDebug, setWrapper
.
setRenderer
.
rendering operations
.
setRenderMode setDebug setWrapper
. . .
4
Example: OpenGL on Android
.
.
All you have to do to initialize a GLSurfaceView is call setRenderer(). However, if desired, you can modify the default behavior of GLSurfaceView by calling one or more of these methods before setRenderer:
setDebug() setChooser() setWrapper()
.
(Android 2.2 API Reference)
.
All you have to do to initialize a GLSurfaceView is call setRenderer(). However, if desired, you can modify the default behavior of GLSurfaceView by calling one or more of these methods before setRenderer:
.
You can optionally modify the behavior of GLSurfaceView by calling one or more of the debugging methods setDe-
bug(), and setWrapper(). These methods may be called be-
fore and/or after setRenderer, …
.
You can optionally modify the behavior of GLSurfaceView by calling one or more of the debugging methods setDe-
bug(), and setWrapper(). These methods may be called be-
fore and/or after setRenderer, …
.
You can optionally modify the behavior of GLSurfaceView by calling one or more of the debugging methods setDe-
bug(), and setWrapper(). These methods may be called be-
fore and/or after setRenderer, …
.
Once the renderer is set, you can control whether the renderer draws continuously or on-demand by calling setRenderMode().
.
.
. raw
.
. ready .
new GLSurfaceView
.
setChooser, setDebug, setWrapper
.
setRenderer
.
rendering operations
.
setRenderMode setDebug, setWrapper
. . .
4
Example: OpenGL on Android
.
.
All you have to do to initialize a GLSurfaceView is call setRenderer(). However, if desired, you can modify the default behavior of GLSurfaceView by calling one or more of these methods before setRenderer:
setDebug() setChooser() setWrapper()
.
(Android 2.2 API Reference)
.
All you have to do to initialize a GLSurfaceView is call setRenderer(). However, if desired, you can modify the default behavior of GLSurfaceView by calling one or more of these methods before setRenderer:
.
You can optionally modify the behavior of GLSurfaceView by calling one or more of the debugging methods setDe-
bug(), and setWrapper(). These methods may be called be-
fore and/or after setRenderer, …
.
You can optionally modify the behavior of GLSurfaceView by calling one or more of the debugging methods setDe-
bug(), and setWrapper(). These methods may be called be-
fore and/or after setRenderer, …
.
You can optionally modify the behavior of GLSurfaceView by calling one or more of the debugging methods setDe-
bug(), and setWrapper(). These methods may be called be-
fore and/or after setRenderer, …
.
Once the renderer is set, you can control whether the renderer draws continuously or on-demand by calling setRenderMode().
.
.
. raw
.
. ready .
new GLSurfaceView
.
setChooser, setDebug, setWrapper
.
setRenderer
.
rendering operations
.
setRenderMode, setDebug, setWrapper
. . .
4
Example: OpenGL on Android
.. .
All you have to do to initialize a GLSurfaceView is call setRenderer(). However, if desired, you can modify the default behavior of GLSurfaceView by calling one or more of these methods before setRenderer:
.
(Android 2.2 API Reference)
.
All you have to do to initialize a GLSurfaceView is call setRenderer(). However, if desired, you can modify the default behavior of GLSurfaceView by calling one or more of these methods before setRenderer:
setDebug() setChooser() setWrapper()
.
You can optionally modify the behavior of GLSurfaceView by calling one or more of the debugging methods setDe-
bug(), and setWrapper(). These methods may be called be-
fore and/or after setRenderer, …
.
You can optionally modify the behavior of GLSurfaceView by calling one or more of the debugging methods setDe-
bug(), and setWrapper(). These methods may be called be-
fore and/or after setRenderer, …
.
You can optionally modify the behavior of GLSurfaceView by calling one or more of the debugging methods setDe-
bug(), and setWrapper(). These methods may be called be-
fore and/or after setRenderer, …
.
Once the renderer is set, you can control whether the renderer draws continuously or on-demand by calling setRenderMode().
