Functional Java - mimuw.edu.plczarnik/sliwowica/... · Outline Java JavaFun Semantical properties Type system Towards soundness proof Motivations I Problems with applications in Java:

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JavaFunSemantical properties

Type systemTowards soundness proof

Functional Java

Aleksy Schubert and Jacek ChrząszczInstitute of InformaticsWarsaw Universityul. Banacha 202–097 WarsawPoland

November 24, 2008

Aleksy Schubert and Jacek Chrząszcz Institute of Informatics Warsaw University ul. Banacha 2 02–097 Warsaw PolandFunctional Java

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Java

JavaFunSyntax of JavaFunSemantics of JavaFun

Semantical propertiesImmutable objectsObservable contextsRepresentationFunctional methodLocal-state method

Type system

Towards soundness proof


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Motivations

I Problems with applications in Java:

I state scattered through the programI unpredictable state changesI frequent state inconsistencies

I

A

Solution:

I state changes done in a small fragment of the applicationI the rest of the application must be “functional”

— impractical

I but functional programs can internally use imperative features

— practical

I We want to prove that these parts are “functional”


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Motivations

I Problems with applications in Java:I state scattered through the program

I unpredictable state changesI frequent state inconsistencies

I

A

Solution:


— impractical


— practical



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Motivations

I Problems with applications in Java:I state scattered through the programI unpredictable state changes

I frequent state inconsistenciesI

A

Solution:


— impractical


— practical



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Motivations

I Problems with applications in Java:I state scattered through the programI unpredictable state changesI frequent state inconsistencies

I

A

Solution:


— impractical


— practical



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Motivations


I

A

Solution:


— impractical


— practical



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Motivations


I

A

Solution:I state changes done in a small fragment of the application

I the rest of the application must be “functional”

— impractical


— practical



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Motivations


I

A

Solution:I state changes done in a small fragment of the applicationI the rest of the application must be “functional”

— impracticalI but functional programs can internally use imperative features

— practical



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Motivations


I

A

Solution:I state changes done in a small fragment of the applicationI the rest of the application must be “functional” — impractical


— practical



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Motivations


I

A

Solution:I state changes done in a small fragment of the applicationI the rest of the application must be “functional” — impracticalI but functional programs can internally use imperative features

— practical



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Motivations


I

A

Solution:I state changes done in a small fragment of the applicationI the rest of the application must be “functional” — impracticalI but functional programs can internally use imperative features— practical



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Motivations


I A Solution:I state changes done in a small fragment of the applicationI the rest of the application must be “functional” — impracticalI but functional programs can internally use imperative features— practical



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Motivations


I A Solution:I state changes done in a small fragment of the applicationI the rest of the application must be “functional” — impracticalI but functional programs can internally use imperative features— practical



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Java

I Huge industrial language

I Requires modellingI We model

I classes and inheritanceI protected fieldsI named public constructorsI named public methodsI control flow (including recursion, but no loops)I exceptions


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Java

I Huge industrial languageI Requires modelling

I We model



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Java

I Huge industrial languageI Requires modellingI We model



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Java


I classes and inheritance

I protected fieldsI named public constructorsI named public methodsI control flow (including recursion, but no loops)I exceptions


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Java


I classes and inheritanceI protected fields

I named public constructorsI named public methodsI control flow (including recursion, but no loops)I exceptions


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Java


I classes and inheritanceI protected fieldsI named public constructors

I named public methodsI control flow (including recursion, but no loops)I exceptions


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Java


I classes and inheritanceI protected fieldsI named public constructorsI named public methods

I control flow (including recursion, but no loops)I exceptions


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Java


I classes and inheritanceI protected fieldsI named public constructorsI named public methodsI control flow (including recursion, but no loops)

I exceptions


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Java




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Java

I Huge industrial languageI Requires modelling of sublanguagesI We do not model

I packagesI interfacesI visibility featuresI static membersI dynamic loadingI multithreading


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Java


I packages

I interfacesI visibility featuresI static membersI dynamic loadingI multithreading


