Monday, November 1, 2010

Categorical Compositionality, Prototype Theory, Cognitive Science, Human-Computer Interaction, Social Genres, Noetic Communes, Universal Grammar

(Social and Cognitive Processes in Knowledge Acquisition)

Figure 8 Dimensions of evaluation of a knowledge-based system
The interactions of the system with some actual world in which it is intended to perform decision or control tasks introduce two dimensions:

• Correspondence–to what extent does the knowledge in the system correspond to actuality?
• Application–to what extend does the system perform the intended task?

These are major dimensions in conventional systems analysis and in the evaluation of a scientific or technological system. The assumption in expert systems development is that these may be difficult, or impossible, criteria to apply because of the nature of the problem. For knowledge-based systems in general, however, these conventional standards are still very important.
The interactions of the system with some mental world, for example that of the clients, for whom it is intended to perform decision or control tasks introduce two dimensions:

• Comprehension–to what extent is the system able to understand information and queries?
• Explanation–to what extend is the system able to explain its activities?

These are important dimensions in any system supporting human activities. There have been attempts to satisfy them in most knowledge-based system development from MYCIN/TEIRESIAS onwards. Conventional data-processing has often been weak in these areas.
The interactions of the system with some knowledge world underlying its performance of decision or control tasks introduce two dimensions:

• Derivation–to what extent can the system derive its required knowledge from well-established sources?
• Creation–to what extend is the system able to generate new knowledge?

The rational reconstruction of knowledge to show, regardless of its source, how it could have been derived from well-established sources is important in expert systems development. We would prefer the minimal content to be dependent on the authority of experts as opposed to their use of professional knowledge. Similarly, while the creation of new knowledge is not the primary focus of system development, it is one of the most valued achievements–"we now think about the problem in different terms" is a significant positive comment resulting from a knowledge-based system development–it would be even more significant if it were a result of system activity.

There are three ancillary dimensions in Figure 8 relating to interactions between the worlds. Expert performance corresponds to a relation between world 2 and one of the others–the link shown is to world 1. We can evaluate the behavior of the system against the behavior of experts. This is a pragmatic evaluation based on the known utility of that expert behavior.
Expert knowledge corresponds to a relation between world 2 and world 3. We can evaluate the rationale of the system against the rationale of experts. This is a hermeneutic evaluation based on respect for the conceptual framework in which the experts perceive themselves as operating.
The objective knowledge of positive science is a relation between world 1 and world 2 that combines correspondence, derivation and explanation. It is usually not applicable in the situations for which expert systems are being developed. However, the concepts and techniques underlying knowledge-based systems in general should have a seamless interface to this highly refined form of knowledge.

The Structures of Computation and the Mathematical Structure of Nature
Michael S. Mahoney
Princeton University

In 1948 John von Neumann expressed the need for a theory of automata to take account of the possibilities and limits of the electronic digital computer, but he could not specify the mathematical form of such a theory might take. Over the next two decades, theoretical computer science emerged from the intersection of the research agendas of a wide range of established fields, including mathematics. By 1970 theoretical computer science had its own rubric in Mathematical Reviews, its own textbooks, and its own research agenda, and its concepts and techniques were becoming a resource for other sciences. The talk will describe the main lines of this process, which took place in two stages corresponding to a continuing division in the subject. Beginning in the late 1950s, automata theory took shape as the quasi-algebraic study of formal languages and the machines that recognize them, and of computational complexity as measured by bounded Turing machines. From the early 1960s formal semantics grew out of concern with extensible programming languages, which brought out in striking form the defining characteristic of the stored-program computer, namely that functions and values occupy the same data space. In the process, theoretical computer science gave meaning to the seemingly paradoxical notion of "applied abstract algebra", as it brought the most advanced concepts of twentieth-century mathematics to bear on what has become the defining technology of our time.

