On the Computation of Boolean Functions by Analog Circuits of Bounded Fan-In

Journal of Computer and System Sciences

Volumn 54, Issue 1, 1997, Pages 199-212

  Turán, György ^a,b   Vatan, Farrokh ^a,c  

^a UNIVERSITY OF ILLINOIS AT CHICAGO   (United States)

^b INSTITUTE OF EXPERIMENTAL MEDICINE   (Hungary)

^c University of California   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

COMPUTATIONAL COMPLEXITY; DIGITAL ARITHMETIC; MATHEMATICAL MODELS; NEURAL NETWORKS; PIECEWISE LINEAR TECHNIQUES;

ANALOG CIRCUITS; ANALOG GATES; REAL VALUED FUNCTIONS;

BOOLEAN FUNCTIONS;

EID: 0031079277     PISSN: 00220000     EISSN: None     Source Type: Journal    
DOI: 10.1006/jcss.1997.1480     Document Type: Article

References (46)

1. M. Ben-Or, Lower bounds for algebraic computation trees, in "Proceedings, 15th ACM Symposium on Theory of Computing, 1983," pp. 80-86.
2. L. Blum, M. Shub, and S. Smale, On a theory of computation and complexity over the real numbers: NP-completeness, recursive functions and universal machines, Bull. Amer. Math. Soc. 21 (1989), 1-46.
3. R. B. Boppana and M. Sipser, The complexity of finite functions, in "Handbook of Theoretical Computer Science," Vol. A, "Algorithms and Complexity" (J. van Leeuwen, Ed.), pp. 757-804, Elsevier Science, New York, MIT Press, Cambridge, MA, 1990.
4. ℝ, J. Complexity 8 (1992), 230-238.
5. F. Cucker, On the complexity of quantifier elimination: The structural approach, Comput. J. 36 (1993), 400-408.
6. F. Cucker and D. Grigoriev, On the power of real Turing machines over binary inputs, manuscript, 1994.
7. F. Cucker, M. Karpinski, P. Koiran, T. Lickteig, and K. Werther, On real Turing machines that toss coins, in "Proceedings, 27th ACM Symposium on Theory of Computing, 1995," pp. 335-342.
8. F. Cucker, M. Shub, and S. Smale, Separation of complexity classes in Koiran's weak model, Theor. Comput. Sci. 133 (1994), 3-14.
9. F. Cucker and A. Torrecillas, Two P-complete problems in the theory of reals, J. Complexity 8 (1992), 454-466.
10. G. Cybenko, Approximation by superposition of a sigmoidal function, Math. Control Signals, Systems 2 (1989), 303-314.
11. B. DasGupta and G. Schnitger, "Efficient Approximation with Neural Networks: A Comparison of Gate Functions," Technical Report TR CS-92-14, Penn. State Univ., Dept. C. S., June 1992.
12. R. L. Devaney, "An Introduction to Chaotic Dynamical Systems," Benjamin/Cummings, Redwood City, CA, 1986.
13. V. Dičiunas, On the positive and the inversion complexity of Boolean functions, Inform. Theor. Appl./Theor. Inform. Appl. 27 (1993), 283-293.
14. S. B. Gashkov, The complexity of the realization of Boolean functions by networks of functional elements and by formulas in bases whose elements realize continuous functions, Prob. Kibernet. 37 (1980), 52-118. [Russian]
15. P. Goldberg and M. Jerrum, Bounding the Vapnik-Chervonenkis dimension of concept classes parameterized by real numbers, Mach. Learning 18 (1995), 131-148.
16. H. D. Gröger and Gy. Turán, A linear lower bound for the size of threshold circuits, Bull. EATCS 50 (1993), 220-222.
17. A. Hajnal, W. Maass, P. Pudlák, M. Szegedy, and Gy. Turán, Threshold circuits of bounded depth, J. Comput. System Sci. 46 (1993), 129-154.
18. R. Impagliazzo, R. Paturi, and M. E. Saks, "Size-Depth Trade-offs for Threshold Circuits," Technical Report TR CS-92-253, Univ. of California at San Diego, Comput. Sci. and Eng., July 1992.
19. M. Karpinski and A. Macintyre, Quadratic bounds for the VC dimension of sigmoidal neural networks, in "Proceedings, 27th ACM Symposium on Theory of Computing, 1995," pp. 200-208.
20. P. Koiran, Computing over the reals with addition and order, Theor. Comput. Sci. 133 (1994), 35-47.
21. P. Koiran, "A Weak Version of the Blum, Shub, & Smale Model," Technical Report NC-TR-94-5, NeuroCOLT Technical Reports Series, August 1994. A preliminary version appeared in "Proceedings, 34th IEEE Symposium on the Foundations of Computer Science, 1993," pp. 486-495.
