Oracle

Turing

Machine

a Neural Network Approach

Contents

  • Oracle Turing Machines
  • Feed Forward Neural Networks
  • Implementation
  • Results

Oracle Turing Machine

Regular Turing machines are defined as a 7-tuple

\{Q, \Gamma, b, \Sigma, \delta, q_0, F\}
{Q,Γ,b,Σ,δ,q0,F}\{Q, \Gamma, b, \Sigma, \delta, q_0, F\}
  • Q is a finite not empty set of states.
  • Gamma is a finite not empty set of alphabet symbols.
  • b is the blank symbol.
  • Sigma is the set of input symbols.
  • q0 is the initial state.
  • F is the final states.
  • delta is a transition function

Turing [1939]  very briefly introduced the notion of an oracle machine

 

A Turing machine which could consult an oracle tape (database), but he did not develop the idea

Turing Machine

Input Tape Head

Oracle Tape Head

Turing's oracle machine concept lay dormant for years until Emil Post's magnificentpaper [1944] revived it, greatly expanded it, and cast the subject in an informal, intuitive light.

Regular Turing machines are defined as a 8-tuple

\{Q, \Gamma, b, \Sigma, \delta, q_0, F, \zeta\}
{Q,Γ,b,Σ,δ,q0,F,ζ}\{Q, \Gamma, b, \Sigma, \delta, q_0, F, \zeta\}
  • Q is a finite not empty set of states.
  • Gamma is a finite not empty set of alphabet symbols.
  • b is the blank symbol.
  • Sigma is the set of input symbols.
  • q0 is the initial state.
  • F is the final states.
  • Zeta is an oracle function to send queries to
  • delta is a transition function

Feed Forward Neural Networks

The architecture of choice for this work was a two layered feed forward neural network

x_t = f_t(w_t^Tx^*_{t-1})
xt=ft(wtTxt−1∗) x_t = f_t(w_t^Tx^*_{t-1})
y = f_N(w_N^Tx^*_{N-1})
y=fN(wNTxN−1∗) y = f_N(w_N^Tx^*_{N-1})

In order to train our neural network we are going to use stochastic gradient descent

w_{t+1} = w_t - \gamma \frac{1}{n}\displaystyle\sum_{i=1}^n \nabla_w l(x_i, w_t)
wt+1=wt−γ1n∑i=1n∇wl(xi,wt) w_{t+1} = w_t - \gamma \frac{1}{n}\displaystyle\sum_{i=1}^n \nabla_w l(x_i, w_t)

with n = 1

Implementation

 

Results

 

 

Thanks

 

 

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