.
.
. raw
.
. ready .
new GLSurfaceView
.
setChooser, setDebug, setWrapper
.
setRenderer
.
rendering operations
.
setRenderMode, setDebug, setWrapper
. . .
4
Stateful Programming
. .
.
. raw
.
. ready .
new GLSurfaceView
.
setChooser, setDebug, setWrapper
.
setRenderer
.
rendering operations
.
setRenderMode, setDebug, setWrapper
.
.
.
Initial
.
.
Error
.
.
Initialized
.
.
DataSourceConfigured
.
.
Prepared
.
.
Recording
.
.
Released
.
error
.
reset()
.
setAudioSource(), setVideoSource()
.
r e l e a s e ( )
.
setAudioSource(), setVideoSource()
.
reset()
.
setOutputFormat()
.
reset()
.
setAudioEncoder(), setVideoEncoder(), setOutputFile(), setVideoSize(), setVideoFrameRate(), setPreviewDisplay()
.
prepare()
.
r e s e t ( )
.
start()
.
reset(), stop()
..
..
..
.
.
.
.
.. vertex float list . vertex .vertex list . (vertex process) list .. .
The malloc(), calloc(), valloc(), realloc(), and reallocf() functions allocate memory. The free() function frees al- locations that were created via the preceding allocation functions. .
.
connection established
.
.
server greeting
.
.
not authenticated
.
.
authenticated
.
.
selected
.
.
logout
.
.
connection closed
5
Stateful Programming
.
.
. raw
.
. ready .
new GLSurfaceView
.
setChooser, setDebug, setWrapper
.
setRenderer
.
rendering operations
.
setRenderMode, setDebug, setWrapper
. .
.
.
Initial
.
.
Error
.
.
Initialized
.
.
DataSourceConfigured
.
.
Prepared
.
.
Recording
.
.
Released
.
error
.
reset()
.
setAudioSource(), setVideoSource()
.
r e l e a s e ( )
.
setAudioSource(), setVideoSource()
.
reset()
.
setOutputFormat()
.
reset()
.
setAudioEncoder(), setVideoEncoder(), setOutputFile(), setVideoSize(), setVideoFrameRate(), setPreviewDisplay()
.
prepare()
.
r e s e t ( )
.
start()
.
reset(), stop()
..
..
..
.
.
.
.
.. vertex float list . vertex .vertex list . (vertex process) list .. .
The malloc(), calloc(), valloc(), realloc(), and reallocf() functions allocate memory. The free() function frees al- locations that were created via the preceding allocation functions. .
.
connection established
.
.
server greeting
.
.
not authenticated
.
.
authenticated
.
.
selected
.
.
logout
.
.
connection closed
5
Stateful Programming
.
.
. raw
.
. ready .
new GLSurfaceView
.
setChooser, setDebug, setWrapper
.
setRenderer
.
rendering operations
.
setRenderMode, setDebug, setWrapper
.
.
.
Initial
.
.
Error
.
.
Initialized
.
.
DataSourceConfigured
.
.
Prepared
.
.
Recording
.
.
Released
.
error
.
reset()
.
setAudioSource(), setVideoSource()
.
r e l e a s e ( )
.
setAudioSource(), setVideoSource()
.
reset()
.
setOutputFormat()
.
reset()
.
setAudioEncoder(), setVideoEncoder(), setOutputFile(), setVideoSize(), setVideoFrameRate(), setPreviewDisplay()
.
prepare()
.
r e s e t ( )
.
start()
.
reset(), stop()
.
..
..
..
.
.
⊕
.
.
⊕
..
(vertex × float) list
. vertex .vertex list . (vertex × process) list .
ε
.
ε
.
ε
.
The malloc(), calloc(), valloc(), realloc(), and reallocf() functions allocate memory. The free() function frees al- locations that were created via the preceding allocation functions. .
.
connection established
.
.
server greeting
.
.
not authenticated
.
.
authenticated
.
.
selected
.
.
logout
.
