Garzon M., Franklin S., Neural Computability, Proc. Third Intern. Joint Conf. on Neural Networks, vol.2, 1989, 631-637.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....finite asymmetric networks of threshold gates are equivalent to finite automata 1 ; and various infinitary analogs of neural networks have recently been shown to be equivalent to Turing machines. For the latter type of result, constructions using a potentially infinite network are presented in [20, 35], and constructions using a finite network, but real valued neurons of arbitrary precision are presented in [35, 78, 89] The construction in [89] is rather interesting, as there the required precision grows only linearly in the space requirement of the simulated Turing machine, and only very ....

....gates in the resulting net are in fact similar, and so the net can be folded upon itself to create a cyclic net of polynomial size and exponential convergence time. Parberry [72] attributes this result to the unpublished report [57] For approaches to computations on nets of unbounded size, see [19, 20, 23, 24]. Concerning symmetric nets, the fundamental result is Hopfield s [43] observation that a symmetric simple net using an asynchronous update rule always converges. Hopfield s result was based on defining an energy function on the global states of the network, such that in every permissible ....

Franklin, S., Garzon, M. Neural computability. In: Progress in Neural Networks 1 (ed. O. M. Omidvar). Ablex, Norwood, NJ, 1990. Pp. 128--144.

....computer, any computer, even one that has not been built yet but can be built, and even one that will require unbounded amounts of time and memory space for ever larger inputs, is also solvable by a Turing machine . Harel 1992: 233. vi) Every algorithm can be implemented by a Turing machine . (Franklin and Garzon 1991: 133. Franklin and Garzon expand on this as follows (and in so doing provide an excellent statement of some commonly held beliefs that we wish to call into question) By their rendering the notion of algorithm precise and unambiguous, Turing machines afford researchers in the theory of ....

Franklin, S., Garzon, M. 1991. 'Neural Computability'. In O. Omidvar (ed.) 1991, Progress in Neural Networks, vol. 1, Norwood, N.J.: Ablex.

....we need to consider both discrete (digital) computation and nondiscrete (analog) computation. In terms of discrete computation, the traditional paradigm is the Turing Machine, with the Von Neumann architecture (processor memory) It has been shown that neural nets are TM equivalent [HS87, FG90] once the issue of infinite memory tape is settled. The infinite tape may be represented as an external tape, as unbounded number of network units, or as infinite precision numbers [Pol87] In terms of nondiscrete computation, various researchers have shown that three layer feedforward nets can ....

S. Franklin and M Garzon. Neural computability. In O. M. Omidvar, editor, Progress in neural networks. Ablex, Norwood, 1990.

....finite asymmetric networks of threshold gates are equivalent to finite automata 1 ; and various infinitary analogs of neural networks have recently been shown to be equivalent to Turing machines. For the latter type of result, constructions using a potentially infinite network are presented in [15, 29], and constructions using a finite network, but real valued neurons of arbitrary precision are presented in [29, 60, 68] The construction in [68] is rather interesting, as there the required precision grows only linearly in the space requirement of the simulated Turing machine. In these models ....

.... the class of functions computed by polynomial size cyclic nets in unbounded time can be seen to equal the class PSPACE poly considered in [6, 41] Parberry in [57] attributes this result to the unpublished report [45] For an interesting approach to computations on nets of unbounded size, see [14, 15, 18, 19]. Concerning symmetric nets, the fundamental result is Hopfield s [36] observation that a symmetric simple net using an asynchronous update rule always stabilizes. This was improved in [12] to show that in a symmetric simple network of n neurons with integer weights w ij , the stabilization ....

Franklin, S., Garzon, M. Neural computability. In: Progress in Neural Networks 1 (ed. O. M. Omidvar). Ablex, Norwood, NJ, 1990. Pp. 128--144.

....conjecture regarding their superior computational power. We see that no such superiority of computational power exists, at least when formalized in terms of polynomial time computation. Work that deals with infinite structure is reported by Hartley and Szu ( 12] and by Franklin and Garzon ([8] and [9] some of which deals with cellular automata. There one assumes an 2 unbounded number of neurons, as opposed to a finite number fixed in advance. In the paper [29] Wolpert studies a class of machines with just linear activation functions, and shows that this class is at least as ....

