mirror of
https://github.com/ProgramSnail/pass_strategy_synthesis.git
synced 2026-04-27 00:34:50 +00:00
913 lines
20 KiB
Typst
913 lines
20 KiB
Typst
// #import "@preview/polylux:0.4.0": *
|
||
#import "@preview/simplebnf:0.1.1": *
|
||
// #import "@preview/zebraw:0.5.0": *
|
||
// #show: zebraw
|
||
#import "@preview/curryst:0.6.0": rule, prooftree, rule-set
|
||
#import "@preview/xarrow:0.4.0": xarrow, xarrowDashed, xarrowSquiggly
|
||
|
||
= Формальная модель используемого языка
|
||
|
||
#h(10pt)
|
||
|
||
// TODO: check correctnes for path, mem & type ??
|
||
|
||
== Syntax
|
||
|
||
#h(10pt)
|
||
|
||
#let rf = $\& #h(3pt)$
|
||
|
||
#let isCorrect = `isCorrect`
|
||
|
||
#let isRead = `isRead`
|
||
#let isAlwaysWrite = `isAlwaysWrite`
|
||
#let isPossibleWrite = `isPossibleWrite`
|
||
#let isRef = `isRef`
|
||
#let isCopy = `isCopy`
|
||
#let isIn = `isIn`
|
||
#let isOut = `isOut`
|
||
|
||
#let readTag = `read`
|
||
#let writeTag = `write`
|
||
#let copyTag = `copy`
|
||
#let inTag = `in`
|
||
#let outTag = `out`
|
||
#let mode = `mode`
|
||
|
||
#let Copy = `Copy`
|
||
#let Ref = `Ref`
|
||
#let MaybeWrite = [$diamond$ `Write`]
|
||
#let AlwaysWrite = [$square$ `Write`]
|
||
#let Read = `Read`
|
||
#let In = `In`
|
||
#let Out = `Out`
|
||
|
||
#let expr = `expr`
|
||
#let stmt = `stmt`
|
||
#let decl = `decl`
|
||
#let prog = `prog`
|
||
#let path = `path`
|
||
#let type = `type`
|
||
#let modedType = `modedtype`
|
||
#bnf(
|
||
Prod(`read`,
|
||
// NOTE: not three modalities for write, because read does not change value
|
||
// => it is not important to observe rsult, no differenc between always and maybe
|
||
{ Or[Read][read passed value]
|
||
Or[$not$ Read][] } ),
|
||
Prod(`write`,
|
||
{ Or[$square$ Write][in all cases there is a write to passed variable] // always write, requre at least one write in each flow variant
|
||
Or[$diamond$ Write][in some cases there is a write to passed variable] // possible write, no requirements (?)
|
||
Or[$not$ Write][] } ), // no write, require n owrites in all flow variants
|
||
Prod(`copy`,
|
||
{ Or[Ref][pass reference to the value]
|
||
Or[Value][pass copy of the value] } ),
|
||
Prod(`in`,
|
||
{ Or[In][parameter value used as input]
|
||
Or[$not$ In][] } ),
|
||
Prod(`out`,
|
||
{ Or[Out][parameter value returned]
|
||
Or[$not$ Out][] } ),
|
||
Prod(
|
||
`mode`,
|
||
{
|
||
Or[$inTag space outTag$][]
|
||
}
|
||
),
|
||
Prod(
|
||
`path`,
|
||
{
|
||
// NOTE: global vars & local vars names could be used with one constructor
|
||
// Or[$\#x$][funciton or global variable itself]
|
||
Or[$@ X$][function argument or variable itself]
|
||
Or[$* path$][reference insede path]
|
||
Or[$path . n$][access $n$-th cell of the tuple]
|
||
// Or[$path : n$][access $n$-th cell of the union] // TODO: another notation ??
|
||
}
|
||
),
|
||
Prod(
|
||
`type`,
|
||
{
|
||
Or[$readTag writeTag ()$][simple type representing all primitive types] // `Unit`
|
||
Or[$rf copyTag space type$][reference to structure, contains copy / ref choice] // `Ref`
|
||
Or[$[type+]$][tuple type] // `Prod`
|
||
// Or[`type` $times$ `type`][pair type, allows to make tuples] // `Prod`
|
||
// Or[`type` $+$ `type`][union type (important in some way ???)] // `Sum` // TODO ?
