Turing´s analysis of computation and artificial neural network

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(1)Universidade Federal de Pernambuco Centro de Informática Departamento de Ciência da Computa¸cão. Pós-gradua¸cão em Ciência da Computa¸cão. TURING’S ANALYSIS OF COMPUTATION AND ARTIFICIAL NEURAL NETWORKS Wilson Rosa de Oliveira Júnior Ph.D Thesis. Recife February 2004.

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(3) Universidade Federal de Pernambuco Centro de Informática Departamento de Ciência da Computa¸cão. Wilson Rosa de Oliveira Júnior TURING’S ANALYSIS OF COMPUTATION AND ARTIFICIAL NEURAL NETWORKS. Thesis presented in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the P´ osgradua¸ c˜ ao em Ciˆ encia da Computa¸ c˜ ao of the Centro de Inform´ atica at Universidade Federal de Pernambuco.. Adviser: Prof. Marc´ılio Carlos Pereira de Souto. Recife 19th of February, 2004.

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(5) Acknowledgments In my academic life I have had the pleasure of having been supported, taught and influenced by so many friends. I am very grateful indeed for all that they have done. In chronological order of their appearance in my life I list: Ivan Pedro, M´ ucio Gomes da Silva Queiroz, Antônio Carlos Pavão, Teresa Bernarda Ludermir, Sóstenes Lu´ıs Soares Lins, Rildo Pragana, Benedito de Melo Acioly, Manoel Agamêmnon Lopes, Em´ılio de Barros Lucena, Michael Berry Smyth, Pietro Di Gianantônio, Dieter Spreen, S´ılvio Romero Lemos Meira e Marc´ılio Carlos Pereira de Souto. I would like to thanks the support of my families in all phases of my life, for all that we have shared, for being understanding that so many times when I was absent from their company for doing my work and for having influenced making myself what I am. My parents: Wilson Rosa de Oliveira and Glauce Tavares de Oliveira. My first wife: Teresa Bernarda Ludermir. My two sons: Artur Ludermir de Oliveira and Rodrigo Ludermir de Oliveira. My wife: S´ılvia Carla Felipe da Costa. In the course of my life I have had many dog-friends all having their share in my heart. In chronological order: Preta, Cacilda, Ieda, Plutão, Bruna, Laila, Julie and Lambão. I am grateful to all of them for their love and friendship. I would like to thanks Prof. Marc´ılio Carlos Pereira de Souto for having suggested to extend an earlier old result of mine from weightless to weighted neural networks which ended up in this thesis. For achieving my results I had the institutional support of Prof.Teresa Bernarda Ludermir. The systematics of requiring that the student must defend the thesis proposal is very illuminating. The examiners back then - Clylton Galamba Fernades, Edson Carvalho de Barros Filho e Gérson Zaverucha - made fruitful and decisive suggestions. The examiners for the viva voce - besides the three, Benjamin Callejas Bedegral and Francisco de Assis Tenório - have also their share in the contribution for the improvement of this final version of the Thesis. Special gratitude to Gérson Zaverucha for many suggestions in the proposal. Feeling overwhelmed I must admit that his suggestions made the Thesis much clearer. He also made me see my work in a better perspective. i.

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(7) Abstract A. simulation. of. Turing. Machines. (TMs). by. Artificial. Neural. Networks (ANNs) is presented, inspired by a suggestion of McCulloch and Pitts in their seminal paper. In contrast to previous work, such a simulation is in agreement with the correct interpretation of Turing’s analysis of computation; compatible with the current approaches to analyze cognition as an interactive agent-environment process; and physically realizable since it does not use connection weights with unbounded precision. A full description of an implementation of a universal TM into a recurrent sigmoid ANN focusing on the TM finite state control is given. The tape, an infinite resource, is left out of the encoding as an external non-intrinsic feature. This resulting network will be called a Neural Turing Machine. The classical model of computation Turing Machine = Tape + Finite State Automaton (FSA) is replaced by the neural model of computation Neural Turing Machine (NTM) = Tape + Artificial Neural Network (ANN) Arguments for cognitive and physical plausibility of this approach are given and the mathematical consequences of them are investigated. iii.

