Continuity of Semantic Operators in Logic Programming and Their Approximation by Artificial Neural Networks

One approach to integrating first-order logic programming and neural network systems employs the approximation of semantic operators by feedforward networks. For this purpose, it is necessary to view these semantic operators as continuous functions on the reals. This can be accomplished by endowing the space of all interpretations of a logic program with topologies obtained from suitable embeddings. We will present such topologies which arise naturally out of the theory of logic programming, discuss continuity issues of several well-known semantic operators, and derive some results concerning the approximation of these operators by feedforward neural networks.

[1]  T. Metin Sezgin,et al.  Continuous Models of Computation for Logic Programs: Importing Continuous Mathematics into Logic Programming's Algorithmic Foundations , 1999, The Logic Programming Paradigm.

[2]  Jack Minker,et al.  Logic-Based Artificial Intelligence , 2000 .

[3]  Pascal Hitzler,et al.  Generalized metrics and uniquely determined logic programs , 2003, Theor. Comput. Sci..

[4]  A. Seda.  Topology And The Semantics Of Logic Programs , 1995 .

[5]  Antony Browne,et al.  Connectionist inference models , 2001, Neural Networks.

[6]  Ken-ichi Funahashi,et al.  On the approximate realization of continuous mappings by neural networks , 1989, Neural Networks.

[7]  Samson Abramsky,et al.  Domain theory , 1995, LICS 1995.

[8]  John Wylie Lloyd,et al.  Foundations of Logic Programming , 1987, Symbolic Computation.

[9]  P. Hitzler.  Generalized Metrics and Topology in Logic Programming Semantics , 2001 .

[10]  Pascal Hitzler,et al.  A Note on the Relationships Between Logic Programs and Neural Networks , 2000, IWFM.

[11]  Melvin Fitting,et al.  Fixpoint Semantics for Logic Programming a Survey , 2001, Theor. Comput. Sci..

[12]  P. Hitzler,et al.  Semantic Operators and Fixed-Point Theory in Logic Programming , 2001 .

[13]  Pascal Hitzler,et al.  Characterizations of Classes of Programs by Three-Valued Operators , 1999, LPNMR.

[14]  Kurt Hornik,et al.  Multilayer feedforward networks are universal approximators , 1989, Neural Networks.

[15]  R. Bartle.  The elements of integration and Lebesgue measure , 1995 .

[16]  Melvin Fitting,et al.  A Kripke-Kleene Semantics for Logic Programs , 1985, J. Log. Program..

[17]  Heekuck Oh,et al.  Neural Networks for Pattern Recognition , 1993, Adv. Comput..

[18]  Krzysztof R. Apt,et al.  Logic Programming , 1990, Handbook of Theoretical Computer Science, Volume B: Formal Models and Sematics.

[19]  S. Ginsburg,et al.  Acyclic logic programs and the completeness of SLDNF-resolution , 1991 .

[20]  Melvin Fitting,et al.  Metric Methods Three Examples and a Theorem , 1994, J. Log. Program..

[21]  Victor W. Marek,et al.  The Logic Programming Paradigm: A 25-Year Perspective , 2011 .

[22]  Steffen Hölldobler.  Challenge problems for the integration of logic and connectionist systems , 2000, WLP.

[23]  Victor W. Marek,et al.  Approximations, stable operators, well-founded fixpoints and applications in nonmonotonic reasoning , 2000 .

[24]  Steffen Hölldobler,et al.  Approximating the Semantics of Logic Programs by Recurrent Neural Networks , 1999, Applied Intelligence.

[25]  Lawrence Cavedon,et al.  Acyclic Logic Programs and the Completeness of SLDNF-Resolution , 1991, Theor. Comput. Sci..