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Java


I packagesI interfaces

I visibility featuresI static membersI dynamic loadingI multithreading


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Java


I packagesI interfacesI visibility features

I static membersI dynamic loadingI multithreading


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Java


I packagesI interfacesI visibility featuresI static members

I dynamic loadingI multithreading


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Java


I packagesI interfacesI visibility featuresI static membersI dynamic loading

I multithreading


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Java


I packagesI interfacesI visibility featuresI static membersI dynamic loadingI multithreading


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Syntax of JavaFunSemantics of JavaFun

JavaFun - Goal

I We want a formal definition of “functional”

I of “local-state”I both syntactic — type system, and semantic — with heaps,locations and values

I and a proof that our type system yields “local-state” and“functional” methods


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JavaFun - Goal

I We want a formal definition of “functional”I of “local-state”

I both syntactic — type system, and semantic — with heaps,locations and values



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JavaFun - Goal

I We want a formal definition of “functional”I of “local-state”I both syntactic — type system, and semantic — with heaps,locations and values



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JavaFun - Goal

I We want a formal definition of “functional”I of “local-state”I both syntactic — type system, and semantic — with heaps,locations and values



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JavaFun - a bit of “history”

I Genrap developed in Comarch R&D Center (AGS and friends)

I ad hoc syntactic rulesI a checker which enforced these rules in the build processI no proofsI Kotek project Comarch-UW-EU (SPO WKP)I start of formalization (syntactic rules −→ a type system),semantics

I now: (beginning of) proof


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I Genrap developed in Comarch R&D Center (AGS and friends)I ad hoc syntactic rules

I a checker which enforced these rules in the build processI no proofsI Kotek project Comarch-UW-EU (SPO WKP)I start of formalization (syntactic rules −→ a type system),semantics



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I Genrap developed in Comarch R&D Center (AGS and friends)I ad hoc syntactic rulesI a checker which enforced these rules in the build process

I no proofsI Kotek project Comarch-UW-EU (SPO WKP)I start of formalization (syntactic rules −→ a type system),semantics



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I Genrap developed in Comarch R&D Center (AGS and friends)I ad hoc syntactic rulesI a checker which enforced these rules in the build processI no proofs

I Kotek project Comarch-UW-EU (SPO WKP)I start of formalization (syntactic rules −→ a type system),semantics



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I Genrap developed in Comarch R&D Center (AGS and friends)I ad hoc syntactic rulesI a checker which enforced these rules in the build processI no proofsI Kotek project Comarch-UW-EU (SPO WKP)

I start of formalization (syntactic rules −→ a type system),semantics



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I Genrap developed in Comarch R&D Center (AGS and friends)I ad hoc syntactic rulesI a checker which enforced these rules in the build processI no proofsI Kotek project Comarch-UW-EU (SPO WKP)I start of formalization (syntactic rules −→ a type system),semantics



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I Genrap developed in Comarch R&D Center (AGS and friends)I ad hoc syntactic rulesI a checker which enforced these rules in the build processI no proofsI Kotek project Comarch-UW-EU (SPO WKP)I start of formalization (syntactic rules −→ a type system),semantics



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JavaFun

I Fragment of Java, including:



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JavaFun

I Fragment of Java, including:I classes and inheritance

I protected fieldsI named public constructorsI named public methodsI control flow (including recursion, but no loops)I exceptions


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JavaFun

I Fragment of Java, including:I classes and inheritanceI protected fields

I named public constructorsI named public methodsI control flow (including recursion, but no loops)I exceptions


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JavaFun

I Fragment of Java, including:I classes and inheritanceI protected fieldsI named public constructors

I named public methodsI control flow (including recursion, but no loops)I exceptions


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JavaFun

I Fragment of Java, including:I classes and inheritanceI protected fieldsI named public constructorsI named public methods

I control flow (including recursion, but no loops)I exceptions


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JavaFun

I Fragment of Java, including:I classes and inheritanceI protected fieldsI named public constructorsI named public methodsI control flow (including recursion, but no loops)

I exceptions


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JavaFun

I Fragment of Java, including:I classes and inheritanceI protected fieldsI named public constructorsI named public methodsI control flow (including recursion, but no loops)I exceptions


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Grammar of JavaFun

C ::= cdecl+

cdecl ::= class [cmod] C1 [ext C2] {F K M}cmod ::= immutable | funcmmod ::= lstate | funckmod ::= lstatefmod ::= repC ::= . . . class name . . .

