Analysis of Boolean Equation Systems through Structure Graphs

We analyse the problem of solving Boolean equation systems through the use of structure graphs. The latter are obtained through an elegant set of Plotkin-style deduction rules. Our main contribution is that we show that equation systems with bisimilar structure graphs have the same solution. We show that our work conservatively extends earlier work, conducted by Keiren and Willemse, in which dependency graphs were used to analyse a subclass of Boolean equation systems, viz., equation systems in standard recursive form. We illustrate our approach by a small example, demonstrating the effect of simplifying an equation system through minimisation of its structure graph

Modelling & analysis of hybrid dynamic systems using a bond graph approach

Hybrid models are those containing continuous and discontinuous behaviour. In constructing dynamic systems models, it is frequently desirable to abstract rapidly changing, highly nonlinear behaviour to a discontinuity. Bond graphs lend themselves to systems modelling by being multi-disciplinary and reflecting the physics of the system. One advantage is that they can produce a mathematical model in a form that simulates quickly and efficiently. Hybrid bond graphs are a logical development which could further improve speed and efficiency. A range of hybrid bond graph forms have been proposed which are suitable for either simulation or further analysis, but not both. None have reached common usage. A Hybrid bond graph method is proposed here which is suitable for simulation as well as providing engineering insight through analysis. This new method features a distinction between structural and parametric switching. The controlled junction is used for the former, and gives rise to dynamic causality. A controlled element is developed for the latter. Dynamic causality is unconstrained so as to aid insight, and a new notation is proposed. The junction structure matrix for the hybrid bond graph features Boolean terms to reflect the controlled junctions in the graph structure. This hybrid JSM is used to generate a mixed-Boolean state equation. When storage elements are in dynamic causality, the resulting system equation is implicit. The focus of this thesis is the exploitation of the model. The implicit form enables application of matrix-rank criteria from control theory, and control properties can be seen in the structure and causal assignment. An impulsive mode may occur when storage elements are in dynamic causality, but otherwise there are no energy losses associated with commutation because this method dictates the way discontinuities are abstracted. The main contribution is therefore a Hybrid Bond Graph which reflects the physics of commutating systems and offers engineering insight through the choice of controlled elements and dynamic causality. It generates a unique, implicit, mixed-Boolean system equation, describing all modes of operation. This form is suitable for both simulation and analysis

Structural Analysis of Boolean Equation Systems

We analyse the problem of solving Boolean equation systems through the use of structure graphs. The latter are obtained through an elegant set of Plotkin-style deduction rules. Our main contribution is that we show that equation systems with bisimilar structure graphs have the same solution. We show that our work conservatively extends earlier work, conducted by Keiren and Willemse, in which dependency graphs were used to analyse a subclass of Boolean equation systems, viz., equation systems in standard recursive form. We illustrate our approach by a small example, demonstrating the effect of simplifying an equation system through minimisation of its structure graph

Construction and analysis of causally dynamic hybrid bond graphs

Engineering systems are frequently abstracted to models with discontinuous behaviour (such as a switch or contact), and a hybrid model is one which contains continuous and discontinuous behaviours. Bond graphs are an established physical modelling method, but there are several methods for constructing switched or ‘hybrid’ bond graphs, developed for either qualitative ‘structural’ analysis or efficient numerical simulation of engineering systems. This article proposes a general hybrid bond graph suitable for both. The controlled junction is adopted as an intuitive way of modelling a discontinuity in the model structure. This element gives rise to ‘dynamic causality’ that is facilitated by a new bond graph notation. From this model, the junction structure and state equations are derived and compared to those obtained by existing methods. The proposed model includes all possible modes of operation and can be represented by a single set of equations. The controlled junctions manifest as Boolean variables in the matrices of coefficients. The method is more compact and intuitive than existing methods and dispenses with the need to derive various modes of operation from a given reference representation. Hence, a method has been developed, which can reach common usage and form a platform for further study

Boolean delay equations on networks: An application to economic damage propagation

We introduce economic models based on Boolean Delay Equations: this formalism makes easier to take into account the complexity of the interactions between firms and is particularly appropriate for studying the propagation of an initial damage due to a catastrophe. Here we concentrate on simple cases, which allow to understand the effects of multiple concurrent production paths as well as the presence of stochasticity in the path time lengths or in the network structure. In absence of flexibility, the shortening of production of a single firm in an isolated network with multiple connections usually ends up by attaining a finite fraction of the firms or the whole economy, whereas the interactions with the outside allow a partial recovering of the activity, giving rise to periodic solutions with waves of damage which propagate across the structure. The damage propagation speed is strongly dependent upon the topology. The existence of multiple concurrent production paths does not necessarily imply a slowing down of the propagation, which can be as fast as the shortest path.Comment: Latex, 52 pages with 22 eps figure

Invariant Generation through Strategy Iteration in Succinctly Represented Control Flow Graphs

We consider the problem of computing numerical invariants of programs, for instance bounds on the values of numerical program variables. More specifically, we study the problem of performing static analysis by abstract interpretation using template linear constraint domains. Such invariants can be obtained by Kleene iterations that are, in order to guarantee termination, accelerated by widening operators. In many cases, however, applying this form of extrapolation leads to invariants that are weaker than the strongest inductive invariant that can be expressed within the abstract domain in use. Another well-known source of imprecision of traditional abstract interpretation techniques stems from their use of join operators at merge nodes in the control flow graph. The mentioned weaknesses may prevent these methods from proving safety properties. The technique we develop in this article addresses both of these issues: contrary to Kleene iterations accelerated by widening operators, it is guaranteed to yield the strongest inductive invariant that can be expressed within the template linear constraint domain in use. It also eschews join operators by distinguishing all paths of loop-free code segments. Formally speaking, our technique computes the least fixpoint within a given template linear constraint domain of a transition relation that is succinctly expressed as an existentially quantified linear real arithmetic formula. In contrast to previously published techniques that rely on quantifier elimination, our algorithm is proved to have optimal complexity: we prove that the decision problem associated with our fixpoint problem is in the second level of the polynomial-time hierarchy.Comment: 35 pages, conference version published at ESOP 2011, this version is a CoRR version of our submission to Logical Methods in Computer Scienc