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simplest_model_with_mods: first draft of the formal semantics
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396
simplest_model_with_mods/model.typ
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396
simplest_model_with_mods/model.typ
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// #import "@preview/polylux:0.4.0": *
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#import "@preview/simplebnf:0.1.1": *
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// #import "@preview/zebraw:0.5.0": *
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// #show: zebraw
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#import "@preview/curryst:0.6.0": rule, prooftree, rule-set
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#import "@preview/xarrow:0.4.0": xarrow, xarrowDashed
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= Формальная модель используемого языка
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*TODO: проблемы с добавлением if в будущем: как записать write and not write ?*
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== Синтаксис
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#h(10pt)
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#let isCorrect = `isCorrect`
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#let isRead = `isRead`
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#let isWrite = `isWrite`
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#let isRef = `isRef`
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#let isCopy = `isCopy`
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#let isIn = `isIn`
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#let isOut = `isOut`
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#let tag = `tag`
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#let value = `value`
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#let stmt = `stmt`
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#let decl = `decl`
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#let prog = `prog`
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#bnf(
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Prod(`read`,
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{ Or[Read][]
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Or[Not Read][] } ),
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Prod(`write`,
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{ Or[Write][]
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Or[Not Write][] } ),
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Prod(`copy`,
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{ Or[Ref][]
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Or[Value][] } ),
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Prod(`in`,
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{ Or[In][]
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Or[Not In][] } ),
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Prod(`out`,
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{ Or[Out][]
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Or[Not Out][] } ),
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Prod(
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`tag`,
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{
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Or[`read` #h(3pt) `write` #h(3pt) `copy` #h(3pt) `in` #h(3pt) `out`][]
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}
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),
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Prod(
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`value`,
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{
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Or[$0$][cell with some value]
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Or[$bot$][spoiled cell]
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}
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),
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// Prod(
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// `arg`,
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// {
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// Or[$0$][new value, no associated variable]
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// Or[$ amp d$][value from some variable]
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// }
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// ),
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Prod(
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`stmt`,
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{
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Or[`CALL` $f space overline(x)$][call function by id]
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Or[`WRITE` $x$][write to variable]
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Or[`READ` $x$][read from variable]
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}
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),
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Prod(
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`decl`,
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{
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Or[ovreline(stmt)][function body]
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Or[$lambda #[`tag`] a.$ `decl`][with argument pass strategy annotation]
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}
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),
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Prod(
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`prog`,
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{
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Or[`decl`][main function]
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Or[`decl` `prog`][with supplimentary funcitons]
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}
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),
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)
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== Семантика статического интерпретатора
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*TODO: исправить с обычного интерпретатора*
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#h(10pt)
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$V := value$ - значения памяти
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$L := NN$ - позиции в памяти
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$X$ - можество переменных
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*TODO: специфицировать доступ*
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*TODO: формально описать isCorrect для tag-а*
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*TODO: формально описать accessor-ы tag*
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$sigma : X -> tag times L$ - #[ позиции памяти, соответстующие переменным контекста,
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частично определённая функция ]
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$mu : NN -> V$ - память, частично определённая функция
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$l in NN$ - длина используемого фрагмента памяти
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$DD : NN -> decl$ - определения функций, частично определённая функция
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$d in decl, s in stmt, f in NN, x in X, a in NN$
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$d space @ space overline(x)$ - запись применения функции (вида #decl) к аргументам
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#let args = `args`
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#[
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#let ref = `ref`
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#let copy = `copy`
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#let read = `read`
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#let cl = $chevron.l$
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#let cr = $chevron.r$
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// #align(center, grid(
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// columns: 3,
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// gutter: 5%,
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// align(bottom, prooftree(
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// ...
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// )),
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// align(bottom, prooftree(
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// ...
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// )),
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// align(bottom, prooftree(
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// ...
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// )),
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// ))
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// TODO: introduce spep env argument ??
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#align(center, prooftree(
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vertical-spacing: 4pt,
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rule(
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name: [ is correct],
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$isOut tag -> isWrite tag$, // NOTE; strong requirment should write
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$isRead tag -> isIn tag$,
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$isWrite tag -> isWrite sigma(x)$, // NOTE: may tag => should sigma(x)
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$isRead tag -> mu (sigma(x)) != bot$, // NOTE: may tag -> ...