In 1984 Springer Verlag added to its series, Undergraduate Texts in Mathematics, a work with the seemingly oxymoronic title Applied Abstract Algebra, by Rudolf Lidl and Gunter Pilz. To the community of practitioners the work may have seemed, as the cliché puts it, "long overdue"; indeed, it was one of a spate of similar works that appeared at the time, evidently filling a perceived need. To the historian of modern mathematics, it appears remarkable on several counts, not least that it was meant for undergraduates. Less than fifty years earlier, Garrett Birkhoff's father had asked him what use his new theory of lattices might possibly have, and many non-algebraists had wondered the same thing about the subject as a whole. Applied Abstract Algebra opens with a chapter on lattices, which form a recurrent theme throughout the book. Subsequent chapters present rings, fields, and semigroups. Interspersed among those topics and motivating their organization are the applications that give the book its title and purpose. They include switching circuits, codes, cryptography, automata, formal languages: in short, problems arising out of computing or intimately related to it.

The Import and Export of Cognitive Science

From its inception, a large part of the motivation for Cognitive Science has been the need for an interdisciplinary journal for the study of minds and intelligent systems. In the inaugural editorial for the journal, Allan Collins (1977) wrote “Current journals are fragmented along old disciplinary lines, so there is no common place for workers who approach these problems from different disciplines to talk to each other” (p. 1). The interdisciplinarity of the journal has served a valuable cross-fertilization function for those who read the journal to discover articles written for and by practitioners across a wide range of fields. The challenges of building and understanding intelligent systems are sufficiently large that they will most likely require the skills of psychologists, computer scientists, philosophers, educators, neuroscientists, and linguists collaborating and coordinating their efforts.

One threat to the interdisciplinarity of Cognitive Science, both the field and journal, is that it may become, or already be, too dominated by psychologists (Schunn, Crowley, & Okada, 1998; Von Eckardt, 2001). One piece of evidence supporting this contention is that many of the manuscripts submitted to Cognitive Science are given “psychology” as field keyword by their authors. In 2005, psychology was a keyword for 51% of submissions, followed distantly by linguistics (17%), artificial intelligence (13%), neuroscience (10%), computer science (9%), and philosophy (8%) (these percentages sum to more than 100% because authors are not restricted to designating only a single field).

Another quantitative way to assess the interdisciplinarity of Cognitive Science as well as its general intellectual niche is to analyze aggregated journal-journal citations. The Institute for Scientific Information (ISI) gathers data not only on how individual articles cite one another, but also on macroscopic citation patterns among journals. Journals or sets of journals can be considered as proxies for fields. As fields become established, they often create journals (Leydesdorff, Cozzens, & Van den Besselaar, 1994). As Collins (1977) wrote when launching Cognitive Science, “In starting the journal we are just adding another trapping in the formation of a new discipline” (p. 1). By studying the patterns of citations among journals that cite and are cited by Cognitive Science, we can better: 1) appreciate the scholarly ecology surrounding the journal and the journal’s role within this ecology, 2) establish competitor and alternate journals, and 3) determine the natural clustering of fields related to cognitive science (Leydesdorff, 2006; forthcoming).

Modeling Cognitive Systems with Category Theory:

ACCAT was the name of a working group established during my time at RISC-Linz, Univ. Linz (1990 - 1996). The origins were two courses, given in the frame of the RISC-Linz curriculum: "Categories, Fiberings, Sheaves, and Topoi" ; "Computational Category Theory (with ML)". CAT abbreviates Category Theory and since I was always interested in applications and in computational aspects I introduced the title Applied and Computational Category Theory (ACCAT), this reflects what I am finally interested in.

Applied and Computational Category Theory (ACCAT): Its Origins and Perspectives Jochen Pfalzgraf

"What Might Category Theory do for Artificial Intelligence and Cognitive Science?

Apr 25, 2010

In August 2008, I posted a series of answers to the question "Why should we be interested in category theory?" on the A Categorical Manifesto thread in the n-Category Café blog. Category theory is a mathematical tool often used to elucidate similarities between apparently unrelated pieces of mathematics. I suggested it could do the same for AI and cognitive science, and discussed examples that include neural nets, holographic reduced representations, Prolog-stye unification, analogical reasoning, and understanding metaphors. Here is the same posting, with an informal explanation of category theory added, and the rest made intelligible (I hope) to non-category-theorists."

