Grid Module Interface Specification

(0) CHARACTERISTICS

-type specified: grid

-features: deterministic, non-parameterized

(1) SYNTAX

ACCESS-PROGRAMS

Program Name	Arg#1	Arg#2	Arg#3	Arg#4	Arg#5	Result Type
INIT	status:O	grid:VO	index:V
DESTROY	status:O	grid:VO
GET_INDEX	status:O	grid:V				index
PUT_INFO	status:O	grid:VO	index:V	info:V
GET_INFO	status:O	grid:V	index:V			info
PUT_CLUSTER	status:O	grid:VO	int:V	int:V	int:V
DELETE_CLUSTER	status:O	grid:VO	int:V	int:V	int:V
PUT_SLICE	status:O	grid:VO	int:V	int:V
DELETE_SLICE	status:O	grid:VO	int:V	int:V
COPY_SLICE	status:O	grid:VO	int:V	int:V	int:V
COPY	status:O	grid:V	grid:VO

(2) CANONICAL TRACES

canonical(T)

T = _

T = INIT(*1, *, xind).[PUT_INFO(*1, *, x[i], e[i])]i=1..nb_els where xind:<index>; nb_els:<int>; x[1]..x[nb_els]:<index>; e[1]..e[nb_els]:<info> [Źindex::isNil(xind)

i:<int>1

index::nbElems(xind) [index::getInt(xind, i)

nb_els = product(xind, index::nbElems(xind))

i:<int>1

index::nbElems(xind) [index::getInt(x[1], i) = 1 ]

i:<int> [1

i < nb_els

index::isLt(x[i], x[i + 1])]

xind = x[nb_els] ]

AUXILIARY FUNCTIONS

copySlice: <grid> × <int> × <int> × <int> <grid>

copySlice(g, d, offset1, offset2) (g _ (1 d d nbDim(g)) (1 offset1 offset1 nthDim(g, d)) (1 offset2 offset2 < nthDim(g, d))) = new_g where new_g:<grid>; m:<int>; x[1]..x[m]:<index>; xind:<index>; e[1]..e[m]:<info>; f[1]..f[m]:<info> [g = INIT(*1, *, xind).[PUT_INFO(*1, *, x[i], e[i])]i=1..m canonical(new_g) new_g = INIT(*1, *, xind).[PUT_INFO(*1, *, x[i], f[i])]i=1..m i:<int>1 i i m [(index::getInt(x[i], d) offset2 f[i] = e[i] ) (index::getInt(x[i], d) = offset2 f[i] = getInfo(g, index::putInt(x[i], d, offset1)) )]]

deleteCluster: <grid> × <int> × <int> × <int> <grid>

deleteCluster(g, d, offset, nb) (g _ (1 d d nbDim(g)) (1 offset offset nthDim(g, d)) (1 nb nb < nthDim(g, d)) offset + nb - 1 nthDim(g, d)) =

Condition	Value
nb = 1	deleteSlice(g, d, offset)
nb > 1	deleteCluster(deleteSlice(g, d, offset), d, offset, nb - 1)

deleteSlice: <grid> × <int> × <int> <grid>

deleteSlice(g, d, offset) (g _ (1 d d nbDim(g)) (1 offset offset nthDim(g, d)) nthDim(g, d) > 1) = new_g where new_g:<grid>; m,n:<int>; x[1]..x[m], y[1]..y[n]:<index>; xind,yind:<index>; e[1]..e[m], f[1]..f[n]:<info> [g = INIT(*1, *, xind).[PUT_INFO(*1, *, x[i], e[i])]i=1..m canonical(new_g) new_g = INIT(*1, *, yind).[PUT_INFO(*1, *, y[i], f[i])]i=1..n nthDim(new_g, d) = nthDim(g, d) - 1 i:<int>1 i i nbDim(g) i d [nthDim(new_g, i) = nthDim(g, i) ] i,v:<int>1 i i n [index::getInt(y[i], d) = v (v < offset k:<int> [y[i] = x[k] f[i] = e[k] ]) (v offset k:<int> [index::putInt(y[i], d, v + 1) = x[k] f[i] = e[k] ])]]

getIndex: <grid> <index>

getIndex(g) (g _) = x where x:<index> [g = INIT(*1, *, x) ]

getInfo: <grid> × <index> <info>

getInfo(g, x) (isValidIndex(g, x)) = e where S1,S2:<grid>; e:<info> [g = S1.PUT_INFO(*1, *, x, e).S2 ]

isValidIndex: <grid> × <index> <bool>

isValidIndex(g, x) (g _) = nbDim(g) = index::nbElems(x) i:<int>1 i i nbDim(g) [1 index::getInt(x, i) index::getInt(x, i) nthDim(g, i)]

nbDim: <grid> <int>

nbDim(g) (g _) = index::nbElems(x) where x:<index>; T:<grid> [g = INIT(*1, *, x).T ]

nthDim: <grid> × <int> <int>

nthDim(g, d) (g _ (1 d d nbDim(g))) = index::getInt(x, d) where x:<index>; T:<grid> [g = INIT(*1, *, x).T ]

product: <index> × <int> <int>

product(y, j) (y _ j 1) =

Condition	Value
j = 1	index::getInt(y, j)
j > 1	index::getInt(y, j) * product(y, j - 1)

putCluster: <grid> × <int> × <int> × <int> <grid>

putCluster(g, d, offset, nb) (g _ (1 d d nbDim(g)) (0 offset offset nthDim(g, d)) nb > 0) =

