Simulating Problem Difficulty in Arithmetic Cognition Through Dynamic Connectionist Models

The present study aims to investigate similarities between how humans and connectionist models experience difficulty in arithmetic problems. Problem difficulty was operationalized by the number of carries involved in solving a given problem. Problem difficulty was measured in humans by response time, and in models by computational steps. The present study found that both humans and connectionist models experience difficulty similarly when solving binary addition and subtraction. Specifically, both agents found difficulty to be strictly increasing with respect to the number of carries. Another notable similarity is that problem difficulty increases more steeply in subtraction than in addition, for both humans and connectionist models. Further investigation on two model hyperparameters --- confidence threshold and hidden dimension --- shows higher confidence thresholds cause the model to take more computational steps to arrive at the correct answer. Likewise, larger hidden dimensions cause the model to take more computational steps to correctly answer arithmetic problems; however, this effect by hidden dimensions is negligible.Comment: 7 pages; 15 figures; 5 tables; Published in the proceedings of the 17th International Conference on Cognitive Modelling (ICCM 2019

The Challenge of Modeling the Acquisition of Mathematical Concepts

As a full-blown research topic, numerical cognition is investigated by a variety of disciplines including cognitive science, developmental and educational psychology, linguistics, anthropology and, more recently, biology and neuroscience. However, despite the great progress achieved by such a broad and diversified scientific inquiry, we are still lacking a comprehensive theory that could explain how numerical concepts are learned by the human brain. In this perspective, I argue that computer simulation should have a primary role in filling this gap because it allows identifying the finer-grained computational mechanisms underlying complex behavior and cognition. Modeling efforts will be most effective if carried out at cross-disciplinary intersections, as attested by the recent success in simulating human cognition using techniques developed in the fields of artificial intelligence and machine learning. In this respect, deep learning models have provided valuable insights into our most basic quantification abilities, showing how numerosity perception could emerge in multi-layered neural networks that learn the statistical structure of their visual environment. Nevertheless, this modeling approach has not yet scaled to more sophisticated cognitive skills that are foundational to higher-level mathematical thinking, such as those involving the use of symbolic numbers and arithmetic principles. I will discuss promising directions to push deep learning into this uncharted territory. If successful, such endeavor would allow simulating the acquisition of numerical concepts in its full complexity, guiding empirical investigation on the richest soil and possibly offering far-reaching implications for educational practice

Building expert systems: cognitive emulation.

Chapter 1 briefly introduces the concept of cognitive emulation, and outlines its current status. Chapter 2 reviews psychological research on human expert thinking. First, the study of expert thinking is placed in the context of modern cognitive psychology. Next, the principal methods and techniques employed by psychologists examining expert cognition are examined. The remainder of the chapter is given over to a review of the published literature on the nature and development of human expertise. Chapter 3 reviews the main arguments for and against cognitive emulation in expert system design. The tentative conclusion reached is that a significant degree of emulation is inevitable, but that a pure, unselective strategy of emulation is neither realistic nor desirable. Chapter 4 examines the prospects for cognitive emulation from a more pragmatic angle. Several factors are identified that represent constraints on the usefulness of a cognitive approach. However, a second set of factors is identified which should facilitate an emulation strategy - especially in the longer term. Some guidance is given on when to seriously consider adopting an emulation strategy. Chapter 5 presents a critical survey of expert system research that has already addressed the emulation issue. Six basic approaches to cognitive emulation are distinguished and evaluated. This helps draw out in more detail the implications of an emulation strategy for knowledge acquisition, knowledge representation and system architecture. The chapter concludes by discussing the issues that arise when different approaches to emulation are combined. Some guidance is offered on how this might be achieved. Chapter 6 summarizes the main themes and issues to have emerged, the design advice contained in the thesis, and the original contributions made by the thesis

Formal Modeling of Connectionism using Concurrency Theory, an Approach Based on Automata and Model Checking

This paper illustrates a framework for applying formal methods techniques, which are symbolic in nature, to specifying and verifying neural networks, which are sub-symbolic in nature. The paper describes a communicating automata [Bowman & Gomez, 2006] model of neural networks. We also implement the model using timed automata [Alur & Dill, 1994] and then undertake a verification of these models using the model checker Uppaal [Pettersson, 2000] in order to evaluate the performance of learning algorithms. This paper also presents discussion of a number of broad issues concerning cognitive neuroscience and the debate as to whether symbolic processing or connectionism is a suitable representation of cognitive systems. Additionally, the issue of integrating symbolic techniques, such as formal methods, with complex neural networks is discussed. We then argue that symbolic verifications may give theoretically well-founded ways to evaluate and justify neural learning systems in the field of both theoretical research and real world applications

Automatic Generation of Cognitive Theories using Genetic Programming

Cognitive neuroscience is the branch of neuroscience that studies the neural mechanisms underpinning cognition and develops theories explaining them. Within cognitive neuroscience, computational neuroscience focuses on modeling behavior, using theories expressed as computer programs. Up to now, computational theories have been formulated by neuroscientists. In this paper, we present a new approach to theory development in neuroscience: the automatic generation and testing of cognitive theories using genetic programming. Our approach evolves from experimental data cognitive theories that explain “the mental program” that subjects use to solve a specific task. As an example, we have focused on a typical neuroscience experiment, the delayed-match-to-sample (DMTS) task. The main goal of our approach is to develop a tool that neuroscientists can use to develop better cognitive theories