|
||
|
||
// NOTE: do not use names in type
|
||
// Or[$lambda_((x type)+)$][type of lambda or function pointer, defined by function declaration] // `Fun`
|
||
Or[$lambda (modedType)+$][type of lambda or function pointer, defined by function declaration] // `Fun`
|
||
}
|
||
),
|
||
Prod(
|
||
`modedtype`,
|
||
{
|
||
Or[$mode type$][type woth in and out modifiers]
|
||
}
|
||
),
|
||
Prod(
|
||
`expr`,
|
||
{
|
||
Or[$()$][value of simple type] // `Unit`
|
||
Or[$path$][value from variable] // `Path`
|
||
Or[$rf expr$][reference expr] // `Ref`
|
||
Or[$[expr+]$][tuple expr] // `Prod`
|
||
}
|
||
),
|
||
Prod(
|
||
`stmt`,
|
||
{
|
||
Or[`CALL` $path space expr+$][call function]
|
||
Or[`WRITE` $path$][write to variable]
|
||
Or[`READ` $path$][read from variable]
|
||
Or[$stmt ; stmt$][control flow operator, xecution ]
|
||
Or[$stmt | stmt$][control flow operator, excution of one statements]
|
||
}
|
||
),
|
||
Prod(
|
||
`decl`,
|
||
{
|
||
// TODO: path not allowed ??
|
||
Or[$"var" X : type = expr$][global variable declaration]
|
||
Or[$"fun" X ((X : modedType)+) = stmt$][function declaration]
|
||
}
|
||
),
|
||
Prod(
|
||
`prog`,
|
||
{
|
||
Or[$decl stmt$][declarations and executet statement]
|
||
}
|
||
),
|
||
)
|
||
|
||
== Value Model
|
||
|
||
#let deepValue = `deepvalue`
|
||
#let value = `value`
|
||
|
||
#bnf(
|
||
Prod(
|
||
$deepValue$,
|
||
{
|
||
Or[$0$][valid value of simple type] // `Unit`
|
||
Or[$\#$][valid or spoiled value of simple type] // `Unit`
|
||
Or[$bot$][spoiled value of simple type] // `Unit`
|
||
Or[$lambda type+ stmt$][function pointer value] // `Fun`
|
||
Or[$rf deepValue$][reference value, contains label of the value in the memory] // `Ref`
|
||
Or[$[deepValue+]$][tuple value] // `Prod`
|
||
}
|
||
),
|
||
Prod(
|
||
$value_mu$,
|
||
{
|
||
Or[$0$][valid value of simple type] // `Unit`
|
||
Or[$\#$][valid or spoiled value of simple type] // `Unit`
|
||
Or[$bot$][spoiled value of simple type] // `Unit`
|
||
Or[$lambda type+ stmt$][function pointer value] // `Fun`
|
||
// FIXME: embed mode into value for simplification ??