The Institute for Research in Cognitive Science at the University of Pennsylvania fosters the development of a science of the human mind through the interaction of investigators from the disciplines of Linguistics, Mathematical Logic, Philosophy, Computer Science, and Neuroscience.

"Category theory also bears on more general philosophical questions. From the foregoing disussion, it should be obvious that category theory and categorical logic ought to have an impact on almost all issues arising in philosophy of logic: from the nature of identity criteria to the question of alternative logics, category theory always sheds a new light on these topics. Similar remarks can be made when we turn to ontology, in particular formal ontology: the part/whole relation, boundaries of systems, ideas of space, etc. Ellerman (1988) has bravely attempted to show that category theory constitutes a theory of universals, one having properties radically different from set theory, which is also seen as a theory of universals. Moving from ontology to cognitive science, MacNamara & Reyes (1994) have tried to employ categorical logic to provide a different logic of reference. In particular, they have attempted to clarify the relationships between count nouns and mass terms. Other researchers are using category theory to study complex systems, cognitive neural networks, and analogies. (See, for instance, Ehresmann & Vanbremeersch 1987, 2007, Healy 2000, Healy & Caudell 2006, Arzi-Gonczarowski 1999.)"

"Prototype theory is a mode of graded categorization in cognitive science, where some members of a category are more central than others. For example, when asked to give an example of the concept furniture, chair is more frequently cited than, say, stool. Prototype theory also plays a central role in linguistics, as part of the mapping from phonological structure to semantics.As formulated in the 1970s by Eleanor Rosch and others, prototype theory was a radical departure from traditional necessary and sufficient conditions as in Aristotelian logic, which led to set-theoretic approaches of extensional or intensional semantics. Thus instead of a definition based model - e.g. a bird may be defined as elements with the features [+feathers], [+beak] and [+ability to fly], prototype theory would consider a category like bird as consisting of different elements which have unequal status - e.g. a robin is more prototypical of a bird than, say a penguin. This leads to a graded notion of categories, which is a central notion in many models of cognitive science and cognitive semantics, e.g. in the work of George Lakoff (Women, Fire and Dangerous Things, 1987) or Ronald Langacker (Foundations of Cognitive Grammar, vol. 1/2 1987/1991)."

Piagetian Roboethics via Category Theory: Moving Beyond Mere Formal Operations to Engineer Robots Whose Decisions are Guaranteed to be Ethically Correct ∗

"My research is in the mathematical semantics of biological and computational systems. The mathematical discipline involved is principally category theory, together with formal logic and topology in the categorical context. With the exception of formal logic, which is applied in the form of theorem-proving software and other formal methods by computer scientists, the use of these disciplines outside mathematics itself is still regarded as a novelty. Yet, they are an excellent fit for studying the structure of the knowledge underlying the structure and processing flow in biological and computational systems. This knowledge can be regarded as an expression of the semantics of systems."

Toward a Category Theory Design of Ontological Knowledge Bases:

"We are a group of people who have been deeply influenced by the works of Robert Rosen* in theoretical biology, systems science and epistemology. Constructivism is a somewhat more familiar approach to those in the social sciences. Located... at VCU we had discussions, graduate seminars, and honors modules focusing on Rosen's work in the main. This stage of activity has now become dormant with the retirement of Don Mikulecky. If we seek to explore these ideas further, it will be necessary to continue here in cyberspace or in the new Complexity Center at VCU. We hope to achieve synthesis wherever possible, and when necessary to become informed critics. As time has allowed us, we have begun to understand Rosen's work more thoroughly, and, in particular, his definition of complexity which is closely captured by the quote above. We now seek to move ahead and extend the horizons established by Rosen and others. We invite others to join us in this venture."

