# All Questions

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### Counting words accepted by a regular grammar

Given a regular language (NFA, DFA, grammar, or regex), how can the number of accepting words in a given language be counted? Both "with exactly n letters" and "with at most n letters" are of ...
1k views

### Class of functions computable by Coq

Since it does not allow nonterminating computation, Coq is necessarily not Turing-complete. What is the class of functions that Coq can compute? (is there an interesting characterization thereof?)
817 views

### Formal Semantics of Programming Languages

I'm new to programming languages theory and I'm seeking for a good resource on a resource for formal semantics of programming languages. Specifically looking for structural operational semantics. I ...
5k views

### What are some real world applications for genetic algorithms?

What are some real world problems that have been solved using a genetic algorithm? What is the problem? What is the fitness test used to solve this problem?
911 views

### What classes of mathematical programs can be solved exactly or approximately, in polynomial time?

I am rather confused by the continuous optimization literature and TCS literature about which types of (continuous) mathematical programs (MPs) can be solved efficiently, and which cannot. The ...
1k views

### Kolmogorov's conjecture that $P$ has linear-size circuits

In his book, Boolean Function Complexity, Stasys Jukna mentions (page 564) that Kolmogorov believed that every language in P has circuits of linear size. No reference is mentioned and I couldn't find ...
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### NP-intermediate problems with efficient quantum solutions

Peter Shor showed that two of the most important NP-intermediate problems, factoring and the discrete log problem, are in BQP. In contrast, the best known quantum algorithm for SAT (Grover's search) ...
1k views

### A data structure for minimum dot product queries

Consider $\mathbb{R}^n$ equipped with the standard dot product $\langle \cdot, \cdot \rangle$ and $m$ vectors there: $v_1, v_2, \ldots, v_m$. We want to build a data structure that allows queries of ...
393 views

### Lower bounds on the size of CFGs for specific finite languages

Consider the following natural question: Given a finite language $L$, what is the smallest context-free grammar generating $L$? We can make the question more interesting by specifying a sequence of ...
1k views

### Computational Power of Neural Networks?

Let's say we have a single-layer feed forward neural network with k inputs and one output. It calculates a function from $\lbrace 0,1\rbrace ^{n}\rightarrow\lbrace 0,1\rbrace$, it's fairly easy to ...
464 views

### Which results in complexity theory make essential use of uniformity?

A complexity class separation proof uses uniformity of complexity classes essentially if the proof does not prove the result for nonuniform version, for example proofs based on diagonalization (like ...
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### Why do we use single tape Turing machines for time complexity?

As you know there are many anomolies for the single tape Turing machines when the time is $o(n^2)$: multi-tape TM simulation, simulation of larger tape alphabet with just $\{0,1,b\}$, time ...
717 views

### Reading up on $BQP = BPP^{BQNC}$

What should I read to understand this problem? The power of small-depth quantum circuits. Is $BQP = BPP^{BQNC}$? In other words, can the "quantum" part of any quantum algorithm be compressed ...
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### Integer programming with a fixed number of variables

The famous 1983 paper by H. Lenstra Integer Programming With A Fixed Number Of Variables states that integer programs with a fixed number of variables are solvable in time polynomial in the length of ...
480 views

### Benefits for syntactic and semantic classes

This is a post separated from Consequences of UP equals NP, and also a follow-up question to Semantic vs. Syntactic Complexity Classes. In the above post we learned about the semantic and syntactic ...
475 views

### Natural candidate against the Isomorphism Conjecture?

The famous Isomorphism Conjecture of Berman and Hartmanis says that all $NP$-complete languages are polynomial time isomorphic (p-isomorphic) to each other. The key significance of the conjecture is ...
1k views

### “All-different hypergraph coloring” - known problem?

I am interested in the following problem: Given a set X and subsets X_1, ..., X_n of X, find a coloring of the elements of X with k colors such that the elements in each X_i are all differently ...
384 views

### Poly time superset of NP complete language with infinitely many strings excluded from it

For any arbitrary NP complete language is there always a polytime superset the complement of which is also infinite? A trivial version which does not stipulate the superset to have infinite ...
3k views

### Monotone bijections between lists of intervals

I have the following problem: Input: two sets of intervals $S$ and $T$ (all endpoints are integers). Query: is there a monotone bijection $f:S \to T$? The bijection is monotone w.r.t. the set ...
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Could anyone please point to one or more websites where is possible to download a working implementation of a #SAT solver? I'm interested in those returning the exact solution count, not an ...
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### What does 'gadget' mean in NP-hard reduction?