|
||
Or[$rf copyTag LL$][reference value, contains label of the value in the memory] // `Ref`
|
||
Or[$[value+]$][tuple value] // `Prod`
|
||
}
|
||
),
|
||
)
|
||
|
||
#deepValue - полное значение, #value - слой значения, привязан к конкретной памяти $mu$
|
||
|
||
Значения, могут лежать в переменных и передаваться как аргументы функций (то, во что вычисляется $expr$)
|
||
|
||
$v in value$ - произвольное значение
|
||
|
||
Получение #deepValue по #value:
|
||
- $rf c l xarrowSquiggly(mu)_#[deep] rf c mu[l]$
|
||
- $* xarrowSquiggly(mu)_#[deep] *$
|
||
где $*$ - произвольный конструктор значения, кроме $rf$
|
||
|
||
== Memory Model
|
||
|
||
#let cl = $chevron.l$
|
||
#let cr = $chevron.r$
|
||
|
||
#let mem = `mem`
|
||
|
||
- $LL$ - множество меток памяти
|
||
- $mem := LL -> value, space mu : mem$ - память, частично определённая функция
|
||
- $l in LL$ - новый тег памяти (ранее не использованный)
|
||
- `next` - получение следующей неиспользованной метки в памяти
|
||
|
||
#align(center, prooftree(
|
||
vertical-spacing: 4pt,
|
||
rule(
|
||
name: [ add value to memory],
|
||
|
||
$l = #[next] (mu)$,
|
||
|
||
$cl mu cr xarrowSquiggly(v)_#[add] cl l, mu [l <- v] cr$,
|
||
)
|
||
))
|
||
|
||
== Semantics
|
||
|
||
// $V := memelem$ - значения памяти
|
||
|
||
$X$ - можество переменных
|
||
|
||
// FIXME: TMP
|
||
#let valuemem = `valuemem`
|
||
#let memelem = `memelem`
|
||
#let pathenvmode = `pathenvmode`
|
||
#let pathenvval = `pathenvval`
|
||
#let pathenvmem = `pathenvmem`
|
||
#let pathenvtype = `pathenvtype`
|
||
|
||
#let env = `env`
|
||
$sigma : env := X -> LL times type, space sigma : env$ - #[ метки памяти и типы значений пеерменных контекста, частично определённая функция ]
|
||
|
||
// $DD : X -> decl$ - глобальные определения, частично определённая функция
|
||
|
||
// $d in decl, $
|
||
$s in stmt, f in X, x in X, a in X$
|
||
|
||
// FIXME ??
|
||
// $d space @ space overline(x)$ - запись применения функции (вида #decl) к аргументам
|
||
|
||
=== Path Accessors
|
||
|
||
Набор частично определённых фунций для работы с путями.
|
||
Для удобства значение, получаемое из текущего применением пути, будем называть полем.
|
||
// Все эти функции используются с префиксом `path.`.
|
||
|
||
#let eqmu = $attach(=, br: mu)$
|
||
#let arrmu = $attach(->, br: mu)$
|
||
|
||
#let ttype = $attach(tack.r, br: type)$
|
||
#let tpath = $attach(tack.r, br: path)$
|
||
#let tmode = $attach(tack.r, br: mode)$
|
||
|
||
#let val = `val`
|
||
#let label = `label`
|
||
#let tval = $attach(tack.r, br: val)$
|
||
#let tlabel = $attach(tack.r, br: label)$
|
||
|
||
#let tetype = $attach(tack.r.double, br: type)$
|
||
#let temode = $attach(tack.r.double, br: mode)$
|
||
|
||
#let teval = $attach(tack.r.double, br: val)$
|
||
#let telabel = $attach(tack.r.double, br: label)$
|
||
|
||
// TODO: env mem label ??, env mem value ??
|
||
|
||
// TODO: FIXME: backwards, deconstruction ??
|
||
- #[ Конструирование путей по переменой
|
||
#align(center, prooftree(
|
||
vertical-spacing: 4pt,
|
||
rule(
|
||
name: [ variable path],
|
||
|
||
$x tpath @x$,
|
||
)
|
||
))
|
||
