Mercurial > hg > Papers > 2015 > kono-lola
changeset 2:2e752ec70578
fix
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Mon, 11 May 2015 21:31:34 +0900 |
| parents | 755cf49d4bcc |
| children | 259facc91c65 |
| files | fig/Duality.graffle fig/Duality.svg fig/IdDataGear.graffle fig/IdDataGear.svg fig/codeGear.graffle fig/codeGear.svg fig/dataGear.graffle fig/dataGear.svg fig/dataGearComposition.graffle fig/dataGearComposition.svg gears.ind |
| diffstat | 11 files changed, 189 insertions(+), 16 deletions(-) [+] |
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--- a/gears.ind Mon May 11 17:12:14 2015 +0900 +++ b/gears.ind Mon May 11 21:31:34 2015 +0900 @@ -5,19 +5,192 @@ --Reliable computation and predictablity Various softwares are used in real world. Each of them have to work in a reliable way. A peice of devie contains -millons of lines of code. These programs are written in C, Haskell and so on. The unit of computation of -conventional programming language is of course a function call. -To assure its reliability, the computation of a function should be predictable, but unfortunately -a function may call another functions and it comes back to the original function. -Proof theory such as Hoare Logic or model checking method is used to make the prediction, such as -zero divide is never occured in a function. +millons of lines of code. These programs are written in C, Haskell and so on. +To assure its reliability, the computation of a function should be predictable. +The correctness of the predecion should be assured by measurements, model checkings or proofs. + +% The unit of computation of conventional programming language is of course a function call. +% a function may call another functions and it comes back to the original function. +% Proof theory such as Hoare Logic or model checking method is used to make the prediction, such as +% zero divide is never occured in a function. We propose new unit of computation, data segments and code segments. Computations in these segments are finite and -predictable. We sometimes call these gears. A set of gears make programming system. This unit is larger then -an machine instruction and smaller than a function call. Data segments and code segments are connected by -meta data segments and meta code segments. The idea is construct system as a set of predictable unit of -computation. +predictable. We sometimes call these gears. + +A set of gears makes a programming system. + +% This unit is larger then an machine instruction and smaller than a function call. + +Data segments and code segments are connected by meta data segments and meta code segments. +The idea is construct system as a set of predictable unit of computation. --Definition of data segment and code segment -We implment our Gears language in LLVM\cite{LLVM}, +Actually we implment our Gears language in LLVM\cite{LLVM}, but we can think both code and data are System F\cite{} term. As usuall, Types are +defined starting from type variables X,Y,Z and is generated by two operations: + + 1. if U and V are types, then U → V is a type. + 2. if V i a type, and X a type variable, then ΠX.V is a type. + +Terms are constructed by six schemes + + 1. variable: $x^T, y^T, z^T,...$ of type $T$, + 2. application: $tu$ ov type $V$, where $t$ is of type $U→V$ and $u$ is of type $U$, + 3. λ-abstraction: $λx^U.v$ of type $U→V$, where $x^U$ is a variable of type $U$ and $v$ is of type $V$, + 4. universal abstraction: if $v$ is a term type V, then we can form $ΛX.v$ of type $ΠX.V$, so long as the variable $X$ is not free in the type of a free variable of $v$. + 5. universal application: if $t$ is a term of type $ΠX.V$ and $U$ is a type, then %tU$ is a tem of type $V[U/X]$. + +and usuall conversions, + + 1. (λx.v)u 〜> v[u/x] + 2. (ΛX.v)U 〜> v[U/X] + +Code segments of type $C$ accepts data segments of type $I$ and generates data segments of type $O$. + + C : I → O + +Domain of C is I and codemain of C is O. + +--Meta computation of Gears + +Computation of a code is limited in the inputs and the outputs and it makes the compuation of the code predictable, but its data are +ususally connected to other data. The code has its continuation also. These connections are out of the scope of the code. +We think these connections are made by a meta computation, such as monads\cite{}. A monad is a data structure with monad laws and +after an execution of a code, monad's join code is called to handle meta data structure. With monad $T$, + + C' : I → T O + +is a meta compuation. $C$ has one to one correcpondence with $C'$. + +Parallel execution or IO handling are represented as a monad in our scheme. + +--A cateogy of codes + +Types of code segments and data segments