recursion tree

递归树是一种图形模型，用于分析递归算法的运行过程。当我们使用递归算法时，可以将递归调用抽象成一棵树，这棵树就是递归树。递归树可以帮助我们更好地理解递归算法的运行过程，并且可以帮助我们分析算法的时间复杂度和空间复杂度。通过观察递归树的结构和特点，我们可以发现算法的重复计算、重复访问的情况，从而优化算法的效率和代码的复杂度。通过分析递归树，我们可以更好地理解递归算法的工作原理，从而更好地设计和优化递归算法。

相关问题

Use the ALC-Worlds algorithm to decide the satisfiability of the concept name B0 w.r.t. the simple Tbox，Draw the recursion tree of a successful run and of an unsuccessful run. Does the algorithm return a positive or negative result on this input? B0 ≡ B1\bigsqcapB2 B1 ≡ ∃r.B3 B2 ≡ B4 \bigsqcap B5 B3 ≡ P B4 ≡ ∃r.B6 B5 ≡ B7 \sqcap B8 B6 ≡ Q B7 ≡ ∀r.B4 B8 ≡ ∀r.B9 B9 ≡ ∀r.B10 B10 ≡ ¬P

To use the ALC-Worlds algorithm, we first need to convert the Tbox and the concept name B0 into their ALC syntax:

B0 ≡ B1 ⊓ B2
B1 ≡ ∃r.B3
B2 ≡ B4 ⊓ B5
B3 ≡ P
B4 ≡ ∃r.B6
B5 ≡ B7 ⊓ B8
B6 ≡ Q
B7 ≡ ∀r.B4
B8 ≡ ∀r.B9
B9 ≡ ∀r.B10
B10 ≡ ¬P

Now, we can start the algorithm:

1. Initialize the set of worlds W to contain the empty interpretation {}.
2. For each concept name and role name, initialize its interpretation in each world to the empty set.
3. For each concept name A that appears in the Tbox, add the pair (A, {}) to the interpretation of A in each world.
4. For each role name r that appears in the Tbox, add the pair (r, {}) to the interpretation of r in each world.
5. For each concept name A that appears in the definition of a concept or role in the Tbox, propagate its interpretation to all worlds that satisfy that concept or role.
6. For each concept name A that appears in the definition of a concept or role in the Tbox, repeat step 5 until no more interpretations can be propagated.
7. For each world w in W, check if w satisfies B0. If so, return "satisfiable" and the model consisting of all interpretations in w. If not, mark w as unsatisfiable.
8. If all worlds are marked as unsatisfiable, return "unsatisfiable".

Recursion tree for a successful run:

W={}
 | 
W1={r={}, B3={}}
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W2={r={}, B3={}, B6={}}
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W3={r={}, B3={}, B6={Q}}
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W4={r={}, B3={P}, B6={Q}}
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W5={r={r1:W4}, B3={P}, B6={Q}}
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W6={r={r1:W4, r2:W5}, B3={P}, B6={Q}}
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W7={r={r1:W4, r2:W5, r3:W6}, B3={P}, B6={Q}}
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W8={r={r1:W4, r2:W5, r3:W6, r4:W7}, B3={P}, B6={Q}}
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W9={r={r1:W4, r2:W5, r3:W6, r4:W7, r5:W8}, B3={P}, B6={Q}}
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W10={r={r1:W4, r2:W5, r3:W6, r4:W7, r5:W8, r6:W9}, B3={P}, B6={Q}}
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W11={r={r1:W4, r2:W5, r3:W6, r4:W7, r5:W8, r6:W9}, B3={}, B6={Q}}
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W12={r={r1:W4, r2:W5, r3:W6, r4:W7, r5:W8, r6:W9}, B3={P}, B6={}}
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W13={r={r1:W4, r2:W5, r3:W6, r4:W7, r5:W8, r6:W9, r7:W12}, B3={P}, B6={}}
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W14={r={r1:W4, r2:W5, r3:W6, r4:W7, r5:W8, r6:W9, r7:W12, r8:W13}, B3={P}, B6={}}
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W15={r={r1:W4, r2:W5, r3:W6, r4:W7, r5:W8, r6:W9, r7:W12, r8:W13, r9:W14}, B3={P}, B6={}}
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W16={r={r1:W4, r2:W5, r3:W6, r4:W7, r5:W8, r6:W9, r7:W12, r8:W13, r9:W14, r10:W15}, B3={P}, B6={¬P}}
 
The algorithm returns a positive result (i.e., "satisfiable") with the model consisting of all interpretations in W16.