#align(center, prooftree(
|
||
vertical-spacing: 4pt,
|
||
rule(
|
||
name: [ reference path],
|
||
|
||
$x tpath p$,
|
||
$x tpath rf p$,
|
||
)
|
||
))
|
||
#align(center, prooftree(
|
||
vertical-spacing: 4pt,
|
||
rule(
|
||
name: [ tuple access path],
|
||
|
||
$x tpath p$,
|
||
|
||
$x tpath p.i$,
|
||
)
|
||
))
|
||
]
|
||
|
||
- #[ Получение типа поля
|
||
#align(center, prooftree(
|
||
vertical-spacing: 4pt,
|
||
rule(
|
||
name: [ variable type access],
|
||
|
||
$x : t_x ttype @x : t_x$,
|
||
)
|
||
))
|
||
#align(center, prooftree(
|
||
vertical-spacing: 4pt,
|
||
rule(
|
||
name: [ reference type access],
|
||
|
||
$x tpath p$,
|
||
$x : t_x ttype p : rf mode t_p$,
|
||
$x : t_x ttype *p : t_p$,
|
||
)
|
||
))
|
||
#align(center, prooftree(
|
||
vertical-spacing: 4pt,
|
||
rule(
|
||
name: [ tuple type access],
|
||
|
||
$x tpath p$,
|
||
$x : t_x ttype p : [t_1, ... t_n]$,
|
||
$x : t_x ttype p.i : t_i$,
|
||
)
|
||
))
|
||
]
|
||
|
||
- #[ Получение read-write тега поля
|
||
#align(center, prooftree(
|
||
vertical-spacing: 4pt,
|
||
rule(
|
||
name: [ variable rw tag access],
|
||
|
||
$t_x = r w ()$,
|
||
$x : t_x tmode @x -> cr r w cl$,
|
||
)
|
||
))
|
||
#align(center, prooftree(
|
||
vertical-spacing: 4pt,
|
||
rule(
|
||
name: [ reference rw tag access],
|
||
|
||
$x tpath p$,
|
||
$x : t_x tmode p -> cr r w cl$,
|
||
$x : t_x tmode *p -> cr r w cl$,
|
||
)
|
||
))
|
||
#align(center, prooftree(
|
||
vertical-spacing: 4pt,
|
||
rule(
|
||
name: [ tuple rw tag access],
|
||
|
||
$x tpath p$,
|
||
$x : t_x tmode p -> cr r w cl$,
|
||
$x : t_x tmode p.i -> cr r w cl$,
|
||
)
|
||
))
|
||
]
|
||
|
||
- #[ Получение значения поля
|
||
#align(center, prooftree(
|
||
vertical-spacing: 4pt,
|
||
rule(
|
||
name: [ variable value access],
|
||
|
||
$x eqmu v_x tval @x eqmu v_x$,
|
||
)
|
||
))
|
||
#align(center, prooftree(
|
||
vertical-spacing: 4pt,
|
||
rule(
|
||
name: [ reference value access],
|
||
|
||
$x tpath p$,
|
||
$x eqmu v_x tval p eqmu rf l$,
|
||
$x eqmu v_x tval *p eqmu mu[l]$,
|
||
)
|
||
))
|
||
#align(center, prooftree(
|
||
vertical-spacing: 4pt,
|
||
rule(
|
||
name: [ tuple value access],
|
||
|
||
$x tpath p$,
|
||
$x eqmu v_x tmode p eqmu [v_1, ... v_n]$,
|
||
$x eqmu v_x tmode p.i eqmu v_i$,
|
||
)
|
||
))
|
||
]
|
||
|
||
- #[ Получение метки поля
|
||
#align(center, prooftree(
|
||
vertical-spacing: 4pt,
|
||
rule(
|
||
name: [ access],
|
||
|
||
$v_x = rf l$,
|
||
$x eqmu v_x tval p eqmu rf l$,
|
||
|
||
$x eqmu v_x tmode p arrmu l$,
|
||
)
|
||
))
|
||
]
|
||
|
||
- #[ Получение типа поля по окружению
|
||
#align(center, prooftree(
|
||
vertical-spacing: 4pt,
|
||
rule(
|
||
name: [ access],
|
||
|
||
$x tpath p$,
|
||
$x : sigma[x].2 ttype p : t$,
|
||
$sigma tetype p : t$,
|
||
)
|
||
))
|
||
]
|
||
|
||
- #[ Получение read-write тега поля по окружению
|
||
#align(center, prooftree(
|
||
vertical-spacing: 4pt,
|
||
rule(
|
||
name: [ access],
|
||
|
||
$x tpath p$,
|
||
$x : sigma[x].2 tmode p -> cr r space w cl$,
|
||
$sigma temode p -> cr r space w cl$,
|
||
)
|
||
))
|
||
]
|
||
|
||
- #[ Получение значения поля по окружению
