30

There's an algorithm for multiplying an $N \times N^{0.172}$ matrix with an $N^{0.172} \times N$ matrix in $N^2 \operatorname{polylog}\left(N\right)$ arithmetic operations. The main identity used for it comes from Coppersmith's paper "Rapid multiplication of rectangular matrices", but the explanation for why it leads to $N^2 \operatorname{polylog}\left(N\...


12

Well, one thing is I think that all the constructions we know of - and even the families of potential constructions that people have proposed (e.g., Cohn-Umans approaches, generalizations of Coppersmith-Winograd) - would "simply" produce a family of algorithms $A_\epsilon$ running in time $O(n^{2+\epsilon})$. So to have a single algorithm which ran in $O(n^2 ...


11

Josh Alman showed some cool lower bound results of MM, which won CCC 2019 best student paper award! http://drops.dagstuhl.de/opus/volltexte/2019/10834/pdf/LIPIcs-CCC-2019-12.pdf


8

Based on a fairly cursory inspection of their paper, IBM is clearly aware of the spatial locality. They seem to in fact have taken spatial locality into account when designing their simulation. They use spatial locality to put the states of local systems into primary memory, and record the whole state in secondary memory. If you can't contain the whole state ...


4

I think that different mathematical models of computation capture different aspects of physical reality. Similar to models of solid state physics (say), these mathematical models may be largely incomparable. Think of analog computers, where the model may not even be described with discrete math. When it comes to automata and formal languages (of finite words ...


4

As proved in this paper: http://www.cs.technion.ac.il/users/wwwb/cgi-bin/tr-get.cgi/1991/CS/CS0699.revised.pdf If P != NP can be shown to be independent of Peano Arithmetic, then NP has extremely-close-to-polynomial deterministic time upper bounds. In particular, in such a case, there is a DTIME(n^1og*(n)) algorithm that computes SAT correctly on ...


3

Codes/Lattices are certain combinatorial objects that are commonly used within TCS. A basic question for both of them is finding "short" codewords/lattice points, known as the Minimum Distance problem/Shortest Vector problem (MDP/SVP). Both have been known to be NP-hard under randomized reductions for 20+ years. Roughly 10 years ago, the NP hardness proof ...


3

As others have pointed out, "model of computation" is an open-ended concept that can hardly be captured by a single defintion. A similar example in traditional mathematics is "space". However, this should not prevent us from giving precise definitions of "model of computation". As our understanding and motivations change, so will the definitions. And keep ...


2

The scaled down version of $\mathsf{PH}$ versus $\mathsf{PP}$ is $\mathsf{AC}^0$ versus $MAJ \circ \mathsf{AC}^0$, and we know that for the latter there is an exponential separation. Of course, this separation doesn't propagate exponentially up, but you could take this as philosophical evidence that $\mathsf{PH}$ is different enough from $\mathsf{PP}$ that ...


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