Information Theory Method for Flexible Network Synthesis

1. Information Theory Method for Flexible Network Synthesis Center of Information Resources and Communications Belarusian State University BELARUS{cheushev@bsu.unibel.by} Department of Computer Science University of Dortmund GERMANY {moraga@cs.uni-dortmund.de} V. Cheushev, C. Moraga, S. Yanushkevich, V. Shmerko, J. Kolodziejczyk Faculty of Computer Science & Information Systems Technical University of Szczecin POLAND / State University of Informatics & Radioelectronics Minsk, BELARUS {yanushkevich,shmerko@wi.ps.pl } Faculty of Computer Science & Information Systems Technical University of Szczecin, POLAND jkolodziejczyk@wi.ps.pl

2. Evolutionary synthesiser of combinational multilevel circuits Related fields Circuit Design Artificial Evolution Theory of Information

3. Related works T. Aoki, N. Homma, T. Higuchi, Evolutionary Design of Arithmetic network H. Iba, M. Iwata, T. Higuchi, Machine Learning Approach to Gate Level Evolvable Hardware S. Yamashita, H. Sawada, A. Nagoya,SPFD: A Method to Express Functional Flexibility S. Muroga, Y. Kambayashi, H. C. Lai, J.N. Culliney,The Transduction Method-Design of Logic Networks Based onPermissible Function

4. Traditional method Is the circuitwith output g a solution? „yes” if gf evolutionary synthesis „no” otherwise Is the Net a solution of the task or not? Net Problem formulation f is the given function

5. Functional search space Single solution: gf "Extended" solution: gT(f) Basic idea If g can be mapped into f , we can correct the Net networkto achieve the given functionality. The g is a neighbour of the given function f

6. Is the Net a solution? Definition of the neighbourhood neighbourhood of the function f „yes” if g T(f) „no” otherwise T(f)= E(f)V(f) the class of functions that can be corrected with a variable j ( g , v) = f vis primary input or internal signal of Net The class of functions that can be corrected with a constant j( g ) = f

7. g1 fsidentical mapping j = [012] g2fsbijective mapping j = [201] g3fcsurjective mapping j = [100] Class E( f ) of neighbour functions corrected with constant

8. g f g f g f 0 0 0 2 0 0 1 0 1 0 1 1 2 1 2 1 2 2 H( f | g) = 0 H(f | g) = 0 H(f | g) = 0 H(g | f) = 0 H(g | f) = 0,2 H(g | f ) = 0 Detection that gE( f ) using Information Theoretical Measures g f identical mapping gf bijective mapping gf surjective mapping

9. Equality H(f | g) = 0 is the necessaryand sufficient condition for g to belong to the class E(f) If H(f | g) = 0, then there exists a logic function jsuch that f= j( g ) In evolutionary synthesizer the entropy H(f | g) is the fitness-function ''Information Portrait'' of the class E( f )

10. Efficient Technique for Entropy Computation We use a non-probabilistic approach andcompute an entropy in ``integrated`` form instead of an averagedone. Thanks to that, only addition and look-up-table operationsare used to compute the entropy H(f | g) = Q(g) - Q(f,g) Relation to Shannon entropy: H(*) = K·H(*); where K is the total number of assignments (K = m nfor m-valuedfunctions of n arguments)

11. 0 f • Adding multiplexer 0 1 Net g • Adding gate with the constant 2 f gate Net g Network transformations to achieve the desired functionality f gE(f) j = [001]

12. Net1 X 1 g X 1 TSUM MAX X 2 Net2 X 1 g MAX X 2 H( f| g2) = 0 j = [120] Evolutionary synthesis look-up-table X 1 g MAX f X MOD 2 1 SUM Example H(f | g) = 1.74

13. H( f | g )  0 H( f | g ) = 0 gE(f) gV(f) primary inputs internal outputs gate gate gate gate suspect, that gate gate Detection of class gV( f )using Information Theoretical Measures v– variable source H(f | g ,v)= 0 If H(f | g ,v)= 0 , then there exists a logic function jsuch that f= j( g, v)

14. X 1 g H( f | g) = 10.63 Net MIN X 2 H( f | g,x1) = 0 H(f | g, x2) = 7.38 look-up-table Evolutionary synthesis X X 1 MOD 1 f g SUM MIN X 2 Example

15. Î Î class g E(f) class g V(f) approach test I/O K P[%] #G CPU P[%] #G CPU P[%] #G CPU c2a 100 0.1 11/1 7 72 9 42 7 No solution c4a 99 6 14/1 7 76 13 45 10 No solution mm3 67 5/1 3 55 25 44 21 43 24 28 63 monks2te 3 6/1 4 100 5 13 100 5 80 4 115 lensesmv 100 1 4/1 3 87 7 23 5 73 6 38 shuttlem 98 21 6/1 4 80 10 118 7 22 10 251 Experimental resultsfor POLO* benchmarks Proposed approach Traditional CPU-average time in minutes #G – the best solution in 100 runs *Portland Logic Optimization Group, Electrical Engineering Department, Portland State University, OR, USA, http://www.ee.pdx.edu/polo

16. instead of looking for a network that exactly implements the target function, we look for neighbournetwork, the neighbournetwork can be corrected with a simple regular methodto achieve the given functionality Conclusions We achievedthe flexibility of network synthesis:

17. Extreme reduction of the search space 100% effectiveness achieved (in most cases) Better possibility to obtain good solutions Reducing the time consumption Conclusions cont. Applying this approach under a method based on evolution paradigm, we can obtain the following advantages:

18. Circuit matching based on Information Measures More powerful apparatus of Information Measures Regular methods of corrections Interpretation of the concept of the Set ofPairs of Function to be Distinguished(SPFD)in terms of Information Measures (approach by S. Yamashita, H. Sawada, A. Nagoya) Future Work