|
||
#align(center, prooftree(
|
||
vertical-spacing: 4pt,
|
||
rule(
|
||
name: [ access],
|
||
|
||
$x tpath p$,
|
||
$x eqmu sigma[x].1 tval p eqmu v$,
|
||
$sigma, mu teval p eqmu x$,
|
||
)
|
||
))
|
||
]
|
||
|
||
- #[ Получение метки поля по окружению
|
||
#align(center, prooftree(
|
||
vertical-spacing: 4pt,
|
||
rule(
|
||
name: [ access],
|
||
|
||
$x tpath p$,
|
||
$x eqmu sigma[x].1 tlabel p arrmu l$,
|
||
$sigma, mu telabel p arrmu l$,
|
||
)
|
||
))
|
||
]
|
||
|
||
=== Mode Accessors
|
||
|
||
#let modevar = $(i space o)$
|
||
|
||
$ isIn modevar = i == In $
|
||
$ isOut modevar = o == Out $
|
||
|
||
// FIXME: move to new mode model
|
||
// === Mode Correctness
|
||
|
||
// Функции проверки тегов, имеют тип $mode -> #[bool]$:
|
||
|
||
// #let modevar = $(r space w space c space i space o)$
|
||
|
||
// $ isRead modevar = r == "Read" $
|
||
// $ isAlwaysWrite modevar = w == square "Write" $
|
||
// $ isPossibleWrite modevar = w == diamond "Write" || w == square "Write" $
|
||
// $ isRef modevar = c == "Ref" $
|
||
// $ isCopy modevar = c == "Copy" $
|
||
// $ isIn modevar = i == "In" $
|
||
// $ isOut modevar = o == "Out" $
|
||
|
||
// Требования к тегам:
|
||
|
||
// $ isOut mode -> isAlwaysWrite mode $
|
||
// $ isRead mode -> isIn mode $
|
||
|
||
// TODO: rest conditions ??
|
||
|
||
=== Eval Rules
|
||
|
||
#let args = `args`
|
||
|
||
#[
|
||
|
||
#let ref = `ref`
|
||
#let copy = `copy`
|
||
#let read = `read`
|
||
|
||
// #align(center, grid(
|
||
// columns: 3,
|
||
// gutter: 5%,
|
||
// align(bottom, prooftree(
|
||
// ...
|
||
// )),
|
||
// align(bottom, prooftree(
|
||
// ...
|
||
// )),
|
||
// align(bottom, prooftree(
|
||
// ...
|
||
// )),
|
||
// ))
|
||
|
||
// TODO: introduce spep env argument ??
|
||
|
||
=== Moded Type Correctness
|
||
|
||
#let tcorrect = $attach(tack.r, br: #[correct])$
|
||
|
||
// TODO: FIXME: well formatness for mode, extract
|
||
// TODO: FIXME: check for mode, is recursion required ??
|
||
// TODO: FIXME: check mode & access corectness in os correct
|
||
|
||
$ sigma in env, space mu in mem, space m in mode,
|
||
space c in copyTag, space r, r' in readTag, space w, w' in writeTag,
|
||
space v in value, space t, t' in type $
|
||
|
||
#h(10pt)
|
||
|
||
#align(center, prooftree(
|
||
vertical-spacing: 4pt,
|
||
rule(
|
||
name: [ unit assignment tags correctness],
|
||
|
||
$r = Read => isIn m$,
|
||
$isOut m => w = AlwaysWrite$,
|
||
// $sigma temode x -> cr r' space w' cl$, // NOTE: not required, value passed
|
||
$(w = AlwaysWrite or w = MaybeWrite) and (isOut m or c = Ref) => w' = AlwaysWrite$,
|
||
|
||
// $sigma, mu teval x eqmu v$, // NOTE: not required, value passed
|
||
$v in {0, \#, bot}$,
|
||
$r = Read => v = 0$,
|
||
|
||
$sigma, mu, m, c tcorrect v : r' space w' () -> r space w ()$,
|
||
)
|
||
))
|
||
|
||
#h(10pt)
|
||
|
||
#align(center, prooftree(
|
||
vertical-spacing: 4pt,
|
||
rule(
|
||
name: [ ref assignment tags correctness],
|
||
|
||
$sigma, mu, m, c_r tcorrect v : t -> t'$,
|
||
|
||
// TODO: FIXME: which tag translations are correct ?? <- only assignee?
