Questions tagged [big-picture]

The big picture tag is for a "broad, overall view or perspective of an issue or problem."

Filter by
Sorted by
Tagged with
2 votes
0 answers
61 views

On mod $p$ constructions related to determinant

Given an $m\times m$ matrix $M\in\mathbb Z^{m\times m}$ and a prime $p$, is it possible to construct in $Logspace$ another matrix $t_1(M)$ whose determinant is guaranteed to be determinant $Det(t_1(M))...
1 vote
0 answers
151 views

Relation between $k$-sum failure and $P=NP$

If $P=NP$ then $W[1]=FPT$ holds. Hence $k$-sum conjecture fails at a finite $k$. What can we say about the time complexity of $SAT$ and the lowest $k$ at which $k$-sum conjecture fails? In particular, ...
5 votes
1 answer
338 views

How does one "understand" complexity theory?

There's a famous quote by Bob Thomason about Grothendieck that he tried to understand algebraic geometry whilst everyone else was super-fixated on trying to prove theorems. Complexity theory, as one ...
3 votes
2 answers
328 views

What are some "who ordered that?" moments in theoretical computer science?

I was recently listening to Sean Carroll's interview with Scott Aaronson, and the two of them briefly talked about surprises in their respective fields (theoretical and particle physics, and ...
22 votes
3 answers
769 views

Bigger picture behind the choice of matrices in the Strassen algorithm

In the Strassen algorithm, to compute the product of two matrices $\mathbf{A}$ and $\mathbf{B}$, the matrices $\mathbf{A}$ and $\mathbf{B}$ are divided into $2 \times 2$ block matrices and the ...
28 votes
2 answers
2k views

Why is there an enormous difference between SAT solvers?

SAT solvers are very important in algebraic attacks, for example walksat and minisat. However, when solving the benchmark problems available here there is an enormous performance difference between ...
34 votes
3 answers
3k views

Is Descriptive Complexity dead?

I recently started reading about Descriptive Complexity, the branch of Complexity Theory studying the logic languages needed to express complexity classes. The main milestone in the area seems to be ...
4 votes
1 answer
153 views

Are there any problems in $\mathsf{BPP}$ that are known to be $\mathsf{RP}$-hard or $\mathsf{coRP}$-hard?

It's suspected that probabilistic complexity classes such as $\mathsf{RP}$ or $\mathsf{BPP}$ don't have complete problems. Of course, their promise counterparts have complete problems, but I am not ...
10 votes
0 answers
272 views

Is there a text that discusses both the “lambda cube” of pure type theories and Martin-Löf's intuitionistic type theories, and compares them?

I am lost in a maze of twisty little type theories, all different. There are a number of works (textbooks and papers) that discuss pure type theories, and specifically the ones constituting the ...
7 votes
0 answers
117 views

Why is showing lower bounds for AM communication complexity difficult?

One of the major open problems in communication complexity is to show interesting lower bounds for the Arthur-Merlin (AM) communication complexity of some natural problems (i.e., lower bounds of the ...
6 votes
1 answer
155 views

Problems which will be in $NC$ if fixed dimension Linear Integer Programming in $NC$

We know if fixed dimension linear integer programming is in $NC$ then integer $GCD$ is in $NC$. Is this the only non-trivial implication of fixed dimension linear integer programming in $NC$?
44 votes
3 answers
6k views

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

It is commonly believed that for all $\epsilon > 0$, it is possible to multiply two $n \times n$ matrices in $O(n^{2 + \epsilon})$ time. Some discussion is here. I have asked some people who are ...
30 votes
4 answers
5k views

Implications of unprovability of $P\neq NP$

I was reading "Is P Versus NP Formally Independent?" but I got puzzled. It is widely believed in complexity theory that $\mathsf{P} \neq \mathsf{NP}$. My question is about what if this is ...
3 votes
0 answers
99 views

Why do some problems seem to admit a richer family of algorithms than others?

Let's take integer multiplication and comparison sorting as examples. Despite being roughly comparable in terms of computational complexity, if we look at the set of algorithms which solve each ...
19 votes
5 answers
2k views

Why is it important to prove that a problem is NP-complete?

Am I correct in understanding that proving a problem NP complete is a research success? If so why?
80 votes
14 answers
23k views

Uses of algebraic structures in theoretical computer science

I'm a software practitioner and I'm writing a survey on algebraic structures for personal research and am trying to produce examples of how these structures are used in theoretical computer science (...
11 votes
1 answer
5k views

Why study type theory?