F ::= [F ; ]∗

F ::= C [fmod] xK ::= [K ; ]∗

K ::= [kmod] k(args) [throws Exc] {E}args ::= ε | C x | C x , argsExc ::= C | C ,Exc


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Grammar of JavaFun

C ::= cdecl+

cdecl ::= class [cmod] C1 [ext C2] {F K M}

cmod ::= immutable | funcmmod ::= lstate | funckmod ::= lstatefmod ::= repC ::= . . . class name . . .

F ::= [F ; ]∗

F ::= C [fmod] xK ::= [K ; ]∗



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Grammar of JavaFun

C ::= cdecl+

cdecl ::= class [cmod] C1 [ext C2] {F K M}cmod ::= immutable | funcmmod ::= lstate | funckmod ::= lstatefmod ::= rep

C ::= . . . class name . . .

F ::= [F ; ]∗

F ::= C [fmod] xK ::= [K ; ]∗



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Grammar of JavaFun

C ::= cdecl+

cdecl ::= class [cmod] C1 [ext C2] {F K M}cmod ::= immutable | funcmmod ::= lstate | funckmod ::= lstatefmod ::= repC ::= . . . class name . . .

F ::= [F ; ]∗

F ::= C [fmod] xK ::= [K ; ]∗



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Grammar of JavaFun

M ::= [M; ]∗

M ::= C [mmod] m(args) [throws Exc] {E}fieldref ::= x1.x2 | this.xvarref ::= x | this | fieldrefx ::= ε | x | x , xE ::= new C .k(x) |

let C x = E1 in E2 |if E1 then E2 else E3 |x .m(x) |fieldref = E |varref |throw x |try {E1} catch (C x) {E2} |null


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Grammar of JavaFun

M ::= [M; ]∗

M ::= C [mmod] m(args) [throws Exc] {E}

fieldref ::= x1.x2 | this.xvarref ::= x | this | fieldrefx ::= ε | x | x , xE ::= new C .k(x) |



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Grammar of JavaFun

M ::= [M; ]∗

M ::= C [mmod] m(args) [throws Exc] {E}fieldref ::= x1.x2 | this.xvarref ::= x | this | fieldrefx ::= ε | x | x , x

E ::= new C .k(x) |let C x = E1 in E2 |if E1 then E2 else E3 |x .m(x) |fieldref = E |varref |throw x |try {E1} catch (C x) {E2} |null


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Grammar of JavaFun

M ::= [M; ]∗




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Grammar of JavaFun

M ::= [M; ]∗


let C x = E1 in E2 |

if E1 then E2 else E3 |x .m(x) |fieldref = E |varref |throw x |try {E1} catch (C x) {E2} |null


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Grammar of JavaFun

M ::= [M; ]∗


let C x = E1 in E2 |if E1 then E2 else E3 |

x .m(x) |fieldref = E |varref |throw x |try {E1} catch (C x) {E2} |null


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Grammar of JavaFun

M ::= [M; ]∗


let C x = E1 in E2 |if E1 then E2 else E3 |x .m(x) |

fieldref = E |varref |throw x |try {E1} catch (C x) {E2} |null


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Grammar of JavaFun

M ::= [M; ]∗


let C x = E1 in E2 |if E1 then E2 else E3 |x .m(x) |fieldref = E |

varref |throw x |try {E1} catch (C x) {E2} |null


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Grammar of JavaFun

M ::= [M; ]∗


let C x = E1 in E2 |if E1 then E2 else E3 |x .m(x) |fieldref = E |varref |

throw x |try {E1} catch (C x) {E2} |null


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Grammar of JavaFun

M ::= [M; ]∗


let C x = E1 in E2 |if E1 then E2 else E3 |x .m(x) |fieldref = E |varref |throw x |

try {E1} catch (C x) {E2} |null


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Grammar of JavaFun

M ::= [M; ]∗


let C x = E1 in E2 |if E1 then E2 else E3 |x .m(x) |fieldref = E |varref |throw x |try {E1} catch (C x) {E2} |

null


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Grammar of JavaFun

M ::= [M; ]∗




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Objects

I Objects: (Ident⇀ Loc)× Cnamewhere

I Loc – abstract set of locations,I Ident – possible fields identifiers,

The set of all objects is Obj.