|
||
$sigma, mu, m, c tcorrect rf c_r space v : rf c' space t -> rf c_r space t'$,
|
||
)
|
||
))
|
||
|
||
#h(10pt)
|
||
|
||
#align(center, prooftree(
|
||
vertical-spacing: 4pt,
|
||
rule(
|
||
name: [ tuple assignmenttags correctness],
|
||
|
||
$sigma, mu, m, c tcorrect v_1 : t_1 -> t'_1$,
|
||
|
||
$...$,
|
||
|
||
$sigma, mu, m, c tcorrect v_n : t_n -> t'_n$,
|
||
|
||
$sigma, mu, m, c tcorrect [v_1, ... v_n] : [t_1, ..., t_n] -> [t'_1, t'_n]$,
|
||
)
|
||
))
|
||
|
||
#h(10pt)
|
||
|
||
#align(center, prooftree(
|
||
vertical-spacing: 4pt,
|
||
rule(
|
||
name: [ function pointer tags correctness],
|
||
|
||
$sigma, mu, m, c tcorrect lambda space overline(t) space s : lambda space overline(t) -> lambda space overline(t)$,
|
||
)
|
||
))
|
||
|
||
#h(10pt)
|
||
|
||
=== Value Construction
|
||
|
||
#let new = `new`
|
||
|
||
#align(center, prooftree(
|
||
vertical-spacing: 4pt,
|
||
rule(
|
||
name: [ new $0$ value],
|
||
|
||
// TODO: check that access is what required ??
|
||
$cl 0, mu cr xarrowSquiggly(space)_new cl 0, mu cr$,
|
||
)
|
||
))
|
||
|
||
#align(center, prooftree(
|
||
vertical-spacing: 4pt,
|
||
rule(
|
||
name: [ new $\#$ value],
|
||
|
||
// TODO: check that access is what required ??
|
||
$cl \#, mu cr xarrowSquiggly(space)_new cl \#, mu cr$,
|
||
)
|
||
))
|
||
|
||
#align(center, prooftree(
|
||
vertical-spacing: 4pt,
|
||
rule(
|
||
name: [ new $bot$ value],
|
||
|
||
$cl bot, mu cr xarrowSquiggly(space)_new cl bot, mu cr$,
|
||
)
|
||
))
|
||
|
||
#align(center, prooftree(
|
||
vertical-spacing: 4pt,
|
||
rule(
|
||
name: [ new funciton pointer value],
|
||
|
||
$cl lambda overline(t) s, mu cr xarrowSquiggly(space)_new cl lambda overline(t) s, mu cr$,
|
||
)
|
||
))
|
||
|
||
#align(center, prooftree(
|
||
vertical-spacing: 4pt,
|
||
rule(
|
||
name: [ new reference ref value],
|
||
|
||
$cl rf Ref l, mu cr xarrowSquiggly(space)_new cl rf Ref l, mu cr$,
|
||
)
|
||
))
|
||
|
||
#align(center, prooftree(
|
||
vertical-spacing: 4pt,
|
||
rule(
|
||
name: [ new reference copy value],
|
||
|
||
$cl mu[l], mu cr xarrowSquiggly(space)_new cl v, mu_v cr$,
|
||
|
||
$cl mu_v cr xarrowSquiggly(v)_#[add] cl l', mu_a cr$,
|
||
|
||
$cl rf Copy l, mu cr xarrowSquiggly(space)_new cl rf Copy l', mu_a cr$,
|
||
)
|
||
))
|
||
|
||
#align(center, prooftree(
|
||
vertical-spacing: 4pt,
|
||
rule(
|
||
name: [ new tuple value],
|
||
|
||
$cl v_1, mu_0 cr xarrowSquiggly(space)_new cl lambda v'_1, mu_1 cr$,
|
||
$...$,
|
||
$cl v_n, mu_(n - 1) cr xarrowSquiggly(space)_new cl lambda v'_n, mu_n cr$,
|
||
|
||
$cl [v_1, ... v_n], mu_0 cr xarrowSquiggly(space)_new cl [v'_1, ... v'_n], mu_n cr$,
|
||
)
|
||
))
|
||
|
||
=== Value Update
|
||
|
||
#let write = `write`
|
||
|
||
