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Counting complexity classes for numeric computations II: Algebraic and semialgebraic sets (Extended Abstract)
 J. COMPL
, 2004
"... We define counting #P classes #P ¡ and in the BlumShubSmale setting of computations over the real or complex numbers, respectively. The problems of counting the number of solutions of systems of polynomial inequalities over ¢ , or of systems of polynomial equalities over £ , respectively, turn ou ..."
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We define counting #P classes #P ¡ and in the BlumShubSmale setting of computations over the real or complex numbers, respectively. The problems of counting the number of solutions of systems of polynomial inequalities over ¢ , or of systems of polynomial equalities over £ , respectively, turn out to be natural complete problems in these classes. We investigate to what extent the new counting classes capture the complexity of computing basic topological invariants of semialgebraic sets (over ¢ ) and algebraic sets (over £). We prove that the problem to compute the (modified) Euler characteristic of semialgebraic sets is FP #P¤complete, and that the problem to compute the geometric degree of complex algebraic sets is FP #P¥complete. We also define new counting complexity classes GCR and GCC in the classical Turing model via taking Boolean parts of the classes above, and show that the problems to compute the Euler characteristic and the geometric degree of (semi)algebraic sets given by integer polynomials are complete in these classes. We complement the results in the Turing model by proving, for all k ¦ ∈ , the FPSPACEhardness of the problem of computing the kth Betti number of the set of real zeros of a given integer polynomial. This holds with respect to the singular homology as well as for the BorelMoore homology.
Formal proof—theory and practice
 Notices AMS
, 2008
"... Aformal proof is a proof written in a precise artificial language that admits only a fixed repertoire of stylized steps. This formal language is usually designed so that there is a purely mechanical process by which the correctness of a proof in the language can be verified. Nowadays, there are nume ..."
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Aformal proof is a proof written in a precise artificial language that admits only a fixed repertoire of stylized steps. This formal language is usually designed so that there is a purely mechanical process by which the correctness of a proof in the language can be verified. Nowadays, there are numerous computer programs known as proof assistants that can check, or even partially construct, formal proofs written in their preferred proof language. These can be considered as practical, computerbased realizations of the traditional systems of formal symbolic logic and set theory proposed as foundations for mathematics. Why should we wish to create formal proofs?
Qualitative Reasoning beyond the Physics Domain: The Density Dependence Theory of Organizational Ecology
 Proceedings of QR95
, 1995
"... Abstract: Qualitative reasoning is traditionally associated with the domain of physics, although the domain of application is, in fact, much broader. This paper investigates the application of qualitative reasoning beyond the domain of physics. It presents a case study of application in the social s ..."
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Abstract: Qualitative reasoning is traditionally associated with the domain of physics, although the domain of application is, in fact, much broader. This paper investigates the application of qualitative reasoning beyond the domain of physics. It presents a case study of application in the social sciences: the density dependence theory of organizational ecology. It discusses how the different nature of soft science domains will complicate the process of model building. Furthermore, it shows that the “model building ” process can also help making theoretically important decisions, and, as a result, have an impact on the original theory. This will require a shift in focus from the “model simulation ” process towards the “model building ” process. 1
Variations by complexity theorists on three themes of Euler, . . .
 COMPUTATIONAL COMPLEXITY
, 2005
"... This paper surveys some connections between geometry and complexity. A main role is played by some quantities —degree, Euler characteristic, Betti numbers — associated to algebraic or semialgebraic sets. This role is twofold. On the one hand, lower bounds on the deterministic time (sequential and pa ..."
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Cited by 11 (3 self)
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This paper surveys some connections between geometry and complexity. A main role is played by some quantities —degree, Euler characteristic, Betti numbers — associated to algebraic or semialgebraic sets. This role is twofold. On the one hand, lower bounds on the deterministic time (sequential and parallel) necessary to decide a set S are established as functions of these quantities associated to S. The optimality of some algorithms is obtained as a consequence. On the other hand, the computation of these quantities gives rise to problems which turn out to be hard (or complete) in different complexity classes. These two kind of results thus turn the quantities above into measures of complexity in two quite different ways.
CHIRON: Planning in an Opentextured Domain
, 1994
"... Most work in artificial intelligence and law has concentrated on modelling the type of reasoning done by trial lawyers. In fact, most lawyers' work involves planning  for example, wills and trusts, real estate deals, and business mergers and acquisitions. Certain planning issues, such as the ..."