.
connection closed
5
Stateful Programming
.
.
. raw
.
. ready .
new GLSurfaceView
.
setChooser, setDebug, setWrapper
.
setRenderer
.
rendering operations
.
setRenderMode, setDebug, setWrapper
.
.
.
Initial
.
.
Error
.
.
Initialized
.
.
DataSourceConfigured
.
.
Prepared
.
.
Recording
.
.
Released
.
error
.
reset()
.
setAudioSource(), setVideoSource()
.
r e l e a s e ( )
.
setAudioSource(), setVideoSource()
.
reset()
.
setOutputFormat()
.
reset()
.
setAudioEncoder(), setVideoEncoder(), setOutputFile(), setVideoSize(), setVideoFrameRate(), setPreviewDisplay()
.
prepare()
.
r e s e t ( )
.
start()
.
reset(), stop()
..
..
..
.
.
⊕
.
.
⊕
..
(vertex × float) list
. vertex .vertex list . (vertex × process) list .
ε
.
ε
.
ε
.. .
The malloc(), calloc(), valloc(), realloc(), and reallocf() functions allocate memory. The free() function frees al- locations that were created via the preceding allocation functions. .
.
connection established
.
.
server greeting
.
.
not authenticated
.
.
authenticated
.
.
selected
.
.
logout
.
.
connection closed
5
Stateful Programming
.
.
. raw
.
. ready .
new GLSurfaceView
.
setChooser, setDebug, setWrapper
.
setRenderer
.
rendering operations
.
setRenderMode, setDebug, setWrapper
.
.
.
Initial
.
.
Error
.
.
Initialized
.
.
DataSourceConfigured
.
.
Prepared
.
.
Recording
.
.
Released
.
error
.
reset()
.
setAudioSource(), setVideoSource()
.
r e l e a s e ( )
.
setAudioSource(), setVideoSource()
.
reset()
.
setOutputFormat()
.
reset()
.
setAudioEncoder(), setVideoEncoder(), setOutputFile(), setVideoSize(), setVideoFrameRate(), setPreviewDisplay()
.
prepare()
.
r e s e t ( )
.
start()
.
reset(), stop()
..
..
..
.
.
⊕
.
.
⊕
..
(vertex × float) list
. vertex .vertex list . (vertex × process) list .
ε
.
ε
.
ε
.
The malloc(), calloc(), valloc(), realloc(), and reallocf() functions allocate memory. The free() function frees al- locations that were created via the preceding allocation functions.
.
.
.
connection established
.
.
server greeting
.
.
not authenticated
.
.
authenticated
.
.
selected
.
.
logout
.
.
connection closed
5
Stateful Programming
.
.
. raw
.
. ready .
new GLSurfaceView
.
setChooser, setDebug, setWrapper
.
setRenderer
.
rendering operations
.
setRenderMode, setDebug, setWrapper
.
.
.
Initial
.
.
Error
.
.
Initialized
.
.
DataSourceConfigured
.
.
Prepared
.
.
Recording
.
.
Released
.
error
.
reset()
.
setAudioSource(), setVideoSource()
.
r e l e a s e ( )
.
setAudioSource(), setVideoSource()
.
reset()
.
setOutputFormat()
.
reset()
.
setAudioEncoder(), setVideoEncoder(), setOutputFile(), setVideoSize(), setVideoFrameRate(), setPreviewDisplay()
.
prepare()
.
r e s e t ( )
.
start()
.
reset(), stop()
..
..
..
.
.
⊕
.
.
⊕
..
(vertex × float) list
. vertex .vertex list . (vertex × process) list .
ε
.
ε
.
ε
.
The malloc(), calloc(), valloc(), realloc(), and reallocf() functions allocate memory. The free() function frees al- locations that were created via the preceding allocation functions. .
.
connection established
.
.
server greeting
.
.
not authenticated
.
.
authenticated
.
.
selected
.
.
logout
.
.
connection closed
.
5
. Theoretical . Practical . Special Resources . General Resources
.
