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Inspection and Selection of Representations
Abstract
We present a novel framework for inspecting representations and encoding their formal properties. This enables us to assess and compare the informational and cognitive value of different representations for reasoning. The purpose of our framework is to automate the process of representation selection, taking into account the candidate representation’s match to the problem at hand and to the user’s specific cognitive profile. This requires a language for talking about representations, and methods for analysing their relative advantages. This foundational work is first to devise a computational end-to-end framework where problems, representations, and user’s profiles can be described and analysed. As AI systems become ubiquitous, it is important for them to be more compatible with human reasoning, and our framework enables just that.
... Being able to answer these questions will allow us to study representations more systematically than is possible currently, and in the future to ask hard questions such as: How can we choose representations to suit individuals with different levels of domain knowledge and experience of representations, for specific problems, in particular domains [24]? How can we systematically invent novel representations [7]? ...
A structural theory of visual representations and an accompanying modelling notation are outlined. The theory identifies three types of fundamental representational components, specified as cognitive schemas, that span internal mental and external physical aspects of representations. The structure of diverse and complex example representations are analyzed. Twenty-three requirements that a general theory of representations must address are presented. The new theory satisfies the large majority of them. The theory and notation may provide a basis for future methods to predict the efficacy of representations.
A cognitive theory of the interpretive structure of visual representations (RIST) was proposed by Cheng (2020), which identified four classes of schemas that specify how domain concepts are encoded by graphical objects. A notation (RISN) for building RIST models as networks of these schemas was also introduced. This paper introduces common RIST/RISN network structures – idioms – that occur across varied representations. A small-scale experiment is presented in which three participants successfully modelled their own interpretation of three diverse representations using RIST/RISN and idioms.KeywordsCognitionRepresentationsInterpretationSchemasIdioms
In this chapter, I give a personal account of my experience in Alan Bundy’s DReaM group in the Department of Artificial Intelligence at the University of Edinburgh between the years of 1995 and 1998. Of course, the impact of this experience has been profound and long-lasting to this day. The culture and the nature of research work, the collaborations, the interests and the connections have endured, evolved and multiplied throughout this time. My own work in the DReaM group started by investigating human “informal” reasoning and formalising it in a diagrammatic theorem prover. After leaving Edinburgh, this work naturally evolved into combining diagrams with other representations in a uniform framework, as well as applying visual representations in other domains, such as reasoning with ontologies. But one of the fundamental questions remained unanswered, namely, how do we choose the right representation of a problem and for a particular user in the first place?
Choosing effective representations for a problem and for the person solving it has benefits, including the ability or inability to solve it. We previously devised a novel framework consisting of a language to describe representations and computational methods to analyse them in terms of their formal and cognitive properties. In this paper we demonstrate the application of this framework to a variety of notations including natural languages, formal languages, and diagrams. We show how our framework, and the analysis of representations that it enables, gives us insight into how and why we can select representations which are appropriate for both the task and the user.
In order to effectively communicate information, the choice of representation is important. Ideally, a chosen representation will aid readers in making desired inferences. In this paper, we develop the theory of observation: what it means for one statement to be observable from another. Using observability, we give a formal characterization of the observational advantages of one representation of information over another. By considering observational advantages, people will be able to make better informed choices of representations of information. To demonstrate the benefit of observation and observational advantages, we apply these concepts to set theory and Euler diagrams. In particular, we can show that Euler diagrams have significant observational advantages over set theory. This formally justifies Larkin and Simon’s claim that “a diagram is (sometimes) worth ten thousand words”.
Representation determines how we can reason about a specific problem. Sometimes one representation helps us to find a proof more easily than others. Most current automated reasoning tools focus on reasoning within one representation. There is, therefore, a need for the development of better tools to mechanise and automate formal and logically sound changes of representation. In this paper we look at examples of representational transformations in discrete mathematics, and show how we have used tools from Isabelle’s Transfer package to automate the use of these transformations in proofs. We give an overview of a general theory of transformations that we consider appropriate for thinking about the matter, and we explain how it relates to the Transfer package. We show a few reasoning tactics we developed in Isabelle to improve the use of transformations, including the automation of search in the space of representations. We present and analyse some results of the use of these tactics.
