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Characterizing traces of processes defined by precedence and response constraints: an order theory approach

Mark Dukes, Anton Sohn

TL;DR

Characterizes traces of declarative processes under precedence and response constraints using order-theoretic tools. The authors model the constraint structure with an implied-occurrence relation $occ$ and an order-preserving relation $ord$, showing that first-passage traces correspond to linear extensions of induced posets. The main theoretical contribution is a complete classification: $Traces(D) = \bigcup_{I \in PossIm(D)} \mathcal{L}((I, ord_I))$, where $PossIm(D)$ is the down-sets of $(\Sigma, occ)$ with antisymmetric $ord_I$. They also provide implementation guidance, discuss special cases, and connect the framework to stakeholder utility metrics, enabling exact trace counts and comparisons in scheduling, databases, and verification contexts.

Abstract

In this paper we consider a general system of activities that can, but do not have to, occur. This system is governed by a set containing two types of constraints: precedence and response. A precedence constraint dictates that an activity can only occur if it has been preceded by some other specified activity. Response constraints are similarly defined. An execution of the system is a listing of activities in the order they occur and which satisfies all constraints. These listings are known as traces. Such systems naturally arise in areas of theoretical computer science and decision science. An outcome of the freedom with which activities can occur is that there are many different possible executions, and gaining a combinatorial insight into these is a non-trivial problem. We characterize all of the ways in which such a system can be executed. Our approach uses order theory to provide a classification in terms of the linear extensions of posets constructed from the constraint sets. This characterization is essential in calculating the stakeholder utility metrics that have been developed by the first author that allow for quantitative comparisons of such systems/processes. It also allows for a better understanding of the theoretical backbone to these processes.

Characterizing traces of processes defined by precedence and response constraints: an order theory approach

TL;DR

Characterizes traces of declarative processes under precedence and response constraints using order-theoretic tools. The authors model the constraint structure with an implied-occurrence relation and an order-preserving relation , showing that first-passage traces correspond to linear extensions of induced posets. The main theoretical contribution is a complete classification: , where is the down-sets of with antisymmetric . They also provide implementation guidance, discuss special cases, and connect the framework to stakeholder utility metrics, enabling exact trace counts and comparisons in scheduling, databases, and verification contexts.

Abstract

In this paper we consider a general system of activities that can, but do not have to, occur. This system is governed by a set containing two types of constraints: precedence and response. A precedence constraint dictates that an activity can only occur if it has been preceded by some other specified activity. Response constraints are similarly defined. An execution of the system is a listing of activities in the order they occur and which satisfies all constraints. These listings are known as traces. Such systems naturally arise in areas of theoretical computer science and decision science. An outcome of the freedom with which activities can occur is that there are many different possible executions, and gaining a combinatorial insight into these is a non-trivial problem. We characterize all of the ways in which such a system can be executed. Our approach uses order theory to provide a classification in terms of the linear extensions of posets constructed from the constraint sets. This characterization is essential in calculating the stakeholder utility metrics that have been developed by the first author that allow for quantitative comparisons of such systems/processes. It also allows for a better understanding of the theoretical backbone to these processes.
Paper Structure (15 sections, 14 theorems, 28 equations, 2 figures, 1 table)

This paper contains 15 sections, 14 theorems, 28 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Preorders and partial orders
  4. Linear extensions
  5. Independence systems
  6. Precedence-and-response-only constraint sets
  7. Implementation discussion
  8. Preorders and posets
  9. Generating $\mathsf{PossIm}(D)$
  10. Generating the linear extensions
  11. Further remarks
  12. Special Cases
  13. Precedence-only constraint sets
  14. Response-only constraint sets
  15. Successor-only constraint sets

Key Result

Lemma 3.7

If $\tau \in \mathsf{Traces}(D)$ then $\mathrm{im}(\tau)$ is a down-set of $(\Sigma, {\overset{occ}{\lesssim} })$.

Figures (2)

  • Figure 1: Hasse diagram for the preordered set from Example \ref{['prec:resp:example']}
  • Figure 2: Hasse diagram of the preordered set from Example \ref{['prec_example']}

Theorems & Definitions (43)

  • Example 2.1
  • Example 2.2
  • Example 3.1
  • Definition 3.2
  • Definition 3.4
  • Example 3.6
  • Lemma 3.7
  • proof
  • Definition 3.8
  • Lemma 3.9
  • ...and 33 more