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Cited by 11 (4 self)
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Most work in artificial intelligence and law has concentrated on modelling the type of reasoning done by trial lawyers. In fact, most lawyers' work involves planning  for example, wills and trusts, real estate deals, and business mergers and acquisitions. Certain planning issues, such as the use of underspecified, or "opentextured" rules, are illustrated especially clearly in this domain. In this thesis, I set forth the characteristic features of planning in law, place it in the context of past artificial intelligence work in both law and planning, and describe CHIRON, a system that I have developed implementing my theory of opentextured planning in the domain of personal income tax law.
On dynamically presenting a topology course
 Annals of Mathematics and Artificial Intelligence
, 2001
"... www.cs.mdx.ac.uk/imp Authors of traditional mathematical texts often have difficulty balancing the amount of contextual information and proof detail. We propose a simple hypermedia framework that can assist in the organisation and presentation of mathematical theorems and definitions. We describe th ..."
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www.cs.mdx.ac.uk/imp Authors of traditional mathematical texts often have difficulty balancing the amount of contextual information and proof detail. We propose a simple hypermedia framework that can assist in the organisation and presentation of mathematical theorems and definitions. We describe the application of the framework to convert an existing course in general topology to a webbased set of materials. A pilot study of the materials indicated a high level of user satisfaction. We discuss further lines of investigation, in particular, the presentation of larger bodies of work. 1
Paradox and theorizing within the resourcebased view
 Academy of Management Review
, 2006
"... By working with and through the paradoxes present in the resourcebased view (RBV) of strategic management, scholars can advance understanding concerning the contradictions and tensions inherent in creating and sustaining superior firm performance. We identify and discuss various RBV paradoxes, il ..."
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By working with and through the paradoxes present in the resourcebased view (RBV) of strategic management, scholars can advance understanding concerning the contradictions and tensions inherent in creating and sustaining superior firm performance. We identify and discuss various RBV paradoxes, illustrating how paradoxical thinking can enhance theorizing and open up new vistas for knowing and understanding. Finally, we discuss the utility of the paradoxical perspective in furthering RBV scholarship. Paradox, “the dynamic tensions of juxtaposed opposites ” (Rosen, 1994: xvii), underpins much organization and management scholarship and practice. The use of paradox can promote divergent or “oppositional ” thinking (Cameron, 1986;
Simulation modeling in organizational and management research
 ACADEMY OF MANAGEMENT REVIEW
, 2007
"... Simulation modeling provides a powerful methodology for advancing theory and research on complex behaviors and systems, yet it has been embraced more slowly in management than in some associated social science disciplines. We suspect that part of the reason is that simulation methods are not well un ..."
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Simulation modeling provides a powerful methodology for advancing theory and research on complex behaviors and systems, yet it has been embraced more slowly in management than in some associated social science disciplines. We suspect that part of the reason is that simulation methods are not well understood. We therefore aim to promote understanding of simulation methodology and to develop an appreciation of its potential contributions to management theory by describing the nature of simulations, its attractions, and its special problems, as well as some uses of computational modeling in management research.
Collaborative knowledge building and integral theory. Integral Review: A Transdisciplinary and Transcultural
 Journal for New Thought, Research and Praxis
, 2006
"... Abstract: Uncertainty in knowing and communicating affect all aspects of modern life. Ubiquitous and inevitable uncertainty, including ambiguity and paradox, is particularly salient and important in knowledge building communities. Because knowledge building communities represent and evolve knowledge ..."
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Cited by 8 (5 self)
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Abstract: Uncertainty in knowing and communicating affect all aspects of modern life. Ubiquitous and inevitable uncertainty, including ambiguity and paradox, is particularly salient and important in knowledge building communities. Because knowledge building communities represent and evolve knowledge explicitly, the causes, effects, and approaches to this “epistemological indeterminacy ” can be directly addressed in knowledge building practices. Integral theory's approach (including “methodological pluralism”) involves accepting and integrating diverse perspectives in ways that transcend and include them. This approach accentuates the problems of epistemological indeterminacy and highlights the general need to deal creatively with it. This article begins with a cursory analysis of textual dialogs among integral theorists, showing that, while integral theory itself points to leadingedge ways of dealing with epistemological indeterminacy, the knowledge building practices of integral theorists, by and large, exhibit the same limitations as traditional intellectual discourses. Yet, due to its values and core methods, the integral theory community is in a unique position to develop novel and more adequate modes of inquiry and dialog. This text explores how epistemological