Sledgehammer integrates automatic theorem provers in the proof assistant Isabelle/HOL. A key component, the relevance filter, heuristically ranks the thousands of facts available and selects a subset, based on syntactic similarity to the current goal. We introduce MaSh, an alternative that learns from successful proofs. New challenges arose from our "zero-click" vision: MaSh should integrate seamlessly with the users' workflow, so that they benefit from machine learning without having to install software, set up servers, or guide the learning. The underlying machinery draws on recent research in the context of Mizar and HOL Light, with a number of enhancements. MaSh outperforms the old relevance filter on large formalizations, and a particularly strong filter is obtained by combining the two filters.
The success of machine learning algorithms generally depends on data representation, and we hypothesize that this is because different representations can entangle and hide more or less the different explanatory factors of variation behind the data. Although specific domain knowledge can be used to help design representations, learning with generic priors can also be used, and the quest for AI is motivating the design of more powerful representation-learning algorithms implementing such priors. This paper reviews recent work in the area of unsupervised feature learning and deep learning, covering advances in probabilistic models, autoencoders, manifold learning, and deep networks. This motivates longer term unanswered questions about the appropriate objectives for learning good representations, for computing representations (i.e., inference), and the geometrical connections between representation learning, density estimation, and manifold learning.
Heterogeneous specification becomes more and more importan t because complex sys- tems are often specified using multiple viewpoints, involvi ng multiple formalisms. Moreover, a formal software development process may lead to a change of formalism during the development. However, current research in integrated formal methods only deals with ad-hoc integrations of different formalisms. The heterogeneous tool set (HETS) is a parsing, static analysis and proof management tool com- bining various such tools for individual specification lang uages, thus providing a tool for hetero- geneous multi-logic specification. H ETS is based on a graph of logics and languages (formalized as so-called institutions), their tools, and their transla tions. This provides a clean semantics of heterogeneous specifications, as well as a corresponding pr oof calculus. For proof management, the calculus of development graphs (known from other large-scale proof management systems) has been adapted to heterogeneous specification. Developme nt graphs provide an overview of the (heterogeneous) specification module hierarchy and the cur rent proof state, and thus may be used for monitoring the overall correctness of a heterogeneous development. We illustrate the approach with a sample heterogeneous proof proving the correctness of the com- position table of a qualitative spatial calculus. The proof involves two different provers and logics: an automated first-order prover solving the vast majority of the goals, and an interactive higher- order prover used to prove a few bridge lemmas.
This chapter provides information to notational systems that range from complex products, such as visual-programming languages, to embedded functionality, such as central-heating controls. Notational systems have four components: an interaction language or notation; an environment for editing the notation; a medium of interaction; and two kinds of subdevices-that is, helper devices and redefinition devices. The notation is what the user sees and edits-letters, musical notes, and graphic elements in computer-aided design. Some word processors have commands operating on paragraphs, sentences, words, and letters, whereas others recognize only individual characters. Some editors allow access to history, and others deny it. Typical media are paper, visual displays, and auditory displays, and the important attributes are persistence/transience and constrained-order/free-order.
This paper presents the Dr.Doodle system, an interactive theorem prover that uses diagrammatic representations. The assumption underlying this project is that, for some domains (principally geometry) , diagrammatic reasoning is easier to understand than conventional algebraic approaches -- at least for a significant number of people. The Dr.Doodle system was developed for the domain of metric-space analysis (a geometric domain, but traditionally taught using a dry algebraic formalism). Pilot experiments were conducted to evaluate its potential as the basis of an educational tool, with encouraging results.
. Theorems in automated theorem proving are usually proved by formal logical proofs. However, there is a subset of problems which humans can prove by the use of geometric operations on diagrams, so called diagrammatic proofs. Insight is often more clearly perceived in these proofs than in the corresponding algebraic proofs; they capture an intuitive notion of truthfulness that humans find easy to see and understand. We are investigating and automating such diagrammatic reasoning about mathematical theorems. Concrete, rather than general diagrams are used to prove particular concrete instances of the universally quantified theorem. The diagrammatic proof is captured by the use of geometric operations on the diagram. These operations are the "inference steps" of the proof. An abstracted schematic proof of the universally quantified theorem is induced from these proof instances. The constructive !-rule provides the mathematical basis for this step from schematic proofs to theoremhood. In ...
