n improving the reliability of mathematical results. Several Automath system have been
非同期通信するプログラムを形式的証明から抽出してバグを防ぐ研究
ementedand used to formalize mathematics. We will discuss some crucial aspects of theems that had an influence on other systems.

受付番号6978 平井洋一

Proofsas objects,
formulas as types:
In the Automathsystems the idea of treating
既存プログラム抽出
History,ideas
and future9
マルチコア時代の非同期プログラム抽出[PPL2009]

fs as first class objects in a formal language, at the same footing as other terms, occurs
中間論理
sed firstaccurately,in thesensethat
linguisticcorrectness
非同期通信用の直観主義知識論理
直観主義論理hetime. Inlogic,
thisidea
is known
as the implies
Curry–Howard formulas-as-types isohis language should be computer checkable and [NASSLLI
it should be help2010 (Howardstudent session]
phism,
for the first time written up in 1968
by Howard1980), going Kripkeモデルの全順序性
back to構成主義
ity of mathematical results. Several Automath system have been(ϕ ⊃ ψ) ∨ (ψ ⊃ ϕ)smalizeof Currywho
hadnoticed
thatthetypesof
thecombinators
areexactly
theaxioms
Brouwer-Heytingmathematics. We will discuss some crucial aspects of theilbert
stylededuction. De Bruijn reinvented
the idea, emphasizing the proofs-as-objects
Kolmogorov解釈
知識中間論理のKripkeモデルとe on othersystems.
非同期知識様相を加えて
ct, whichcomes down to the following: There is an isomorphism T between formulas
Kripkeモデル
1. (Kx ϕ ⊃ Ky ψ) ∨ (Ky ψ ⊃ Kx ϕ)
共有メモリの実行を比較
mulas
as types:
the Automath
the萩typesof theirInproofs
giving systems
rise to the idea of treating
persistency
in a formal
language, at the same footing as other terms, occurs 修士論文・研究科長賞
2. (Kx ϕ ⊃ Kx ψ) ∨ (Kx ψ ⊃ Kx ϕ)
谷[LPAR-16 (LNAI
his idea is known as the Curry–Howard formulas-as-typesiso- 6355, Springer)]! !logic ϕ if and only if ! !type theory M : T (ϕ),
3. (Ka Kx ϕ ⊃ Ka Ky ψ) ∨ (Ka Ky ψ ⊃ Ka Kx ϕ)e written
up in 1968 by Howard (Howard 1980),
going back[FoSSaCS
to有限モデル性2011 投稿済]
研
ticed that the types of the combinators are exactly the axioms
re Bruijn
M isCurry-Howard同型
areinvented
direct encoding(as a λ-term)of the deduction of ϕ from !. In logic, ! justCurry-Howard同型De
the idea, emphasizing
the proofs-as-objects! alsoformulas
contains the declarations x : A of theains
assumptions,in type theory,
to thethefollowing:
There is but
an isomorphism
T between
多くの既存手法
1.sequential consistency
共有メモリ付λ計算
giving rise
tovariables
occurringintheformulas.
Theformulas-as-types
correspondence
goes even
型付λ計算
separation logic
2. cache consistency
y : T (ψ) (we assume a hypothetical ‘proof’ y ofher: assumptions in ! are of the form [Baaz,03]の演繹体系を利用
fractional typeonly if ! !type theory M : T (ϕ),
3. processor consistency
さらに部分的な同期の導入(y is a name for thend
proven lemmas are definitions recorded
in ! as y := p : T (ψ)線形λ計算
共有メモリの整合性
fngp(asofaψ).
λ-term) of the deduction of ϕ from !. In logic, ! justFastTrack＝実行の全順序性
nutinteresting
consequenceof thisanalogy is xthat
typechecking’.
So,! also contains
the declarations: A‘proofof the checking =in type theory,
Goldilocks
非同期プログラム抽出器
高階化
theformulas.
Theformulas-as-types
correspondence
goeseven型理論
pe checking
algorithm suffices to satisfy the De Bruin criterion of 非決定性を排除
the previous section. [Steinke, 2004]
e of the証明=プログラム
formthey : type
T (ψ)theory,(we assume
hypotheticaly ofdifficult. The original Automath
非決定性も利用可
endingon
this acanbemore‘proof’or lessname for thedefinitions recorded in ! as y := p : T (ψ) バグが無く共有メモリ型
ems
hadプログラム抽出
a small kernel, so for those it(y isis arathersimple.
Later developments based on
非決定性
same
idea
areNuPrl,theis systems(Harperet alchecking’.
1993, Twelf
ce of thisanalogy
that...
‘proofLF
checking=高速である
type
So, (Twelf), Lego (Luo & Pollack
Coq,分散計算2),
Alf to(Magnusson&Bruin
NordströmAgda(Agda),et al 1986) and
非決定性
並列ソフト
[平井,山本ufficessatisfy the投稿済JFP]
De
criterion1994),of theprevious
section.NuPrl (Constable論
理計
算
応
用

対応を発見