22. W. Maass, Bounds for the computational power and learning complexity of analog neural nets, in "Proceedings, 25th ACM Symposium on Theory of Computing, 1993," pp. 335-344.
23. W. Maass, Neural nets with superlinear VC dimension, Neural Comput. 6 (1994), 877-884.
24. W. Maass, G. Schnitger, and E. D. Sontag, A comparison of the computational power of sigmoid and Boolean threshold circuits, in "Theoretical Advances in Neural Computation and Learning" (V. Roychowdhury, K.-Y. Siu, and A. Orlitsky, Eds.), pp. 127-151, Kluwer, Amsterdam, 1994; a preliminary version appeared in "Proceedings, 32nd IEEE Symposium on the Foundations of Computer Science, 1991," pp. 767-776.
25. W. Maass, G. Schnitger, and E. D. Sontag, A comparison of the computational power of sigmoid and Boolean threshold circuits, in "Theoretical Advances in Neural Computation and Learning" (V. Roychowdhury, K.-Y. Siu, and A. Orlitsky, Eds.), pp. 127-151, Kluwer, Amsterdam, 1994; a preliminary version appeared in "Proceedings, 32nd IEEE Symposium on the Foundations of Computer Science, 1991," pp. 767-776.
26. A. Macintyre and E. D. Sontag, Finiteness results for sigmoidal "neural" networks, in "Proceedings, 25th ACM Symposium on Theory of Computing, 1993," pp. 325-334.
27. J. L. McClelland and D. E. Rumelhart, "Parallel Distributed Processing," Vol. 2, MIT Press, Cambridge, MA, 1986.
28. J. Milnor, On the Betti numbers of real varieties, Proc. Amer. Math. Soc. 15 (1964), 275-280.
29. J. L. Montaña and L. M. Pardo, Lower bounds for arithmetic networks, Appl. Algebra Eng. Commun. Comput. 4 (1993), 1-24.
30. N. Nisan, The communication complexity of threshold gates, in "Combinatorics, Paul Erdös is Eighty, Vol. 1," pp. 301-315, Bolyai Soc. Math. Studies, 1993.
31. P. Pudlák, Lower bounds for resolution and cutting plane proofs and monotone computations, preliminary draft, 1995.
32. J. Renegar, On the computational complexity and geometry of the first-order theory of the reals, Part I, J. Symbolic Comput. 13 (1992), 255-299.
33. V. P. Roychowdhury, K. Y. Siu, and A. Orlitsky, Lower bounds on threshold and related circuits via communication complexity, IEEE Trans. Inf. Theory 40 (1994), 467-474.
34. D. E. Rumelhart and J. L. McClelland, "Parallel Distributed Processing," Vol. 1, MIT Press, Cambridge, MA, 1986.
35. M. Saks, Slicing the hypercube, in "Surveys in Combinatorics," Cambridge Univ. Press, Cambridge, 1993.
36. H. T. Siegelmann and E. D. Sontag, Analog computation via neural networks, Theor. Comput. Sci. 131 (1994), 331-360.
37. H. T. Siegelmann and E. D. Sontag, On the computational power of neural nets, J. Comput. System Sci. 50 (1995), 132-150.
38. K. Y. Siu, V. P. Roychowdhury, and T. Kailath, Computing with almost optimal size threshold circuits, in "Proceedings, IEEE Int. Symp. Inf. Theory, 1991," pp. 370.
39. E. D. Sontag, Feedforward nets for interpolation and classification, J. Comput. System Sci. 45 (1992), 20-48.
40. V. Strassen, Algebraic complexity theory, in "Handbook of Theoretical Computer Science," Vol. A, "Algorithms and Complexity" (J. van Leeuwen, Ed.), pp. 633-672, Elsevier Science, Amsterdam/MIT Press, Cambridge, MA, 1990.
41. Gy. Turán and F. Vatan, A size-depth trade-off for the analog computation of Boolean functions, Inform. Proc. Lett. 59 (1996), 251-254.
42. F. Vatan, Some lower and upper bounds for algebraic decision trees and the separation problem, in "Proceedings, 7th Ann. IEEE Structure in Complexity Theory, 1992," pp. 295-304.
43. A. G. Vitushkin, On representation of functions by means of super-positions and related topics, Enseign. Math. 23 (1977), 255-320.
44. H. E. Warren, Lower bounds for approximation by non-linear manifolds, Trans. Amer. Math. Soc. 133 (1968), 167-178.
45. L. Wegener, "The Complexity of Boolean Functions," Teubner-Wiley, New York, 1987.
46. L. Wegener, On the complexity of encoding in analog circuits, Inform. Proc. Lett. 60 (1996), 49-52.