This question may not be technical. As a non-native speaker and a TA for algorithm class, I always wondered what gadget means in 'clause gadget' or 'variable gadget'. The dictionary says a gadget is a ...
2k views

### Transitivity check vs. Transitive Closure

Is checking transitivity of a digraph not easier than (in terms of asymptotic complexity) taking the transitive closure of the digraph? Do we know any lower bound better than $\Omega(n^2)$ to ...
1k views

### P vs. NP and Pseudorandom Bit Generators

According to an article on pseudorandom number generators (PRNG) by Jeff Lagarias, he states that trying to prove that a PRNG is unpredictable (secure) is just "as hard" as trying to prove that P!=NP. ...
990 views

### Checking equivalence of two polytopes

Consider a vector of variables $\vec{x}$, and a set of linear constraints specified by $A\vec{x}\leq b$. Furthermore, consider two polytopes \begin{align*} P_1&=\{(f_1(\vec{x}), \cdots, f_m(\...
528 views

### techniques or examples of analyzing a series of graphs

Let there be a sequence of graphs $G_1, G_2, G_3, ...$ constructed using some particular approach or algorithm. in this particular case $G_n$ is constructed by modifying $G_{n-1}$ in some "systematic" ...
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### Subgraph containing all nodes and edges that are part of length-limited simple s-t paths in an undirected graph

Quite similar to my previously posted question. This time however, the graph is undirected. Given An undirected graph $G$ with no multiple-edges or loops, A source vertex $s$, A target vertex $t$, ...
241 views

### Notion of associativity for ternary relations?

Consider the problem $\def\GEN{\mathrm{GEN}}\GEN$: given a ternary relation $R \subseteq X^3$ and a subset $S \subseteq X$, is the closure of $S$ with respect to $R$ equal to the whole set $X$? By ...
541 views

### State of the art for SAT solvers [duplicate]

Possible Duplicate: Best Upper Bounds on SAT I'm working on the obstruction-set-free grid coloring problem; a specific instance of it is described in this previous question on coloring 17x17 ...
96k views

### What is the enlightenment I'm supposed to attain after studying finite automata?

I've been revising Theory of Computation for fun and this question has been nagging me for a while (funny never thought of it when I learnt Automata Theory in my undergrad). So "why" exactly do we ...
46k views

### How do the state-of-the-art pathfinding algorithms for changing graphs (D*, D*-Lite, LPA*, etc) differ?

A lot of pathfinding algorithms have been developed in recent years which can calculate the best path in response to graph changes much faster than A* - what are they, and how do they differ? Are ...
117k views

### Is Norbert Blum's 2017 proof that $P \ne NP$ correct?

Norbert Blum recently posted a 38-page proof that $P \ne NP$. Is it correct? Also on topic: where else (on the internet) is its correctness being discussed? Note: the focus of this question text has ...
8k views

### Examples of the price of abstraction?

Theoretical computer science has provided some examples of "the price of abstraction." The two most prominent are for Gaussian elimination and sorting. Namely: It is known that Gaussian elimination ...
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### What tools do you use to write papers?

What tools do you use to write papers? From the little experience that I have, theoreticians spend a large amount of time writing and refining papers, besides actually being creative. That is, ...
8k views

### How do I referee a paper?

Updated below We all know the critical importance of peer-review. It is the main form of quality control and feedback on research. However, to an early-stage researcher (like me), it can sometimes ...
3k views

### Good examples for how to write well in TCS

I was editing a student manuscript. The student remarked that it would be nice to see examples of quality writing in published work, and I realized that I couldn't really come up with good examples ...
10k views

### One Stack, Two Queues

background Several years ago, when I was an undergraduate, we were given a homework on amortized analysis. I was unable to solve one of the problems. I had asked it in comp.theory, but no ...
4k views

### Shallow versus Deep Embeddings

When encoding a logic into a proof assistant such as Coq or Isabelle, a choice needs to be made between using a shallow and a deep embedding. In a shallow embedding logical formulas are written ...
2k 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 ...
6k views

### Is the Chomsky-hierarchy outdated?

The Chomsky(–Schützenberger) hierarchy is used in textbooks of theoretical computer science, but it obviously only covers a very small fraction of formal languages (REG, CFL, CSL, RE) compared to the ...
4k views

### Wikipedia-style explanation of Geometric Complexity Theory

Can someone provide a concise explanation of Mulmuley's GCT approach understandable by non-experts? An explanation that would be suitable for a Wikipedia page on the topic (which is stub at the moment)...
4k views

### What hierarchies and/or hierarchy theorems do you know?

I am currently writing a survey on hierarchy theorems on TCS. Searching for related papers I noticed that hierarchy is a fundamendal concept not only in TCS and mathematics, but in numerous sciences, ...
4k views

### Dinner-table description of theoretical computer science?

I'm often asked what a theoretical computer scientist does. It would be great to have some nice responses to this question. I tend to fall back to technical jargon and people's eyes usually glaze ...
4k views

### Applications of representation theory of the symmetric group

Inspired by this question and in particular the final paragraph of Or's answer, I have the following question: Do you know of any applications of the representation theory of the symmetric group in ...