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Publications about 'theory of computing'
Articles in journal or book chapters
  1. B. Dasgupta, P. Berman, and E.D. Sontag. Computational complexities of combinatorial problems with applications to reverse engineering of biological networks. In D. Liu and F-Y. Wan, editors, Advances in Computational Intelligence: Theory & Applications, pages 303-316. World Scientific, Hackensack, 2006. Keyword(s): systems biology, biochemical networks, gene and protein networks, reverse engineering, systems identification, theory of computing and complexity.


  2. B. Dasgupta, G.A. Enciso, E.D. Sontag, and Y. Zhang. Algorithmic and complexity results for decompositions of biological networks into monotone subsystems. In C. Ālvarez and M. Serna, editors, Lecture Notes in Computer Science: Experimental Algorithms: 5th International Workshop, WEA 2006, pages 253-264. Springer-Verlag, 2006. Note: (Cala Galdana, Menorca, Spain, May 24-27, 2006). Keyword(s): systems biology, biochemical networks, monotone systems, theory of computing and complexity.


  3. P. Kuusela, D. Ocone, and E.D. Sontag. Learning Complexity Dimensions for a Continuous-Time Control System. SIAM J. Control Optim., 43(3):872-898, 2004. [PDF] [doi:http://dx.doi.org/10.1137/S0363012901384302] Keyword(s): theory of computing and complexity, VC dimension.
    Abstract:
    This paper takes a computational learning theory approach to a problem of linear systems identification. It is assumed that input signals have only a finite number k of frequency components, and systems to be identified have dimension no greater than n. The main result establishes that the sample complexity needed for identification scales polynomially with n and logarithmically with k.


  4. B. DasGupta, H.T. Siegelmann, and E.D. Sontag. On the complexity of training neural networks with continuous activation functions. IEEE Trans. Neural Networks, 6:1490-1504, 1995. [PDF] Keyword(s): neural networks, analog computing, theory of computing, neural networks, computational complexity, machine learning.
    Abstract:
    Blum and Rivest showed that any possible neural net learning algorithm based on fixed architectures faces severe computational barriers. This paper extends their NP-completeness result, which applied only to nets based on hard threshold activations, to nets that employ a particular continuous activation. In view of neural network practice, this is a more relevant result to understanding the limitations of backpropagation and related techniques.


  5. H. T. Siegelmann and E.D. Sontag. On the computational power of neural nets. J. Comput. System Sci., 50(1):132-150, 1995. [PDF] [doi:http://dx.doi.org/10.1006/jcss.1995.1013] Keyword(s): neural networks, recurrent neural networks, analog computing, theory of computing, neural networks, computational complexity, super-Turing computation.
    Abstract:
    This paper deals with finite size networks which consist of interconnections of synchronously evolving processors. Each processor updates its state by applying a "sigmoidal" function to a rational-coefficient linear combination of the previous states of all units. We prove that one may simulate all Turing Machines by such nets. In particular, one can simulate any multi-stack Turing Machine in real time, and there is a net made up of 886 processors which computes a universal partial-recursive function. Products (high order nets) are not required, contrary to what had been stated in the literature. Non-deterministic Turing Machines can be simulated by non-deterministic rational nets, also in real time. The simulation result has many consequences regarding the decidability, or more generally the complexity, of questions about recursive nets.


  6. E.D. Sontag. On some questions of rationality and decidability. J. Comput. System Sci., 11(3):375-381, 1975. [PDF] Keyword(s): theory of computing and complexity.
    Abstract:
    Some results are given in the theory of rational power series over a broad class of semirings. In particular, it is shown that for unambiguous sets the notion of rationality is independent of the semiring over which representations are defined. The undecidability of the rationality of probabilistic word functions is also established.


