Virtually indexed physically tagged cache Synonym

Question

I am not able to entirely grasp the concept of synonyms or aliasing in VIPT caches.

Consider the address split as:-

Here, suppose we have 2 pages wi

Answer 1

The page offset is bits 0-11, not 0-13. Look at your bottom diagram: the page offset is the low 12 bits, so you have 4k pages (like x86 and other common architectures).

If any of the index bits come from above the page offset, VIPT no longer behaves like a PIPT with free translation for the index bits. That's the case here.

A process can have the same physical page (frame) mapped to 2 different virtual pages.

Your claim that The pageno part of VA (bits 13-39) which are different gets translated to PFN of PA(bits 12-35) and the PFN remains same for both the VA's is totally bogus. Translation can change bit #12. So one of the index bits really is virtual and not also physical, so two entries for the same physical line can go in different sets.

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I think my main confusion is regarding the page offset range. Is it the same for both PA and VA (that is 0-11) or is it 0-12 for VA and 0-11 for PA? Will they always be same?

It's always the same for PA and VA. The page offset isn't marked on the VA part of your diagram, only the range of bits used as the index.

It wouldn't make sense for it to be any different: virtual and physical memory are both byte-addressable (or word-addressable). And of course a page frame (physical page) is the same size as a virtual page. Right or left shifting an address during translation from virtual to physical would make no sense.

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As discussed in comments:

I did eventually find http://www.cse.unsw.edu.au/~cs9242/02/lectures/03-cache/node8.html (which includes the diagram in the question!). It says the same thing: physical tagging does solve the cache homonym problem as an alternative to flushing on context switch.

But not the synonym problem. For that, you can have the OS ensure that bit 12 of every VA = bit 12 of every PA. This is called page coloring.

Page coloring would also solve the homonym problem without the hardware doing overlapping tag bits, because it gives 1 more bit that's the same between physical and virtual address. phys idx = virt idx. (But then the HW would be relying on software to be correct, if it wanted to depend on this invariant.)

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Another reason for having the tag overlap the index is write-back during eviction:

Outer caches are almost always PIPT, and memory itself obviously needs the physical address. So you need the physical address of a line when you send it out the memory hierarchy.

A write-back cache needs to be able to evict dirty lines (send them to L2 or to physical RAM) long after the TLB check for the store was done. Unlike a load, you don't still have the TLB result floating around unless you stored it somewhere. How does the VIPT to PIPT conversion work on L1->L2 eviction

Having the tag include all the physical address bits above the page offset solves this problem: given the page-offset index bits and the tag, you can construct the full physical address.

(Another solution would be a write-through cache, so you do always have the physical address from the TLB to send with the data, even if it's not reconstructable from the cache tag+index. Or for read-only caches, e.g. instruction caches, there is no write-back; eviction = drop.)