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

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0answers
49 views

Greatest evidences for different variations of complexity class separations

There are many classes which are conjecturally separate in complexity theory? For instance, the famous $P$ versus $NP$ problem. What are famous examples of such classes which are conjecturally ...
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1answer
119 views

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

$\mathsf{P/Poly}$ captures those problems that could be solved in polynomial time given some precomputed polynomial number of constants. Is there an analogous complexity class in randomized world ...
1
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0answers
39 views

Analogues of different complexity classes in various models

We suspect following relation: $$TC^0\subsetneq NC^1\subsetneq L\subsetneq NL\subsetneq AC^1\subsetneq NC^2\subsetneq P\subsetneq NP\subsetneq PH\subsetneq PSPACE$$ in Turing/boolean circuit ...
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1answer
73 views

what can be said about complexity of “typical” supercomputing programs/ applications? any NP hard?

supercomputers have risen dramatically in their computational powers last few decades due to Moore's law & also increasing parallelism technology in hardware and software. many different types of ...
3
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1answer
222 views

Analogies between VNP and NP

Valiant introduced the class VNP with respect to "arithmetic circuits" over 35 years ago in a "rough" analogy to NP. Recently, there have been major advances in the area of arithmetic circuits eg as ...
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1answer
134 views

A trivial question on hierarchy [closed]

According to wiki we know that $\mathsf{ACC^0\subseteq TC^0\subseteq NC^1\subseteq L\subseteq P\subseteq NP\subsetneq NEXPTIME}.$ Class $\mathsf{ACC^0}$ is included in $\mathsf{TC^0}$ is in ...
-1
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1answer
198 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? ...
3
votes
3answers
221 views

Distinguishing semantics vs syntactic techniques and the syntax of your semantic domains

Consider a denotational semantics from simply-typed $\lambda$-calculus into dependent type theory. Is that actually a (trivial) term transformation into that dependent type theory? After all, type ...
9
votes
1answer
774 views

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

I've been reading up on Intuitionistic Type Theory (ITT) and it does make sense. But what I'm struggling to understand is "why" was it created in the first place? Intuitionistic Logic (IL) and ...
5
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1answer
296 views

How would a theory of computation course that culminated in lambda-calculus as “the” model of computation, instead of Turing machines, look like?

Currently, our ToC (Theory of Computation) courses are designed with the following progression of topics: Finite automata and regular languages Pushdown automata and context-free languages Turing ...
5
votes
1answer
179 views

possible bridge between group growth theory and complexity theory?

RJ Lipton conjectures a link between group growth theory and complexity theory. Group growth theory has undergone rapid advance in the last decade and has many surface similarities/ parallels with ...
1
vote
3answers
366 views

Why is lambda calculus so “function” oriented?

I've always had this question nagging at me subconsciously but have never been able to intuitively grasp it. Why does $\lambda$-calculus have a functional notation? Why is everything a function? It ...
5
votes
3answers
175 views

What is the relationship between intuitionistic logic, combinatory logic and lambda calculus?

I've been reading Lectures on the Curry-Howard Isomorphism and it talks about intuitionistic/constructive logic (IL) , combinatory logic (CL) and lambda calculus ($\lambda$c) before moving on to the ...
20
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2answers
2k views

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

I've read that initially Church proposed the $\lambda$-calculus as part of his Postulates of Logic paper (which is a dense read). But Kleene proved his "system" inconsistent after which, Church ...
2
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0answers
43 views

Modern tools deterministic communication applications

Partition number, Fooling-set method along with rank method provide important tools to identify deterministic communication complexity of a function. These techniques were identified some decades ...
7
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0answers
108 views

Examples of open problems solved through application of a theorem already known

Are there good examples of reasonable open problems in TCS that had an 'obvious' solution via application of a theorem found in mathematics probably found a few decades earlier but went unnoticed in ...
8
votes
2answers
193 views

Sensitivity-Block sensitivity conjecture - Implications

Let $f$ be a boolean function with sensitivity $s(f)$ and block sensitivity $bs(f)$. The Sensitivity-Block sensitivity conjecture conjecture states that there is a $c>0$ such that $\forall ...
2
votes
1answer
214 views

