------
## 最终结论
```python
def replace_successors(dictionary, sentence):
    words = sentence.split()
    result = []
    
    for word in words:
        # Check for possible roots that can replace the current word
        possible_roots = [root for root in dictionary if word.startswith(root)]
        
        if possible_roots:
            # Find the shortest root
            shortest_root = min(possible_roots, key=len)
            result.append(shortest_root)
        else:
            result.append(word)
    
    # Join the result list into a single string and return the final sentence
    return ' '.join(result)

# Example usage
dict = ["cat", "bat", "rat"]
sentence = "the cattle was rattled by the battery"
output = replace_successors(dict, sentence)
print(output)  # Output: "the cat was rat by the bat"
```