.
ILL
.
.
URAL
.
.
session types calculi
.
.
region calculi
.
.
Sing#
.
.
Vault
.
.
Cyclone
.
.
.
.
(Fluet 2007; Pucella and Heller 2008)
6
. Theoretical . Practical . Special Resources . General Resources
.
.
ILL
.
. λURAL
.
.
session types calculi
.
.
region calculi
.
.
Sing#
.
.
Vault
.
.
Cyclone
.
.
.
.
(Fluet 2007; Pucella and Heller 2008)
6
. Theoretical . Practical . Special Resources . General Resources
.
.
ILL
.
. λURAL
.
.
session types calculi
.
.
region calculi
.
.
Sing#
.
.
Vault
.
.
Cyclone
.
.
.
.
(Fluet 2007; Pucella and Heller 2008)
6
. Theoretical . Practical . Special Resources . General Resources
.
.
ILL
.
. λURAL
.
.
session types calculi
.
.
region calculi
.
.
Sing#
.
.
Vault
.
.
Cyclone
.
.
.
.
(Fluet 2007; Pucella and Heller 2008)
6
. Theoretical . Practical . Special Resources . General Resources
.
.
ILL
.
. λURAL
.
.
session types calculi
.
.
region calculi
.
.
Sing#
.
.
Vault
.
.
Cyclone
.
.
.
.
(Fluet 2007; Pucella and Heller 2008)
6
What We’ve Done
A language design (like Ocaml, but with affine types) A prototype implementation (with libraries and examples) A core model (with nice theorems)
7
What We’ve Done
A language design (like Ocaml, but with affine types) A prototype implementation (with libraries and examples) A core model (with nice theorems)
7
OpenGL in Alms
module PrimGLSurface : sig type glSurface val create
: unit → glSurface
val setChooser : glSurface → unit val setRenderer : glSurface → unit val setMode
: glSurface → unit
val setDebug
: glSurface → unit
end
9
An OpenGL Client
. . .
..
ready
.
setDebug(), setMode()
.
.
raw
.
setDebug(), setChooser()
.
setRenderer()
.
create()
let newSurface () = let surface = create () in . raw setChooser surface; raw . setRenderer surface; raw. ready . setMode surface; ready setDebug surface; ready surface .
10
An OpenGL Client
. . .
..
ready
.
setDebug(), setMode()
.
.
raw
.
setDebug(), setChooser()
.
setRenderer()
.
create()
let newSurface () = let surface = create () in
(∗ .
raw ∗) setChooser surface;
(∗ raw ∗)
. setRenderer surface;
(∗ raw.
ready ∗) . setMode surface;
(∗ ready ∗)
setDebug surface;
(∗ ready ∗)
surface .
10
An OpenGL Client
. . .
..
ready
.
setDebug(), setMode()
.
.
raw
.
setDebug(), setChooser()
.
setRenderer()
.
create()
let newSurface () = let surface = create () in
(∗ .
raw ∗) setChooser surface;
(∗ raw ∗)
. setMode surface;
(∗ ready? ∗)
. setRenderer surface;
(∗ raw.
ready ∗) setDebug surface;
(∗ ready ∗)
surface .
10
OpenGL in Alms: Take 2
. . .
..
ready
.
setDebug(), setMode()
.
.
raw
.
setDebug(), setChooser()
.
setRenderer()
.
create()
module type GL SURFACE = sig type raw type ready type β glSurface val create
: unit → raw glSurface
val setChooser : raw glSurface → unit val setRenderer : raw glSurface → ready glSurface val setMode
: ready glSurface → unit
val setDebug
:
A
β. β glSurface → unit
end
11
An OpenGL Client: Take 2
. . .
..
ready
.
setDebug(), setMode()
.
.
raw
.
setDebug(), setChooser()
.
setRenderer()
.
create()
let newSurface () = let surface = create () in
(∗ .
raw ∗) setChooser surface;
(∗ ready ∗)
. let surface = setRenderer surface in
(∗ raw.
ready ∗) . setMode surface;
(∗ ready ∗)
setDebug surface;
(∗ ready ∗)
surface .