Important papers and open problems in Boolean functions

I am interested in Boolean function polynomial/rational exact/approximate/one sided approximate representation, relation to circuit/communication complexity, tools utilized to study Boolean functions ...
25
votes
3answers
3k views

Impact of Grothendieck's program on TCS

Grothendieck has passed away. He had massive impact on 20th century mathematics continuing into the 21st century. This question is asked somewhat in the style/spirit, for example, of Alan Turing's ...
1
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0answers
171 views

intuition that VP=?VNP is (not?) connected to P=?NP

recently there has been major progress into the VP=?VNP problem for algebraic circuits originated by Valiant, inspiring some optimistic outlook on its eventual or imminent resolution.[1] what is ...
1
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0answers
136 views

proving speedup phenomenon does not apply to any open complexity class separations

Aaronson recently wrote a blog refuting the idea that there could be some "glitch" in the formulation of the P vs NP conjecture[1] which reminds me of this following question. the Blum speedup ...
3
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0answers
81 views

minimal languages that “cover” grammar productions

this question is based on generalizing two somewhat similar questions that recently appeared on the "sister" beta site cs.se (now with more questions than this one!) and which seems theoretically ...
4
votes
1answer
106 views

any relation/ overlap between small world graphs, scale free graphs, and expander graphs?

small world graphs (eg Watts-Strogatz model & others) and scale free graphs are a relatively recently discovered graph type via mainly empirical analysis of large real-world graphs (eg via Big ...
30
votes
5answers
3k views

The unreasonable power of non-uniformity

From the common sense point of view, it is easy to believe that adding non-determinism to $\mathsf{P}$ significantly extends its power, i.e., $\mathsf{NP}$ is much larger than $\mathsf{P}$. After ...
4
votes
2answers
225 views

TCS oriented refs/survey on group theoretic word problem

The word problem for groups was shown to be Turing-complete in 1955 but has many decidable subcases. This problem arose more in mathematical group theory than in theoretical computer science, but now ...
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votes
3answers
196 views

What structures allow a notion of 'strictness', 'weakness' and 'mildness'?

This is more of a 'meta' question as I cannot give a precise formulation of my question. Consider for example the category of total quasi-orders: we can then distinguish between a 'strict' order ...
4
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0answers
128 views

Interaction between metamathematics and combinatorics?

Metamathematics started in the 19th century with the discovery of paradoxes intrinsic to certain axiom systems involving infinite objects; attempts to resolve these paradoxes led to the formulation of ...
13
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5answers
660 views

Should reductions make us more or less optimistic for the tractability of a problem?

It seems to me that most complexity theorists generally believe the following philosophical rule: If we can't figure out an efficient algorithm for problem $A$, and we can reduce problem $A$ to ...
64
votes
7answers
22k views

What is the contribution of lambda calculus to the field of theory of computation?

I'm just reading up on lambda calculus to "get to know it". I see it as an alternate form of computation as opposed to the Turing Machine. It's an interesting way of doing things with ...
0
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0answers
126 views

Observable correlation vs non-observable causality

It is frequent to find statistical studies observing some correlation between two events; this correlation is often interpreted by the medias as a causality relation (e.g. "blue-eyed people have more ...
3
votes
0answers
118 views

Randomly Discovered Algorithm/Counterexample

I was reading Scott Aaronson's blog, and one of the comments sparked a question. "if P!=NP, this would be a general, conceptual result, so you’d expect the proof to be explanatory and in particular ...
8
votes
1answer
474 views

Connections between the Erdos Discrepancy Problem and (Theoretical) CS?