This paper presents a taxonomy of 19 cognitive criteria for judging what constitutes effective representational systems, particularly for knowledge rich topics. Two classes of cognitive criteria are discussed. The first concerns access to concepts by reading and making inferences from external representations. The second class addresses the generation and manipulation of external representations to fulfill reasoning or problem solving goals. Suggestions for the use of the classification are made. Examples of conventional representations and Law Encoding Diagrams for the conceptual challenging topic of particle collisions are provided throughout.
Heterogeneous reasoning refers to theorem proving with mixed diagrammatic and sentential languages and inference steps. We introduce a heterogeneous logic that enables a simple and flexible way to extend logics of existing general-purpose theorem provers with representations from entirely different and possibly not formalised domains. We use our heterogeneous logic in a framework that enables integrating different reasoning tools into new heterogeneous reasoning systems. Our implementation of this framework is MixR – we demonstrate its flexibility and extensibility with a few examples.
Quotients, subtypes, and other forms of type abstraction are ubiquitous in formal reasoning with higher-order logic. Typically, users want to build a library of operations and theorems about an abstract type, but they want to write definitions and proofs in terms of a more concrete representation type, or “raw” type. Earlier work on the Isabelle Quotient package has yielded great progress in automation, but it still has many technical limitations.
We present an improved, modular design centered around two new packages: the Transfer package for proving theorems, and the Lifting package for defining constants. Our new design is simpler, applicable in more situations, and has more user-friendly automation.
In this paper we give a short introduction in first-order theorem proving and the use of the theorem prover Vampire. We discuss the superposition calculus and explain the key concepts of saturation and redundancy elimination, present saturation algorithms and preprocessing, and demonstrate how these concepts are implemented in Vampire. Further, we also cover more recent topics and features of Vampire designed for advanced applications, including satisfiability checking, theory reasoning, interpolation, consequence elimination, and program analysis.
In this paper, we introduce Speedith which is a diagrammatic theorem prover for the language of spider diagrams. Spider diagrams are a well-known logic for which there is a sound and complete set of inference rules. Speedith provides a way to input diagrams, transform them via the diagrammatic inference rules, and prove diagrammatic theorems. It is designed as a program that plugs into existing general purpose theorem provers. This allows for seamless formal verification of diagrammatic proof steps within established proof assistants such as Isabelle. We describe the general structure of Speedith, the diagrammatic language, the automatic mechanism that draws the diagrams when inference rules are applied on them, and how formal diagrammatic proofs are constructed.
The representational epistemic approach to the design of visual displays and notation systems advocates encoding the fundamental conceptual structure of a knowledge domain directly in the structure of a representational system. It is claimed that representations so designed will benefit from greater semantic transparency, which enhances comprehension and ease of learning, and plastic generativity, which makes the meaningful manipulation of the representation easier and less error prone. Epistemic principles for encoding fundamental conceptual structures directly in representational schemes are described. The diagrammatic recodification of probability theory is undertaken to demonstrate how the fundamental conceptual structure of a knowledge domain can be analyzed, how the identified conceptual structure may be encoded in a representational system, and the cognitive benefits that follow. An experiment shows the new probability space diagrams are superior to the conventional approach for learning this conceptually challenging topic.
A grammar can be regarded as a device that enumerates the sentences of a language. We study a sequence of restrictions that limit grammars first to Turing machines, then to two types of system from which a phrase structure description of the generated language can be drawn, and finally to finite state Markov sources (finite automata). These restrictions are shown to be increasingly heavy in the sense that the languages that can be generated by grammars meeting a given restriction constitute a proper subset of those that can be generated by grammars meeting the preceding restriction. Various formulations of phrase structure description are considered, and the source of their excess generative power over finite state sources is investigated in greater detail.
