<p>A <strong>boolean expression</strong> is an expression that evaluates to either <code>true</code> or <code>false</code>. It can be in one of the following shapes:</p>

<ul>
	<li><code>&#39;t&#39;</code> that evaluates to <code>true</code>.</li>
	<li><code>&#39;f&#39;</code> that evaluates to <code>false</code>.</li>
	<li><code>&#39;!(subExpr)&#39;</code> that evaluates to <strong>the logical NOT</strong> of the inner expression <code>subExpr</code>.</li>
	<li><code>&#39;&amp;(subExpr<sub>1</sub>, subExpr<sub>2</sub>, ..., subExpr<sub>n</sub>)&#39;</code> that evaluates to <strong>the logical AND</strong> of the inner expressions <code>subExpr<sub>1</sub>, subExpr<sub>2</sub>, ..., subExpr<sub>n</sub></code> where <code>n &gt;= 1</code>.</li>
	<li><code>&#39;|(subExpr<sub>1</sub>, subExpr<sub>2</sub>, ..., subExpr<sub>n</sub>)&#39;</code> that evaluates to <strong>the logical OR</strong> of the inner expressions <code>subExpr<sub>1</sub>, subExpr<sub>2</sub>, ..., subExpr<sub>n</sub></code> where <code>n &gt;= 1</code>.</li>
</ul>

<p>Given a string <code>expression</code> that represents a <strong>boolean expression</strong>, return <em>the evaluation of that expression</em>.</p>

<p>It is <strong>guaranteed</strong> that the given expression is valid and follows the given rules.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<pre>
<strong>Input:</strong> expression = &quot;&amp;(|(f))&quot;
<strong>Output:</strong> false
<strong>Explanation:</strong> 
First, evaluate |(f) --&gt; f. The expression is now &quot;&amp;(f)&quot;.
Then, evaluate &amp;(f) --&gt; f. The expression is now &quot;f&quot;.
Finally, return false.
</pre>

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> expression = &quot;|(f,f,f,t)&quot;
<strong>Output:</strong> true
<strong>Explanation:</strong> The evaluation of (false OR false OR false OR true) is true.
</pre>

<p><strong class="example">Example 3:</strong></p>

<pre>
<strong>Input:</strong> expression = &quot;!(&amp;(f,t))&quot;
<strong>Output:</strong> true
<strong>Explanation:</strong> 
First, evaluate &amp;(f,t) --&gt; (false AND true) --&gt; false --&gt; f. The expression is now &quot;!(f)&quot;.
Then, evaluate !(f) --&gt; NOT false --&gt; true. We return true.
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= expression.length &lt;= 2 * 10<sup>4</sup></code></li>
	<li>expression[i] is one following characters: <code>&#39;(&#39;</code>, <code>&#39;)&#39;</code>, <code>&#39;&amp;&#39;</code>, <code>&#39;|&#39;</code>, <code>&#39;!&#39;</code>, <code>&#39;t&#39;</code>, <code>&#39;f&#39;</code>, and <code>&#39;,&#39;</code>.</li>
</ul>

class Solution {
public:
 unordered_map<string, bool> memo;
    bool func(int i, int j, string& exp) {
        string key = to_string(i) + "," + to_string(j) + "," + exp.substr(i, j - i + 1);
        if (memo.find(key) != memo.end()) return memo[key];
        if (i == j) return exp[i] == 't';
        bool result = false;
        if (exp[i] == '!') {
            result = !func(i + 2, j - 1, exp);
        } 
        else if (exp[i] == '&') {
            result = true;
            int open = 0, start = i + 2;
            for (int k = i + 2; k < j; k++) {
                if (exp[k] == '(') open++;
                if (exp[k] == ')') open--;
                if (exp[k] == ',' && open == 0) {
                    result = result && func(start, k - 1, exp);
                    start = k + 1;
                }
            }
            result = result && func(start, j - 1, exp);
        } 
        else if (exp[i] == '|') {
            result = false;
            int open = 0, start = i + 2;
            for (int k = i + 2; k < j; k++) {
                if (exp[k] == '(') open++;
                if (exp[k] == ')') open--;
                if (exp[k] == ',' && open == 0) {
                    result = result || func(start, k - 1, exp);
                    start = k + 1;
                }
            }
            result = result || func(start, j - 1, exp);
        }
        memo[key] = result;
        return result;
    }

    bool parseBoolExpr(string exp) {
        int n = exp.size();
        // vector<vector<int>> dp(n, vector<int> (n, -1)); 
        return func(0, n-1, exp);
    }
};
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