Type error at <opengl.alms> (line 4, columns 13-20): In application, operand type not in operator’s domain: actual: raw glSurface expected: ready glSurface
12
An OpenGL Client: Take 2
. . .
..
ready
.
setDebug(), setMode()
.
.
raw
.
setDebug(), setChooser()
.
setRenderer()
.
create()
let newSurface () = let surface = create () in
(∗ .
raw ∗) setChooser surface;
(∗ ready ∗)
. setMode surface in
(∗ ready? ∗)
. let surface = setRenderer surface;
(∗ raw.
ready ∗) setDebug surface;
(∗ ready ∗)
surface .
Type error at <opengl.alms> (line 4, columns 13-20): In application, operand type not in operator’s domain: actual: raw glSurface expected: ready glSurface
12
An OpenGL Client: Take 2
. . .
..
ready
.
setDebug(), setMode()
.
.
raw
.
setDebug(), setChooser()
.
setRenderer()
.
create()
let newSurface () = let surface = create () in
(∗ .
raw ∗) . setChooser surface;
(∗ raw ∗)
. let surface = setRenderer surface in
(∗ raw.
ready ∗) setMode surface;
(∗ ready ∗)
setDebug surface;
(∗ ready ∗)
surface .
Type error at <opengl.alms> (line 4, columns 16-23): In application, operand type not in operator’s domain: actual: ready glSurface expected: raw glSurface
13
An OpenGL Client: Take 2
. . .
..
ready
.
setDebug(), setMode()
.
.
raw
.
setDebug(), setChooser()
.
setRenderer()
.
create()
let newSurface () = let surface = create () in
(∗ .
raw ∗) . let surface = setRenderer surface;
(∗ raw.
ready ∗) . setChooser surface in
(∗ raw? ∗)
setMode surface;
(∗ ready ∗)
setDebug surface;
(∗ ready ∗)
surface .
Type error at <opengl.alms> (line 4, columns 16-23): In application, operand type not in operator’s domain: actual: ready glSurface expected: raw glSurface
13
An OpenGL Client: Take 2
. . .
..
ready
.
setDebug(), setMode()
.
.
raw
.
setDebug(), setChooser()
.
setRenderer()
.
create()
let newSurface () = let surface = create () in
(∗ .
raw ∗) . let surface′ = setRenderer surface;
(∗ raw.
ready′ ∗) . setChooser surface in
(∗ raw ∗)
setMode surface′;
(∗ ready′ ∗)
setDebug surface′;
(∗ ready′ ∗)
surface′ .
Type error at <opengl.alms> (line 4, columns 16-23): In application, operand type not in operator’s domain: actual: ready glSurface expected: raw glSurface
13
OpenGL in Alms: Take 3
. . .
..
ready
.
setDebug(), setMode()
.
.
raw
.
setDebug(), setChooser()
.
setRenderer()
.
create()
module type GL SURFACE = sig type raw type ready type β glSurface : A val create
: unit → raw glSurface
val setChooser : raw glSurface → raw glSurface val setRenderer : raw glSurface → ready glSurface val setMode
: ready glSurface → ready glSurface
val setDebug
:
A
β. β glSurface → β glSurface
end
14
An OpenGL Client: Take 3
. . .
..
ready
.
setDebug(), setMode()
.
.
raw
.
setDebug(), setChooser()
.
setRenderer()
.
create()
let newSurface () = let surface = create () in
(∗ .
raw ∗) . let surface = setChooser surface in
(∗ raw ∗)
. let surface = setRenderer surface in
(∗ raw.
ready ∗) let surface = setMode surface in
(∗ ready ∗)
let surface = setDebug surface in
(∗ ready ∗)
surface .
Type error at <opengl.alms> (line 2, col. 7 to line 7, col. 12): Affine variable ‘surface’ of type ‘raw glSurface’ duplicated in match or let.