narually compose a category of function and types. Codes and data are interconnected one by one. + + Object : f, g, h ... + Arrows : a, b, c ... + +An arrow has its domain object and codomain object. + +There is compositon of arrows, + + a o b + +and it satisfies the compisition law. + + (a o b) o c = a o (b o c ) + +It also have identity arrow $id f$. + + a o id (domain a) = a = id (domain a) o a + +--A cateogy of data + +Is there any duality in codes and data? Ususall answer is no, since we cannot simply combine data segments and code segments cannot be objects +of a category. Here we show a trivial extension of data segments which makes data of category from category of function and types. + +The problem of data segments compisition is that it forgets about later computation. We can simply store it as a continuation in the +data segment. Data segments now have a continuation, which is a code segment. It is a part of meta computation. + +Data category C is constructed from a category C. Objects of Data category C are arrows of category C. An arrow of Data category C from $a$ to $b$ is +a data segment $f$ with continuation $n$, $a'$ is a intermidiate data segment. + + data(f,n) + f : codomain a → a' + n : a' → domain b + +If $f$ and $g$ has same codomain $b$, equality of $data(f,n)$ and $data(g,m)$ is defined as follows, + + data(f,n) = data(g,m) iff b o n = b o m + +where $b$ is a codomain of both $data(f,n)$ and $data(g,m)$. + +To make an identity arrow in the Data category, if $a$ is a codomain of $f$, we need a reverse arrow of $a$, $a'$. + + data(f,(codomain f)') + +Compisition of arrows of Data category is defined as follows. + + data(f,n) ∙ data(g,m) = data(f,n o g o m) + +It is easy to see its composition lows. + + data(f,n) ∙ ( data(g,m) ∙ data(g,m) ) = (data(f,n) ∙ data(g,m) ) ∙ data(g,m) + +To check $data(f,a')$ is an identity, + + data(f,(codomain f)') ∙ data(g,n) = data(g,n) + +it is true if and only if + + codmain f o ( (codmain f)' o g ) o n = g o n + +since $(codmain f)'$ is a reverse of $codomain f$, it holds. Right identity law holds the same way. + + +--A trivial duality of code and data + +Data category is a data segment with continuation, which is one step behind, so there is a trivial one to one corespondence. +It is also easy to show C and Data category C is an adjuntion pair. + +Data segments in a Data category has an arrow in an original category C, so its corespondence is trivial. Acturally, we have a map F, + + F : Obj C → Obj (DataCategory C) + F f = data(f, id f) + +It has identity continuation, behave as a data segemtn with no continuation. Functor U : (DataCategory C) C is defined as follows + + U f = codomain f + U data(f,n) = b o n + +wehre $b$ is a codomain of $data(f,n)$. +Object $f$ in Data Category C is an allow of C and it has a codomain. Arrow $data(f,n)$ in Data Category C has +original arrow $f$ and continuation $n$. + +With a mapping transformation $η$, + + η : Obj C → Arrow C + η f = id f + +We can define a solution of universal mapping problem, + + * f = b' o f + +and + + U ( *f ) o (η a) = f + +is directry checke and $*f$ is unique, that is + + if there is an arrow g $U g o (η a) = f$ then $g = *f$. + +This is also trivial. + +--Comparison + +Meta compuation in Haskell\cite{} is defined as set of explict monads. In Gears, meta computation is unique among the system, +which is something like operating system kernel or libraries. + +In Open CL\cite{} or Cuda\cite{}, there is a code segment which is called kernel. The kernel runs parallelly in a GPU. The kernel +is very similar to our code segment. + +Object oriented languages, such as Smalltalk-80 or Java , has meta compuation as a virtual machines. It is very different from +monad or meta computation of our Gears system. Data segment has no idenity such as Smalltalk's self, it can be copied easily, +which is a very good property in a parallel compuation environment. + +Code segment can be seen as a kind of Typed assembly language, which has typed input and predictable behaviour. + +In old age, there is a software design method called data flow diagram. Category of code and data is very much like +data flow diagram approach. It can be seen as a rivival of main frame technologies. + + +--Conclusion + +As usual categorrical result, trivial duality of category and Data category means nothing itself. +During the design of gears system, we see similarities of code segment and data segment. It has +meta segments and contnuations. We think the duality gives us some guidance to design software +system such as an operating system or libraries. + +We have implemented Gears system on top of LLVM\cite{} and hope it can be used as real sysetm description language. +