A Process Modelling Framework Based on Point Interval Temporal Logic with an Application to Modelling Patient Flows

This thesis considers an application of a temporal theory to describe and model the patient journey in the hospital accident and emergency (A&E) department. The aim is to introduce a generic but dynamic method applied to any setting, including healthcare. Constructing a consistent process model can be instrumental in streamlining healthcare issues. Current process modelling techniques used in healthcare such as flowcharts, unified modelling language activity diagram (UML AD), and business process modelling notation (BPMN) are intuitive and imprecise. They cannot fully capture the complexities of the types of activities and the full extent of temporal constraints to an extent where one could reason about the flows. Formal approaches such as Petri have also been reviewed to investigate their applicability to the healthcare domain to model processes. Additionally, to schedule patient flows, current modelling standards do not offer any formal mechanism, so healthcare relies on critical path method (CPM) and program evaluation review technique (PERT), that also have limitations, i.e. finish-start barrier. It is imperative to specify the temporal constraints between the start and/or end of a process, e.g., the beginning of a process A precedes the start (or end) of a process B. However, these approaches failed to provide us with a mechanism for handling these temporal situations. If provided, a formal representation can assist in effective knowledge representation and quality enhancement concerning a process. Also, it would help in uncovering complexities of a system and assist in modelling it in a consistent way which is not possible with the existing modelling techniques. The above issues are addressed in this thesis by proposing a framework that would provide a knowledge base to model patient flows for accurate representation based on point interval temporal logic (PITL) that treats point and interval as primitives. These objects would constitute the knowledge base for the formal description of a system. With the aid of the inference mechanism of the temporal theory presented here, exhaustive temporal constraints derived from the proposed axiomatic system’ components serves as a knowledge base. The proposed methodological framework would adopt a model-theoretic approach in which a theory is developed and considered as a model while the corresponding instance is considered as its application. Using this approach would assist in identifying core components of the system and their precise operation representing a real-life domain deemed suitable to the process modelling issues specified in this thesis. Thus, I have evaluated the modelling standards for their most-used terminologies and constructs to identify their key components. It will also assist in the generalisation of the critical terms (of process modelling standards) based on their ontology. A set of generalised terms proposed would serve as an enumeration of the theory and subsume the core modelling elements of the process modelling standards. The catalogue presents a knowledge base for the business and healthcare domains, and its components are formally defined (semantics). Furthermore, a resolution theorem-proof is used to show the structural features of the theory (model) to establish it is sound and complete. After establishing that the theory is sound and complete, the next step is to provide the instantiation of the theory. This is achieved by mapping the core components of the theory to their corresponding instances. Additionally, a formal graphical tool termed as point graph (PG) is used to visualise the cases of the proposed axiomatic system. PG facilitates in modelling, and scheduling patient flows and enables analysing existing models for possible inaccuracies and inconsistencies supported by a reasoning mechanism based on PITL. Following that, a transformation is developed to map the core modelling components of the standards into the extended PG (PG*) based on the semantics presented by the axiomatic system. A real-life case (from the King’s College hospital accident and emergency (A&E) department’s trauma patient pathway) is considered to validate the framework. It is divided into three patient flows to depict the journey of a patient with significant trauma, arriving at A&E, undergoing a procedure and subsequently discharged. Their staff relied upon the UML-AD and BPMN to model the patient flows. An evaluation of their representation is presented to show the shortfalls of the modelling standards to model patient flows. The last step is to model these patient flows using the developed approach, which is supported by enhanced reasoning and scheduling

Efficient Algorithms for Asymptotic Bounds on Termination Time in VASS

Vector Addition Systems with States (VASS) provide a well-known and fundamental model for the analysis of concurrent processes, parameterized systems, and are also used as abstract models of programs in resource bound analysis. In this paper we study the problem of obtaining asymptotic bounds on the termination time of a given VASS. In particular, we focus on the practically important case of obtaining polynomial bounds on termination time. Our main contributions are as follows: First, we present a polynomial-time algorithm for deciding whether a given VASS has a linear asymptotic complexity. We also show that if the complexity of a VASS is not linear, it is at least quadratic. Second, we classify VASS according to quantitative properties of their cycles. We show that certain singularities in these properties are the key reason for non-polynomial asymptotic complexity of VASS. In absence of singularities, we show that the asymptotic complexity is always polynomial and of the form $\Theta(n^k)$, for some integer $k\leq d$, where $d$ is the dimension of the VASS. We present a polynomial-time algorithm computing the optimal $k$. For general VASS, the same algorithm, which is based on a complete technique for the construction of ranking functions in VASS, produces a valid lower bound, i.e., a $k$ such that the termination complexity is $\Omega(n^k)$. Our results are based on new insights into the geometry of VASS dynamics, which hold the potential for further applicability to VASS analysis.Comment: arXiv admin note: text overlap with arXiv:1708.0925

Unboundedness Problems for Languages of Vector Addition Systems

A vector addition system (VAS) with an initial and a final marking and transition labels induces a language. In part because the reachability problem in VAS remains far from being well-understood, it is difficult to devise decision procedures for such languages. This is especially true for checking properties that state the existence of infinitely many words of a particular shape. Informally, we call these unboundedness properties. We present a simple set of axioms for predicates that can express unboundedness properties. Our main result is that such a predicate is decidable for VAS languages as soon as it is decidable for regular languages. Among other results, this allows us to show decidability of (i) separability by bounded regular languages, (ii) unboundedness of occurring factors from a language K with mild conditions on K, and (iii) universality of the set of factors

A unified view of parameterized verification of abstract models of broadcast communication

We give a unified view of different parameterized models of concurrent and distributed systems with broadcast communication based on transition systems. Based on the resulting formal models, we discuss related verification methods and tools based on abstractions and symbolic state exploration

Parameterized verification

The goal of parameterized verification is to prove the correctness of a system specification regardless of the number of its components. The problem is of interest in several different areas: verification of hardware design, multithreaded programs, distributed systems, and communication protocols. The problem is undecidable in general. Solutions for restricted classes of systems and properties have been studied in areas like theorem proving, model checking, automata and logic, process algebra, and constraint solving. In this introduction to the special issue, dedicated to a selection of works from the Parameterized Verification workshop PV \u201914 and PV \u201915, we survey some of the works developed in this research area

Decidability Issues for Petri Nets

This is a survey of some decidability results for Petri nets, covering the last three decades. The presentation is structured around decidability of specific properties, various behavioural equivalences and finally the model checking problem for temporal logics