# All Questions

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### Evidence that PPAD is hard?

There is often-quoted philosophical justification for believing that P != NP even without proof. Other complexity classes have evidence that they are distinct, because if not, there would be "...
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### For which algorithms is there a large gap between the theoretical analysis and reality?

Two ways of analyzing the efficiency of an algorithm are to put an asymptotic upper bound on its runtime, and to run it and collect experimental data. I wonder if there are known cases where there ...
4k views

### Kolmogorov complexity applications in computational complexity

Informally speaking, Kolmogorov complexity of a string $x$ is a length of a shortest program that outputs $x$. We can define a notion of 'random string' using it ($x$ is random if $K(x) \geq 0.99 |x|$)...
2k views

### Is APX contained in NP?

A problem P is said to be in APX if there exists some constant c > 0 such that a polynomial-time approximation algorithm exists for P with approximation factor 1 + c. APX contains PTAS (seen by ...
• 398
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### Surprising Results in Complexity (Not on the Complexity Blog List)

What were the most surprising results in complexity? I think it would be useful to have a list of unexpected/surprising results. This includes both results that were surprising and came out of ...
4k views

### History of recursion

Who introduced the idea of recursion? Can someone explain where it came from and how it impacted computer science?
387 views

### Is there a definitive reference for Turing machines with multiple oracle tapes?

Most of the literature seems to be concerned with machines with single oracles for specific problems, however there appear to be a few papers that consider machines with multiple oracles. Is there a ...
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### Are lambda calculus and combinatory logic the same?

I am currently reading "Lambda-Calculus and Combinators" by Hindley and Seldin. I'm not an expert, but have always taken an interest in lambda calculus because of involvement with functional ...
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604 views

### How hard is it to reduce termination to partial correctness?

If you are familiar with program verification you are likely to prefer reading the Question before the Background. If you are not familiar with program verification then you may still be able to ...
• 4,746
361 views

### Finding small sets of integers in which every element is a sum of two others

This is a follow-up to this question on math.stackexchange. Let us say that a non-empty set S ⊆ ℤ is self-supporting if for every a ∈ S, there exist distinct ...
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423 views

### Property testing in other metrics?

There is a large literature on "property testing" -- the problem of making a small number of black box queries to a function $f\colon\{0,1\}^n \to R$ to distinguish between two cases: $f$ is a ...
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### Shuffling of tokens on a graph using local swaps

Let $G= (V, E)$ be a non-regular connected graph whose degree is bounded. Suppose that each node contain a unique token. I want to uniformly shuffle the tokens amongst the graph using only local ...
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### Extensions of beta-theory of lambda calculus

The beta-eta-theory of the lambda-calculus is Post-complete. Can additional rules be added to extend the beta-theory of the lambda-calculus to get confluent theories other than the beta-eta theory? ...
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### References for TCS proof techniques

Are there any references (online or in book form) that organize and discuss TCS theorems by proof technique? Garey and Johnson do this for the various kinds of widget constructions needed for NP-...
551 views

### Parallel Dynamic Search

Is there a natural parallel analog to red-black trees with similar or even not-terribly-worse properties for updates while being reasonably work-efficient ? More generally, what's the best we can do ...
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### Chernoff bound for weighted sums

Consider $X = \sum_i \lambda_i Y_i^2$, where $\lambda_i$ > 0 and $Y_i$ is distributed as a standard normal. What kind of concentration bounds can one prove on $X$, as a function of the (fixed) ...
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### Explain Gurvits's tensor-rank interpretation of Deolalikar's paper

[Note: I believe this question in no way hinges on the correctness or incorrectness of Deolalikar's paper.] On Scott Aaronson's blog Shtetl Optimized, in the discussion about Deolalikar's recent ...
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759 views

### Algorithm for inverting a bijective function.

Does there exist a generalized algorithm for finding the inverse function of an arbitrary bijective function? In order for this algorithm to be useful, it must eventually halt once the correct answer ...
265 views

### Can someone suggest a recent survey on product form Markov chains?

I'm especially interested in their use in model checking applications. I have Open, Closed and Mixed Networks of Queues with Different Classes of Customers by Baskett et al. Any other suggestions ...
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807 views

### Lower bounds on Gaussian complexity

Define the Gaussian complexity of an $n \times n$ matrix to be the minimal number of elementary row and column operations required to bring the matrix into upper-triangular form. This is a quantity ...
• 1,704
769 views

### Best known joint containments for/by NP and Parity-P?

Parity-P is the set of languages recognized by a non-deterministic Turing machine which can only distinguish between an even number or odd number of "acceptance" paths (rather than a zero or non-zero ...
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### Translating SAT to HornSAT

Is it possible to translate a boolean formula B into an equivalent conjunction of Horn clauses? The Wikipedia article about HornSAT seems to imply that it is, but I have not been able to chase down ...
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### Compared growth of the number of syntactic classes and Nerode classes.