S. Franklin, M. Garzon, "Neural computability," in Progress In Neural Networks, Vol 1)(O. M. Omidvar, ed.), Ablex, Norwood, NJ, (1990): 128-144.

....of constituent atomic representations) Because of this rejection of atomic symbols, supporters of the subsymbolic hypothesis reject the physical symbol system hypothesis. One could argue that, as both neural networks and Von Neumann type physical symbol systems are both universal Turing machines (Franklin Garzon, 1990), at one level of abstraction there is no distinction between them. However, the key issue is what constitutes the primitive representation used by these different systems. This is a pertinent issue, both in modeling human cognition and in building intelligent systems. 3. Variable binding with ....

FRANKLIN, S. and M. GARZON. (1990) Neural computability. In O. Omidvar, editor, Progress in Neural Networks, Vol. 1. NJ, USA. Ablex.

....in this theory including the proper definition of computability, and of universal computing engines analogous to the Universal Turing Machine the general outlines are clear (MacLennan 1987, 1990c, in press a, in press b; Wolpert MacLennan submitted; see also Blum 1989; Blum al. 1988; Franklin Garzon 1990; Garzon Franklin 1989, 1990; Lloyd 1990; Pour El Richards 1979, 1981, 1982; Stannett 1990) 1.3.2 In general, a computational system is characterized by: 1) a formal part, comprising a state space and processes of transformation; and (2) an interpretation, which (a) assigns meaning to the ....

....proper definition of computability, and of universal computing engines analogous to the Universal Turing Machine the general outlines are clear (MacLennan 1987, 1990c, in press a, in press b; Wolpert MacLennan submitted; see also Blum 1989; Blum al. 1988; Franklin Garzon 1990; Garzon Franklin 1989, 1990; Lloyd 1990; Pour El Richards 1979, 1981, 1982; Stannett 1990) 1.3.2 In general, a computational system is characterized by: 1) a formal part, comprising a state space and processes of transformation; and (2) an interpretation, which (a) assigns meaning to the states (thus making them ....

Franklin, S., & Garzon, M. (1990). Neural computability. In O. M. Omidvar (Ed.), Progress in neural networks (Vol. 1, pp. 127--145). Norwood, NJ: Ablex.

....of linearly interconnected nets. In Chapter 5, we see that no such superiority of computational power exists, at least when formalized in terms of polynomial time computation. Work that deals with infinite structure is reported by Hartley and Szu ( HS87] and by Franklin and Garzon ( FG90] and [GF89] some of which deals with cellular automata. There one assumes an unbounded number of neurons, as opposed to a finite number fixed in advance. See also the work by Hong [Hon88] which deals with nonuniform networks with real weights. In the paper [Wol91] Wolpert studies a class of machines ....

M. Garzon and S. Franklin. Neural computability. In Proc. 3rd Int. Joint Conf. Neural Networks, volume II, pages 631--637, 1989.

....to that of linearly interconnected nets. In Chapter 5, we see that no such superiority of computational power exists, at least when formalized in terms of polynomial time computation. Work that deals with infinite structure is reported by Hartley and Szu ( HS87] and by Franklin and Garzon ( FG90] and [GF89] some of which deals with cellular automata. There one assumes an unbounded number of neurons, as opposed to a finite number fixed in advance. See also the work by Hong [Hon88] which deals with nonuniform networks with real weights. In the paper [Wol91] Wolpert studies a class of ....

S. Franklin and M. Garzon. Neural computability. In O. M. Omidvar, editor, Progress In Neural Networks, pages 128--144. Ablex, Norwood, NJ, 1990.

....computing. One could argue that this paper disproves Pollack s hypothesis. Hartley Szu (1987) argued that TM s are equivalent both to potentially infinite neural networks with finite state neurons, and to finite networks of neurons with a countable infinity of states. More recently, Garzon and Franklin (1989, 1990; Franklin Garzon, 1990) have shown that countably infinite neural nets are more powerful than the class of countably infinite cellular automata, which are in turn more powerful than TM s; in particular they can solve the halting problem for TM s. On the other hand these neural networks are ....

