在臨界值邏輯的理論中，臨界值函數的辨別方法是一個基本卻很重要的工作，它可以決定一個布林函數是否可以被單一臨界值邏輯閘呈現。在此理論中，藉由建立非冗餘的不等式系統與全面性權重分配，我們提出一個更有效率與效能的臨界值邏輯函數辨別演算法。這是第一個可以辨別所有8輸入臨界值函數的非整數線性規劃的演算法。從實驗結果中，我們顯示出提倡的方法比所有現存的非整數線性規劃方法更加有效，以及提倡的方法產生出的臨界值邏輯閘有將近100%的機率會是最佳的。對於9到15輸入臨界值函數，提倡的方法也可以在合理的執行時間內，辨別出所有由隨機產生的100,000個臨界值函數。

The identification of threshold function, which determines whether a Boolean function can be represented by an LTG or not, is a fundamental but important task in the theories of threshold logic. In this thesis, we propose a more efficient and effective algorithm of threshold function identification by constructing the system of irredundant inequalities and adjusting the weight assignment comprehensively. This is the first non-ILP-based approach that is able to identify all the 8-input threshold functions. The experimental results demonstrated that the proposed approach is more effective than all the existing non-ILP-based approaches and the LTGs obtained by the proposed approach are optimal for near 100\% cases. For threshold functions with 9 to 15 inputs, the proposed approach can identify 100,000 randomly generated threshold functions as well in a reasonable CPU time.

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 Linear Threshold Gate . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Threshold Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 Irredundant Sum-of-Products . . . . . . . . . . . . . . . . . . . . . . 6
2.4 Modified Chow’s Parameter . . . . . . . . . . . . . . . . . . . . . . . 6
3 Algorithm 8
3.1 The Review of the State-of-the-art . . . . . . . . . . . . . . . . . . . 8
3.2 Generation of the System of Irredundant Inequalities . . . . . . . . . 11
3.2.1 Redundant Weighted Summation Removal . . . . . . . . . . . 12
3.2.2 Redundant Inequality Removal . . . . . . . . . . . . . . . . . 12
3.3 Weight Assignment Procedure . . . . . . . . . . . . . . . . . . . . . . 14
3.3.1 The Review of the State-of-the-art [15] . . . . . . . . . . . . . 14
3.3.2 Proposed Weight Assignment Method . . . . . . . . . . . . . . 17
3.4 Threshold Value Computation . . . . . . . . . . . . . . . . . . . . . . 21
3.5 Overall Flowchart of TF Identification Algorithm . . . . . . . . . . . 21
4 Experimental Results 23
4.1 Effectiveness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.2 Optimality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.3 Applicability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
5 Conclusion 28

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