Condition	Value
nb = 1	putSlice(g, d, offset)
nb > 1	putCluster(putSlice(g, d, offset), d, offset, nb - 1)

putInfo: <grid> × <index> × <info> <grid>

putInfo(g, y, e) (g _ isValidIndex(g, y)) = S1.PUT_INFO(*1, *, y, e).S2 where S1,S2:<grid>; f:<info> [g = S1.PUT_INFO(*1, *, y, f).S2 ]

putSlice: <grid> × <int> × <int> <grid>

putSlice(g, d, offset) (g _ (1 d d nbDim(g)) (0 offset offset nthDim(g, d))) = new_g where new_g:<grid>; xind,yind:<index>; m,n,v:<int>; x[1]..x[m]:<index>; e[1]..e[m]:<info>; y[1]..y[n]:<index>; f[1]..f[n]:<info> [g = INIT(*1, *, xind).[PUT_INFO(*1, *, x[i], e[i])]i=1..m canonical(new_g) new_g = INIT(*1, *, yind).[PUT_INFO(*1, *, y[i], f[i])]i=1..n nthDim(new_g, d) = nthDim(new_g, d) + 1 i:<int>1 i i nbDim(g) i d [nthDim(new_g, i) = nthDim(g, i) ] i:<int>1 i i n [index::getInt(y[i], d) = v (v offset k:<int> [y[i] = x[k] f[i] = e[k] ]) (v = offset + 1 f[i] = _ ) (v > offset + 1 k:<int> [index::putInt(y[i], d, v - 1) = x[k] f[i] = e[k] ])]]

(3) SEMANTICS

ACCESS-PROGRAMS

Legality(INIT(s, (a, g), x)) =

Condition	Value
g _	%nonempty_trace%
g = _ x = _	%no_init%
g = _ (x _ (Źindex::isNil(x) i:<int>1 i i index::nbElems(x) [index::getInt(x, i) 1] = false ))	%bad_index%
g = _ (x _ (Źindex::isNil(x) i:<int>1 i i index::nbElems(x) [index::getInt(x, i) 1] = true ))	%legal%

INIT(s , (a, g), x) =

Condition	Value
g _	status::NONEMPTY()
g = _ x = _	status::INIT()
g = _ x _ Ź(Źindex::isNil(x) i:<int>1 i i index::nbElems(x) [index::getInt(x, i) 1])	status::INDEX()
g = _ x _ (Źindex::isNil(x) i:<int>1 i i index::nbElems(x) [index::getInt(x, i) 1])	status::LEGAL()

INIT(s, (a, g) , x) = G where G:<grid>; n:<int>; y[1]..y[n]:<index> [G = INIT(*1, *, x).[PUT_INFO(*1, *, y[i], _)]i=1..n ]

Legality(DESTROY(s, (a, g))) =

Condition	Value
g = _	%no_init%
g _	%legal%

DESTROY(s , (a, g)) =

Condition	Value
g = _	status::INIT()
g _	status::LEGAL()

DESTROY(s, (a, g) ) = _

Legality(GET_INDEX(s, g)) =

Condition	Value
g = _	%no_init%
g _	%legal%

GET_INDEX(s , g) =

Condition	Value
g = _	status::INIT()
g _	status::LEGAL()

GET_INDEX(s, g) = getIndex(g)

Legality(PUT_INFO(s, (a, g), x, e)) =

Condition	Value
g = _	%no_init%
g _ ŹisValidIndex(g, x)	%bad_access%
g _ isValidIndex(g, x)	%legal%

PUT_INFO(s , (a, g), x, e) =

Condition	Value
g = _	status::INIT()
g _ ŹisValidIndex(g, x)	status::ACCESS()
g _ isValidIndex(g, x)	status::LEGAL()

PUT_INFO(s, (a, g) , x, e) = putInfo(g, x, e)

Legality(GET_INFO(s, g, x)) =

Condition	Value
g = _	%no_init%
g _ ŹisValidIndex(g, x)	%bad_access%
g _ isValidIndex(g, x)	%legal%

GET_INFO(s , g, x) =

Condition	Value
g = _	status::INIT()
g _ ŹisValidIndex(g, x)	status::ACCESS()
g _ isValidIndex(g, x)	status::LEGAL()

GET_INFO(s, g, x) = getInfo(g, x)