Vancouver Studies in Cognitive Science:

A New Perspective on "Community" and its Implications for Computer-Mediated Communication Systems:

Scholars have long argued about the nature of “community,” and the growth of Internet-based communication and “online communities” has intensified this debate. This paper argues that a newperspective on the concept “community” can shed lighton the subject. Ideas from cognitive science, particularly category theory, can help. I suggest thatcommunity can be viewed as a prototype-based category. Prototype-based categories are defined not by simple rules of inclusion and exclusion, but instead by their prototypical members–a robin is a betterexample of a bird than an emu or a penguin. Items in a category are better or worse examples of the category depending on their degree of similarity to the prototypical members. I will argue that these theoretical insights can help resolve debates about thenature of community, and also can help guide designers of computer-mediated communication (CMC) systems.

Keywords Community, online communities, virtual communities, cognitive science, category theory.

‎"Radial Categories and GenreMany concepts resist definition with one set of prototypes. For example, Lakoff details how complicated it is to define the category “mother.” Beyond the central case of “a mother who has alwaysbeen female, and ...who gave birth to the child, supplied her half of the child’s genes, nurtured the child, is married to the father, is one generation older than the child, and is the child’s legal guardian” [5, p. 83], we must also account for the idea of a stepmother, adoptive mother, birth mother, etc. He notes that “the point is that the central case does not productively generate all these subcategories. Instead, the subcategories are defined by convention as variations on the central case.” The category “mother” is, in Lakoff’s terminology, a “radial category” [5, p. 83-84]. Similarly, community is also a radial category. Our notion of the basic concept “community” for manyconjures up idealized images of small-town life. However, the base-level concept does not explain subcategories like corporations, work groups, army platoons, and daycare centers. A corporation has many aspects that can be understood through the lens of the idea of community. How do employees support one another? What are the patterns of social relation that emerge among them? Can employees successfully leverage both strong and weak ties to others in theorganization to help them accomplish goals? These are excellent questions to ask—the idea of community can be a powerful one for understanding organizations.However, it’s more productive to ask “How is this corporate community like and unlike IBM?” than “How is this like and unlike the town square?” We can better understand this example within the subcategory. Wecan think of the subcategories as genres of community. In HCI design work, we need to understand our deliberately engineered social groups each within their intended genre(s)."

One aspect of the study of natural language is the quest for a universal grammar, a system that would explain conclusively the way all languages are organized and function. This is in part a search for an underlying principle of organization. This paper contends that a universal grammar is not to be found by the usual method of studying words, abstracting categories of words, and overlaying a structure based on those categories. We should discard this approach altogether and adopt a different technique that emphasizes the biology of language: language as a product of the brain. Grammar, and generally the organizing principles of language, can be seen as sharing biological rules of constraint and are subject to evolutionary principles. Further, language may itself be an expression of similar rules that inform the material processes of evolution; one thus may be able to understand information transfer in living organisms (genotype to phenotype, synaptic chemical communication) by an analysis of natural language.

That fractals are not irrelevant to semantics is evident, not only from research work conducted by university and government scientist but also by private research firms. In Taking QuickPlace to the Next Level of Collaborative Knowledge Sharing (10), Bian McKay, Executive Vice President and Chief Scientist of CIRI Lab Inc. put it in this way

"Unlike first generation Knowledge Management technology, second generation doesn't use complex mathematics to calculate patterns from information because it needs to be dynamically adaptive with real-time responses. 2G-KM uses adaptive processing methods like those found in genetic algorithms with a focus on pattern-based Darwinian survival techniques in high-dimensional fractal semantic space to recognize emerging patterns of significance, facilitating early reactions to opportunities and threats to improve business health. Through a high-dimensional spatial continuum index, it turns the digital computing metaphor into an analog spatial semantic pattern metaphor, delivering a host of powerful automatic assimilation derivatives. These are based on concepts like semantic proximity, semantic convergence, recursive containment, and containment propagations that emulate machine learning, facilitated by abductive, inductive, and deductive inferencing with causal reasoning to aid the user in knowledge assimilation through concept usage cause-effect-impact analysis."

