What is a structural representation

We outline a formal foundation for a “structural” (or “symbolic”) object/event representation, the necessity of which is acutely felt in all sciences, including mathematics and computer science. The proposed foundation incorporates two hypotheses: 1) the object’s formative history must be an integral part of the object representation and 2) the process of object construction is irreversible, i.e. the “trajectory” of the object’s formative evolution does not intersect itself. The last hypothesis is equivalent to the generalized axiom of (structural) induction. Some of the main difficulties associated with the transition from the classical numeric to the structural representations appear to be related precisely to the development of a formal framework satisfying these two hypotheses. The concept of (inductive) class representation—which has inspired the development of this approach to structural representation—differs fundamentally from the known concepts of class. In the proposed, evolving transformations system (ETS), model, the class is defined by the transformation system—a finite set of weighted transformations acting on the class progenitor— and the generation of the class elements is associated with the corresponding generative process which also induces the class typicality measure. Moreover, in the ETS model, a fundamental role of the object’s class in the object’s representation is clarified: the representation of an object must include the class. From the point of view of ETS model, the classical discrete representations, e.g. strings and graphs, appear now as incomplete special cases, the proper completion of which should incorporate the corresponding formative histories, i.e. those of the corresponding strings or graphs.

[1]  Robin I. M. Dunbar The Trouble With Science , 1995 .

[2]  M. A. Aiserman REMARKS ON TWO PROBLEMS CONNECTED WITH PATTERN RECOGNITION , 1969 .

[3]  Lev Goldfarb,et al.  What is distance and why do we need the metric model for pattern learning? , 1992, Pattern Recognit..

[4]  Thomas G. Dietterich What is machine learning? , 2020, Archives of Disease in Childhood.

[5]  R. Feynman QED: The Strange Theory of Light and Matter , 1985 .

[6]  Alfred W. Crosby,et al.  The Measure Of Reality , 1996 .

[7]  L. Goldfarb,et al.  Inductive learning with the evolving tree transformation system , 1996 .

[8]  K. Parthasarathy Introduction to Probability and Measure , 1979 .

[9]  J. Bransford,et al.  Abstraction of visual patterns. , 1971, Journal of experimental psychology.

[10]  Jean Alexandre Dieudonné,et al.  The Work of Nicholas Bourbaki , 1970 .

[11]  James Jeans,et al.  The new background of science , 1933 .

[12]  Edmund Landau,et al.  Foundations of analysis , 2001 .

[13]  M. Leyton Symmetry, Causality, Mind , 1999 .

[14]  Erwin Schrödinger,et al.  'Nature and the Greeks' and 'Science and Humanism' , 1996 .

[15]  King-Sun Fu,et al.  Syntactic Pattern Recognition And Applications , 1968 .

[16]  A. G. Kurosh,et al.  Lectures on general algebra , 1966 .

[17]  Noam Chomsky Knowledge of Language , 1986 .

[18]  Georges Ifrah,et al.  The Universal History of Numbers , 1998 .

[19]  Lev Goldfarb,et al.  What is a symbolic measurement process? , 1997, 1997 IEEE International Conference on Systems, Man, and Cybernetics. Computational Cybernetics and Simulation.

[20]  Massimo Piattelli-Palmarini,et al.  Language and Learning: The Debate Between Jean Piaget and Noam Chomsky , 1980 .

[21]  Gordon L. Kane The particle garden : our universe as understood by particle physicists , 1995 .

[22]  Oleg Golubitsky,et al.  On the generating process and the class typicality measure , 2002 .

[23]  Lev Goldfarb,et al.  Why Classical Models for Pattern Recognition are Not Pattern Recognition Models , 1999 .

[24]  Virendrakumar C. Bhavsar,et al.  Can a vector space based learning model discover inductive class generalization in a symbolic environment? , 1995, Pattern Recognit. Lett..

[25]  Frank Mueller,et al.  Preface , 2009, 2009 IEEE International Symposium on Parallel & Distributed Processing.

[26]  M. Ridley Triumph of the embryo? , 1992, Nature.

[27]  J. Losee A historical introduction to the philosophy of science , 1972 .

[28]  L. Goldfarb,et al.  Ets learning of kernel languages , 2003 .

[29]  C. Morris,et al.  Philosophy of Science and Science of Philosophy , 1935, Philosophy of Science.

[30]  Douglas Brown A Dictionary of Philosophy 3rd edition , 1997 .

[31]  Noam Chomsky Knowledge of language: its nature, origin, and use , 1988 .

[32]  Lev Goldfarb,et al.  On the foundations of intelligent processes - I. An evolving model for pattern learning , 1990, Pattern Recognit..

[33]  A. D. Ritchie The Dictionary of Philosophy , 1945, Nature.

[34]  R. Tennant Algebra , 1941, Nature.

[35]  Richard Schlegel,et al.  Time and the physical world , 1968 .

[36]  Stefan Wermter,et al.  A Novel Modular Neural Architecture for Rule-Based and Similarity-Based Reasoning , 1998, Hybrid Neural Systems.