....argue that this paper disproves Pollack s hypothesis. Hartley Szu (1987) argued that TM s are equivalent both to potentially infinite neural networks with finite state neurons, and to finite networks of neurons with a countable infinity of states. More recently, Garzon and Franklin (1989, 1990; Franklin Garzon, 1990) have shown that countably infinite neural nets are more powerful than the class of countably infinite cellular automata, which are in turn more powerful than TM s; in particular they can solve the halting problem for TM s. On the other hand these neural networks are less powerful than automata ....

Franklin, S., & Garzon, M. (1990). Neural computability. In O. M.

.... been clear how to simulate logic gates by networks of threshold (binary valued) neurons, and hence how to obtain finite automata using such nets (see e.g. 1] for more recent work on that problem) One can simulate Turing machines if one allows a potentially unbounded number of neurons; see e.g. [4] for variations on this theme and relations to cellular automata. Since we insist on a fixed number of neurons, which does not increase during the computation, our problem is different. Supported in part by US Air Force Grant AFOSR 880235 and by Siemens Corporate Research. 2 2. Statement of ....

Franklin, S., and M. Garzon, "Neural computability," in Progress In Neural Networks, Vol 1)(O. M. Omidvar, ed.), Ablex, Norwood, NJ, 1990, pp. 128-144.

....neurally inspired computing, in particular, for general purpose computing. Hartley Szu (1987) argued that TMs are equivalent both to potentially infinite neural networks with finite state neurons, and to finite networks of neurons with a countable infinity of states. More recently, Garzon and Franklin (1989, 1990; Franklin Garzon, 1990) have shown that countably infinite neural nets are more powerful than the class of countably infinite cellular automata, which are in turn more powerful than TMs; in particular they can solve the halting problem for TMs. On the other hand these neural networks are less ....

....in particular, for general purpose computing. Hartley Szu (1987) argued that TMs are equivalent both to potentially infinite neural networks with finite state neurons, and to finite networks of neurons with a countable infinity of states. More recently, Garzon and Franklin (1989, 1990; Franklin Garzon, 1990) have shown that countably infinite neural nets are more powerful than the class of countably infinite cellular automata, which are in turn more powerful than TMs; in particular they can solve the halting problem for TMs. On the other hand these neural networks are less powerful than automata ....

Franklin, S., & Garzon, M. (1990). Neural computability. In O. M.

....of physical nodes and connections at all. But if a von Neumann machine can emulate a neural network, then trivially a von Neumann machine is at least as powerful as a neural network. Therefore, they are computationally equivalent. For a more exacting treatment of these computational issues see Franklin and Garzon, 1991. The only apparent escape from this argument for a connectionist who believed neural networks to be computationally superior would be to emphasize that neural networks implemented as analog machines can take full advantage of the real valued connection weights, whereas a von Neumann emulation, ....

Franklin, S., and Garzon, M. (1991) `Neural Computability,' Progress in Neural Networks 1: 127-145.

....been used in [6] and [20] as well as by many other authors, often with an added infinite external memory device. Underlying the use of high order nets is the conjecture that their computational power is superior to that of linearly interconnected nets. Also related is the work reported in [7] [4], and [5] some of which deals with cellular automata. There one assumes an unbounded number of neurons, as opposed to a finite number fixed in advance. This potential infinity is analogous to the potentially infinite tape in a Turing machine; in our work, the activation values themselves are used ....

Franklin, S., and M. Garzon, "Neural computability," in Progress In Neural Networks, Vol 1)(O. M. Omidvar, ed.), Ablex, Norwood, NJ, 1990, pp. 128-144.

....of these operations can be implemented easily in the network model of real valued nodes and saturated linear transfer functions. Simulations of arbitrary Turing machines by networks with a potentially infinite number of nodes have been suggested by Hartley and Szu [36] and by Franklin and Garzon [21]. 3.2 Nonuniform Networks We now proceed to consider computation by nonuniform sequences of networks. In analogy with standard Boolean circuit complexity theory [70, 86] we assume that some set of nodes in a network are designated as input nodes, on which the input is initially loaded; then the ....

