數學歸納分析程序正確性

{"type":"doc","content":[{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","marks":[{"type":"strong","attrs":{}}],"text":"1 數學歸納","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"先看個數學證明問題:","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"    s(n) = 1+2+3+...n = ((n+1) * n) / 2, 證明公式對任意整數n(n >= 0)都成立","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","marks":[{"type":"strong","attrs":{}}],"text":"證明：","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","marks":[{"type":"strong","attrs":{}}],"text":"設置斷言 G(n): 0到n的整數之和s(n) = ((n + 1) * n) / 2","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","marks":[{"type":"strong","attrs":{}}],"text":" ","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","marks":[{"type":"strong","attrs":{}}],"text":"步驟1： 基底的證明","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","marks":[{"type":"strong","attrs":{}}],"text":"s(0) = ((0+1)*0) / 2 = 0，G(0)成立","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","marks":[{"type":"strong","attrs":{}}],"text":"步驟2：歸納的證明 ","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","marks":[{"type":"strong","attrs":{}}],"text":"先假設 G(k)成立，可得到：","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","marks":[{"type":"strong","attrs":{}}],"text":"    s(k) = 0+1+2...+k = ((k+1) * k) / 2","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","marks":[{"type":"strong","attrs":{}}],"text":"則：","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","marks":[{"type":"strong","attrs":{}}],"text":"    s(k+1) = s(k) + k+1 = ((k+1) * k) / 2 + ((k+1) * 2) / 2 = ((k+1) * (k+2)) / 2","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","marks":[{"type":"strong","attrs":{}}],"text":"等式成立，推出G(k) 成立，則G(k+1)也成立","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","marks":[{"type":"strong","attrs":{}}],"text":"通過步驟1，步驟2證得斷言G(n)是正確的","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"上面求證過程，就運用到了","attrs":{}},{"type":"text","marks":[{"type":"strong","attrs":{}}],"text":"數學歸納","attrs":{}},{"type":"text","text":"思想。","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","marks":[{"type":"strong","attrs":{}}],"text":"2 運用數學歸納分析計算機程序正確性","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"求fibonacci數列([0, 1 , 1,  2,  3,....ai, a(i+1), a(i)+a(i+1)])第i項(i >= 0)","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"常規寫法如下","attrs":{}}]},{"type":"codeblock","attrs":{"lang":"java"},"content":[{"type":"text","text":"static int fib(int i) {\n    if (i == 0) {\n      return 0;\n    }\n    if (i == 1) {\n      return 1;\n    }\n\n    int tmp;\n    int a = 0;\n    int b = 1;\n    int $i = 2;\n \n    while ($i <= i) {\n      $i++;\n      tmp = a;\n      a = b;\n      b = tmp +b;\n    }\n\n    return b;\n}","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"思考：對任意的輸入i, 怎麼證明代碼輸出結果都是正確的? ......","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"把上面代碼結構調整爲下面這樣,只要水爲了方便分析，邏輯和上面代碼等效","attrs":{}}]},{"type":"codeblock","attrs":{"lang":"java"},"content":[{"type":"text","text":" static int fib(int i) {\n     int[] L = new int[] {0, 0, 1};\n     if (i == 0) {\n       return L[1];\n     }\n     if (i == 1) {\n       return L[2];\n     }\n     while (L[0] <= i) {\n       next(L)；\n     }\n     return L[2];\n  }\n\n  static int next(int[] L) {\n     int tmp;\n     int a = L[1];\n     int b = L[2];\n     tmp = a;\n     a = b;\n     b = b + tmp;\n     // 計算下一次循環需要的變量\n     L[0] = L[0] + 1;\n     L[1] = a;\n     L[2] = b;\n \n     return L[2];\n  }","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"用上面的","attrs":{}},{"type":"text","marks":[{"type":"strong","attrs":{}}],"text":"數學歸納思想","attrs":{}},{"type":"text","text":"，分析下程序是否正確。","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":" 證明：","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","marks":[{"type":"strong","attrs":{}}],"text":"    設置斷言 M(i): L(i) = [i, fib(i-2), fib(i-1)]（i > 2，fib(i) 表示fibonacci第i項）","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","marks":[{"type":"strong","attrs":{}}],"text":"    步驟1： 基底的證明","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","marks":[{"type":"strong","attrs":{}}],"text":"        i = 2時， L(2) = [2, 0, 1]；由fibonacci前3項爲[0, 1, 1], 可知， L(2)成立 ;","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","marks":[{"type":"strong","attrs":{}}],"text":"    步驟2：歸納的證明","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","marks":[{"type":"strong","attrs":{}}],"text":"        由程序代碼可知：L(k+1) = [k+1, L(k)[2], L(k)[1]+L(k)[2]],   假設M(k)成立，則：","attrs":{}}]},{"type":"paragraph","attrs":{"indent":1,"number":0,"align":null,"origin":null},"content":[{"type":"text","marks":[{"type":"strong","attrs":{}}],"text":" L(k) = [k, fib(k-2), fib(k-1)]，所以：","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","marks":[{"type":"strong","attrs":{}}],"text":"        L(k+1)=[k+1, fib(k-1), fib(k-1)+fib(k-2)] = [k+1, fib(k-1), fib(k)], ","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","marks":[{"type":"strong","attrs":{}}],"text":" 推出M(k)成立時，M(k+1)成立","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","marks":[{"type":"strong","attrs":{}}],"text":"       ","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","marks":[{"type":"strong","attrs":{}}],"text":"       所以斷言正確","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","marks":[{"type":"strong","attrs":{}}],"text":"通過上面證明，可以得出，輸入爲i, 當循環結束時 $i = i + 1, 此時 L($i) = [i+1, fib(i-1), fib(i)], ","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","marks":[{"type":"strong","attrs":{}}],"text":"所以L($i)[2] 爲fibonacci第i項, 輸出正確 。","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"問題：證明下面二分查找程序是否正確？","attrs":{}}]},{"type":"codeblock","attrs":{"lang":"java"},"content":[{"type":"text","text":"static boolean search(int[] a, int target) {\n    int left = 0;\n    int right = a.length - 1;\n    while (left < right) {\n      int mid = (left + right) / 2;\n      if (a[mid] > target) {\n        right = mid + 1;\n      }\n      if(a[mid] < target) {\n        left = mid + 1;\n      } else {\n        return true;\n      }\n    }\n    return false;\n  }","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}}]}