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Computational Complexity of Alignments

Christopher T. Schwanen, Wied Pakusa, Wil M. P. van der Aalst

TL;DR

This work proves the existence of optimal alignments of polynomial length which positions the alignment problem in NP for live, bounded, free-choice systems and establishes NP-completeness on several related classes as well, including acyclic systems.

Abstract

In process mining, alignments quantify the degree of deviation between an observed event trace and a business process model and constitute the most important conformance checking technique. We study the algorithmic complexity of computing alignments over important classes of Petri nets. First, we show that the alignment problem is PSPACE-complete on the class of safe Petri nets and also on the class of safe and sound workflow nets. For live, bounded, free-choice systems, we prove the existence of optimal alignments of polynomial length which positions the alignment problem in NP for this class. We further show that computing alignments is NP-complete even on basic subclasses such as process trees and T-systems. We establish NP-completeness on several related classes as well, including acyclic systems. Finally, we demonstrate that on live, safe S-systems the alignment problem is solvable in P and that both assumptions (liveness and safeness) are crucial for this result.

Computational Complexity of Alignments

TL;DR

This work proves the existence of optimal alignments of polynomial length which positions the alignment problem in NP for live, bounded, free-choice systems and establishes NP-completeness on several related classes as well, including acyclic systems.

Abstract

In process mining, alignments quantify the degree of deviation between an observed event trace and a business process model and constitute the most important conformance checking technique. We study the algorithmic complexity of computing alignments over important classes of Petri nets. First, we show that the alignment problem is PSPACE-complete on the class of safe Petri nets and also on the class of safe and sound workflow nets. For live, bounded, free-choice systems, we prove the existence of optimal alignments of polynomial length which positions the alignment problem in NP for this class. We further show that computing alignments is NP-complete even on basic subclasses such as process trees and T-systems. We establish NP-completeness on several related classes as well, including acyclic systems. Finally, we demonstrate that on live, safe S-systems the alignment problem is solvable in P and that both assumptions (liveness and safeness) are crucial for this result.
Paper Structure (15 sections, 40 theorems, 14 equations, 4 figures, 3 tables)

This paper contains 15 sections, 40 theorems, 14 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Preliminaries
  4. Alignments
  5. Petri Nets as Language Acceptors, Workflow Nets, and the Soundness Property
  6. Synchronous Product
  7. Reachability as a Lower Bound for Alignments
  8. An Upper Bound for Alignments on Safe Petri Nets
  9. Alignments on Safe and Sound Workflow Nets
  10. An Upper Bound for Alignments on Live, Bounded, Free-Choice Systems
  11. Membership as a Lower Bound for Alignments
  12. Alignments on Process Trees
  13. Alignments on S-Systems and T-Systems
  14. Alignments on Acyclic Petri Nets
  15. Conclusion

Key Result

Proposition 4.1

Given a trace $\sigma$ and an accepting system $S$ as process model, complete firing sequences of their synchronous product correspond to alignments between $\sigma$ and $S$, i.e., $\Gamma_{\sigma,S}=\lang{\mathcal{T}(\sigma)\otimes S}$.

Figures (4)

  • Figure 1: Example of a Petri net (sound free-choice workflow net)
  • Figure 2: Two possible alignments for $w=\seq{a,b,a,a}$
  • Figure 4: Workflow net representations of different process trees $T$.
  • Figure 5: Example of safe T-systems and a bounded S-system composed of trace systems of the words $w_1=\text{PETRI}$, $w_2=\text{TURING}$, and $w_3=\text{GOEDEL}$.

Theorems & Definitions (75)

  • Definition 3.1: Multiset
  • Definition 3.2: Sequence, Permutation
  • Definition 3.3: Alphabet, Word, Language
  • Definition 3.4: Petri Net
  • Definition 3.5: Free-Choice Petri Net
  • Definition 3.6: Conflict-Free Petri Net
  • Definition 3.7: Marking, System, Firing Rule
  • Definition 3.8: Boundedness, Safeness
  • Definition 3.9: (Quasi-)Liveness
  • Definition 3.10: Home Marking, Cyclic System
  • ...and 65 more