Computing temporal data from temporal data

◮

_{medical applications}

◮

_{bioinformatics}

◮

_animation

◮

_simulation

◮

_{multimedia applications}

◮

_{digital libraries}

◮

_{web applications}

◮

_{videos, scenaries}

◮

_{personal information management}

◮

_transport

◮

_{sensor applications}

◮

_...

languages based on linear

[Prior 57]

, and extensions

[Wolper 83]

,

µ-

[Vardi88]

:

-

,

-

,

RNTL

...

[Chomicki, Toman 98], [Abiteboul,Herr,Van den Busche 99], [Bidoit, De Amo 99], [Bidoit, Objois 05]

◮

_{temporal database = time-stamped database}

Time-stamped calculus (

-

) and extensions

-

,

-

...,

languages based on linear

[Prior 57]

, and extensions

[Wolper 83]

,

µ-

[Vardi88]

:

-

,

-

,

RNTL

...

[Chomicki, Toman 98], [Abiteboul,Herr,Van den Busche 99], [Bidoit, De Amo 99], [Bidoit, Objois 05]

◮

_{temporal database = time-stamped database}

Time-stamped calculus (

-

) and extensions

-

,

-

...,

[Abiteboul,Herr,Van den Busche 99],[Bidoit, Objois 05]

•

temporal queries :

TDB

−→

BD

”temporal”

to

”static”

•

temporal queries :

TDB

−→

BD

”temporal”

to

”static”

•

Example : elements whose color appears everywhere.

Temporal input

•

◮

◮

•

•

7−→

Static output

•

◮

Goal

•

temporal query :

TDB

−→

TDB

”temporal”

to

”temporal”

”sequence”

to

”sequence”

Goal

•

temporal query :

TDB

−→

TDB

”temporal”

to

”temporal”

”sequence”

to

”sequence”

•

Example : color based spliting.

Temporal input

•

◮

◮

•

•

Goal

•

temporal query :

TDB

−→

TDB

”temporal”

to

”temporal”

”sequence”

to

”sequence”

•

Example : color based spliting.

Temporal input

•

◮

◮

•

Goal

•

temporal query :

TDB

−→

TDB

”temporal”

to

”temporal”

”sequence”

to

”sequence”

•

Example : color based spliting.

Temporal input

•

◮

◮

•

Goal

•

temporal query :

TDB

−→

TDB

”temporal”

to

”temporal”

”sequence”

to

”sequence”

•

Example : color based spliting.

Temporal input

•

◮

◮

•

•

Temporal output

•

◮

◮

•

•

Goal

•

temporal query :

TDB

−→

TDB

”temporal”

to

”temporal”

”sequence”

to

”sequence”

•

Example : Canon

Musical score

≈

temporal instance

Goal

•

temporal query :

TDB

−→

TDB

”temporal”

to

”temporal”

”sequence”

to

”sequence”

•

Example : Coalescing: collapse consecutive and identical states into a single one.

Goal

temporal query = ”temporal” to ”temporal” mappings

What are ”temporal” to ”temporal” queries

Relational Temporal Machines

Goal

temporal query = ”temporal” to ”temporal” mappings

What are ”temporal” to ”temporal” queries

Relational Temporal Machines

joint work with Franc¸ois Hantry

(TIME 2007)

What languages for defining ”temporal” to ”temporal” queries / computations

֒

→

Relational Machine, Loosely coupled Generic Machine

[Abiteboul,Vianu 95]

•

Toward a ”normal form” of Relational Temporal Machines

Which components are really useful ?

◮

_{Extended one tape}

◮

_{One tape}

•

complexity classes and temporal languages

Space complexity = size of auxiliary tapes

S

J

J

←

output tape

· · ·

S

J

J

J

m nm

S

J

J

←

output tape

· · ·

S

J

J

J

m nm

k registers

[] [] []

S

J

J

←

output tape

· · ·

S

J

J

J

k registers

[] [] []

Transitions

S

J

J

←

output tape

· · ·

S

J

J

J

k registers

[] [] []

Transitions

S

J

J

←

output tape

· · ·

S

J

J

J

k registers

[] [] []

Transitions

S

J

J

←

output tape

· · ·

S

J

J

J

k registers

[] [] []

Transitions

S

J

J

←

output tape

· · ·

S

J

J

J

k registers

[] [] []

Transitions

S

J

J

←

output tape

· · ·

S

J

J

J

k registers

[] [] []

Transitions

S

J

J

←

output tape

· · ·

S

J

J

J

k registers

[] [] []

Transitions

S

J

J

∅

←

output tape

· · ·

S

J

J

J

k registers

[] [] []

ϕ

(

~

x

)

Transitions

S

J

J

←

output tape

· · ·

S

J

J

J

֒

→

Relational Machines, Loosely coupled Generic Machine

[Abiteboul, Vinau 95]

An example of

: color split

◮ ◮

•

s

s

start

f

s

s

s

s

¬

last

(0)

/create

last

(0)

/

red/right

(1)

/cpred

blue/create

/right

(1)

/cpblue

¬

red/

/right

(0)

queries and actions

last(0)

checks end of input

red

∃

xP

(

x, red

)

blue

∃

xP

(

x, blue

)