After reading the literature on type theory (especially the constructive kind - CTT) I'm left wondering "why" should one study type theory, specifically within the confines of "computing" in general? ...
11 votes
7 answers
3k views

How does Theoretical Computer Science relate to security?

When I think of software that is insecure I think that it is "too useful" and can be abused by an attacker. So in a sense securing software is the process of making software less useful. In ...
9 votes
3 answers
3k views

How/Why are linear systems so crucial to computer science?

I've begun to get involved with Mathematical Optimization quite recently and am loving it. It seems a lot of optimization problems can be easily expressed and solved as linear programs (e.g. network ...
32 votes
15 answers
13k views

Complex analysis in theoretical computer science

There are many applications of real analysis in theoretical computer science, covering property testing, communication complexity, PAC learning, and many other fields of research. However, I can't ...
0 votes
0 answers
76 views

Algorithms with advices of huge precomputed data

My main interest is complexity theory, and I'm studying the large or huge advice of Turing machines in the ongoing work. As related to the study, I'm wondering what's known about "precomputation&...
0 votes
1 answer
50 views

Baud rates applied to human communication

The systems that I design always include large arrays of data acqusistion channels, implemented with an assortment of communication protocols, all running at different speeds. I have always wondered: ...
45 votes
4 answers
2k views

Why have we not been able to develop a unified complexity theory of distributed computing?

The field of distributed computing has fallen woefully short in developing a single mathematical theory to describe distributed algorithms. There are several 'models' and frameworks of distributed ...
3 votes
0 answers
93 views

Complete problems for fast-growing hierarchy classes

I need examples of natural complete problems in classes $\textbf{F}_\alpha$, definition of $\textbf{F}_\alpha$ can be found here. Also in section 6 there are examples for $\omega$, $\omega^\omega$, $\...
5 votes
0 answers
186 views

Dequantumizability known and unknown?

Dequantumizable problems have been taking some headlines these days (for example this blog post by Scott Aaronson and this article in Quantum Magazine). What are some problems that are currently ...
23 votes
4 answers
2k views

Communication Complexity ...Classes?

Discussion: I've been spending some personal time lately learning various things in communication complexity. For instance, I've re-familiarized myself with the relevant chapter in Arora/Barak, ...
59 votes
6 answers
3k views

Theoretical explanations for practical success of SAT solvers?

What theoretical explanations are there for the practical success of SAT solvers, and can someone give a "wikipedia-style" overview and explanation tying them all together? By analogy, the smoothed ...
0 votes
0 answers
223 views

$CH=UL$ and partial breaking of transitive closure bottleneck problem and Savitch's theorem?

Let $L^t=DSPACE[O(\log n)^t]$, $NL^t=NSPACE[O(\log n)^t]$ and $UL^t=USPACE[O(\log n)^t$. Savitch provides $NL\subseteq L^{2}$. If $CH=UL$ we clearly got rid of the transitive closure bottleneck ...
4 votes
0 answers
218 views

What does width $4$ permutation branching program correspond to?

$L$ can be computed by a family of programs over $S_3$ of polynomial length if and only if $L$ can be computed by a family of $MOD3 ◦ MOD2$ circuits of polynomial size. $L$ can be computed by a family ...
2 votes
0 answers
213 views

Simultaneous evidence for $L\neq NL$ and $P\neq NP$

We believe $L\neq NL$ and $P\neq NP$. Is there any evidence which simultaneously imply $L\neq NL$ and $P\neq NP$?
2 votes
1 answer
251 views

DFSA and NFSA intersection problem

Given $k$ deterministic FSAs of $n$ states the intersection of their languages is empty is decidable in $n^{o(k)}$ time is an open problem. For unbounded $k$ it is known the problem is $PSPACE$ ...
-9 votes
2 answers
696 views

Is every "nontrivial" algorithm Turing-complete?

Recently there was a big response here to a question relating to the Church-Turing (from now on referred to as "CT") thesis [1]. this is another question that has nagged at me for close to ...
3 votes
1 answer
135 views

What is the best simulation of majority utilizing $\bmod\{2,3,\dots,p\}$ gates?

It is known $AC^0[2]$ cannot get majority function. Is there a literature on simulation of $MAJ$ function utilizing $AC^0[2,3,\dots,p]$ gates for a finite fixed set of primes $2$ to $p=O(1)$? What is ...
4 votes
1 answer
224 views

Is $GCT$ necessarily a negative result program?