Legality(PUT_CLUSTER(s, (a, g), d, nb, offset)) =

Condition	Value
g = _	%no_init%
g _ 1 d d nbDim(g) nb > 0 (0 offset offset nthDim(g, d)) = false	%bad_access%
g _ 1 d d nbDim(g) nb > 0 (0 offset offset nthDim(g, d)) = true	%legal%

PUT_CLUSTER(s , (a, g), d, nb, offset) =

Condition	Value
g = _	status::INIT()
g _ Ź(1 d d nbDim(g) nb > 0 (0 offset offset nthDim(g, d)))	status::ACCESS()
g _ (1 d d nbDim(g) nb > 0 (0 offset offset nthDim(g, d)))	status::LEGAL()

PUT_CLUSTER(s, (a, g) , d, nb, offset) = putCluster(g, d, offset, nb)

Legality(DELETE_CLUSTER(s, (a, g), d, nb, offset)) =

Condition	Value
g = _	%no_init%
g _ 1 d d nbDim(g) (1 nb nb nthDim(g, d)) (1 offset offset nthDim(g, d)) offset + nb - 1 nthDim(g, d) = false	%bad_access%
g _ 1 d d nbDim(g) (1 nb nb nthDim(g, d)) (1 offset offset nthDim(g, d)) offset + nb - 1 nthDim(g, d) = true	%legal%

DELETE_CLUSTER(s , (a, g), d, nb, offset) =

Condition	Value
g = _	status::INIT()
g _ Ź(1 d d nbDim(g) (1 nb nb < nthDim(g, d)) (1 offset offset nthDim(g, d)) offset + nb - 1 nthDim(g, d))	status::ACCESS()
g _ (1 d d nbDim(g) (1 nb nb < nthDim(g, d)) (1 offset offset nthDim(g, d)) offset + nb - 1 nthDim(g, d))	status::LEGAL()

DELETE_CLUSTER(s, (a, g) , d, nb, offset) = deleteCluster(g, d, offset, nb)

Legality(PUT_SLICE(s, (a, g), d, offset)) =

Condition	Value
g = _	%no_init%
g _ 1 d d nbDim(g) (0 offset offset nthDim(g, d)) = false	%bad_access%
g _ 1 d d nbDim(g) (0 offset offset nthDim(g, d)) = true	%legal%

PUT_SLICE(s , (a, g), d, offset) =

Condition	Value
g = _	status::INIT()
g _ Ź(1 d d nbDim(g) 0 offset offset nthDim(g, d))	status::ACCESS()
g _ (1 d d nbDim(g) 0 offset offset nthDim(g, d))	status::LEGAL()

PUT_SLICE(s, (a, g) , d, offset) = putCluster(g, d, offset, 1)

Legality(DELETE_SLICE(s, (a, g), d, offset)) =

Condition	Value
g = _	%no_init%
g _ 1 d d nbDim(g) 1 < nthDim(g, d) (1 offset offset nthDim(g, d)) = false	%bad_access%
g _ 1 d d nbDim(g) 1 < nthDim(g, d) (1 offset offset nthDim(g, d)) = true	%legal%

DELETE_SLICE(s , (a, g), d, offset) =

Condition	Value
g = _	status::INIT()
g _ Ź(1 d d nbDim(g) 1 < nthDim(g, d) 1 offset offset nthDim(g, d))	status::ACCESS()
g _ (1 d d nbDim(g) 1 < nthDim(g, d) 1 offset offset nthDim(g, d))	status::LEGAL()

DELETE_SLICE(s, (a, g) , d, offset) = deleteCluster(g, d, offset, 1)

Legality(COPY_SLICE(s, (a, g), d, offset1, offset2)) =

Condition	Value
g = _	%no_init%
g _ 1 d d nbDim(g) (1 offset1 offset1 nthDim(g, d)) (1 offset2 offset2 nthDim(g, d)) = false	%bad_access%
g _ 1 d d nbDim(g) (1 offset1 offset1 nthDim(g, d)) (1 offset2 offset2 nthDim(g, d)) = true	%legal%

COPY_SLICE(s , (a, g), d, offset1, offset2) =

Condition	Value
g = _	status::INIT()
g _ Ź(1 d d nbDim(g) 1 offset1 offset1 nthDim(g, d) 1 offset2 offset2 nthDim(g, d))	status::ACCESS()
g _ (1 d d nbDim(g) 1 offset1 offset1 nthDim(g, d) 1 offset2 offset2 nthDim(g, d))	status::LEGAL()

COPY_SLICE(s, (a, g) , d, offset1, offset2) = copySlice(g, d, offset1, offset2)

Legality(COPY(s, g, (a, h))) =

Condition	Value
g = _ h = _	%no_init%
g _ h _	%legal%

COPY(s , g, (a, h)) =

Condition	Value
g = _ h = _	status::INIT()
g _ h _	status::LEGAL()

COPY(s, g, (a, h) ) = g