structured control for active tree the decidability of axml
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Structured Control for Active Tree
The Decidability of AXML
AXML (on 1 peer)last@UCI
trophyage
last@UCI: query last trophys of name_of_a_child, and append it under trophy
F.Landis
Paris-Nice
26
last@UCI
trophyage
F.Landis
Paris-Nice
35
Tour de France
M.Indurain M.Indurain
- Confluence: does asking first Landis or Indurain result in same doc?
- Termination: Is there no infinite sequence of fireable services?
- Reachability: Can some configuration be reached?
Yes
No
???
invoc. ofservice =
rewritingrule
Positive AXML
last@UCI: query last trophys of name_of_a_child, and append it under trophy
Positive AXML: If a service invocation is possible some day, it is possible forever. => Services can only add, never delete. Services cannot stop.
Ex:
Non Positive: last@UCI can also delete trophy if doping.
Confluence:
- Termination:- Positive Term.: Any sequence ultimately stays in same equivalence class
- Reachability:
Always Yes
Always Nodecidable
???
Over Positivity?
Positive AXML: Services can only add, never delete. Services cannot change.
What if things can be changed?
Termination/Confluence becomes non trivial, interesting under simple queries
Distributed Tennis Fields
Federer
Play
Roland Garros
S.Lenglen Central
Booked BookedS. Lenglen
Federer Federer
Fields act independently, can book themselves if find a request (2 can be booked for the same player!)
Free Free
LeaveRequest
root
X(playing)
Y(playing)
Rules = Tree Transformations
Free
root
Booked
root
Request
root
player
$
Booked
root
player
court
Request
root
player
root
$
player
play
Query one-in
root
Leave
root
player
Free
root
player
Booked
Query not-empty
play
root
playerroot
Query all
root
$
Variable of query
changed
One answer of query, createddeleted
X+subtree deleted All playing players
created
ancestor
variable
Rules = Tree Transformations
Court
Free
Request
root
Court
Booked
Play
root
Court
No queryplayer player
player
We can also do it in one step:
Rules = Tree Transformations
Query one-in is not needed, can be done in Tree pattern
Free
Request
root
BookedRequest
root
No query
player
player
player
Free
rootroot
Request
root
player
$
Query one-in
Booked
Rules = Tree Transformations
Query-all+ guard counting number answers
Tree Pattern T Tree Pattern T’
Nodes in T’ and not in T are createdNodes in T and not in T’ are deleted + its subtree deletedNodes in T and T’ are conserved with its subtree (can be moved)$ in T’ is replaced by the forest of results of query.
Rule = (T,query,guard,T’):
Rewriting Step
Court
Free
Request
root
Court
Booked
Play
root
Court
No queryplayer player
playerFederer Roland Garros
S.Lenglen Central
Booked
Federer
Free
Request
Federer
Play
Roland Garros
S.Lenglen Central
Booked BookedCentral
Federer Federer
injectiveinjective
Document
New Document
Rewriting Rule
Formatting of the query
play
root
player
Query = 2 Tree Pattern, transformation as before to format result
player
court
court
playing
Simple query: use variables General query: use same name of nodes (copy subtree)
play
root
Moya
Central
play
Nadal
Lenglen
play
Federer
Lenglen
Nadal
Lenglen
playing
Federer
Lenglen
Moya
Central
query
Possible Options
Depth of Tree (Bounded/unbounded)
Degree of Tree (Bounded/unbounded)
Successors on brothers
Number of data type (finite/infinite)
Service can only delete itself or not, or nothing deleted
Well structured query (cannot test non existence of a TP, or at most…)
Options and UndecidabilityThe following leads to undecidability:
Non positive query + any infiniteness (unbounded depth or degree or data type is |N)
(2 counters machine)or
New (Last time): positive query + service can only delete itself + any linear order (successor on brothers or unbounded tree or data type is |N). Unbounded degree does not suffice (coding of turing machine on words-rewriting with query)
or use Loeding’s Thesis and rewriting on trees
Options and DecidabilityThe following leads to some decidability:
Depth and Degree of Tree Bounded + finite set of data type(finite state systems)
or Service cannot move/delete (monotonic systems)
or
New: Depth of Tree Bounded + no Successors on brothers + finite number of data type + Positive guards.
(Well Structured Transition System)
Allow : Unbounded Degree of Tree Service can delete,move anything
WSTS
the following < is a well quasi order:A< B if A can be injectively send on B (son/label preserved). Then, In any infinite sequence, there exists u_i > u_j with i>j
last@UCI
trophy26
F.Landis
TdF
M.Indurain
trophy42
TdE TdF TdF
WSTS for well quasi order <finite degree/number of rules
If X Y and X ->* X’ thenY’ with X’ Y’ and Y ->* Y’
X < Y
X’ < Y’
WSTS
WSTS for well quasi order <finite degree/number of rules
If X Y and X ->* X’ thenY’ with X’ Y’ and Y ->* Y’
X < Y
X’ < Y’
-Build the transition system TS- Do not extend Y with X -> Y and Y X.- Mark such Y.
Prop: TS has finite number of states contradiction: Koenig withFinite degree, finite number of initial states,infinite number of states: inifinite path.
With Extended Dickson: there is X<Y on that path and Y is extended, contradiction.
false with guards « less than »…
Relevant Properties
Finite State = is there finite number of documents
Termination: is there no cycle nor marked states
Reachability: can i reach doc D. more complicated than for Petri Nets.
Confluence: not clear how to separete even and odd inifinite sequence
Weak reachability: Given D, can i reach D’ D. Backward methods exist.
Weak confluence: all reachable documents s,s’, can reach respectively some t,t’ with t>t’ = Is there a unique maximal strongly connected component in abst. graph of docs
Complexity: probably tower of exponential wrt depth of tree. Lower bound? what if we assume that a service can only close itself?
-Do not extend Y with X -> Y and Y X, Mark such Y.
(Un)Decidability
Strict = no deletion allowed, but moves are allowed.
Not strict, we can have (delete subtree)
X Y and X ->* X’ and
Y’ with X’ Y’ and Y ->* Y’
Strict, we always have
If X Y and X ->* X’ then
Y’ with X’ Y’ and Y ->* Y’
Harder than reachability
Discussion
Cannot handle optimization: no guards « less than » (think Dell supply chain, if less than 3 items in revolver, order something)
So far, set of labels is finite (we know set of players, fields beforehand).Might work with inifinite set (generator of new players, fields, open systems).
We have bag semantics, it makes sense with finite set of labels abstracting inifinite set.
players
Federer Moya Nadal
players
player player player
Abstracted in
Weak reachability = regular well structured properties.We can know whether there exists a path wtih (TP1 or TP2) Until (TP3) (no negation).(add one node = state + make new rules updating state depending on TP_i)
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