$GCT$ is a candidate program to separate permanent and determinant through symmetries. If indeed permanent and determinant can be handled in similar complexity class would $GCT$ be a program which can ...
6 votes
2 answers
546 views

(concise?) definition of thread safety

Wikipedia has the following definition: Thread safety is a computer programming concept applicable in the context of multi-threaded programs. A piece of code is thread-safe if it only ...
-2 votes
1 answer
187 views

What evidences are there that $PP$ is in $BQP$ and $PP$ is not in $BQP$?

Unlike hierarchy collapse arguments for classical complexity we have that quantum complexity is different. What evidences are there that $PP$ is in $BQP$? What evidences are there that $PP$ is not ...
17 votes
3 answers
923 views

How Hard is Exact Simulation of Algorithms, and a Related Operation on Complexity Classes

Teaser Since the problem is longish, here is a special case that captures its essence. Problem: Let A be a deterministic algorithm for 3-SAT. Is the problem of completely simulating the ...
1 vote
1 answer
236 views

Analytic Number theory in TCS [closed]

Are there any applications of analytic number theory in TCS?
14 votes
0 answers
435 views

What percentage of SODA papers are galactic algorithms?

Consider papers published in major theoretical CS conferences during the last 5 year, where the main result is that there exists an algorithm with some time or space complexity to solve some problem. ...
2 votes
3 answers
498 views

what is a model of computation, mathematically? [closed]

Where can I find a mathematical definition for "model of computation"? https://en.m.wikipedia.org/wiki/Model_of_computation doesn't provide a precise definition for "model of computation"--it doesn't ...
5 votes
1 answer
239 views

Qubit gates in google supremacy

The gates in quantum supremacy experiment are nearest-neighbor and have spatial locality. Would this additional information help bolster IBM's argument to perhaps simulate quantum supremacy experiment ...
9 votes
1 answer
338 views

Evidence for $\mathsf{P} \neq \mathsf{PP}$ if the polynomial hierarchy collapses?

We think that $\mathsf{PH}$ does not collapse, and that $\mathsf{PP}$ is not in $\mathsf{P}$. Suppose on the contrary that $\mathsf{PH}$ does collapse, say even $\mathsf{P}= \mathsf{NP}$. $\mathsf{...
1 vote
0 answers
194 views

P=BPP and derandomizing Vazirani-Valiant?

Vazirani-Valiant reduction is a randomized reduction from $SAT$ to unambiguous $SAT$. 1. Is $P=BPP$ strong enough to derandomize Vazirani-Valiant reduction? 2. If not what other ingredients are ...
7 votes
0 answers
199 views

Geometric Intuition behind Locally testable codes

Conventional coding theory provides a good geometric picture behind linear error correction codes in terms of Hamming distance. What additional geometric requirement one should add to make a code ...
9 votes
3 answers
874 views

Why exactly are complexity theorists interested in closed timelike curves?

Context: There are several papers that study the implications of closed timelike curves (CTCs) to quantum complexity. In 2008, Aaronson and Watrous published their famous paper on this topic which ...
2 votes
1 answer
159 views

Lower bound on alternations needed in $BQP$ versus $PH$ result?

What is the fastest $f(n)$ the relatively new result of oracle separation of $\mathsf{BQP}$ from $\mathsf{PH}$ provides such that ${\#\mathsf{SAT}}\not\subseteq\mathsf{FP}^{\mathsf{PH}[O(f(n))]}$ ...
2 votes
0 answers
471 views

Is Combinatory Logic (CL) still relevant for programming language theory?

I've been reading up on R. Smullyan's "To Mock a Mockingbird" and Hindley's "Lambda-Calculus and Combinators: An Introduction". I've even read Schonfinkel's 1924 paper introducing the idea of ...
13 votes
1 answer
4k views

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

I've been interested in various topics like Combinatory Logic, Lambda Calculus, Functional Programming for a while and have been studying them. However, unlike the "Theory of Computation" which ...
1 vote
0 answers
276 views

On $BPP$ in $P^{NP}$ and $SETH$

It is believed showing $BPP$ in $P$ involves good $PRG$s and faces lower bound barriers. Does showing $BPP$ in $P^{NP}$ which would mean $BPP\neq EXP^{NP}$ face similar $PRG$ and give lower bounds? ...
71 votes
7 answers
4k views

Which interesting theorems in TCS rely on the Axiom of Choice? (Or alternatively, the Axiom of Determinacy?)

Mathematicians sometimes worry about the Axiom of Choice (AC) and Axiom of Determinancy (AD). Axiom of Choice: Given any collection ${\cal C}$ of nonempty sets, there is a function $f$ that, given a ...

1
2 3 4 5