"Rather than indexing bulk content (i.e., all the words in documents) as does fulltext indexing, 2G-KM indexes multi-dimensional hyperspatial surfaces of emerging semantic patterns as determined through the usage of common vocabulary, tracing concept boundaries in hypercube fractal fashion. This facilitates the implementation of a high-dimensional, scalable, active-reflexive, associative semantic environment that can deliver on the need for real-time adaptive knowledge management services to responsively drive collaborative knowledge sharing environments."

"Such an index can adaptively map onto and extend the solution provided by first generation Knowledge Management techniques such as fulltext indexing, neural network, Baysean-belief Trees, Taxonomies and Classifications to bring them to the next level of productivity enhancement for corporations."

If by now, you are not convinced that fractals are relevant to the nature of semantics, please keep reading. Like a painter refining his masterpiece, McKay beautifully explains

"In this model, documents contain fractal patterns as signatures that emulate semantic hypercubes, traced by their common vocabulary, dynamically joining documents in collections to reflect Knowledge ViewPoints called kThreads. KThreads provide a similar interpretative value-added to dynamically adaptive knowledge management modeling, as do Eigen Vectors used in physical Engineering models. In 2G-KM adaptive models, kThreads intersect to emulate inferencing patterns, inferencing patterns intersect to emulate meta-inferencing patterns, and so forth into higher-orders of automated intelligence, introspecting fractal semantic patterns that trace through document collections as the knowledge concept use and reusage vehicles of business and governance."

"The effect is that, by fractally distilling or factoring fractal semantic patterns as common vocabulary surfaces out of unstructured documents, much like first order predicate calculus, you get highly operative semantic fractal patterns as first order Knowledge Operators that can be reused to generate higher order semantic assemblies as component-based concepts and next-level semantic operators based on common context."

"By automatically deriving fractal semantic patterns as simplicity from the apparent complexity of unstructured documents, you get the ability to "mix and match" combinations of knowledge concepts contained in and across documents, as they apply to corporate problems and corporate opportunities. This results in the ability to explore and analyze the use of corporate knowledge across corporations to business problems."

Categorial Compositionality: A Category Theory Explanation for the Systematicity of Human Cognition

"Classical and Connectionist theories of cognitive architecture seek to explain systematicity (i.e., the property of human cognition whereby cognitive capacity comes in groups of related behaviours) as a consequence of syntactically and functionally compositional representations, respectively. However, both theories depend on ad hoc assumptions to exclude specific instances of these forms of compositionality (e.g. grammars, networks) that do not account for systematicity. By analogy with the Ptolemaic (i.e. geocentric) theory of planetary motion, although either theory can be made to be consistent with the data, both nonetheless fail to fully explain it. Category theory, a branch of mathematics, provides an alternative explanation based on the formal concept of adjunction, which relates a pair of structure-preserving maps, called functors. A functor generalizes the notion of a map between representational states to include a map between state transformations (or processes). In a formal sense, systematicity is a necessary consequence of a higher-order theory of cognitive architecture, in contrast to the first-order theories derived from Classicism or Connectionism. Category theory offers a re-conceptualization for cognitive science, analogous to the one that Copernicus provided for astronomy, where representational states are no longer the center of the cognitive universe—replaced by the relationships between the maps that transform them.

Author Summary

Our minds are not the sum of some arbitrary collection of mental abilities. Instead, our mental abilities come in groups of related behaviours. This property of human cognition has substantial biological advantage in that the benefits afforded by a cognitive behaviour transfer to a related situation without any of the cost that came with acquiring that behaviour in the first place. The problem of systematicity is to explain why our mental abilities are organized this way. Cognitive scientists, however, have been unable to agree on a satisfactory explanation. Existing theories cannot explain systematicity without some overly strong assumptions. We provide a new explanation based on a mathematical theory of structure called Category Theory. The key difference between our explanation and previous ones is that systematicity emerges as a natural consequence of structural relationships between cognitive processes, rather than relying on the specific details of the cognitive representations on which those processes operate, and without relying on overly strong assumptions."

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