I o.f is π1(o)(f )I and typeof(o) is π2(o).


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Objects


I Loc – abstract set of locations,

I Ident – possible fields identifiers,




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Objects






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Objects






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Objects



The set of all objects is Obj.I o.f is π1(o)(f )

I and typeof(o) is π2(o).


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Objects



The set of all objects is Obj.I o.f is π1(o)(f )I and typeof(o) is π2(o).


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Heap

I Heap h : Loc⇀ Obj, which

I is undefined for nullI has finite domain

The set of all heaps is Heap.


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Heap

I Heap h : Loc⇀ Obj, whichI is undefined for null

I has finite domain



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Heap

I Heap h : Loc⇀ Obj, whichI is undefined for nullI has finite domain



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Heap

I Heap h : Loc⇀ Obj, whichI is undefined for nullI has finite domain



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The reduction relation — contexts

A ::= C | ∅ (labels)

C ::= J KlA | (contexts)new C .k(x1, C, x2) |new C .k(C) |let C x = C in E |if C then E1 else E2 |x .m(x1, C, x2) |x .m(C) |try {C} catch (C x) {E}

Context expressions:

C{JEKlA/J KlA} or C{Jl ′KlA/J KlA},


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A ::= C | ∅ (labels)C ::= J KlA | (contexts)

new C .k(x1, C, x2) |new C .k(C) |let C x = C in E |if C then E1 else E2 |x .m(x1, C, x2) |x .m(C) |try {C} catch (C x) {E}




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The form of the reduction relation

C , h, C1JE1Kl1A1 :: · · · :: CnJEnKlnAn → h′, C′1JE1K

l ′1A′1:: · · · :: C′

mJEmKl′1A′1

I h — heapI C1JE1K :: · · · :: CnJEnK — frame stack


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l ′1A′1:: · · · :: C′

mJEmKl′1A′1

I h — heap

I C1JE1K :: · · · :: CnJEnK — frame stack


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l ′1A′1:: · · · :: C′

mJEmKl′1A′1

I h — heapI C1JE1K :: · · · :: CnJEnK — frame stack


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An example of a reduction rule

n ≥ 0 alloc(h) = (l0, h′′) E0 = body(C ,C , k)l1, . . . , ln ∈ Loc

params(C ,C , k) = (y1, . . . , yn) E = E0{l1/y1, . . . , ln/yn}

h′(l) =

{h′′(l) dla l 6= l0emptyC ,C dla l = l0

C , h, C :: CJnew C .k(l1, . . . , ln)Kl∅ →h′, C :: CJnew C .k(l1, . . . , ln)Kl∅ :: JEKl0∅

(newk)


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An example of a reduction rule — a more standard one

n ≥ 0 l ∈ LocC , h, C :: CJnew C .k(l1, . . . , ln)Kl∅ :: JlKl ′∅ → h, C :: CJlKl∅

(newret)


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Immutable objectsObservable contextsRepresentationFunctional methodLocal-state method

Immutable objects

Immutability of h(l):

I Ingredients:

I heap h, location l ∈ Dom(h), and program CI observable contexts C1, C2 such that

C , h, C1 →∗ h′, C2

I Conditions to be met:

I rep(l , h) = rep(l , h′) andI for each l ′ ∈ rep(l , h′) the equality h(l ′) = h′(l ′) holds


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Immutable objects

Immutability of h(l):I Ingredients:


