Simulation and Visualization of Concept Algebra in MATLAB

Simulation and Visualization of Concept Algebra in MATLAB

Xiaoyu Lin (International Institute of Cognitive Informatics and Cognitive Computing (ICIC), Department of Electrical and Computer Engineering, Schulich School of Engineering, University of Calgary, Calgary, Canada & College of Computer and Information, Fujian Agriculture and Forestry University, Fuzhou, Fujian, China) and Yingxu Wang (International Institute of Cognitive Informatics and Cognitive Computing (ICIC), Department of Electrical and Computer Engineering, Schulich School of Engineering, University of Calgary, Calgary, Canada)
DOI: 10.4018/ijssci.2014010103
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Abstract

Concept algebra (CA) is a denotational mathematics for formal knowledge manipulation and natural language processing. In order to explicitly demonstrate the mathematical models of formal concepts and their algebraic operations in CA, a simulation and visualization software is developed in the MATLAB environment known as the Visual Simulator of Concept Algebra (VSCA). This paper presents the design and implementation of VSCA and the theories underpinning its development. Visual simulations for the sets of reproductive and compositional operations of CA are demonstrated by real-world examples throughout the elaborations of CA and VSCA.
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1. Introduction

A concept is a cognitive unit to identify and model a real-world concrete entity and a perceived-world abstract subject. The formal model and its discourse of abstract concepts are perceived widely different in literature (Tarski, 1944; Chomsky, 1956; Montague, 1974; Smith & Medin, 1981; Wille, 1982; Medin & Shoben, 1988; Partee et al., 1990; Murphy, 1993; Crystal, 1995; Codin et al., 1995; Bender, 1996; Pullman, 1997; Zadeh, 1999; Ganter & Wille, 1999; Wilson & Keil, 1999; O’Grady & Archibald, 2000; Taylor, 2002; Wang, 2007c, 2008b, 2009a, 2009b, 2010c, 2012c, 2013c, 2014a, 2014b; Wang et al., 2012; Tain et al., 2009, 2011; Hu et al., 2010) due to the lack of rigorous syntactic, semantic, and mathematical theories on formal concepts. Concepts in linguistics are a noun or noun-phrase that serves as the subject of a to-be statement (O’Grady & Archibald, 2000; Wang, 2007c, 2008b; Wang & Berwick, 2012, 2013). Concepts in cognitive informatics are an abstract structure that carries certain meaning in almost all cognitive processes such as thinking, learning, and reasoning (Wang, 2002, 2003, 2006, 2007b, 2007e, 2008a, 2009c, 2009d, 2009e, 2010a, 2011a, 2011b, 2012b, 2012d, 2012e, 2012f, 2012g, 2012h, 2013b; Wang & Wang, 2006; Wang & Chiew, 2010; Wang & Fariello, 2012; Wang et al., 2006, 2009a, 2009b, 2013). A typical perception on a general concept is a pair of intension and extension (Wille, 1982; O’Grady & Archibald, 2000; Wang, 2008b). Concepts in semantic algebra (Wang, 2013a) are formally modeled as a carrier of semantics (Keenan, 1975; Berners-Lee et al., 2001; Saeed, 2009; Wang, 2010b, 2010c; Wang et al., 2011) for a certain conceptual entity. In knowledge engineering and ontology, the rigorous modeling and treatment of concepts are at the center of theories for knowledge representation and manipulation (Smith & Medin, 1981; Wille, 1982; Murphy, 1993; Codin et al., 1995; Wang, 2009b, Wang & Tian, 2013; Bancroft & Wang. 2011).

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