66 votes
Accepted

The origin of the notion of treewidth

If you really want to know what led Neil Robertson and me to tree-width, it wasn't algorithms at all. We were trying to solve Wagner's conjecture that in any infinite set of graphs, one of them is a ...
44 votes

Real computers have only a finite number of states, so what is the relevance of Turing machines to real computers?

To complete the other answers: I think that Turing Machine are a better abstraction of what computers do than finite automata. Indeed, the main difference between the two models is that with finite ...
  • 7,757
32 votes
Accepted

Real computers have only a finite number of states, so what is the relevance of Turing machines to real computers?

There are two approaches when considering this question: historical that pertains to how concepts were discovered and technical which explains why certain concepts were adopted and others abandoned or ...
  • 3,741
30 votes

Evidence that matrix multiplication is not in $O(n^2\log^kn)$ time

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 ...
29 votes
Accepted

Impact of Grothendieck's program on TCS

Grothendieck's inequality, from his days in functional analysis, was initially proved to relate fundamental norms on tensor product spaces. Grothendieck called the inequality "the fundamental theorem ...
27 votes

Theoretical explanations for practical success of SAT solvers?

I am assuming that you are referring to CDCL SAT solvers on benchmark data sets like those used in the SAT Competition. These programs are based on many heuristics and lots of optimization. There were ...
  • 21.3k
26 votes
Accepted

What was the original intent for the creation of Lambda calculus?

He wanted to create a formal system for the foundations of logic and mathematics that was simpler than Russell's type theory and Zermelo's set theory. The basic idea was to add a constant $\Xi$ to ...
24 votes
Accepted

Why was there a need for Martin-Löf to create intuitionistic type theory?

Very briefly: the simply-typed $\lambda$-calculus does not have dependent types. Dependent types were proposed by de Bruijn and Howard who wanted to extend the Curry-Howard correspondence from ...
22 votes

Theoretical explanations for practical success of SAT solvers?

I am typing this quite quickly due to severe time constraints (and didn't even get to responding earlier for the same reason), but I thought I would try to at least chip in with my two cents. I ...
20 votes

Impact of Grothendieck's program on TCS

Grothendieck's impact can be felt in type theory and logic. For instance, Bart Jacobs' 700+ page volume Categorical Logic and Type Theory gives a uniform treatment of various type theories ($X$-type ...
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18 votes

Theoretical explanations for practical success of SAT solvers?

I'm not an expert in this area, but I think the random SAT / phase transition stuff is more or less completely unrelated to the industrial/practical applications stuff. E.g., the very good solvers ...
18 votes

Theoretical explanations for practical success of SAT solvers?

Let me add my two cents of understanding to this, even though I've never actually worked in the area. You're asking one of two questions, "what are all the known approaches to proving theoretical ...
17 votes

Why study type theory?

Type theories in which every type is inhabited are far from being useless. True enough, through the eyes of logic they are inconsistent, but there are other things in life apart from logic. A general ...
  • 26.8k
17 votes

Status quo of category theory and monads in theoretical computer science research?

There have been a number of developments with regards to the use of monads in the theory of computation since Eugenio Moggi's work. I am not able to give a comprehensive account, but here are some ...
  • 26.8k
17 votes

Theoretical explanations for practical success of SAT solvers?

There is a paper "Relating Proof Complexity Measures and Practical Hardness of SAT" by Matti Järvisalo, Arie Matsliah, Jakob Nordström, and Stanislav Živný in CP '12 that attempts to link the hardness ...
17 votes

Possible to do Complexity theory with only counting and Pigeonhole

If you are looking for non-pigeon-hole type arguments, then there is good news: they exist! The pigeon-hole principle is a certain template for proof by contradiction. There are concepts in TCS which ...
16 votes
Accepted

What CS theories are absolutely paramount for someone new to TCS to understand?

(Disclaimer: this answer has a focus on programming languages theory, which is only one of the many disciplines under the TCS umbrella. Apologies for the length.) A small digression You are asking ...
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16 votes

What is the "question" that programming language theory is trying to answer?

The overall purpose of PLT is to make industrial software engineering (in a general sense) cheaper (also in a general sense), through optimising the most important tool (programming languages) and ...
15 votes
Accepted

Analogies between VNP and NP

The basic idea is that summing over all Boolean strings (VNP) is like counting the solutions to an NP problem. Even from this perspective, one sees that VNP is more like #P than NP. This is also true ...
13 votes

Impact of Grothendieck's program on TCS

Any applications of $p$-adic cohomology, etale cohomology in point counting formulas for algebraic varieties has roots in his work. I am guessing Mulmuley's vision of generalization of Riemann ...
  • 12.5k
13 votes

Sensitivity-Block sensitivity conjecture - Implications

Here is what Scott Aaronson has to say on the subject: What makes this interesting is that block-sensitivity is known to be polynomially related to a huge number of other interesting complexity ...
  • 14.1k
13 votes

Evidence that matrix multiplication is not in $O(n^2\log^kn)$ time

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 ...
11 votes
Accepted

Is there a randomized complexity class analogous to $\mathsf{P/Poly}$?

Suppose you have a circuit which takes as input an advice string and a random string. (So this circuit would be in $BPP/Poly$ or something like that.) You can convert this into a purely deterministic ...
  • 3,468
11 votes

Uncertainties in GCT program

It depends what you count as "the GCT program." Consider the specific suggestion (GCT I, GCT II) to use the vanishing/nonvanishing of certain multiplicities in the orbit closures of the determinant ...
10 votes
Accepted

Consequences of $NP\subseteq P/poly$ to $BQP$

I'm not aware of any direct consequence of $NP\subset P/poly$ for $BQP$. Of course it might lessen the interest in quantum computing, since it would mean that we could do something far more ...
10 votes

Real computers have only a finite number of states, so what is the relevance of Turing machines to real computers?

Andrej Bauer gave one important reason in the comments: Because sometimes $\infty$ is a better approximation to $10000000000000000000000000000000$ than $10000000000000000000000000000000$. Let me ...
9 votes

Consequences of $NP\subseteq P/poly$ to $BQP$

If $\mathsf{NP} \subseteq \mathsf{P/poly}$, then $\mathsf{PH}$ collapses to $\mathsf{\Sigma_2 P}$ (Karp-Lipton), and in fact to $\mathsf{S_2 P}$ (attributed to Sengupta by Cai, FOCS 2001), and even to ...
9 votes

Why exactly are complexity theorists interested in closed timelike curves?

Sorry for the very "big picture" answer from a non-quantum-theorist, but this contrast might help: you could describe algorithms as the study of how to solve computational problems, whereas complexity ...
  • 7,090
8 votes

Uses of algebraic structures in theoretical computer science

Algebra (and algebraic geometry) has had a pretty big role to play in cryptography, with elliptic curve groups, (number-theoretic) lattices, and of course $\mathbb{Z}_p$ being the basis for nearly all ...

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