(8) iv It is widely known in the neurocomputing theory community that not every arbitrary FSA can be implemented in an ANN when noise or limited precision is considered: under these conditions analog systems in general, and ANN in particular, are computationally equivalent to Definite Automata — a very restrict class of FSA. Among the main contribution of the approach is the definition of a new machine model, definite Turing machine(DTM), which arises when noise is taken into account. This result reflects on the second equality described above giving Noisy NTM (ηNTM) = Tape + Noisy ANN(ηANN) with a corresponding Definite Turing Machine = Tape + Definite Finite State Automaton (DFSA) The investigation of the computational abilities of the Definite Turing Machines is another important contribution of the Thesis. It is proved that they compute the class of elementary functions (Brainerd & Landweber, 1974) from Recursion Theory. Keywords: i) Theory of Computing. ii) Artificial Neural Networks - Neurocomputing. iii) Automata Theory - Definite Automata. iv) Recursive Functions - Simple and Elementary..

(9) Resumo Inspirado por uma sugestão de McCulloch e Pitts em seu trabalho pioneiro, uma simula¸cão de Máquinas de Turing (MT) por Redes Neurais Artifiais (RNAs) apresentada. Diferente dos trabalhos anteriores, tal simula¸cão está de acordo com a interpreta¸cão correta da análise de Turing sobre computa¸cão; é compatvel com as abordagens correntes para análise da cogni¸cão como um processo interativo agente-ambiente; e é fisicamente realizável uma vez que não se usa pesos nas conexõs com precisão ilimitada. Uma descri¸cão completa de uma implementa¸cão de uma MT universal em uma RNA recorrente do tipo sigmóide é dada. A fita, um recurso infinito, é deixada fora da codifiçcão como uma caracterstica externa não-intr´ınsica. A rede resultante é chamada de Máquina de Turing Neural. O modelo clássico de computa¸cão Máquina de Turing = Fita + Autômato de Estados Finito (AEF) é trocado pelo modelo de computa¸cão neural Máquina de Turing Neural (MTN) = Fita + Rede Neural Artifial (RNA) Argumentos para plausabilidade f´ısica e cognitiva desta abordagem são fornecidos e as conseq¨ uências matemáticas são investigadas. v.

(10) vi ´ bastante conhecido na comunidade de neurocomputa¸cão teórica, que E um AEF arbitrário não pode ser implementado em uma RNA quando ru´ıdo ou limite de precisão é considerado: sob estas condi¸cões, sistemas analõgicos em geral, e RNA em particular, são computacionalmente equivalentes aos Autˆ omatos Definidos – uma classe muita restrita de AEF. Entre as principais contribui¸cões da abordagem proposta é a defini¸cão de um novo modelo de máquina, M´ aquina de Turing Definida(MTD), que surge quando ru´ıdo é levado em considera¸cão. Este resultado reflete na segunda equa¸cão descrita acima se tornando MTN com ru´ıdo (ηMTN) = Fita + RNA com ru´ıdo(ηRNA) com a equa¸cão correspondente Máquina de Turing Definida = Fita + Autômatos Finitos Definidos (AFD) A investiga¸cão de capacidades computacionais das Máquina de Turing ´ provado que elas Definida é uma outra contribui¸cão importante da Tese. E computam a classe das fun¸cões elementares (Brainerd & Landweber, 1974) da Teoria da Recursão. Palavras-Chave: i) Teoria da computa¸cão. ii) Redes neurais artificiais - Neurocomput¸cão. iii) Teoria dos autômatos - Autômatos definidos. iv) Fun¸cões recursivas - Simples e elementares..

