Decomposton of Boolen Functon Sets for Boolen Neurl Netorks Romn Kohut,, Bernd Stenbch Freberg Unverst of Mnng nd Technolog Insttute of Computer Scence Freberg (Schs), Germn
Outlne Introducton Boolen Neuron AND-Decomposton Trnng of the Boolen Neurl Netork (BNN) Usng the BNN Exmple Concluson Decomposton of Boolen Functon Sets for Boolen Neurl Netorks / 11 Romn Kohut / Bernd Stenbch / 6th Interntonl Workshop on Boolen Problems, Freberg, September 3-4, 004
Introducton The ms: compct representton nd fst computton of Boolen functons for rtfcl neurl netorks The Problem: f : B n B n dfferent bnr vectors Usul ANNs Itertve lgorthms (BP, RBF, ) Long trnng tme Nontertve lgorthms /sequentl trnng lgorthms/ (ETL, FTFS, ) Lrge memor (RAM) sze Decomposton of Boolen Functon Sets for Boolen Neurl Netorks 3 / 11 Romn Kohut / Bernd Stenbch / 6th Interntonl Workshop on Boolen Problems, Freberg, September 3-4, 004
Boolen Neuron Defnton of Boolen Neuron Out Inp f B B f B { x x,..., } 1, B x N x Out {,..., }, B 1 N x Out B - output sgnl f B, Out B {0,1} f ( Inp, ) ( Inp, ) B x B {0,1} {0,1} - Boolen trnsfer functon Advntges of the BN: speedng up of clculton sgnfcntl, reducton of necessr memor sze, possblt to mp the BN nto CLB of FPGAs. Inputs x 1 x x 3 x N Decomposton of Boolen Functon Sets for Boolen Neurl Netorks 4 / 11 Romn Kohut / Bernd Stenbch / 6th Interntonl Workshop on Boolen Problems, Freberg, September 3-4, 004 1 3 n Weghts of snptc connectons f Trnsfer functon Output ( x ) f, Generl structure of Boolen neuron
Boolen Neuron hdden neuron output neuron Out Z [ ] ( [ ] ) n ( ) z [ ] [ ] [ ] f Inp Out f Inp [ z] z [z] Out output sgnl of the neuron th number z, [z] f trnsfer functon of the neuron thnumber z, [z] ndex z 1,..., Z N, Z N number of neurons on the hdden ler, [ ] [ ] f f :, [ 1, ] Z N Decomposton of Boolen Functon Sets for Boolen Neurl Netorks 5 / 11 Romn Kohut / Bernd Stenbch / 6th Interntonl Workshop on Boolen Problems, Freberg, September 3-4, 004 1 [ ] number of neuron on the output ler of the BNN f {AND, equvlence } Zn [ ] 1 ( [ ] Inp ) Out f f {OR, EXOR}
A AND-Decomposton - Trnng of the BNN Trnng lgorthm 1,1,1,1 N x,1 1,,, N x, 1, N, N, N N x, N,1,1 N x,1 z number of hdden lers,, coeffcent of mtrx A, N x number of Boolen vrbles, N number of Boolen functons, m, n eghts vlues, N x, 1, N, N, N N x, N I - ndex of bse ro of mtrx A, N I set of numbers of mnml vlues of the vector m, s element of uxlr vector s, v element of bse ro v of mtrx A, k vlue of trnsfer functon k, N set of column numbers th v 1, n mx(v, N v ), snptc eght, N k0 set of ro numbers th k 0. 1,1 1,, N Nx m Σ, ; n, 1 Σ1 Inumb(mn(m, Nx )) N ( n * ) Decomposton of Boolen Functon Sets for Boolen Neurl Netorks 6 / 11 Romn Kohut / Bernd Stenbch / 6th Interntonl Workshop on Boolen Problems, Freberg, September 3-4, 004 Y s v I, k, N v N k 1, 0 ( z+ 1), z 0 N I 1 ( z), ( z),,, Inumb(mx(s,N I )) k z z+1 m N, 0 1 N