C , h, C1 →∗ h′, C2




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Immutable objects


I heap h, location l ∈ Dom(h), and program C

I observable contexts C1, C2 such that

C , h, C1 →∗ h′, C2




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Immutable objects



C , h, C1 →∗ h′, C2




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Immutable objects



C , h, C1 →∗ h′, C2




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Immutable objects



C , h, C1 →∗ h′, C2

I Conditions to be met:I rep(l , h) = rep(l , h′) and

I for each l ′ ∈ rep(l , h′) the equality h(l ′) = h′(l ′) holds


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Immutable objects



C , h, C1 →∗ h′, C2

I Conditions to be met:I rep(l , h) = rep(l , h′) andI for each l ′ ∈ rep(l , h′) the equality h(l ′) = h′(l ′) holds


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Immutable objects



C , h, C1 →∗ h′, C2



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Observable contexts

C is observable wrt. l in the heap h for C when:

I it returns from a constructor for l :C0 :: CJnew C .k(l)K∅ :: JlK∅

I it refers to a field of lI it assigns to a field of lI it calls a method on lI it returns from a method in l


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Observable contexts

C is observable wrt. l in the heap h for C when:I it returns from a constructor for l :C0 :: CJnew C .k(l)K∅ :: JlK∅



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Observable contexts


I it refers to a field of l

I it assigns to a field of lI it calls a method on lI it returns from a method in l


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Observable contexts


I it refers to a field of lI it assigns to a field of l

I it calls a method on lI it returns from a method in l


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Observable contexts


I it refers to a field of lI it assigns to a field of lI it calls a method on l

I it returns from a method in l


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Observable contexts




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Immutable objects



C , h, C1 →∗ h′, C2



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Immutable objects



C , h, C1 →∗ h′, C2



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Representation

Representation of h(l) (denoted rep(l , h)) is a set of locations suchthat:

I l belongs to thatI we follow rep links

if l ′ ∈ rep(l , h) i C = typeof(h(l ′)) and C1 rep f occurs infields(C ,C ), then h(l ′)(f ) ∈ rep(l , h)


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Representation


I l belongs to that

I we follow rep links



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Representation


I l belongs to thatI we follow rep links



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Equality of immutable objects

h1(l1) is equal to h2(l2) in Cwhen

I both are of the same immutable classI for each field x

I for non-rep fields: h1(l1)(x) = h2(l2)(x),I for rep fields h1(l1)(x)[h1] = h2(l2)(x)[h2]

Notation: l1[h1] =C l2[h2]


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I both are of the same immutable class

I for each field xI for non-rep fields: h1(l1)(x) = h2(l2)(x),I for rep fields h1(l1)(x)[h1] = h2(l2)(x)[h2]



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Functional method

A method is functional when it is

I pure andI extensional.


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Functional method

A method is functional when it isI pure and

I extensional.


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Functional method

A method is functional when it isI pure andI extensional.


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Purity

Consider declaration of m in C :

Cn+1[mmod] m(C1 x1, . . . ,Cn xn) [throws Exc] {E},

m is pure, when

I for all locations l , l1, . . . , ln, l ′ ∈ Loc of appropriate types in hI each sequence of contexts C :: CJK∅ such that

C , h1, C :: CJl .m(l1, . . . , ln)K∅ →(∗) h2, C :: CJl .m(l1, . . . , ln)K∅ :: Jl ′K,

we have h1 v∅ h2.


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Purity



m is pure, whenI for all locations l , l1, . . . , ln, l ′ ∈ Loc of appropriate types in h

I each sequence of contexts C :: CJK∅ such that


we have h1 v∅ h2.


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Purity



m is pure, whenI for all locations l , l1, . . . , ln, l ′ ∈ Loc of appropriate types in hI each sequence of contexts C :: CJK∅ such that


we have h1 v∅ h2.


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Purity



m is pure, whenI for all locations l , l1, . . . , ln, l ′ ∈ Loc of appropriate types in hI each sequence of contexts C :: CJK∅ such that


we have h1 v∅ h2.


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Extensionality

The idea:

I each time a method is called with a given set of argumentsI it returns the same result

Immutability necessary to make comparison possible.Irritating exceptions we think of them as a component of a disjointunion.


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Extensionality

The idea:I each time a method is called with a given set of arguments

I it returns the same result



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Extensionality

The idea:I each time a method is called with a given set of argumentsI it returns the same result



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Extensionality


Immutability necessary to make comparison possible.

Irritating exceptions we think of them as a component of a disjointunion.