(11) Contents. 1 Introduction and overview. 1. 1.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1. 1.2. Artificial neural networks . . . . . . . . . . . . . . . . . . . . .. 2. 1.3. Feedforward and sequential neural networks . . . . . . . . . .. 5. 1.4. Models of artificial neuron . . . . . . . . . . . . . . . . . . . .. 8. 1.5. Recurrent Systems . . . . . . . . . . . . . . . . . . . . . . . . 11. 1.6. Organization of the thesis . . . . . . . . . . . . . . . . . . . . 13. 2 Classical Models. 17. 2.1. Strings, alphabets and languages . . . . . . . . . . . . . . . . 17. 2.2. Finite-state machines and languages . . . . . . . . . . . . . . . 21 2.2.1. 2.3. Finite-state machines . . . . . . . . . . . . . . . . . . . 22. Finite-state machines with finite memory . . . . . . . . . . . . 30 2.3.1. Finite-memory and definite-memory machines . . . . . 30. 2.4. Turing machines and effective computation . . . . . . . . . . . 37. 2.5. Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 42. 3 Neural Networks Models. 45 vii.

(12) viii. CONTENTS 3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45. 3.2. Recurrent weighted neural networks . . . . . . . . . . . . . . . 46 3.2.1. Discrete-time recurrent weighted neural network . . . . 48. 3.2.2. First-order Trains . . . . . . . . . . . . . . . . . . . . . 50. 3.2.3. Second-order DTRNNs. 3.2.4. NARX networks. . . . . . . . . . . . . . . . . . 54. . . . . . . . . . . . . . . . . . . . . . 55. 3.3. Recurrent asynchronous neural networks . . . . . . . . . . . . 58. 3.4. Recurrent networks behaving as FSMs . . . . . . . . . . . . . 61. 3.5. 3.4.1. Recurrent networks based on threshold units . . . . . . 61. 3.4.2. Recurrent networks based on sigmoid functions . . . . 63. 3.4.3. Stable Implementation . . . . . . . . . . . . . . . . . . 66. Simple single-layer sequential WNNs . . . . . . . . . . . . . . 76 3.5.1. Weightless Neural Networks . . . . . . . . . . . . . . . 79. 4 Implementations of TMs in ANNs.. 83. 4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83. 4.2. Weighted Neural Turing Machines . . . . . . . . . . . . . . . . 84. 4.3. 4.4. 4.2.1. Networks with infinite number of neurons. . . . . . . . 85. 4.2.2. Networks with a finite number of neurons. . . . . . . . 86. A Critique. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 4.3.1. Turing’s analysis of computation . . . . . . . . . . . . 90. 4.3.2. Analog noise and ANNs . . . . . . . . . . . . . . . . . 92. 4.3.3. Turing machines and the study of cognition . . . . . . 94. 4.3.4. Objections for super-Turing computation . . . . . . . . 96. TM and Weightless Neural Networks . . . . . . . . . . . . . . 97.

(13) ix. CONTENTS 5 Neural Turing Machines. 101. 5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101. 5.2. Simulating the finite control of a TM in ANN . . . . . . . . . 104. 5.3. Implementing a Universal TM . . . . . . . . . . . . . . . . . . 107. 5.4. A Proof of Universality of ANN . . . . . . . . . . . . . . . . . 110. 6 Turing Machines with Finite Memory. 113. 6.1. Definite Turing Machines . . . . . . . . . . . . . . . . . . . . . 114. 6.2. Simple Functions . . . . . . . . . . . . . . . . . . . . . . . . . 115. 6.3. Finite Memory Machines . . . . . . . . . . . . . . . . . . . . . 117. 7 Conclusions and further directions 7.1. 121. Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 121. A Robinson-Fallside Networks and FSA. 125. A.1 Single Layer Robinson-Fallside Networks . . . . . . . . . . . . 125 A.2 Augmented Robinson-Fallside Networks . . . . . . . . . . . . . 128 B Related Published Papers. 133. B.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 B.2 Turing Machine Simulation by Logical Neural Networks . . . . 133 B.3 Agent-Environment Approach to the Simulation of Turing Machines by Neural Networks . . . . . . . . . . . . . . . . . . . . 134 B.4 Turing Machines with Finite Memory . . . . . . . . . . . . . . 134 B.5 Turing’s Analysis of Computation and Artificial Neural Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 B.6 Computational Complexity and Pyramidal Architectures . . . 135.

