Abstraction and Complexity Measures

Abstraction is fundamental for both human and artificial reasoning. The word denotes different activities and process, but all are intuitively related to the notion of complexity/simplicity, which is as elusive a notion as abstraction. From an analysis of the literature on abstraction and complexity it clearly appears that it is unrealistic to find definitions valid in all disciplines and for all tasks. Hence, we consider a particular model of abstraction, and try to investigate how complexity measures could be mapped to it. Preliminary results show that abstraction and complexity are not monotonically coupled notions, and that complexity may either increase or decrease with abstraction according to the definition of both and to the specificities of the considered domain.

[1]  Fausto Giunchiglia,et al.  A Theory of Abstraction , 1992, Artif. Intell..

[2]  Craig A. Knoblock.  Learning Hierarchies of Abstraction Spaces , 1989, ML.

[3]  P. Pandurang Nayak,et al.  A Semantic Theory of Abstractions , 1995, IJCAI.

[4]  Cosma Rohilla Shalizi,et al.  Methods and Techniques of Complex Systems Science: An Overview , 2003, nlin/0307015.

[5]  Devika Subramanian,et al.  A Theory of Justified Reformulations , 1989, ML.

[6]  Brian Tanner,et al.  Hierarchical Heuristic Search Revisited , 2005, SARA.

[7]  Seth Lloyd,et al.  Information measures, effective complexity, and total information , 1996 .

[8]  Tomasz Imielinski.  Domain Abstraction and Limited Reasoning , 1987, IJCAI.

[9]  Robert C. Holte,et al.  Speeding up Problem Solving by Abstraction: A Graph Oriented Approach , 1996, Artif. Intell..

[10]  Kanad K. Biswas,et al.  Dominant color region based indexing for CBIR , 1999, Proceedings 10th International Conference on Image Analysis and Processing.

[11]  Sheila A. McIlraith,et al.  Towards a practical theory of reformulation for reasoning about physical systems , 2005, Artif. Intell..

[12]  Thomas Ellman.  Synthesis of Abstraction Hierarchies for Constraint Satisfaction by Clustering Approximately Equivalent Objects , 1993, ICML.

[13]  Peter F. Fisher,et al.  Developments in Spatial Data Handling, 11th International Symposium on Spatial Data Handling, Leicester, UK, August 23-25, 2004 , 2005, SDH.

[14]  Moshe Koppel,et al.  Complexity, Depth, and Sophistication , 1987, Complex Syst..

[15]  David A. Plaisted,et al.  Theorem Proving with Abstraction , 1981, Artif. Intell..

[16]  Thomas S. Deisboeck,et al.  Complex systems science in biomedicine , 2006 .

[17]  Earl D. Sacerdoti,et al.  Planning in a Hierarchy of Abstraction Spaces , 1974, IJCAI.

[18]  David H. Wolpert,et al.  Self-dissimilarity: an empirically observable complexity measure , 2000 .

[19]  Scot Anderson,et al.  Verifying the Incorrectness of Programs and Automata , 2005, SARA.

[20]  Michael R. Lowry.  The Abstraction/Implementation Model of Problem Reformulation , 1987, IJCAI.

[21]  D. Benjamin.  Change of Representation and Inductive Bias , 1989 .

[22]  P. Landsberg,et al.  Simple measure for complexity , 1999 .

[23]  Josh D. Tenenberg.  Preserving Consistency Across Abstraction Mappings , 1987, IJCAI.

[24]  Jean-Daniel Zucker,et al.  Semantic Abstraction for Concept Representation and Learning , 2001 .

[25]  J. Crutchfield,et al.  Measures of statistical complexity: Why? , 1998 .

[26]  Pietro Torasso,et al.  Formalizing the Abstraction Process in Model-Based Diagnosis , 2007, SARA.

[27]  Jean-Daniel Zucker,et al.  Abstraction, Reformulation and Approximation, 6th International Symposium, SARA 2005, Airth Castle, Scotland, UK, July 26-29, 2005, Proceedings , 2005, SARA.

[28]  Rolf Herken,et al.  The Universal Turing Machine: A Half-Century Survey , 1992 .

[29]  Jean-Daniel Zucker,et al.  A model of abstraction in visual perception , 2001, Appl. Artif. Intell..

[30]  N. Bredeche,et al.  Perceptual learning and abstraction in machine learning: an application to autonomous robotics , 2006, IEEE Transactions on Systems, Man, and Cybernetics, Part C (Applications and Reviews).

[31]  Robert L. Goldstone,et al.  Reuniting perception and conception , 1998, Cognition.

[32]  Paul M. B. Vitányi,et al.  Meaningful Information , 2001, IEEE Transactions on Information Theory.

[33]  Jean-Daniel Zucker,et al.  Consistency Assessment Between Multiple Representations of Geographical Databases: a Specification-Based Approach , 2004, SDH.

[34]  Charles H. Bennett.  Logical depth and physical complexity , 1988 .