Structured Control for Active Tree The Decidability of AXML

AXML (on 1 peer) trophy age query last trophys of name_of_a_child, and append it under trophy F.Landis Paris-Nice 26 trophy age F.Landis Paris-Nice 35 Tour de France 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. of service = rewriting rule

Positive AXML 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: can also delete trophy if doping. Confluence: - Termination: - Positive Term.: Any sequence ultimately stays in same equivalence class - Reachability: Always Yes Always No decidable ???

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 S. Lenglen Federer Fields act independently, can book themselves if find a request (2 can be booked for the same player!) 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 player root Query all root $ Variable of query changed One answer of query, created deleted X+subtree deleted All playing players created ancestor variable

Rules = Tree Transformations Court Free Request root Court Booked Play root Court No query 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 Booked Request root No query player Free root Request root player $ Query one-in Booked

Rules = Tree Transformations Query-all + guard counting number answers Tree Pattern TTree Pattern T’ Nodes in T’ and not in T are created Nodes in T and not in T’ are deleted + its subtree deleted Nodes 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 query player Federer Roland Garros S.Lenglen Central Booked Federer Free Request Federer Play Roland Garros S.Lenglen Central Booked Central Federer injective Document New Document Rewriting Rule

Formatting of the query play root player Query = 2 Tree Pattern, transformation as before to format result player 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 Undecidability The 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 Decidability The 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 trophy 26 F.Landis TdF M.Indurain trophy 42 TdETdF WSTS for well quasi order < finite degree/number of rules If X  Y and X ->* X’ then  Y’ with X’  Y’ and Y ->* Y’ X<YX<Y X’ <Y’

WSTS WSTS for well quasi order < finite degree/number of rules If X  Y and X ->* X’ then  Y’ with X’  Y’ and Y ->* Y’ X<YX<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 with Finite 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 Abstracted in Weak reachability = regular well structured properties. We can know whether there exists a path wtih (TP 1 or TP 2 ) Until (TP 3 ) (no negation). (add one node = state + make new rules updating state depending on TP_i)