A Schedulability-Preserving Transformation of BDF to Petri Nets

1. A Schedulability-Preserving Transformation of BDF to Petri Nets Cong Liu EECS 290n Class Project December 10, 2004

2. Outline • Motivation • Scheduling Petri Nets (PN) • Scheduling Boolean Dataflow (BDF) • Proposing a Transformation • Extension of Consistence to PN • Future Research Direction

3. Motivation • Both BDF and Petri nets are concurrent models. • Both have the scheduling problem • Existence of Bounded Memory Schedules for BDF is undecidable [Buck 93]. • No algorithms can guarantee to find a schedule if one exists. • Petri nets are not Turing complete [Peterson 81]. • Efficient scheduling algorithms exist [Cortedella 00]. • Can we apply scheduling techniques for PN to scheduling BDF?

4. Motivation • It is known SDF is equivalent to marked graph, a subclass of Petri nets • How about BDF? • If we want to build up equivalence, then what kind of equivalence? • Trace equivalence • Language containment • Simulation relation • What property to preserve during transformation? • Turing-completeness of BDF make transformation applicable to a subclass of BDF

5. SELECT F T SWITCH F T If-Then-Else using BDF IN >0? while (1) { read (IN, x, 1); if (x>0) y = f(x); else y = g(x); write (OUT, y, 1); } f g OUT

6. IN X>0? B C OUT If-Then-Else using PN while (1) { read (IN, x, 1); if (x>0) y = f(x); else y = g(x); write (OUT, y, 1); }

7. D E Boolean Dataflow (BDF) • Consumption/production rate is a two-valued (including zero) function of the value of a Boolean token received by the control port of the actor. • Control ports always transfer exactly one token per execution. • BDF: superset of synchronous dataflow (SDF) [Lee87] A B SWITCH T F T F SELECT F

8. A B T F D E T F F (A,B,b:D,!b:E,F) Schedules of Boolean Dataflow • A schedule is a finite list of guarded firings, where: • there exists a feasible cyclic firing sequence returning to initial state regardless of values of Boolean controls • Bounded length schedule • Length of cyclic firing sequences are bounded by a constant, E.g. repeat (5) times. • Bounded memory schedule • Length of cyclic firing sequences are bounded by a constant, E.g. repeat (x) times.

9. Scheduling Boolean Dataflow • Solving balance equation • Consistency check • Simulation to test firability • Clustering • Dynamic scheduling

10. a p1 b c p2 p3 e d p4 f Petri Nets • A directed graph with two kinds of nodes: place, transition • System state (marking): number of tokens in each place • Transition enabled if enough tokens in all incoming places • Firing a transition consume/produce tokens Free choice set: {b, c} T-invariants: {a, b, d, f}, {a, c, e, f}

11. null a p1 f a c b p2 p3 p1 e b d c p4 p2 p3 e d p4 f Scheduling Petri Nets • A schedule is a rooted tree • Finite • Nodes → reachable markings, (root → initial marking) Edges → transitions • Transitions in a FCS are fired at each node • Each node has a path to root (if add returning arc)

12. null a p1 f a c b A B p2 p3 p1 T F e b d c p4 D E p2 p3 T F e d F p4 (A,B,b:D,!b:E,F) f Comparing BDF and PN schedules fire A; fire B; if (B) { fire D; } else { fire E; } fire F;

13. Challenges to transformation • Petri nets do not distinguish tokens. • Petri nets do not preserve order of tokens. • Proposed solutions: • Use different places to hold tokens with different values • Use synchronization (blocking write) to enforce ordering

14. A B A B A A pF pF pT pT T F F T F T T F C pT’ pF’ pT’ pF’ C B B C C A Transformation • A set of “True”/ ”False” places express controls. • A set of “Acknowledge” places to synchronize the production and consumption of Boolean tokens.

15. A Transformation • Strong synchronization enforces blocking write at Boolean control ports A T F A pT’ pF’ pT’ pF’

16. E E A SWITCH T F T F B C pT1 pF1 F F T SELECT pTa1 paF1 D Handling initial tokens A T F B C T F T F D pT2 pF2 pTa2 paF2

17. Algorithm Transform (BDF) { transform_SDF_actors; transform_BooleanGenerators; transform_SWITCH; transform_SELECT; handeling_initial_tokes; }

18. Propositions • Proposition 1 If the transformed Petri net has a bounded length schedule, the corresponding BDF has a bounded length schedule. Sketch of proof: show that Petri net contains a subset of behaviors of BDF. • Proposition 2 If the BDF has a bounded length schedule, the transformed Petri net has a bounded length schedule. Sketch of proof: • Determine exact times each actor to be fired • Transform BDF to acyclic precedence graph (APG) • Decompose Petri net to marked graph component (MGC) • Build up equivalence between APG and MGC

19. E A SWITCH T F B C F F T SELECT D Extension of Consistence to PN • The restrictions on the values of symbolic variables of a BDF are transformed into the dependence relation of corresponding transitions of the transformed Petri net. For any T-invariants, if it contains transition T of SWITCH, it also contains transition T of SELECT p1 = p2 p1 p2

20. E E A SWITCH T F T F B C pT1 pF1 F F T SELECT pTa1 paF1 D A T F B C T F T F D pT2 pF2 pTa2 paF2

21. Future Research Direction • Does the two propositions also holds for bounded memory schedule? • Can generalize the notion of schedule by assuming some kind of fairness? • Assuming program will always exit iterations • Transform a Petri net to a BDF? • Other equivalence alternatives? • More exploration on schedulability and consistence notion for the transformation