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Models for AXML Keeping Decidability in Mind.. AXML (on 1 peer) felony age query last felonies of name_of_a_child, and append it under.

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Presentation on theme: "Models for AXML Keeping Decidability in Mind.. AXML (on 1 peer) felony age query last felonies of name_of_a_child, and append it under."— Presentation transcript:

1 Models for AXML Keeping Decidability in Mind.

2 AXML (on 1 peer) felony age query last felonies of name_of_a_child, and append it under felony L.Burrows killer 35 felony age L.Burrows killer 35 EFP M.Scoffield - Confluence: does asking first Scoffield or Burrows result in same doc? - Termination: Is there no infinite sequence of fireable services? - Reachability: Can some configuration be reached/be subsumed? Yes No ??? invoc. of service = rewriting rule

3 Positive AXML query last felonies of name_of_a_child, and append it under felony 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 felony from revised trials. Confluence: does asking first Scoffield or Burrows result in same doc? - Termination: Is there no infinite sequence of fireable services? - Reachability: Can some configuration be reached/be subsumed? Always Yes Always No ???

4 “Positive” Termination document A ´ B if exists 2 child-consistent mappings from A to B and from B to A. Ex: felony age L.Burrows killer 35 felony age L.Burrows killer 35 killer M.Scoffield ´ Positive Termination: Is any infinite sequence of service invocation ultimately document-class constant? last felonies of name_of_a_child is a list Ex simple queries: For Positive Systems: Decidable if simple queries (no tree variables in queries) else Undecidable

5 Over Positivity? Positive AXML: Services can only add, never delete. Services cannot stop. What if a service can stop? -Termination/Confluence becomes non trivial, interesting under simple queries -Tennis Court example revisited: Federer request leaveF1 manager Field1: book Field2: free playF2Field1: book Field2: book Federer-F2 request running stopped

6 Rulez for Services Request: if query manager(X(Federer-FX/active)), close and open playFX FieldX,Y:..: if son inactive & query PlayerX(request/active), then close and open state with FieldY booked(PlayerX-FieldY) PlayerX-FieldY: if query PlayerX(PlayFieldY), close PlayFY: if query manager(active(PlayerX-FieldZ/inactive)), close and open leaveFY (or if Z \neq Y, we are talking about another field) FieldX,Y:..: if son inactive & query PlayerX(LeaveY/active), then close and open state with FieldY free(FreePlayerX-FieldY) leaveFY: if query manager(active(FreePlayerX-FieldZ/inactive)),close and open request (or if Z \neq Y, we are talking about another field)

7 Over Positivity? Federer request playF1 manager Field1: book Field2: free playF2Field1: book Field2: book Federer-F2 A service can create new services/data as brother/son. A service is the only one that can decide to close itself. Query can know state of services (but no tree variable) A service should have a knowledge of its neighborhood

8 More distributed Fields Federer PlayLeave Roland Garros S.Lenglen Central Booked S Lenglen Free S Lenglen Federer Fields act independently, can book themselves if find a request (2 can be booked for the same player!) request

9 More distributed Fields Free: if query PlayerX(Request/active), close and open Booked(playerX). PlayerX(Request): if query RG(FieldY(Booked/active(PlayerX))), then close and open Play(FieldY). Play(FieldY): close and open Leave(FieldY). FieldY(Booked(PlayerX)): if query PlayerX is active in Leave or Play(FieldZ), Z  Y then close and open Free. (Finite number of possibilities, hence a negative test is possible) PlayerX(Leave(FieldY)) : if query RG(FieldY) is not active in Play(PlayerX), then close and open Play(FieldY).

10 How soon comes Undecidability? Result: Simulation of a deterministic 2 counters machine, with A service can create new services/data as brother/son. A service is the only one that can decide to close itself. Query can know state of services (but no tree variable) A service should have a knowledge of its 1-neighborhood. Either non bounded depth or ordered trees (brother-successor) Trivial: If depth 1 (word) and unordered trees, decidable (subclass of PN) (Problem : simulating zero-test/unnexistence test. Nota, easily undecidable with order on rules/optimisations)

11 Simulating 2 counters machine Result: Simulation of a deterministic 2 counters (C,D) machine We use 4 pseudo counters (C 0,C 1,D 0,D 1 ), and phases i,j  {0,1}, and states E In phase i, C i is the current counter and C i+1 is the temporary/next counter C 0 = …$ XX YY C 1 = …$ Z T C 0 = …$ XX X Y C 1 = …$ Z T C 0 = …$ XXX Y C 1 = …$ ZZ T X/Z = closed services Y/T = open services X/Y = copy Z/T = paste

12 Simulating 2 counters Machine C 0 = …$ X 0 X 1 Y 0 Y 1 C 1 = …$ Z 0 T 1 C 0 = …$ X 0 X 1 X 0 Y 1 C 1 = …$ Z 0 T 1 C 0 = …$ X 0 X 1 X 0 Y 1 C 1 = …$ Z 0 Z 1 T 0 X k+1 Y k : if query Z k T k+1 then close T k : if query X k+1 Y k then close and open T k We need a parity-bit k

13 Simulating 2 counters machine C 0 = $1.. 1$ … $ l.. l C 1 = $1.. 1$ … $ l’..l’ We also need a coup-bit l  {0,1}: at phase i, C i on coup l copies on C i+1 at coup i+l i =0 i=1 Current counter Old values We cannot mess a couter in coup l with a counter in coup l+1 and we cannot mess a with a wrong state E = (q 0,0,0) … (q,i,j). Old states current state

14 4 Rewriting rulez X i,k-1,l T Y i,k,l : query (p,i’,j’) (q,i,j) and Z i,k-1,l+i X T i,k,l+i and q concerns C then close. i: counter for C i, k phase-bit, l coup-bit, X/Y copy, Z/T paste, p/q states Z i+1,k-1,l Y T i+1,k,l : query (p,i’,j’) (q,i,j) and X i,k-1,l+i T Y i,k,l+i, q concerns C then close and create Y i+1,k,l T i+1,k+1,l. (q,i,j) : query X$T i,0,l and $T i+1,0,l+i, q concerns C then close and create (succ(q),i+1,j). Z i+1,k-1,l Y T i+1,k,l : query (p,i’,j’) (q,i,j) and X i,k-1,l+i T Y i,k,l+i $ q concerns C then close and create Y i+1,k,l $ T i+1,k+1,l. (decrement : create $ T i+1,k+1,l )

15 How soon comes Undecidability? Result: Simulation of a deterministic 2 counters machine, with non bounded depth, create sons machine C0 C1 (q,i,j) D0D1 bounded depth, ordered, create brothers machine C0 C1 (q,i,j) D0 D1

16 How soon comes Undecidability? Result: exact class-reachability undecidable for A service can create services/data as cousin (or can close another service, take common ancestor). A service can close itself. Query can know state of services (but no tree variable) Bounded-depth and unordered trees. machine C0C1 Tokens (C0+nZero) C0 creates data nZero, or create data Zero. Token can copy to cousin with data nZero. C0 can close anytime and open fresh C0 (with opposite bit). Token open under closed service is a proof of bad simulation (tested by reachable class) C0

17 Over Positivity vs Workflow Petri Net community sees workflow as acyclic PN. Problem: Simulate more than one place in the pre of a transition. A1B1 A2B2 t A1B1 A2B2 A3 At Bt B0 A0 Query Bt or B2... (not …B1) Query At or A2... (not …A1) Reachability simulation for acyclic PN (may introduce deadlock) Works also for safe PN (may introduce exponential blow-up) 


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