15
An OpenGL Client: Take 3
. . .
..
ready
.
setDebug(), setMode()
.
.
raw
.
setDebug(), setChooser()
.
setRenderer()
.
create()
let newSurface () = let surface = create () in
(∗ .
raw ∗) . let surface′= setRenderer surface in
(∗ raw.
ready′ ∗) . let
= setChooser surface in (∗ raw ∗)
let surface = setMode surface′ in
(∗ ready′.
ready ∗) let surface = setDebug surface in
(∗ ready ∗)
surface .
Type error at <opengl.alms> (line 2, col. 7 to line 7, col. 12): Affine variable ‘surface’ of type ‘raw glSurface’ duplicated in match or let.
15
An OpenGL Client: Take 3
. . .
..
ready
.
setDebug(), setMode()
.
.
raw
.
setDebug(), setChooser()
.
setRenderer()
.
create()
let newSurface () = let surface = create () in
(∗ .
raw ∗) . let surface′= setRenderer surface in
(∗ raw.
ready′ ∗) . let
= setChooser surface in (∗ raw ∗)
let surface = setMode surface′ in
(∗ ready′.
ready ∗) let surface = setDebug surface in
(∗ ready ∗)
surface .
Type error at <opengl.alms> (line 2, col. 7 to line 7, col. 12): Affine variable ‘surface’ of type ‘raw glSurface’ duplicated in match or let.
15
An OpenGL Client: Take 3
. . .
..
ready
.
setDebug(), setMode()
.
.
raw
.
setDebug(), setChooser()
.
setRenderer()
.
create()
let newSurface () = let surface = create () in
(∗ .
raw ∗) . let surface = setChooser surface in
(∗ raw ∗)
. let surface = setRenderer surface in
(∗ raw.
ready ∗) let surface = setMode surface in
(∗ ready ∗)
let surface = setDebug surface in
(∗ ready ∗)
surface .
Type error at <opengl.alms> (line 2, col. 7 to line 7, col. 12): Affine variable ‘surface’ of type ‘raw glSurface’ duplicated in match or let.
15
An OpenGL Client: Take 3
. . .
..
ready
.
setDebug(), setMode()
.
.
raw
.
setDebug(), setChooser()
.
setRenderer()
.
create()
let newSurface () = let surface = create () in
(∗ .
raw ∗) setChooser surface;
(∗ raw ∗)
setRenderer surface;
(∗ raw.
ready ∗) setMode surface;
(∗ ready ∗)
setDebug surface
(∗ ready ∗)
15
OpenGL Implementation
module GLSurface : GL SURFACE = struct type raw
= unit
type ready
= unit
type β glSurface = PrimGLSurface.glSurface
· · ·
end . . .
module type GL SURFACE = sig type raw type ready type β glSurface : A val create : unit → raw glSurface val setChooser : raw glSurface → raw glSurface val setRenderer : raw glSurface → ready glSurface val setMode : ready glSurface → ready glSurface val setDebug : β glSurface → β glSurface end 16
OpenGL Implementation
module GLSurface : GL SURFACE = struct type raw
= unit
type ready
= unit
type β glSurface = PrimGLSurface.glSurface
· · ·
end . . .
PrimGLSurface.glSurface : U
β glSurface : A . U ⊑ A . . .
module type GL SURFACE = sig type raw type ready type β glSurface : A val create : unit → raw glSurface val setChooser : raw glSurface → raw glSurface val setRenderer : raw glSurface → ready glSurface val setMode : ready glSurface → ready glSurface val setDebug : β glSurface → β glSurface end 16
OpenGL Implementation
module GLSurface : GL SURFACE = struct type raw
= unit
type ready
= unit
type β glSurface = PrimGLSurface.glSurface let create = PrimGLSurface.create let setRenderer (surface: raw glSurface) = PrimGLSurface.setRenderer surface; surface
· · ·
end . . .