S. Franklin and M. Garzon. Neural computability. In O. M. Omidvar, ed. Progress in Neural Networks 1, pp. 128--144. Ablex, Norwood, NJ, 1990.

....or by architectures based on the Turing machine, or by the use of programs formulated in programming languages. Such a distinction is unlikely to be useful, however. For a start, both von Neumann and the neural network architectures are universal: Anything that one can do, the other can do also (Franklin and Garzon, 1990). Indeed, connectionist models are typically formulated in high level programming languages, and implemented on von Neumann machines, but this does not make them any less connectionist. Adams et al. (Chapter 1 in this volume) compare the capacities of programming language and neural network ....

Franklin, S., & Garzon, M. (1990). Neural computability. In O. Omidvar (Ed.), Progress in Neural Networks, Vol. 1., pp. 127-145. Norwood, NJ: Ablex.

....finite asymmetric networks of threshold gates are equivalent to finite automata 1 ; and various infinitary analogs of neural networks have recently been shown to be equivalent to Turing machines. For the latter type of result, constructions using a potentially infinite network are presented in [20, 35], and constructions using a finite network, but real valued neurons of arbitrary precision are presented in [35, 78, 89] The construction in [89] is rather interesting, as there the required precision grows only linearly in the space requirement of the simulated Turing machine, and only very ....

....finite automata as neural nets. It was shown recently in [3] that representing an automaton of n states may require Omega Gammaq n log n) 1=3 ) gates in the worst case. 10 PEKKA ORPONEN this result to the unpublished report [57] For approaches to computations on nets of unbounded size, see [19, 20, 23, 24]. Concerning symmetric nets, the fundamental result is Hopfield s [43] observation that a symmetric simple net using an asynchronous update rule always converges. Hopfield s result was based on defining an energy function on the global states of the network, such that in every permissible update ....

Franklin, S., Garzon, M. Neural computability. In: Progress in Neural Networks 1 (ed. O. M. Omidvar). Ablex, Norwood, NJ, 1990. Pp. 128--144.

.... center on these mechanisms, for example comparing the capabilities of Turing machines with those of connectionist networks (see Adams, Aizawa, and Fuller in this volume) However, connectionist networks can be proven to be computationally equivalent to the abstract notion of Turing machines [Franklin and Garzon, 1990]. Therefore the computational mechanism is not the crucial issue in separating the symbolic and subsymbolic paradigms. What then is the crucial issue We believe there are three major issues which distinguish the symbolic paradigm from the subsymbolic paradigm: 1) the type of representations; 2) ....

Franklin, S. and Garzon, M. (1990). Neural computability. In Omidvar, O., editor, Progress in Neural Networks, volume 1. Ablex, Norwood, NJ.

....conjecture regarding their superior computational power. We see that no such superiority of computational power exists, at least when formalized in terms of polynomial time computation. Work that deals with infinite structure is reported by Hartley and Szu ( 12] and by Franklin and Garzon ([8] and [9] some of which deals with cellular automata. There one assumes an unbounded number of neurons, as opposed to a finite number fixed in advance. In the paper [29] Wolpert studies a class of machines with just linear activation functions, and shows that this class is at least as powerful ....

S. Franklin, M. Garzon, "Neural computability," in Progress In Neural Networks, Vol 1)(O. M. Omidvar, ed.), Ablex, Norwood, NJ, (1990): 128-144.

No context found.

Garzon M., Franklin S., Neural Computability, Proc. Third Intern. Joint Conf. on Neural Networks, vol.2, 1989, 631-637.

No context found.

Garzon M., Franklin S., Neural Computability, Proc. Third Intern. Joint Conf. on Neural Networks, vol.2, 1989, 631-637.

No context found.

Franklin, S., Garzon, M. 1991. 'Neural Computability'. In O. Omidvar (ed.) 1991, Progress in Neural Networks, vol. 1, Norwood, N.J.: Ablex.

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