Recursion tree for an unsuccessful run:

W={}
|
W1={r={}, B3={}}
|
W2={r={}, B3={}, B6={}}
|
W3={r={}, B3={}, B6={Q}}
|
W4={r={}, B3={P}, B6={Q}}
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W5={r={r1:W4}, B3={P}, B6={Q}}
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W6={r={r1:W4, r2:W5}, B3={}, B6={Q}}
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W7={r={r1:W4, r2:W5, r3:W6}, B3={P}, B6={Q}}
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W8={r={r1:W4, r2:W5, r3:W6, r4:W7}, B3={P}, B6={Q}}
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W9={r={r1:W4, r2:W5, r3:W6, r4:W7, r5:W8}, B3={P}, B6={Q}}
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W10={r={r1:W4, r2:W5, r3:W6, r4:W7, r5:W8, r6:W9}, B3={}, B6={Q}}
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W11={r={r1:W4, r2:W5, r3:W6, r4:W7, r5:W8, r6:W9, r7:W10}, B3={P}, B6={Q}}
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W12={r={r1:W4, r2:W5, r3:W6, r4:W7, r5:W8, r6:W9, r7:W10, r8:W11}, B3={P}, B6={Q}}
|
W13={r={r1:W4, r2:W5, r3:W6, r4:W7, r5:W8, r6:W9, r7:W10, r8:W11, r9:W12}, B3={P}, B6={Q}}
|
W14={r={r1:W4, r2:W5, r3:W6, r4:W7, r5:W8, r6:W9, r7:W10, r8:W11, r9:W12, r10:W13}, B3={P}, B6={¬P}}

The algorithm returns a negative result (i.e., "unsatisfiable") as all worlds are marked as unsatisfiable.

Suppose you have a lock that consists of 5 digits that are randomly chosen from the set {0,1,2,3,4,5,6,7,8,9}. Write a C++ program to demonstrate how to find the correct lock combination by using recursion and backtracking. Show step-by-step how to solve the problem. Draw three recursive level calls example for the recursion tree call with input values. Note that, the initial state of the key is {0,0,0,0,0} and the lock combination is randomly chosen {rand(0-9), rand(0-9), rand(0-9), rand(0-9), ), rand(0-9)}.

Sure, I can help you with that! Here's a C++ program that uses recursion and backtracking to find the correct lock combination:

#include <iostream>
#include <vector>
#include <cstdlib>
#include <ctime>

using namespace std;

const int NUM_DIGITS = 5;

void generateRandomCombination(vector<int>& combination) {
    for (int i = 0; i < NUM_DIGITS; i++) {
        combination.push_back(rand() % 10);
    }
}

void printCombination(const vector<int>& combination) {
    for (int i = 0; i < NUM_DIGITS; i++) {
        cout << combination[i];
    }
    cout << endl;
}

bool tryCombination(vector<int>& currentCombination, const vector<int>& targetCombination, int digitIndex) {
    if (digitIndex == NUM_DIGITS) {
        return currentCombination == targetCombination;
    }
    for (int i = 0; i < 10; i++) {
        currentCombination[digitIndex] = i;
        if (tryCombination(currentCombination, targetCombination, digitIndex + 1)) {
            return true;
        }
    }
    return false;
}

int main() {
    srand(time(0));
    vector<int> targetCombination;
    generateRandomCombination(targetCombination);
    cout << "Target combination: ";
    printCombination(targetCombination);
    vector<int> currentCombination(NUM_DIGITS, 0);
    if (tryCombination(currentCombination, targetCombination, 0)) {
        cout << "Found the correct combination: ";
        printCombination(currentCombination);
    } else {
        cout << "Unable to find the correct combination." << endl;
    }
    return 0;
}

The `generateRandomCombination` function generates a random lock combination and stores it in a vector. The `printCombination` function simply prints out a given combination.

The `tryCombination` function takes in the current combination being tested, the target combination to be found, and the current digit index that is being tested. It recursively tries all possible combinations of digits for the current index, and returns true if the correct combination is found.

In the `main` function, we first generate a random target combination, print it out, and initialize the current combination to all zeros. We then call `tryCombination` with the current combination, target combination, and starting index of 0. If the correct combination is found, we print it out. Otherwise, we print a message saying that we were unable to find the correct combination.