module type GL SURFACE = sig type raw type ready type β glSurface : A val create : unit → raw glSurface val setChooser : raw glSurface → raw glSurface val setRenderer : raw glSurface → ready glSurface val setMode : ready glSurface → ready glSurface val setDebug : β glSurface → β glSurface end 16
More Examples
Typestate Socket.accept : α socket → α listening →
(∃β. β socket × β ready) × α listening
Session types Session.send : (!ˆ
α; β) channel → ˆ α A − → β channel
Regions (with adoption/focus) Rgn.adopt : (γ,ˆ
α) rgn → (δ,ˆ α) rgn1 A − → δ ptr A − → γ ptr × (γ,ˆ α) rgn
Strong updates Ref.swap : ˆ
α aref → ˆ β
A
− → ˆ β aref × ˆ α
Fractional capabilities Fractional.split : (β,γ) cap → (β,γ/2) cap × (β,γ/2) cap
17
The Exponential
Linear Logic (Girard 1987):
. . , !B, !B . . . . , !B . . . (Contraction)
19
The Problem
Ocaml:
λf (x,y) → f x y : (α → β → γ) → α × β → γ
ILL (Bierman 1993): f promote f for g in p let derelict p be x y in derelict derelict g x y f p let derelict p be x y in derelict f x y
20
The Problem
Ocaml:
λf (x,y) → f x y : (α → β → γ) → α × β → γ
ILL (Bierman 1993):
λf → promote f for g in λp → let derelict p be x ⊗ y
in derelict (derelict g x) y
: !(α ⊸ !(β ⊸ γ)) ⊸ !(!(α ⊗ β) ⊸ γ) λf p → let derelict p be x ⊗ y
in derelict (f x) y
: (α ⊸ !(β ⊸ γ)) ⊸ !(α ⊗ β) ⊸ γ
20
The Problem
Ocaml:
λf (x,y) → f x y : (α → β → γ) → α × β → γ
ILL to Alms:
λf → promote f for g in λp → let derelict p be x ⊗ y
in derelict (derelict g x) y
: (α → β → γ) A − → α × β → γ λf p → let derelict p be x ⊗ y
in derelict (f x) y
: (α A − → β → γ) A − → α × β
A
− → γ
20
The Problem
Ocaml:
λf (x,y) → f x y : (α → β → γ) → α × β → γ
ILL to Alms:
λf (x,y) → f x y : (α → β → γ) → α × β → γ λf (x,y) → f x y : (α A − → β → γ) → α × β
A
− → γ
20
The Problem
Ocaml:
λf (x,y) → f x y : (α → β → γ) → α × β → γ
Alms:
λf (x,y) → f x y : (α δ − → β → γ) → α × β δ − → γ
f x y f x y
A A 20
Dereliction Subtyping
workerThread : unit U
− → unit
Thread.fork
: (unit A − → unit) U − → thread
unit U unit unit A unit (U A) workerThread unit A unit Thread.fork workerThread thread
21
Dereliction Subtyping
workerThread : unit U
− → unit
Thread.fork
: (unit A − → unit) U − → thread
unit U
− → unit ≤
unit A
− → unit
(U ⊑ A) workerThread : unit A
− → unit
Thread.fork workerThread : thread
21
Principal Promotion
λx → x : α U − → α λf x → f x : (α1
γ
− → α2) U − → α1
γ
− → α2 λf g x → f (g x) : (α2
γ
− → α3) U − → (α1
δ
− → α2) γ − → α1
γ⊔δ
− − → α3
.
.
with least kind for every typable function.
22
Principal Promotion
λx → x : α U − → α λf x → f x : (α1
γ
− → α2) U − → α1
γ
− → α2 λf g x → f (g x) : (α2
γ
− → α3) U − → (α1
δ
− → α2) γ − → α1
γ⊔δ
− − → α3
.
.
with least kind for every typable function.