Recently there have been some new results on computer-based experimental study of the Erdos Discrepancy Problem (EDP) (via SAT solvers, cited below). This problem has been cited and studied by several ...
12
votes
4answers
4k views

Interesting results in TCS which are easily explainable to programmers without technical background

Suppose you're meeting with programmers who have taken some professional programming courses (/ self thought) but didn't study a university level math. In order to show them the beauty of TCS, I'd ...
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0answers
89 views

looking for notable applications of ASP (Answer Set Programming) in TCS

a recent difficult question of interest to the group[1] by GB has possibly led to verification of a new graph property by dspyz, by use of Answer Set Programming/ASP. via sophisticated logic ...
21
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4answers
1k views

Justifying asymptotic worst-case analysis to scientists

I've been working on on introducing some results from computational complexity into theoretical biology, especially evolution & ecology, with the goal of being interesting/useful to biologists. ...
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votes
1answer
178 views

“tree-like” vs “DAG-like” resolution

hi all there seems to be a deep/not-much-explored phenomenon in the way that SAT resolution proofs can define a tree and/or a DAG & its relationship to lower bounds/circuit complexity. could there ...
7
votes
1answer
423 views

Public-key encryption without the assumption that $P \neq NP$

I'm not talking about the RSA, El-gamal, nor any specific encryption scheme. Rather, my question, as related to this and this threads, is why the idea of Public-Key encryption scheme cannot be secure ...
4
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1answer
249 views

Distributive expansion of CNF and implicants

I am looking for references for the following theorems. Theorem 1: Distributive expansion of a CNF formula $P_c$ (product of sums) results in a DNF formula (sum of products) consisting of all prime ...
3
votes
0answers
139 views

sketch of Razborovs paper “on the method of approximations”

(the following question has bothered me for many years.) Razborov seems to have obtained some of the strongest/award winning lower bounds on circuits found in the field over many years, largely ...
249
votes
29answers
157k views

Core algorithms deployed

To demonstrate the importance of algorithms (e.g. to students and professors who don't do theory or are even from entirely different fields) it is sometimes useful to have ready at hand a list of ...
0
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0answers
84 views

Contributions Of Computer Science

I would like to know how is theoretical computer science helping to understand mathematics, physics and biology better in our universe. How is it all merging out and how is theoretical computer ...
13
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2answers
556 views

What is theoretical computer science?

What exactly is theoretical computer science? Is it learning to code in various language and making apps in platforms? Or is it just thinking about faster and faster algorithms so that you can achieve ...
15
votes
1answer
597 views

Career in Theoretical Computer Science

I am currently a high school student, interested in theoretical computer science and applied mathematics. I have self taught myself linear algebra and calculus and concrete mathematics. I have a naive ...
3
votes
3answers
480 views

Why do theoreticians in CS use multiple-letter variables?

This is a "dual" question of a popular post on math.se. Some mathematical objects in computational complexity theory have multiple-letter names. Complexity classes such as $\mathbf{BPP}$ have an ...
48
votes
5answers
4k views

Overarching reasons why problems are in P or BPP

Recently, when talking to a physicist, I claimed that in my experience, when a problem that naively seems like it should take exponential time turns out nontrivially to be in P or BPP, an "overarching ...
10
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6answers
604 views

Geometric Interpretation of Computation

Being from Physics, I have been trained to look into a lot of problems from a geometrical point of view. For example the differential geometry of manifolds in dynamical systems etc. When I read the ...
11
votes
2answers
248 views

Is there an explanation for the difficulty of proving quadratic lower bounds for interesting NP problems?

This is a follow up to my previous question: Best known deterministic time complexity lower bound for a natural problem in NP I find it bewildering that we haven't been able to prove any quadratic ...
10
votes
1answer
305 views

Why does the log-rank conjecture use rank over the reals?

In communication complexity, the log-rank conjecture states that $$cc(M) = (\log rk(M))^{O(1)}$$ Where $cc(M)$ is the communication complexity of $M(x,y)$ and $rk(M)$ is the rank of $M$ (as a ...
5
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0answers
115 views

conceptual tools for illustrating types of computation?

From time to time I come across concepts in programming that take a certain number of exposures to grasp. Things like: tail calls, futures, monads, coroutines, closures, call/cc. The common theme is ...
12
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1answer
643 views

How can I use my computational theory and analysis powers for the greater good?

Outside of academia, what are the uses of my 'powers'? What can I do other than teaching and publishing papers? Where all can I apply my powers? For the sake of argument: Please assume I have a PhD ...