Electrical Engineering and Computer Science Department The University of Toledo Toledo, OH 43606

Transcription

1 Gursel Serpen and David L. Livingston Electrical Engineering and Computer Science Department The University of Toledo Toledo, OH ANNIE 98 PRESENTATION

2 THE RESEARCH WORK A higher-order single-layer relaxation-type recurrent neural network is employed to plan a task in a decomposed state space. The finite state machine model of the state space of the complex task is decomposed into parallel/series combinations of component machines, each of which represents a subtask, using lattice theoretic techniques. Planning the complex task is realized by identifying the transfer sequence, an ordered set of inputs, of the component state machines for the subtasks. A fourth-order stochastic optimization neural network, single-layer relaxation-type recurrent network with simulated annealing, is utilized to perform the search needed to specify the transfer sequence in the state space of the complex task. The effect of decomposition on two essential areas is highlighted: significant reduction in computational resources required to implement the neural algorithm and the need to employ a high order neural network are emphasized. ANNIE 98 Presentation Page 2 of 8

3 TOWERS OF HANOI PROBLEM The problem is moving a stack of disks from one peg to another by moving one disk at a time and never stacking a disk on top of a smaller disk. State 0 State 1 State 2 State 3 State 4 State 5 State 6 State 7 State 8 State Space for the Two-Disk Towers of Hanoi Problem. ANNIE 98 Presentation Page 3 of 8

4 STATE SPACE STATE MACHINE LATTICE OF SUBSTITUTION PROPERTY PARTITIONS PARALLEL OR SERIALLY DECOMPOSED STATE MACHINE ANNIE 98 Presentation Page 4 of 8

5 SERIAL DECOMPOSITION Using substitution property partition π, head component machine, { 0, 3, 6; 1, 4, 7; 2, 5, 8} π = S S S S S S S S S and generating a suitable partition τ, tail component machine, with the constraint = 0 where τ = { S0, S1, S2; S3, S4, S5; S6, S7, S8} π τ a serial decomposition of the task can be obtained. The next step is to assign states, which can be done in an arbitrary fashion, to the blocks of the partitions to facilitate the derivation of state tables for the two component machines. π = S τ = S B1, S, S B 4, S, S B2 ; S, S, S B5 ; S, S, S B3 ; S, S, S B6 ; S, S, S input I Machine π (head) Machine τ (tail) Two Component Serial Decomposition Topology ANNIE 98 Presentation Page 5 of 8

6 N-TH ORDER RELAXATION-TYPE BOLTZMANN MACHINE The n-th order Boltzmann machine is a stochastic single-layer recurrent network and will be employed in relaxation mode to search for the shortest path. The performance function of an n-th order Boltzmann machine is defined as E N 1 = w s s s s b N i i in i i in ik ik i1 i2 in ik = 1 where w i1i2 in is an N-dimensional weight matrix symmetric on all pairs of indices, b ik is the external input to and s ik is the output of the computation node i k with k =1 N. Each computation node output is binary valued, zero or one, and the activation function is defined as ( 1) k P s 1 = = 1 + e i net / T where P ik is the probability that s ik, the output of unit i k, is equal to 1, T is a timevarying computational parameter analogous to temperature and net ik is the input sum to unit i k. i k The term net ik is defined by where i { j j j } k net = w s s s i j j j j j j j j j k 1 2 N N N 1 2 N 1, ANNIE 98 Presentation Page 6 of 8

7 FOURTH-ORDER RELAXATION-TYPE BOLTZMANN MACHINE TOPOLOGY For a two-component decomposition, the M N topology can be partitioned into two arrays with dimensions M 1 N and M N where M 2 1 is the number of states in the first component machine and M 2 is the number of states in the second component machine with M1 + M2 < M B 1 H E A D B 2 B 3 T A I L B 4 B 5 B 6 Neural Network Array for a Solution of the Towers of Hanoi Problem ANNIE 98 Presentation Page 7 of 8

8 DERIVATION OF WEIGHT PARAMETER VALUES A serially decomposed state machine will require a fourth order neural network architecture. The general form of the performance function for a fourth-order Boltzmann machine is 1 E = w s s s s s b 2 i j k l ijkl i j k l i i i where w ijkl is the weight between computation nodes si, s j, sk, and sl. The weight term is defined by w ijkl =g αδ α α ijkl where α is the index over the set of constraints and g α R + if the hypothesis all nodes represent for constraint α are mutually supporting and g α R if the hypothesis are conflicting. The term δ α ijkl is equal to 1 if the hypotheses represented by si, s j, sk, and sl are compatible under constraint α and is equal to 0 otherwise. Bounds on constraint weight parameters can defined to establish the complete set of solutions as stable equilibrium points in the state space of the neural network dynamics as follows: g1 + 2g3 g2 and 2g 1 + g 3 g 4 ANNIE 98 Presentation Page 8 of 8

9 CONCLUSIONS We have demonstrated the use of a high order Boltzmann machine for task planning problem in decomposed state spaces. By using the decomposed form of the state machines that are models of the tasks, the number of processing elements in the Boltzmann machine can be reduced at the expense of high order connections. We have demonstrated the need for high order machines in a class of problems, which may be of practical significance to task planning and execution. In the case of task planning in decomposed state spaces, it was observed that a high-order neural network was necessary to implement the compatibility relations between subtasks. It is still an open research question if the proposed task planning neural network will scale well with the large size state spaces. We have employed a procedure to define the values of constraint weight parameters to establish the stability of problem solutions in the state space of the neural network dynamics. Preliminary simulation results indicated that the high order Boltzmann machine converged to a solution after majority of relaxations. Further research is needed to firmly establish the performance of the high order Boltzmann machine performance. ANNIE 98 Presentation Page 9 of 8