◮ ◮

•

s

s

start

f

s

s

s

s

¬

last

(

0

)

/

create

last

(0)

/

red

/

right

(

1

)

/cpred

blue/create

/right

(1)

/cpblue

¬

red/

/right

(0)

queries and actions

last(0)

checks end of input

red

∃xP

(

x, red

)

◮ ◮

•

s

s

start

f

s

s

s

s

last

(0)

/create

last

(0)

/

red/right

(1)

/

cpred

blue/create

/right

(1)

/cpblue

¬

red/

/right

(0)

queries and actions

last(0)

checks end of input

red

∃

xP

(

x, red

)

blue

∃

xP

(

x, blue

)

◮ ◮

•

s

s

start

f

s

s

s

s

¬

last

(0)

/create

last

(0)

/

red/right

(1)

/cpred

blue

/

create

/

right

(

1

)

/cpblue

¬

red/

/right

(0)

queries and actions

last(0)

checks end of input

red

∃

xP

(

x, red

)

◮ ◮

•

◮

s

s

start

f

s

s

s

s

¬

last

(0)

/create

last

(0)

/

red/right

(1)

/cpred

blue/create

/right

(1)

/

cpblue

¬

red/

/right

(0)

queries and actions

last(0)

checks end of input

red

∃

xP

(

x, red

)

blue

∃

xP

(

x, blue

)

◮ ◮

•

◮

s

s

start

f

s

s

s

s

¬

last

(0)

/create

last

(0)

/

red/right

(1)

/cpred

blue/create

/right

(1)

/cpblue

¬

red/

/

right

(

0

)

queries and actions

last(0)

checks end of input

red

∃

xP

(

x, red

)

◮ ◮

•

◮ ◮

s

s

start

f

s

s

s

s

¬

last

(

0

)

/

create

last

(0)

/

red/right

(1)

/cpred

blue/create

/

right

(

1

)

/

cpblue

¬

red

/

/

right

(

0

)

queries and actions

last(0)

checks end of input

red

∃

xP

(

x, red

)

blue

∃

xP

(

x, blue

)

◮ ◮

•

◮ ◮

•

s

s

start

f

s

s

s

s

¬

last

(

0

)

/

create

last

(0)

/

red

/

right

(

1

)

/cpred

blue/create

/right

(1)

/cpblue

¬

red/

/right

(0)

queries and actions

last(0)

checks end of input

red

∃

xP

(

x, red

)

◮ ◮

•

◮ ◮

•

s

s

start

f

s

s

s

s

¬

last

(0)

/create

last

(

0

)

/

red/right

(1)

/cpred

blue

/

create

/

right

(

1

)

/

cpblue

¬

red/

/

right

(

0

)

queries and actions

last(0)

checks end of input

red

∃

xP

(

x, red

)

blue

∃

xP

(

x, blue

)

•

Toward a ”normal form” of Relational Temporal Machines

Which components are really useful ?

◮

_{Extended one tape}

•

Auxiliary tapes

[

1

result]

RTMm_k

=

Extended one-tape

s

(

– one (temporal) auxiliary tape

– a (static) store









”concatenate” the m auxiliary tapes

−→

one auxiliary tape

problem : m cursors ?

answer : use the store to simulate the (use of the) m cursors









M ∈

_;

N

;

N

;

N

;

N

with

N

∈

Proof : extension of

[Abiteboul, Vinau 95]

proof technique for loosely coupled Generic Machine.























N

:

Encoding of a superset of the auxiliary relations computed when running the

M

.

Computation of a representation of an ordered partition

{

δ

. . .δ

}

of the set of r-ary tuples built

from the active domain of the temporal input.

M

uses a finite number of

queries.

N

:

Encoding on the auxiliary tape of the ordered partition, the action tables and the input instance

The auxiliary tape is then used as a Turing machine tape.























•

complexity classes and temporal languages

Space complexity = size of auxiliary tapes

V

_Frame

_T

•

1

◮

1

◮

2

•

3

•

3

V

Frame

•

◮

V

Frame

◮

V

Frame

•

•

-

≈

+ assignments

+ while loops

-

≈

+ assignments

+ while loops

+ temporal moves

auxiliary tape

J

J

J

J

S

store =

output tape

K

S

cursors on input and auxiliary tapes are

synchronized

the ouput is one of the store relations

configuration = auxiliary tape + store + cursor value

While

Q

Do

Body

iteration

1

−→

configuration

1

· · ·

· · ·

· · ·

•

PSpace

-

=

-















TS

-

⊆

LinSpace-

requires to simulate:

generalized time stamped relations in an implicit manner, and

-

queries by

queries

PSpace-

⊆

-

is based on the converse encoding.















•

-

=

T

-

is

-

restricted to shared auxiliary schema (no temporal auxiliary tape)

extended with temporal variables

t

while

do

end

↓t′_;

while

_do

_end

Constant Space

:

no auxiliary temporal tape , just a store

(right move only, restriction on reversal move, synchronization of input and auxiliary moves)

◮

_{data model}

(relational + abstract data types, XML, specific domains)

◮

_{”internal” query language}

(inflationary,

+ aggregate, Xpath, ...)

֒

→

applications (streaming, multimedia, ...)

•

Defining (declarative) languages