Here are three example recursive level calls for the recursion tree call with input values:

tryCombination({0, 0, 0, 0, 0}, {3, 2, 5, 8, 1}, 0)
├──tryCombination({0, 0, 0, 0, 3}, {3, 2, 5, 8, 1}, 1)
│  ├──tryCombination({0, 0, 0, 2, 3}, {3, 2, 5, 8, 1}, 2)
│  │  ├──tryCombination({0, 0, 5, 2, 3}, {3, 2, 5, 8, 1}, 3)
│  │  │  ├──tryCombination({3, 0, 5, 2, 3}, {3, 2, 5, 8, 1}, 4) // returns true
│  │  │  └──tryCombination({4, 0, 5, 2, 3}, {3, 2, 5, 8, 1}, 4)
│  │  └──tryCombination({0, 1, 5, 2, 3}, {3, 2, 5, 8, 1}, 3)
│  │     ├──tryCombination({3, 1, 5, 2, 3}, {3, 2, 5, 8, 1}, 4) // returns true
│  │     └──tryCombination({4, 1, 5, 2, 3}, {3, 2, 5, 8, 1}, 4)
│  └──tryCombination({0, 2, 5, 2, 3}, {3, 2, 5, 8, 1}, 3)
│     ├──tryCombination({3, 2, 5, 2, 3}, {3, 2, 5, 8, 1}, 4) // returns true
│     └──tryCombination({4, 2, 5, 2, 3}, {3, 2, 5, 8, 1}, 4)
├──tryCombination({0, 0, 0, 0, 4}, {3, 2, 5, 8, 1}, 1)
│  ├──tryCombination({0, 0, 0, 2, 4}, {3, 2, 5, 8, 1}, 2)
│  │  ├──tryCombination({0, 0, 5, 2, 4}, {3, 2, 5, 8, 1}, 3)
│  │  │  ├──tryCombination({3, 0, 5, 2, 4}, {3, 2, 5, 8, 1}, 4) // returns true
│  │  │  └──tryCombination({4, 0, 5, 2, 4}, {3, 2, 5, 8, 1}, 4)
│  │  └──tryCombination({0, 1, 5, 2, 4}, {3, 2, 5, 8, 1}, 3)
│  │     ├──tryCombination({3, 1, 5, 2, 4}, {3, 2, 5, 8, 1}, 4) // returns true
│  │     └──tryCombination({4, 1, 5, 2, 4}, {3, 2, 5, 8, 1}, 4)
│  └──tryCombination({0, 2, 5, 2, 4}, {3, 2, 5, 8, 1}, 3)
│     ├──tryCombination({3, 2, 5, 2, 4}, {3, 2, 5, 8, 1}, 4) // returns true
│     └──tryCombination({4, 2, 5, 2, 4}, {3, 2, 5, 8, 1}, 4)
└──tryCombination({0, 0, 0, 0, 5}, {3, 2, 5, 8, 1}, 1)
   ├──tryCombination({0, 0, 0, 2, 5}, {3, 2, 5, 8, 1}, 2)
   │  ├──tryCombination({0, 0, 5, 2, 5}, {3, 2, 5, 8, 1}, 3)
   │  │  ├──tryCombination({3, 0, 5, 2, 5}, {3, 2, 5, 8, 1}, 4) // returns true
   │  │  └──tryCombination({4, 0, 5, 2, 5}, {3, 2, 5, 8, 1}, 4)
   │  └──tryCombination({0, 1, 5, 2, 5}, {3, 2, 5, 8, 1}, 3)
   │     ├──tryCombination({3, 1, 5, 2, 5}, {3, 2, 5, 8, 1}, 4) // returns true
   │     └──tryCombination({4, 1, 5, 2, 5}, {3, 2, 5, 8, 1}, 4)
   └──tryCombination({0, 2, 5, 2, 5}, {3, 2, 5, 8, 1}, 3)
      ├──tryCombination({3, 2, 5, 2, 5}, {3, 2, 5, 8, 1}, 4) // returns true
      └──tryCombination({4, 2, 5, 2, 5}, {3, 2, 5, 8, 1}, 4)

In this example, we're trying to find the correct combination for the lock with the target combination {3, 2, 5, 8, 1}. The recursion tree shows the different combinations being tried at each recursive call. At each level, the function tries all possible digits for the current index, and recursively calls itself with the next digit index. If a combination is found that matches the target combination, the function returns true and stops recursing.