臨界函數是布林函數的子集，可以被單一臨界邏輯閘呈現的布林函數是臨界函數。在臨界邏輯的研究中，臨界函數的識別方法是一個很基本的工作，它決定一個布林函數是否為臨界函數。在此研究中，我們提出了文獻上第一個很實用的臨界函數之充分必要條件。藉由我們所提出的充分必要條件，我們設計了一個更有效率的臨界函數識別的演算法。根據實驗結果，與最先進的臨界函數識別方法相比，我們所提出的方法在識別所有8輸入臨界函數時節省了93%的時間。此外，通過提出的方法識別的臨界函數所對應臨界邏輯閘，有更多臨界邏輯閘擁有最小的權重和臨界值。

Threshold Function (TF) is a subset of Boolean function that can be represented with a single linear threshold gate (LTG). In the research about threshold logic, the identification of TF is an important task that determines whether a given function is a TF or not. In this paper, we propose the first practical sufficient and necessary condition for a function being a TF in the literature. With the proposed sufficient and necessary condition, we devise a TF identification algorithm. The experimental results show that the proposed approach saves 93% CPU time for identifying all the 8-input NP-class TFs as compared with the state-of-the-art. Furthermore, the LTGs corresponding to the identified TFs obtained by the proposed approach have the minimum weights and threshold value for more TFs than the state-of-the-art.

中文摘要 i
Abstract ii
Acknowledgement iii
Contents iv
List of Tables vi
List of Figures vii
1 Introduction 1
2 Preliminaries 5
2.1 Hyperplane and Half-Space 5
2.2 Chow’s Parameter 5
2.3 Shannon’s Expansion 6
2.4 Equivalence Classes 6
3 Threshold Function Identification 8
3.1 Review of the State-of-the-Art 8
3.2 Sufficient and Necessary Condition for Function Being TF 11
3.3 Composite Inequality System Generation 13
3.4 Weight Assignment and Threshold Value Computation 16
3.5 New Initial Weight Assignment 17
3.6 Overall Flow 19
4 Experimental Results 21
5 Conclusion and Future Work 24

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