22
Usage Kinds
type α list = Nil | Cons of α × α list let rec foldr f z xs = match xs with | Cons(x, xs) → f x (foldr f z xs) | Nil
→ z
int list U raw glSurface list A raw glSurface ref
23
Usage Kinds
type α list = Nil | Cons of α × α list let rec foldr f z xs = match xs with | Cons(x, xs) → f x (foldr f z xs) | Nil
→ z
int list
: U
raw glSurface list : A raw glSurface ref
23
Usage Kinds
type α list = Nil | Cons of α × α list let rec foldr f z xs = match xs with | Cons(x, xs) → f x (foldr f z xs) | Nil
→ z
int list
: U
raw glSurface list : A raw glSurface ref : ?
23
Usage Kinds
type α list = Nil | Cons of α × α list let rec foldr f z xs = match xs with | Cons(x, xs) → f x (foldr f z xs) | Nil
→ z
int list
: U
raw glSurface list : A raw glSurface ref : ?
23
Usage Kinds
type (α:U) list = Nil | Cons of α × α list
listU U U
let rec foldr f z xs = match xs with | Cons(x, xs) → f x (foldr f z xs) | Nil
→ z
type A listA NilA | ConsA of listA
listA A A
let rec foldrA f z xs match xs with | ConsA x xs f x foldrA f z xs | NilA z
23
Usage Kinds
type (α:U) listU = NilU | ConsU of α × α list (∗ listU : U ⇒ U ∗) let rec foldrU f z xs = match xs with | ConsU(x, xs) → f x (foldrU f z xs) | NilU
→ z
type A listA NilA | ConsA of listA
listA A A
let rec foldrA f z xs match xs with | ConsA x xs f x foldrA f z xs | NilA z
23
Usage Kinds
type (α:U) listU = NilU | ConsU of α × α list (∗ listU : U ⇒ U ∗) let rec foldrU f z xs = match xs with | ConsU(x, xs) → f x (foldrU f z xs) | NilU
→ z
type (α:A) listA = NilA | ConsA of α × α listA (∗ listA : A ⇒ A ∗) let rec foldrA f z xs = match xs with | ConsA(x, xs) → f x (foldrA f z xs) | NilA
→ z
23
Dependent Usage Kinds
type α list = Nil | Cons of α × α list (∗ list : Πα. ⟨α⟩ ∗) let rec foldr f z xs = match xs with | Cons(x,xs) → f x (foldr f z xs) | Nil
→ z
23
Dependent Usage Kinds
type α list = Nil | Cons of α × α list (∗ list : Πα. ⟨α⟩ ∗) let rec foldr f z xs = match xs with | Cons(x,xs) → f x (foldr f z xs) | Nil
→ z
. . .
(×) : Πα. Πβ. ⟨α⟩ ⊔ ⟨β⟩ (+) : Πα. Πβ. ⟨α⟩ ⊔ ⟨β⟩
ref
: Πα. U
glSurface :
Πα. A
23
Related Work
λURAL (Ahmed et al. 2005)
“Uniqueness Typing Simplified” (de Vries et al. 2008) System F (Mazurak et al. 2010) Fine (Swamy et al. 2010) Plaid (Aldrich et al. 2009)
25
Related Work
λURAL (Ahmed et al. 2005)
“Uniqueness Typing Simplified” (de Vries et al. 2008) System F◦ (Mazurak et al. 2010) Fine (Swamy et al. 2010) Plaid (Aldrich et al. 2009)
25
Alms: Practical Affine Types
Affine types:
Paper: more examples and our model Online: prototype implementation and extended paper
26
Thank You
Affine types:
Paper: more examples and our model Online: prototype implementation and extended paper
26
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27
Why Affine Types?
Control.
. E[callcc v] −
→ .
E[v (λx.abort . E[x])] .
28
Why Affine Types?
Control.
. E[callcc v] −
→ .
E[v (λx.abort . E[x])] .
28
Why Affine Types?
Control.
E[C v] −
→ v (λx. abort E[x])
. E abort e e.
28
Why Affine Types?
Control.
E[C v] −
→ v (λx. abort E[x])
. E[abort e] −
→ e.
.
28