Characterising a problem in terms of a system of equations is common to many branches of science and engineering. Due to their size, such systems are often described in a modular fashion by composition of individual equation system fragments. Checking the balance between the number of variables (unknowns) and equations is a common approach to early detection of mistakes that might render such a system unsolvable. However, current approaches to modular balance checking have a number of limitations. This paper investigates a more flexible approach that makes it possible to treat equation system fragments as true first-class entities. Furthermore, the approach handles so-called structurally dynamic systems, systems whose behaviour changes discretely and abruptly over time. The central idea is to record balance information in the type of an equation fragment. This information can then be used to determine if individual fragments are well formed, and if composing fragments preserves this property. The type system presented in this paper is developed in the context of Functional Hybrid Modelling (FHM). However, the key ideas are in no way specific to FHM, but should be applicable to any language featuring a notion of modular systems of equations, including systems with first-class components and structural dynamism.

Modelling and simulation languages are evolving rapidly to support modelling of systems of ever increasing size and complexity. A relatively recent development in the area of physical modelling is the noncausal modelling languages. They support a declarative, highly modular modelling approach, promoting the reuse of components. Modelica is a prime example of this class of languages. However, the mainstream representatives of this class of languages provide limited support for higher-order modelling and structurally dynamic systems. Moreover, the semantics of this class of languages remains a relatively unexplored area. Functional Hybrid Modelling (FHM) is a novel approach to noncausal, hybrid modelling that aims to address these concerns. In this paper, we give a semantics to the discrete part of a simple FHM language expressed in dependent type theory. We use Normalisation by Evaluation to produce a type-preserving and terminating normalisation procedure, the latter property being particularly important for FHM as highly structurally dynamic systems are supported by computing new system configurations during simulation. Furthermore, our implementation has been carefully structured to allow continuous aspects of the semantics to be described separately, in whatever way is deemed appropriate, while retaining the ability to describe precisely how a system evolves in response to discrete events.

This progress report is about equation-based modelling languages. In particular, the report discusses work [4] on formalising semantic aspects of Functional Hybrid Modelling (FHM) [12] – a hybrid, structurally dynamic, non-causal modelling language – in the dependently-typed programming language Agda [13]. Section 2 gives a preliminary outline of equation-based languages and FHM. Section 3 gives an overview on the current status of the aforementioned work being conducted by Capper and Nilsson. Finally, Sect. 4 discusses potential future work.

Describing a problem in terms of a system of equations is common to many branches of science and engineering. Due to their size and complexity, constructing and solving such systems often warrants the use of computers. A class of equation-based languages has grown out of the desire to describe equation systems using the wellunderstood and mature techniques from conventional programming language theory. Just as with many conventional programming language, the correctness of equation-based programs can be, to at least some degree, mechanically verified by ensuring certain simple invariants are upheld. The existence of certain structural properties related to solvability are of particular interest. In this report we present a preliminary investigation into the role types can play in equation-based languages. We also discuss ongoing work into a novel refinement type system that is able to capture a number of simple structural properties that might be indicative of problems in a modular system of equations.

Characterising a problem in terms of a system of equations is common to many branches of science and engineering. Due to their size, such systems are often described in a modular fashion by composition of individual equation system fragments. Checking the balance between the number of variables (unknowns) and equations is a common approach to early detection of mistakes that might render such a system unsolvable. However, current approaches to modular balance checking have a number of limitations. This paper investigates a more flexible approach that in particular makes it possible to treat equation system fragments as true first-class entities. The central idea is to record balance information in the type of an equation fragment. This information can then be used to determine if individual fragments are well formed, and if composing fragments preserves this property. The type system presented in this paper is developed in the context of Functional Hybrid Modelling (FHM). However, the key ideas are in no way specific to FHM, but should be applicable to any language featuring a notion of modular systems of equations.

Equation-based modelling languages have become a vital tool in many areas of science and engineering. Functional Hybrid Modelling (FHM) is an approach to equation-based modelling that allows the behaviour of a physical system to be expressed as a modular hierarchy of undirected equations. FHM supports a variety of advanced language features — such as higher-order models and variable system structure — that sets it apart from the majority of other modelling languages. However, the inception of these new features has not been accompanied by the semantic tools required to effectively use and understand them. Specifically, there is a lack of static safety assurances for dynamic models and the semantics of the aforementioned language features are poorly understood. Static safety guarantees are highly desirable as they allow problems that may cause an equation system to become unsolvable to be detected early, during compilation. As a result, the use of static analysis techniques to enforce structural invariants (e.g. that there are the same number of equations as unknowns) is now in use in main-stream equation-based languages like Modelica. Unfortunately, the techniques employed by these languages are somewhat limited, both in their capacity to deal with advanced language features and also by the spectrum of invariants they are able to enforce. Formalising the semantics of equation-based languages is also important. Semantics allow us to better understand what a program is doing during execution, and to prove that this behaviour meets with our expectation. They also allow different implementations of a language to agree with one another, and can be used to demonstrate the correctness of a compiler or interpreter. However, current attempts to formalise such semantics typically fall short of describing advanced features, are not compositional, and/or fail to show correctness. This thesis provides two major contributions to equation-based languages. Firstly, we develop a refined type system for FHM capable of capturing a larger number of structural anomalies than is currently possible with existing methods. Secondly, we construct a compositional semantics for the discrete aspects of FHM, and prove a number of key correctness properties.