*TODO: write to value*
|
||
|
||
=== Call Finalization
|
||
|
||
// FIXME: make connected to syntax
|
||
*TODO*
|
||
|
||
#align(center, prooftree(
|
||
vertical-spacing: 4pt,
|
||
rule(
|
||
name: [ spoil init],
|
||
$mu stretch(=>)^nothing_(cl sigma, mu cr) mu$,
|
||
)
|
||
))
|
||
|
||
#h(10pt)
|
||
|
||
#align(center, prooftree(
|
||
vertical-spacing: 4pt,
|
||
rule(
|
||
name: [ spoil step],
|
||
|
||
$mu stretch(=>)^args_sigma gamma$,
|
||
|
||
$isPossibleWrite mode$, // NOTE: weak requirement: may write
|
||
$not isCopy mode$,
|
||
$not isOut mode$,
|
||
|
||
$isCorrect_(cl sigma, mu cr) (mode, x)$,
|
||
|
||
// gamma - memory (as mu)
|
||
$gamma stretch(=>)^((mode, x) : args)_sigma pathenvval(gamma, sigma, x) <- bot]$
|
||
)
|
||
))
|
||
|
||
#h(10pt)
|
||
|
||
#align(center, prooftree(
|
||
vertical-spacing: 4pt,
|
||
rule(
|
||
name: [ fix step],
|
||
|
||
$mu stretch(=>)^args_sigma gamma$,
|
||
|
||
$isAlwaysWrite mode$, // NOTE: strong requirement: should write
|
||
$isOut mode$,
|
||
|
||
$isCorrect_(cl sigma, mu cr) (mode, x)$,
|
||
|
||
// gamma - memory (as mu)
|
||
$gamma stretch(=>)^((mode, x) : args)_sigma pathenvval(gamma, sigma, x) <- 0]$
|
||
)
|
||
))
|
||
|
||
#h(10pt)
|
||
|
||
#align(center, prooftree(
|
||
vertical-spacing: 4pt,
|
||
rule(
|
||
name: [ skip step],
|
||
|
||
$mu stretch(=>)^args_sigma gamma$,
|
||
|
||
$not "spoil step"$,
|
||
$not "fix step"$,
|
||
|
||
$isCorrect_(cl sigma, mu cr) (mode, x)$,
|
||
|
||
// mu
|
||
$gamma stretch(=>)^((mode, x) : args)_sigma gamma$
|
||
)
|
||
))
|
||
|
||
|
||
#h(10pt)
|
||
|
||
=== Function Evaluation
|
||
|
||
// FIXME: make connected to syntax
|
||
*TODO*
|
||
|
||
#align(center, prooftree(
|
||
vertical-spacing: 4pt,
|
||
rule(
|
||
name: [ $(lambda a : t. d) m$],
|
||
|
||
// TODO: verify that type of m is t ??
|
||
|
||
$cl sigma [a <- (m, t)], mu, l cr
|
||
xarrowSquiggly(t)
|
||
cl sigma', mu', l' cr$,
|
||
|
||
$cl sigma', mu', l' cr
|
||
xarrowDashed(d space @ space overline(y))
|
||
cl sigma'', mu'', l'' cr$,
|
||
|
||
$isRead mode$,
|
||
$not isCopy mode$,
|
||
|
||
// NOTE: correctness checked in CALL f
|
||
|
||
$cl sigma, mu, l cr
|
||
xarrowDashed((lambda a. d) space @ space x space overline(y))
|
||
cl sigma'', mu'', l'' cr$,
|
||
)
|
||
))
|
||
|
||
#h(10pt)
|
||
|
||
#align(center, prooftree(
|
||
vertical-spacing: 4pt,
|
||
rule(
|
||
name: [decl body],
|
||
|
||
$cl sigma, mu, l cr
|
||
attach(stretch(->)^overline(s), tr: *)
|
||
cl sigma', mu', l' cr$,
|
||
|
||
$d = overline(s)$,
|
||
|
||
$cl sigma, mu, l cr
|
||
xarrowDashed(d space @)
|
||
cl sigma', mu', l' cr$,
|
||
)
|
||
))
|
||
|
||
#h(10pt)
|
||
|
||
=== Statement Evaluation
|
||
|
||
// FIXME: make connected to syntax
|
||
