Energy functions of Various Logic Networks and their Dynamics Analysis
Project/Area Number  09680382 
Research Category 
GrantinAid for Scientific Research (C).

Section  一般 
Research Field 
Intelligent informatics

Research Institution  Ryukoku University 
Principal Investigator 
KOBUCHI Youichi Faculty of Science and Technology, Ryukoku University, Professor, 理工学部, 教授 (60025450)

Project Fiscal Year 
1997 – 1999

Project Status 
Completed(Fiscal Year 1999)

Budget Amount *help 
¥2,700,000 (Direct Cost : ¥2,700,000)
Fiscal Year 1999 : ¥600,000 (Direct Cost : ¥600,000)
Fiscal Year 1998 : ¥600,000 (Direct Cost : ¥600,000)
Fiscal Year 1997 : ¥1,500,000 (Direct Cost : ¥1,500,000)

Keywords  Logic network / Energy function / Multiple valued logic network / Global state transition graph / Homomorphism / Prime decomposition / Quasiinverse function / Invertible ANN / 論理ネットワーク / エネルギー関数 / 多値論理回路網 / 大域的状態遷移図 / 準同型 / 素分解 / 準逆関数 / 逆論理回路網 / 論理回路網 / 非同期動作 / 論理関数の展開 / 同型 / 周期構造 / 距離を保つ写像 / グレイ符号 / リヤプノフ関数 / 神経回路網 / 状態関数 / Influence system 
Research Abstract 
1. A network of binary logic functions can be expressed as a higherorder neural network. We characterized the behavior of such artificial neural networks (ANN's) in terms of an Energy function. We first treat a special type of ANN's which have Energy functions and give their characterization. This is related with symmetry of the networks. Then we define a partial order on ANN's induced by their dynamic structure. The special type ANN's mentioned above are the maximal elements in the set of ANN's with Energy functions. 2. For a multiple valued logic network in which the state transition of each element occurs in a stepwise fashion, we show that any state function can be described in a standard polynomial form. Then we define a derivative of state functions as if they were defined on real vectors. If a multilinear state function is given, then the network is stable and has an Energy function under asynchronous operation mode. 3. If a multilinear state function has order two, then the corresponding logic network has cycles whose lengths are at most two under synchronous operation mode. 4. We define a homomorphism relation of logic networks and using this basic relation, we analyze the cycle structures of logic network dynamics. 5. We also treat the isomorphism of the dynamics and reveal some basic properties. In doing so, we characterize distance preserving mappings by their generators : transpositions and negation of variables. 6. An ANN is said to be invertible if there exists another ANN of the same order such that whose graph is obtained from the former graph by reversing the direction of all edges disregarding self loops. We characterize the invertible ANN using a new kind of logic function expansion, called prime decomposition. 7. Using prime decomposition, we define a quasiinverse function with respect to a variable. For a given logic network, we can have the inverse logic network by replacing each logic function with its quasiinverse function.

Report
(5results)
Research Output
(17results)