Conference articles
  1. E.D. Sontag. From linear to nonlinear: some complexity comparisons. In Proc. IEEE Conf. Decision and Control, New Orleans, Dec. 1995, IEEE Publications, 1995, pages 2916-2920, 1995. [PDF] Keyword(s): theory of computing and complexity, computational complexity, controllability, observability.
    Abstract:
    This paper deals with the computational complexity, and in some cases undecidability, of several problems in nonlinear control. The objective is to compare the theoretical difficulty of solving such problems to the corresponding problems for linear systems. In particular, the problem of null-controllability for systems with saturations (of a "neural network" type) is mentioned, as well as problems regarding piecewise linear (hybrid) systems. A comparison of accessibility, which can be checked fairly simply by Lie-algebraic methods, and controllability, which is at least NP-hard for bilinear systems, is carried out. Finally, some remarks are given on analog computation in this context.


  2. J. L. Balcázar, R. Gavaldā, H. T. Siegelmann, and E.D. Sontag. Some structural complexity aspects of neural computation. In Proceedings of the Eighth Annual Structure in Complexity Theory Conference (San Diego, CA, 1993), Los Alamitos, CA, pages 253-265, 1993. IEEE Comput. Soc. Press. [PDF] Keyword(s): analog computing, neural networks, computational complexity, super-Turing computation, theory of computing and complexity.
    Abstract:
    Recent work by H.T. Siegelmann and E.D. Sontag (1992) has demonstrated that polynomial time on linear saturated recurrent neural networks equals polynomial time on standard computational models: Turing machines if the weights of the net are rationals, and nonuniform circuits if the weights are real. Here, further connections between the languages recognized by such neural nets and other complexity classes are developed. Connections to space-bounded classes, simulation of parallel computational models such as Vector Machines, and a discussion of the characterizations of various nonuniform classes in terms of Kolmogorov complexity are presented.


  3. A. Macintyre and E.D. Sontag. Finiteness results for sigmoidal neural networks. In STOC '93: Proceedings of the twenty-fifth annual ACM symposium on Theory of computing, New York, NY, USA, pages 325-334, 1993. ACM Press. [PDF] [doi:http://doi.acm.org/10.1145/167088.167192] Keyword(s): neural networks, theory of computing and complexity.
    Abstract:
    This paper deals with analog circuits. It establishes the finiteness of VC dimension, teaching dimension, and several other measures of sample complexity which arise in learning theory. It also shows that the equivalence of behaviors, and the loading problem, are effectively decidable, modulo a widely believed conjecture in number theory. The results, the first ones that are independent of weight size, apply when the gate function is the "standard sigmoid" commonly used in neural networks research. The proofs rely on very recent developments in the elementary theory of real numbers with exponentiation. (Some weaker conclusions are also given for more general analytic gate functions.) Applications to learnability of sparse polynomials are also mentioned.


  4. H.T. Siegelmann, E.D. Sontag, and C.L. Giles. The Complexity of Language Recognition by Neural Networks. In Proceedings of the IFIP 12th World Computer Congress on Algorithms, Software, Architecture - Information Processing '92, Volume 1, pages 329-335, 1992. North-Holland. Keyword(s): neural networks, computational complexity, machine learning, recurrent neural networks, theory of computing and complexity.


  5. W. Maass, G. Schnitger, and E.D. Sontag. On the computational power of sigmoid versus Boolean threshold circuits (extended abstract). In Proceedings of the 32nd annual symposium on Foundations of computer science, Los Alamitos, CA, USA, pages 767-776, 1991. IEEE Computer Society Press. Keyword(s): neural networks, theory of computing and complexity.


  6. E.D. Sontag. Some complexity questions regarding controllability. In Proc. IEEE Conf. Decision and Control, Austin, Dec. 1988, pages 1326-1329, 1988. [PDF] Keyword(s): theory of computing and complexity, computational complexity, controllability, computational complexity.
    Abstract:
    It has been known for a long time that certain controllability properties are more difficult to verify than others. This article makes this fact precise, comparing controllability with accessibility, for a wide class of nonlinear continuous time systems. The original contribution is in formalizing this comparison in the context of computational complexity. (This paper placed here by special request.)



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