The representational analysis and design project is investigating the critical role that representations have on conceptual learning in complex scientific and mathematical domains. The fundamental ideas are that the representations used for learning can substantially determine what is learnt and how easily this occurs, and that to improve conceptual learning effective representations should be found or invented. Through the conceptual analysis and empirical evaluation of a class of representations that appear to be particularly beneficial for conceptual learning, Law Encoding Diagrams (LEDs), the project has identified certain general characteristics of effective representations. In this paper a descriptive model of the components and processes of conceptual learning is presented and used for several purposes: to explain why the nature of representation used for learning is critical; to demonstrate how representations possessing the identified characteristics of effective representations appear to support the major processes of conceptual learning; to consider how computers may further enhance the potential benefit of LEDs for conceptual learning.
Multiple representations and multi-media can support learning in many different ways. In this paper, it is claimed that by identifying the functions that they can serve, many of the conflicting findings arising out of the existing evaluations of multi-representational learning environments can be explained. This will lead to more systematic design principles. To this end, this paper describes a functional taxonomy of MERs. This taxonomy is used to ask how translation across representations should be supported to maximise learning outcomes and what information should be gathered from empirical evaluation in order to determine the effectiveness of multi-representational learning environments.
HOL Light is an interactive proof assistant for classical higher-order logic, intended as a clean and simplified version of
Mike Gordon’s original HOL system. Theorem provers in this family use a version of ML as both the implementation and interaction
language; in HOL Light’s case this is Objective CAML (OCaml). Thanks to its adherence to the so-called ‘LCF approach’, the
system can be extended with new inference rules without compromising soundness. While retaining this reliability and programmability
from earlier HOL systems, HOL Light is distinguished by its clean and simple design and extremely small logical kernel. Despite
this, it provides powerful proof tools and has been applied to some non-trivial tasks in the formalization of mathematics
and industrial formal verification.
In this paper we describe the Openproof heterogeneous reasoning framework. The Openproof framework provides support for the implementation of heterogeneous reasoning environments, i.e., environments for writing arguments or proofs involving a number of different kinds of representation. The resulting environments are in a similar spirit to our Hyperproof program, though the Openproof framework goes beyond Hyperproof by providing facilities for the inclusion of a variety of representation systems in the same environment. The framework serves as the core of a number of widely used educational programs including Fitch .
ACL2 (“A Computational Logic for Applicative Common
Lisp”) is a reimplemented extended version of Boyer and Moore's
(1979, 1988) Nqthm and Kaufmann's (1988, 1990, 1992) Pc-Nqthm, intended
for large-scale verification projects. However, the logic supported by
ACL2 is compatible with the applicative subset of Common Lisp. The
decision to use an “industrial strength” programming
language as the foundation of the mathematical logic is crucial to our
advocacy of ACL2 in the application of formal methods to large systems.
However, one of the key reasons Nqthm has been so successful, we
believe, is its insistence that functions be total. Common Lisp
functions are not total and this is one of the reasons Common Lisp is so
efficient. This paper explains how we scaled up Nqthm's logic to Common
Lisp, preserving the use of total functions within the logic but
achieving Common Lisp execution speeds
We investigate several conceptions of linguistic structure to determine whether or not they can provide simple and "revealing" grammars that generate all of the sentences of English and only these. We find that no finite-state Markov process that produces symbols with transition from state to state can serve as an English grammar. Furthermore, the particular subclass of such processes that produce n -order statistical approximations to English do not come closer, with increasing n , to matching the output of an English grammar. We formalize-the notions of "phrase structure" and show that this gives us a method for describing language which is essentially more powerful, though still representable as a rather elementary type of finite-state process. Nevertheless, it is successful only when limited to a small subset of simple sentences. We study the formal properties of a set of grammatical transformations that carry sentences with phrase structure into new sentences with derived phrase structure, showing that transformational grammars are processes of the same elementary type as phrase-structure grammars; that the grammar of English is materially simplified if phrase structure description is limited to a kernel of simple sentences from which all other sentences are constructed by repeated transformations; and that this view of linguistic structure gives a certain insight into the use and understanding of language.