AND-Decomposton - Usng of the BNN Results of the trnng: unque trnsfer functons of the hdden neurons: k z f (x), eghts z, of the output neurons, the structure of BNN: three-ler rchtecture (fx), N x nput neurons, N output neurons, Z N hdden neurons. Tpe of decomposton: AND Z N ( x) z, k z ( x) z, z 1 ( ) Decomposton of Boolen Functon Sets for Boolen Neurl Netorks 7 / 11 Romn Kohut / Bernd Stenbch / 6th Interntonl Workshop on Boolen Problems, Freberg, September 3-4, 004
Exmple Trnng process 1,...,, 10 - Boolen functons, f ( x1, x, x3) v I, Prmr mtrx A x 1 x x 3 1 3 4 5 6 7 8 9 10 0 0 0 1 1 0 0 1 1 0 0 0 1 0 0 1 0 0 0 0 1 0 0 0 0 1 0 1 0 1 1 0 1 0 0 1 1 0 0 0 1 1 0 0 1 0 1 0 0 1 1 1 1 0 0 1 0 1 1 0 1 0 0 0 1 1 0 1 0 0 0 0 0 0 0 1 0 0 1 1 0 0 0 1 0 1 0 0 1 1 1 1 1 1 1 1 0 1 0 0 1 1 0 0 A 1 N x n, Σ1 5 ( z) ( z+ 1),, ( z), N k, m N, Σ1 v 1 Mtrx A fter the frst ccle of trnng 1 mn(m), I 5 0 mx(n, N v ) 5 v Nk Decomposton of Boolen Functon Sets for Boolen Neurl Netorks 8 / 11 Romn Kohut / Bernd Stenbch / 6th Interntonl Workshop on Boolen Problems, Freberg, September 3-4, 004, 0 k, 0 1 1 3 4 5 6 7 8 9 10 m 1 1 0 0 1 1 0 0 0 1 5 0 0 0 0 0 1 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 10 0 0 1 0 1 0 0 1 1 1 5 1 0 1 1 0 1 0 0 0 1 5 0 0 1 0 1 1 0 1 1 1 6 0 0 1 0 1 0 0, 1 1 1 5 1 1 1 1 1 10 1 1 1 1 10
Exmple Structure of BNN 4 Hdden neurons 1 f ( x1, x, x3) 3 Input neurons k 1 x 1 k N x 3 x x 3 k 3 10 Output neurons N 10 10 k 4 BNN to represent the set of Boolen functon, 1,..., 10 Decomposton of Boolen Functon Sets for Boolen Neurl Netorks 9 / 11 Romn Kohut / Bernd Stenbch / 6th Interntonl Workshop on Boolen Problems, Freberg, September 3-4, 004
Exmple Reconstructon of the set of Boolen functons Out: 1 3 4 5 6 7 8 9 10 1 0 0 1 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 1 0 3 0 1 0 0 1 0 1 1 1 0 4 1 1 0 1 0 1 1 0 0 0 Z N ( x) z, k z ( x) z, z 1 ( ) x 1 x x 3 k 1 k k 3 k 4 1 3 4 5 6 7 8 9 10 0 0 0 1 0 1 1 1 1 0 0 1 1 0 0 0 1 0 0 1 1 0 1 0 0 0 0 0 1 0 0 0 0 1 0 1 0 0 1 1 1 1 1 0 1 0 0 1 1 0 0 0 1 1 1 1 1 0 0 0 1 0 1 0 0 1 1 1 1 0 0 1 1 0 1 1 0 1 1 0 1 0 0 0 1 1 0 1 0 1 1 0 0 0 0 0 0 0 0 1 0 0 1 1 0 1 1 1 0 0 0 1 0 1 0 0 1 1 1 1 1 1 0 1 1 1 1 1 0 1 0 0 1 1 0 0 Decomposton of Boolen Functon Sets for Boolen Neurl Netorks 10 / 11 Romn Kohut / Bernd Stenbch / 6th Interntonl Workshop on Boolen Problems, Freberg, September 3-4, 004
Concluson Results: (1) forml defnton of the Boolen neuron () smple mppng to the CLBs of FPGAs (3) pplcton of Boolen neurons n Boolen neurl netorks (BNN), tht relze set of Boolen functons (4) decomposton of set of Boolen functons nto common bsc functons (5) structures of the BNN for AND, OR, XOR nd equvlence-decomposton of set of Boolen functons Further ork: optmzton of the BNN usng mxed tpes of output neurons developng а progrm tht uses Boolen neurl netorks for compct presentton nd fst clculton of Boolen functons Decomposton of Boolen Functon Sets for Boolen Neurl Netorks 11 / 11 Romn Kohut / Bernd Stenbch / 6th Interntonl Workshop on Boolen Problems, Freberg, September 3-4, 004