Workflow Soundness

Soundness is a notion of correctness of workflow nets

These properties are usually checked with Reachability Graphs


Situations to Avoid

Unboundness

Unboundness means:

  • there is no bound on the number of tokens that a place can hold
  • this always means a problem


Improper Termination

A workflow net satisfies proper completion when

  • if there's a token in the output place, then there are no tokens in other places
  • if this property is not satisfied, then the problem is called improper termination


workflow-nets-unbounded.png

  • firing sequences:
    • $s \to a \to b \to a \to d$
    • $s \to a \to c \to a \to d$
  • there are remaining tokens after the flow ends
    • it means: it stopped, but there is still some work to do
    • clearly it's a problem


Example:

  • workflow-nets-problem.png
  • there are paths that lead to proper termination:
    • card $\to$ charge $\to$ skip $\to$ finish
    • transfer $\to$ receive payment $\to$ NA $\to$ repay
  • but there are problematic firing sequences
    • card $\to$ charge $\to$ NA $\to$ no repay $\to$ finish
    • at the step of no-repay both repay and no repay are available
    • after firing no repay there's one token left in the network
  • workflow-nets-problem-il.png


Deadlocks and Livelocks

Both livelock and deadlock mean that

  • there's a situation when you cannot reach the end


Deadlock

  • workflow-nets-deadlock.png
  • once you fire $a$ nothing is enabled
  • but the workflow in not complete: there's no token in the output place
  • in this case we don't have matching joins - that leads to a deadlock


Livelock

  • workflow-nets-livelock.png
  • $C$ and $D$ can fire forever: they're always enabled
  • but the workflow will never reach the end


Dead Transitions

A dead transition

  • is a transition that can never fire

workflow-nets-dead-transition.png

  • in this case $F$ never gets activated
  • this workflow can finish successfully
  • the only fix - to remove the dead transition


Petri Nets

For Petri Nets that represent infinite processes there is a notion of correctness of there nets:

  • liveness
  • boundness
  • deadlock-free

Used notation:


Liveness

In a live petri net there are no dead transitions

  • a dead transition is a transition that can never fire in any marking reachable from the initial marking

a petri net $N$ with initial marking $M_0$ is live $\iff$

  • $\forall M: M_0 \to^* M, \forall t: \exists M \to^* M'$ in which $t$ is enabled

It reads:

  • for any marking $M$ reachable from the initial marking $M_0$
  • and for every transition $t \in T$
  • there exists a marking $M'$ reachable from $M$ s.t.
  • $t$ is enabled in $M'$


In other words:

  • no matter what happens, for any $t$ there exists a marking $M'$ reachable from the current state $M$
  • such that $t$ is enabled in it


Example

  • petri-net-liveness.png
  • the example on the left is live: no matter what happens the process continues
  • the net on the right is not live: after firing $ccc$ nothing is active anymore


Example 2

  • petri-net-liveness2.png
  • this network is deadlock-free
  • Reachability Graph: petri-net-liveness2-rg.png
  • so whatever we do there are always active transitions
  • but we cannot escape marking $[p_2]$, and $a$ is no longer live
  • petri-net-liveness3.png
  • but if we add another transition $c$ it becomes live:
  • petri-net-liveness3-rg.png


So liveness means:

  • no matter what happens, every transition $t$ should be able to fire at some point in future


Boundness

A petri net $N$ with initial marking $M_0$ is $k$-bounded iff

  • $\forall M: M_0 \to^* M, \forall p \in P: M(p) \leqslant k$
  • there never can appear more than $k$ tokens in any $p \in P$ in any reachable marking $M$

5-bounded net:

  • each $p$ can hold no more than 5 tokens


1-bounded net is called safe


Example

  • petri-net-live-unbound.png
  • in this net $a$ can fire infinitely many number of times
  • so this net is not bounded
  • but it's always live: there exists matchings for which $a$ and $b$ can fire


Example 2

  • petri-net-unboundness.png
  • not bounded: can put as many matchings as we want


Deadlock Free

a petri net $N$ with initial marking $M_0$ is deadlock-free iff

  • $\forall M: M_0 \to^* M \ \exists M': M \to M'$
  • for any reachable marking $M$ there exists another marking $M'$ that can be reached from $M$
  • i.e. every reachable marking enables some transition $t \in T$


petri-net-deadlock.png

  • after $b$ fires nothing can fire anymore
  • $a$ needs tokens from two its input places, but there's only one token


Workflow Nets Soundness

With workflow nets

  • we can decide many things statically, before enacting the workflow
  • there are some desirable properties that help to avoid these problems

A workflow net is sound $\iff$ it has

  • the option to complete
  • proper completion
  • no dead transitions


Notation:


Option to Complete

$\forall M, M_0 \to^* M: \exists M', M \to^* M': M' \geqslant [o]$

  • for all $M$ reachable from the initial marking
  • it should be possible to reach the final marking from $M$


Example:

  • workflow-nets-no-otc.png
  • deadlock = no option to complete
  • marking $[o]$ is not reachable from $[i]$


Livelock Example:

  • consider the net with a livelock
  • workflow-nets-livelock2.png
  • the following is its reachability graph
  • workflow-nets-livelock2-rg.png
  • we see that two nodes are not connected: it's clearly a problem


Proper Completion

$M_0 \to ^* M \land M \geqslant [o] \Rightarrow M \equiv [o]$

  • when $M$ is reachable from the initial marking
  • and it's great or equal to the final marking
  • for proper completion to hold $M$ should be equal to $[o]$


No Dead Transitions

$\forall t \in T, \exists M: M_0 \to^* M$ and $t$ in enabled in $M$

  • for all possible transitions there should exists at least one marking $M$ in which $t$ is enabled


Petri Net Soundness

A sound workflow net can never be live:

  • it should be able to reach the final state
  • and nothing should be able to fire from that state


A workflow net $N = (P, T, F)$ is sound iff

  • a petri net $N' = (P, T \cup \{ \alpha \}, F \cup \{ (o, \alpha), (\alpha, i) \} )$ is sound

So we just connect $o$ and $i$ via some transition $\alpha$

  • and check Petri Net soundness
  • petri-net-soundness.png


Relation:

  • if $N$ doesn't have the proper completion property, $N'$ will not be bounded
  • if there's a deadlock in $N$,
    • there exists a firing sequence in $N'$ s.t.
    • it brings you to some state where you cannot reach $o$
    • i.e. $N'$ is not live
  • dead transitions: same notion in both $N$ and $N'$


Sources

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