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$isCorrect_(cl sigma, mu cr) (tag, x)$,
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)
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))
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#h(10pt)
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#align(center, prooftree(
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vertical-spacing: 4pt,
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rule(
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name: [ spoil init],
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$mu stretch(=>)^nothing_(cl sigma, mu cr) mu$,
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)
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))
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#h(10pt)
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#align(center, prooftree(
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vertical-spacing: 4pt,
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rule(
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name: [ spoil step],
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$mu stretch(=>)^args_sigma gamma$,
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$isWrite tag$, // NOTE: weak requirement: may write
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$not isCopy tag$,
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$not isOut tag$,
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$isCorrect_(cl sigma, mu cr) (tag, x)$,
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// mu
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$gamma stretch(=>)^((tag, x) : args)_sigma gamma[sigma(x) <- bot]$
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)
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))
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#h(10pt)
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#align(center, prooftree(
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vertical-spacing: 4pt,
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rule(
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name: [ fix step],
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$mu stretch(=>)^args_sigma gamma$,
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$isWrite tag$, // NOTE: strong requirement: should write
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$isOut tag$,
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$isCorrect_(cl sigma, mu cr) (tag, x)$,
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// mu
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$gamma stretch(=>)^((tag, x) : args)_sigma gamma[sigma(x) <- 0]$
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)
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))
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#h(10pt)
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#align(center, prooftree(
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vertical-spacing: 4pt,
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rule(
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name: [ skip step],
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$mu stretch(=>)^args_sigma gamma$,
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$not "spoil step"$,
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$not "fix step"$,
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$isCorrect_(cl sigma, mu cr) (tag, x)$,
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// mu
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$gamma stretch(=>)^((tag, x) : args)_sigma gamma$
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)
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))
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#h(10pt)
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#align(center, line())
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#h(10pt)
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#align(center, prooftree(
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vertical-spacing: 4pt,
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rule(
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name: [ $(lambda tag a. d) x, ref + read$],
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$cl sigma, mu, l cr
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xarrowDashed(d space @ space overline(y))
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cl sigma, mu', l' cr$,
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$isRead tag$,
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$not isCopy tag$,
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// NOTE: correctness checked in CALL f
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$cl sigma, mu, l cr
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xarrowDashed((lambda tag a. d) space @ space x space overline(y))
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cl sigma, mu', l' cr$,
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)
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))
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#h(10pt)
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#align(center, prooftree(
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vertical-spacing: 4pt,
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rule(
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name: [ $(lambda tag a. d) x, ref - read$],
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$cl sigma, mu [sigma(x) <- bot], l cr
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xarrowDashed(d space @ space overline(y))
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cl sigma, mu', l' cr$,
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$not isRead tag$,
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$not isCopy tag$,
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// NOTE: correctness checked in CALL f
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$cl sigma, mu, l cr
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xarrowDashed((lambda tag a. d) space @ space x space overline(y))
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cl sigma, mu', l' cr$,
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)
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))
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#h(10pt)
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#align(center, prooftree(
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vertical-spacing: 4pt,
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rule(
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name: [ $(lambda tag a. d) x, copy + read$],
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$cl sigma [a <- l], mu [l <- 0], l + 1 cr
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xarrowDashed(d space @ space overline(y))
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cl sigma', mu', l' cr$,
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$isRead tag$,
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$isCopy tag$,
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// NOTE: correctness checked in CALL f
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$cl sigma, mu, l cr
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xarrowDashed((lambda tag a. d) space @ space x space overline(y))
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cl sigma', mu', l' cr$,
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)
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))
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#h(10pt)
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#align(center, prooftree(
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vertical-spacing: 4pt,
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rule(
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name: [ $(lambda tag a. d) x, copy - read$],
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$cl sigma [a <- l], mu [l <- bot], l + 1 cr
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xarrowDashed(d space @ space overline(y))
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cl sigma', mu', l' cr$,
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$not isRead tag$,
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$isCopy tag$,
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// NOTE: correctness checked in CALL f
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$cl sigma, mu, l cr
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xarrowDashed((lambda tag a. d) space @ space x space overline(y))
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cl sigma', mu', l' cr$,
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)
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))
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#h(10pt)
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#align(center, prooftree(
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vertical-spacing: 4pt,
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rule(
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name: [decl body],
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$cl sigma, mu, l cr
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attach(stretch(->)^overline(s), tr: *)
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cl sigma', mu', l' cr$,
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$d = overline(s)$,
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$cl sigma, mu, l cr
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xarrowDashed(d space @)
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cl sigma', mu', l' cr$,
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)
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))
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#h(10pt)
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#align(center, line())
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#h(10pt)
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#align(center, prooftree(
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vertical-spacing: 4pt,
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rule(
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name: [ CALL $f space overline(x)$],
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$cl [], mu, l cr
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xarrowDashed(d space @ space overline(x))
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cl sigma', mu', l' cr$,
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// TODO: FIXME define args in some way
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$mu attach(stretch(=>)^args_sigma, tr: *) gamma$,
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$DD(f) := d$,
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$cl sigma, mu, l cr
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xarrow("CALL" f space overline(x))
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cl sigma, gamma, l cr$,
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)
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))
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#h(10pt)
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#align(center, prooftree(
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vertical-spacing: 4pt,
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rule(
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name: [ READ $x$],
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$mu(sigma(x)) != bot$,
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$cl sigma, mu, l cr
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xarrow("READ" x)
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cl sigma, mu, l cr$,
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)
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))
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#h(10pt)
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#align(center, prooftree(
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vertical-spacing: 4pt,
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rule(
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name: [ WRITE $x$],
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$isWrite sigma(x)$,
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$cl sigma, mu, l cr
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xarrow("WRITE" x)
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cl sigma, mu[x <- 0], l union {sigma(x)} cr$,
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)
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))
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]
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