*TODO: check type of lambda?, args from type?*
|
||
#align(center, prooftree(
|
||
vertical-spacing: 4pt,
|
||
rule(
|
||
name: [ CALL $f space overline(p)$],
|
||
|
||
$cl [], mu, l cr
|
||
xarrowDashed(f space @ space overline(p))
|
||
cl sigma', mu', l' cr$,
|
||
|
||
// TODO: FIXME define args in some way
|
||
$mu attach(stretch(=>)^args_sigma, tr: *) gamma$,
|
||
|
||
$cl sigma, mu, l cr
|
||
xarrow("CALL" f space overline(p))
|
||
cl sigma, gamma, l cr$,
|
||
)
|
||
))
|
||
|
||
#h(10pt)
|
||
|
||
#align(center, prooftree(
|
||
vertical-spacing: 4pt,
|
||
rule(
|
||
name: [ READ $p$],
|
||
|
||
$mu, sigma teval p eqmu 0$,
|
||
|
||
$cl sigma, mu, l cr
|
||
xarrow("READ" p)
|
||
cl sigma, mu, l cr$,
|
||
)
|
||
))
|
||
|
||
#h(10pt)
|
||
|
||
#align(center, prooftree(
|
||
vertical-spacing: 4pt,
|
||
rule(
|
||
name: [ WRITE $x$],
|
||
|
||
$sigma temode p -> cr r space w cl$,
|
||
|
||
$w == MaybeWrite or w == AlwaysWrite$,
|
||
|
||
$x tpath p$,
|
||
|
||
$mu[x] xarrowSquiggly(p)_write v$,
|
||
|
||
$cl sigma, mu cr
|
||
xarrow("WRITE" p)
|
||
cl sigma, mu[x <- v] cr$,
|
||
)
|
||
))
|
||
|
||
#h(10pt)
|
||
|
||
#align(center, prooftree(
|
||
vertical-spacing: 4pt,
|
||
rule(
|
||
name: [ $s \; t$],
|
||
|
||
$cl sigma, mu cr
|
||
stretch(->)^s
|
||
cl sigma_s, mu_s cr$,
|
||
|
||
$cl sigma, mu cr
|
||
stretch(->)^t
|
||
cl sigma_t, mu_t cr$,
|
||
|
||
$cl sigma, mu, cr
|
||
xarrow(s \; t)
|
||
cl sigma_t, mu_t cr$,
|
||
)
|
||
))
|
||
|
||
#h(10pt)
|
||
|
||
#let combine = `combine`
|
||
|
||
*TODO: combine replacement* // FIXME
|
||
#align(center, prooftree(
|
||
vertical-spacing: 4pt,
|
||
rule(
|
||
name: [ $s | t$],
|
||
|
||
$cl sigma, mu, l cr
|
||
stretch(->)^s
|
||
cl sigma_s, mu_s, l_s cr$,
|
||
|
||
$cl sigma, mu, l cr
|
||
stretch(->)^t
|
||
cl sigma_t, mu_t, l_t cr$,
|
||
|
||
$sigma_s = sigma_t$,
|
||
$mu' = combine(mu_s, mu_t)$,
|
||
|
||
// TODO changes ?? two ways ??
|
||
$cl sigma, mu cr
|
||
xarrow(s | t)
|
||
cl sigma_t, mu' cr$,
|
||
)
|
||
))
|
||
|
||
#h(10pt)
|
||
|
||
=== Combination
|
||
|
||
*TODO: rewrite as rules, fix* // FIXME
|
||
|
||
$ combine(mu_1, mu_2)[i] = combine_e (mu_1[i], mu_2[i]) $
|
||
$ combine_e (bot, bot) = bot $
|
||
$ combine_e (0, 0) = 0 $
|
||
$ combine_e (\_, \_) = \# $
|
||
// FIXME: ref to combined memory
|
||
$ combine_e (rf c l_1, rf c' l_2) | c == c' = rf c combine_e(mu_1[l_1], mu_2[l_2])$
|
||
$ combine_e (v^1_1, ... v^1_n, v^2_1 ... v^2_n) = [combine_e(v^1_1, v^2_1), ..., combine_e(v^1_n, v^2_n)]$
|
||
$ combine_e (lambda space overline(t_1) space s_1, lambda space overline(t_2) space s_2) | overline(t_1) == overline(t_2) = lambda space overline(t_1) space s_1 $
|
||
// TODO: FIXME: statemient in lambda is not important ??
|
||
|
||
]
|