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Extensionality


Immutability necessary to make comparison possible.Irritating exceptions

we think of them as a component of a disjointunion.


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Extensionality




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Local-state



m is local-state, when

I for all locations l , l1, . . . , ln, l ′ ∈ Loc of appropriate types in hI each sequence of contexts C :: CJK∅ such that


we have h1 vR h2 whereR = Rep(h1, l , l1, . . . , ln)


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Local-state



m is local-state, whenI for all locations l , l1, . . . , ln, l ′ ∈ Loc of appropriate types in h

I each sequence of contexts C :: CJK∅ such that




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Local-state



m is local-state, whenI for all locations l , l1, . . . , ln, l ′ ∈ Loc of appropriate types in hI each sequence of contexts C :: CJK∅ such that




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Local-state





we have h1 vR h2 where

R = Rep(h1, l , l1, . . . , ln)


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Local-state







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Different kinds of rules

I cok rulesI mok rulesI type rules


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An example of a typing rule: constructors

C1 ∈ C x : 〈D, τ, nthis〉 ∈ Locxi : 〈Di , τi , nthis〉 ∈ Loc dla i = 1, . . . , kC ; C : cmod; Ex ; Loc ` x .m(x1, . . . , xk) : C1C ; C : func; Ex ; Loc ` x .m(x1, . . . , xk) : cok

(methodfun cok)

C1 ∈ C cmod 6= funcC ; C : cmod; Ex ; Loc ` x .m(x1, . . . , xk) : C1C ; C : cmod; Ex ; Loc ` x .m(x1, . . . , xk) : cok

(methodnfun cok)


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An example of a typing rule: constructors

C1 ∈ C x : 〈D, τ, nthis〉 ∈ Locxi : 〈Di , τi , nthis〉 ∈ Loc dla i = 1, . . . , kC ; C : cmod; Ex ; Loc ` x .m(x1, . . . , xk) : C1C ; C : func; Ex ; Loc ` x .m(x1, . . . , xk) : cok

(methodfun cok)

C1 ∈ C cmod 6= funcC ; C : cmod; Ex ; Loc ` x .m(x1, . . . , xk) : C1C ; C : cmod; Ex ; Loc ` x .m(x1, . . . , xk) : cok

(methodnfun cok)


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An example of a typing rule: methods

modifier(C ,D, k) = lstate∀y ∈ x .Loc `local y : local

C ; C : cmod, k : ∅; Ex ; Loc ` new D.k(x) : D

C ; C : cmod,m : func; Ex ; Loc ` new D.k(x) : mok(funcnew mok)

C1,C2 ∈ C x : 〈C1, τ1, τ2〉 ∈ Locmodifier(C ,C1,m′) = func

C ; C : cmod,m : func; Ex ; Loc ` x .m′(x1, . . . , xk) : C2C ; C : cmod,m : func; Ex ; Loc ` x .m′(x1, . . . , xk) : mok

(funcmeth mok)


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An example of a typing rule: methods

modifier(C ,D, k) = lstate∀y ∈ x .Loc `local y : local

C ; C : cmod, k : ∅; Ex ; Loc ` new D.k(x) : D

C ; C : cmod,m : func; Ex ; Loc ` new D.k(x) : mok(funcnew mok)

C1,C2 ∈ C x : 〈C1, τ1, τ2〉 ∈ Locmodifier(C ,C1,m′) = func

C ; C : cmod,m : func; Ex ; Loc ` x .m′(x1, . . . , xk) : C2C ; C : cmod,m : func; Ex ; Loc ` x .m′(x1, . . . , xk) : mok

(funcmeth mok)


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Soundness proof

I difficult because longI problems with formulation


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Related work

I “immutability” of objects (e.g. C. Haack, E. Poll, J. Schafer,A. Schubert, Immutable Objects for a Java-like Language,ESOP’2007. (ETAPS BEST PAPER AWARD!)

I “who can modify what” approach UTT (e.g. P. Muller,A. Poetzsch-Heffter Universes: A Type System for Alias andDependency Control, 2001

I “how to write a lambda in Java” (a few)


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Thank you!


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