Describe how an operating system that uses paging could make stacks grow dynamically if the original size proves too small. What would the page table data structures look like? Explain what would happen when a thread's stack grows past the area that has been allocated for it (e.g., how would the OS notice this fact, and what would happen inside the OS when it notices this fact). Could a similar effect be achieved with multi-level page tables? Segments? Paged segmentation? Inverted page tables?

An operating system can make stacks grow dynamically in a paging system by using a technique called demand paging. In this approach, the stack is initially allocated with a fixed size, but additional pages are allocated on demand when the stack grows beyond the allocated size.

When a thread's stack grows past the allocated area, the operating system needs to notice this fact and take appropriate actions. This can be achieved by utilizing a page fault mechanism. When the stack grows beyond the allocated area, a page fault will occur because the required stack page has not been allocated. The operating system will then handle this page fault and allocate a new page to the stack.

To implement this in a paging system, the page table data structures would include the following components:

1. Page Table Entry (PTE): Each page table entry would have a flag to indicate whether the page has been allocated or not. This flag can be set to indicate that a page is not present or allocated.

2. Page Fault Handler: The page fault handler in the operating system would be responsible for managing the dynamic growth of the stack. When a page fault occurs for a stack page, the handler would allocate a new physical page and update the page table entry accordingly.

With multi-level page tables, a similar effect can be achieved by allocating new pages in the same manner as in a single-level page table system. The only difference is the structure of the page table itself, with multiple levels of tables to handle larger address spaces.

In a segmented paging system, dynamic stack growth can also be supported. The segment descriptor for the stack would need to be set to allow for dynamic growth, and the page table entries would be updated as needed when new pages are allocated for the stack.

Paged segmentation is another technique where each segment can be divided into pages. The page table in this case would contain entries for each segment, with additional fields to indicate whether the segment is allocated or not. When a stack page grows beyond the allocated segment, a page fault will occur, and the page fault handler would allocate a new segment page and update the page table entry for that segment.

Inverted page tables store mapping information for physical memory rather than virtual memory. Therefore, they are not directly applicable to managing the dynamic growth of a stack.

In summary, an operating system using a paging system can make stacks grow dynamically through demand paging. The page table data structures would involve updating the page table entries and allocating new pages when a page fault occurs. Similar effects can be achieved with multi-level page tables, segmented paging, and paged segmentation, while inverted page tables are not suitable for this purpose.