(14) x. CONTENTS B.7 Synthesis of Probabilistic Automata in p−RAM Neural Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135.

(15) List of Figures 1.1. A typical feedforward neural network . . . . . . . . . . . . . .. 6. 1.2. A typical sequential neural network . . . . . . . . . . . . . . .. 7. 1.3. A typical weighted neuron . . . . . . . . . . . . . . . . . . . . 10. 2.1. The finite-state machine model . . . . . . . . . . . . . . . . . 23. 2.2. A conventional implementation of an FSM . . . . . . . . . . . 24. 2.3. An FSM computing the dual parity of its inputs. 2.4. An implementation of an FMM . . . . . . . . . . . . . . . . . 31. 2.5. An FMM of input-order 2 and output-order 2 . . . . . . . . . 33. 2.6. An implementation of a DMM. 2.7. A DMM of input-memory 1 . . . . . . . . . . . . . . . . . . . 36. 2.8. A Turing machine . . . . . . . . . . . . . . . . . . . . . . . . . 39. 3.1. A schematic view of an DTRNN. 3.2. A schematic view of a Simple Recurrent Network (SRN) . . . 51. 3.3. A schematic view of a Robinson-Fallside Network . . . . . . . 52. 3.4. A schematic view of an augmented Robinson-Fallside Network. 3.5. A second-order architecture . . . . . . . . . . . . . . . . . . . 55. 3.6. An implementation of a second-order DTRNN . . . . . . . . . 56 xi. . . . . . . . 27. . . . . . . . . . . . . . . . . . 35. . . . . . . . . . . . . . . . . 49. 53.

(16) xii. LIST OF FIGURES 3.7. A NARX network with 2 input-memory taps and 3 outputmemory taps . . . . . . . . . . . . . . . . . . . . . . . . . . . 57. 3.8. A TDNN with 3 input-memory taps . . . . . . . . . . . . . . . 58. 3.9. A Typical Hopfield network . . . . . . . . . . . . . . . . . . . 60. 3.10 A Simple Single-layer Sequential WNN (SSSWNN) . . . . . . 78 4.1. A WNN simulating the TM instruction δ(qi , σ) = (qj , σ 0 , m). 5.1. (a)A Turing machine T which multiplies by two a given num-. . 98. ber in unary base. (b)The control of the TM T as a Mealy machine M , as in the text. . . . . . . . . . . . . . . . . . . . . 106 5.2. (a)The split-state split-output Mealy machine S, from M , above. Again pairs been encoded into integers. The encoding takes (q, σ) into q + |Σ| ∗ σ. (b)The Mealy machine R obtained from S above removing the inaccessible sates . . . . . . . . . . 107. 5.3. The control of the TM T as a Mealy machine M , as in the text. The composed output pairs (σ, d), σ ∈ Σ, d ∈ D, has been coded into the integer σ|M | + d, i.e. it is seen as a number in base |D| (which is 3 in this case). . . . . . . . . . . 108. 5.4. The split-state split-output Mealy machine S, from M , above. Again pairs have been encoded into integers. The encoding takes (q, σ) into q + |Σ| ∗ σ. . . . . . . . . . . . . . . . . . . . 109. 5.5. The Mealy machine R obtained from S above removing the inaccessible sates . . . . . . . . . . . . . . . . . . . . . . . . . 110.

(17) LIST OF FIGURES 6.1. xiii. A not definite Turing machine T which finds the GCD of two number in unary notation using the Euclid’s algorithm . . . . 114. A.1 (a) FSA for odd parity, i.e. recognizes the language of 0’s and 1’s in which the number of 1’s is odd. . . . . . . . . . . . . . . 126.

(18) xiv. LIST OF FIGURES.

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