Critical Abstraction : Generating Simplest Models for Causal Explanation

Central to most scientific and engineering tasks design, diagnosis, analogy and design rationale capture are descriptions of a device that capture its salient features with respect to how it works . These descriptions are used as the central focus of many engineering reasoning techniques . Yet little has been said about what constitutes a best, or even adequate, description . Likewise, related questions, such as "what is an interesting qualitative landmark?" and "what constitutes a simplest model?" have been left unanswered . Each is an instance of the more general question : what constitutes a goold abstraction? While current reasoning systems are effective users of qualitative abstractions, future systems must also be able to create these abstractions . We take the perspective that a device works by constructing a topology of interactions between quantities . To construct a parsimonious description of an interaction topologywe introduce the concept of a critical abstractiona most abstract description of a topology relative to a set of queries, that preserves the link between the individual mechanisms of a device, and the behaviors mentioned in the queries . We demonstrate critical abstraction for equational descriptions of interactions . We present a generative modelling technique for computing critical topologies using only the mathematical properties of the interaction topology representation . The use of causal connections during all phases of abstraction highlights the central link between causality and what is interesting . Capturing and reasoning about how devices work are central to most scientific and engineering reasoning . For example, when looking for innovative solutions the designer searches among devices that work in qualitatively different ways[19] . Given a faulty device, a diagnostician hypothesizes mechanisms that could account for qualitative differences between 77 the artifacts observed and expected behavior[6] . And to help account for new phenomena a scientist searches for known mechanisms that work analogously[7] . A central focus of all these tasks are abstract accounts of how a device works. Up to now our reasoning systems have been users of abstractions . Past research has concentrated on the effective use of particular qualitative representations for some task . A key direction for the future are systems that create new abstractions . An account of how a device works can be represented in many ways; for example, as a system of qualitative equations, as a causal diagram or as a time-varying set of histories . Our long term goal is a unified theory of abstraction that builds on our experience with these different types of representations [16, 17, 18, 19] . The theory must provide a criteria for what constitutes a good abstraction and techniques for generating these abstractions for the various representations . In this paper we introduce one such criteria, called critical abstraction, and a generation technique for equational descriptions, called generative modelling . A good abstraction highlights only features of interest, suppressing all superfluous detail . In our experience what is qualitatively and temporally interesting is inextricably tied to the concepts of causality and local interaction . Causality has taken a back seat in most current qualitative reasoning research . In this paper we show that causality plays a central role in all stages of the abstraction process . The remainder of this introduction provides an overview of what constitutes an account of how a device works, and its overall impact on the abstraction process. Such an account establishes the role played by each of a device's internal mechanisms, with respect to achiev ing the behavior of interest . This link is crucial, for example, during diagnosis to pinpoint faulty mechanisms, or during design modification to identify the appropriate mechanisms to be changed. Intuitively, for continuous systems a "device works" by establishing a network of local interactions between quantities, and modulating these interactions over time.' Each of these local interactions are produced by basic mechanisms such as processes, components and their interconnections . And the modulation of interactions produces such effects as raising and lowering signals, changing the operating modes of components, and, more generally, pushing the device between different regions of its state space. We refer to such a network as an interaction topology [19] . What description of an interaction topology is appropriate? A system of quantitative equations, such as those from physical system dynamics, would be overly detailed . To capture how a device works these quantitative interactions must be abstracted in a way that captures exactly those properties that are essential to achieving the behavior of interest . But in a way that assigns responsibility for each abstract interaction to a single mechanism . When all but the essential properties of the interactions are eliminated we say that the topology is critically abstracted, and we refer to the result as a critical topology . A critical topology is by no means an absolute concept . What internal features of interactions are interesting depends on certain external features of interest for particular variables, 'The reason we use the term interaction, for example as opposed to equation, is to distinguish between the representation of an object and the object being represented . An interaction is an entity in the world (or abstraction) that we are trying to model or describe . An equation is one of many possible ways of describincr an interaction .