For a language L ⊆ Σ^*, define the syntactic congruence ≡ of L as the least congruence on Σ^* that saturates L, i.e. : u ≡ v ⇔ (∀ x, y)[xuy ∈ L ↔ xvy ∈ L]. Now define the Nerode equivalence as the ...
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### Bounded-cardinality bounded-frequency set cover: hardness of approximation

Consider the minimum set cover problem with the following restrictions: each set contains at most $k$ elements and each element of the universe occurs in at most $f$ sets. Example: the case $k = 4$ ...
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### Algorithms from the Book

Paul Erdős talked about the "Book" where God keeps the most elegant proof of each mathematical theorem. This even inspired a book (which I believe is now in its 4th edition): Proofs from the ...
6k views

### What's the difference between the Actor Model of Concurrency and Communicating Sequential Processes

I'm trying to wrap my head around what the real differences between the Actor Model of concurrency and Communicating Sequential Processes (CSP) model of concurrency. So far the best that I have ...
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### Parameterized complexity of Hitting Set in finite VC-dimension

I'm interested in the parameterized complexity of what I'll call the d-Dimensional Hitting Set problem: given a range space (i.e. a set system / hypergraph) S = (X,R) having VC-dimension at most d and ...
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### Approximating the sign rank of a matrix

The sign rank of a matrix A with +1,-1 entries is the least rank (over the reals) of a matrix B which has the same sign pattern as A (i.e., $A_{ij}B_{ij}>0$ for all $i,j$). This notion is important in ...
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### What are the consequences of Parity-L = P?

Parity-L is the set of languages recognized by a non-deterministic Turing machine which can only distinguish between an even number or odd number of "acceptance" paths (rather than a zero or ...
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### Major unsolved problems in theoretical computer science?

Wikipedia only lists two problems under "unsolved problems in computer science": P = NP? The existence of one-way functions What are other major problems that should be added to this list? Rules: ...
1k views

### How does the Mulmuley-Sohoni geometric approach to producing lower bounds avoid producing natural proofs (in the Razborov-Rudich sense)?

The exact phrasing of the title is due to Anand Kulkarni (who proposed this site be created). This question was asked as an example question, but I’m insanely curious. I know very little about ...
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### What would be the consequences of factoring being NP-complete?

Are there any references covering this?
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### Can it be determined if language L lies in NP?

Given a language L defined by a Turing Machine that decides it, is it possible to determine algorithmically whether L lies in NP?
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### Is integer factorization an NP-complete problem? [duplicate]

Possible Duplicate: What are the consequences of factoring being NP-complete? What notable reference works have covered this?
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### Finding odd holes in circulant Paley graphs

The Paley graphs Pq are those whose vertex-set is given by the finite field GF(q), for prime powers q≡1 (mod 4), and where two vertices are adjacent if and only if they differ by a2 ...
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385 views

### Computational query complexity of SQ-learning

It is known that for PAC learning, there are natural concept classes (e.g. subsets of decision lists) for which there are polynomial gaps between the sample complexity needed for information theoretic ...
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### DFA intersection in subquadratic space?

The intersection of two (minimal) DFAs with n states can be computed using O(n2) time and space. This is optimal in general, since the resulting (minimal) DFA may have n2 states. However, if the ...
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### Applications of TCS to classical mathematics?

We in TCS often use powerful results and ideas from classical mathematics (algebra, topology, analysis, geometry, etc.). What are some examples of when it has gone the other way around? Here ...
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### Can every non-deterministic finite automate be changed into one with only one acceptance state? [closed]

How can an arbitrary non-deterministic finite automate be converted into one with only one accept stage? If so, what is the proof that this can always be done?
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### What is the advantage of red/black trees in comparison with unbalanced trees? [closed]

In which situations would I use a red/black tree instead of an unbalanced tree?
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### Many-one reductions vs. Turing reductions to define NPC

Why do most people prefer to use many-one reductions to define NP-completeness instead of, for instance, Turing reductions?
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379 views

### Cliquewidth of Almost Cographs

(I posted this question to MathOverflow two weeks ago, but so far without a rigorous answer) I have a question about graph width measures of undirected simple graphs. It is well-known that cographs (...
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### Using error-correcting codes in theory

What are applications of error-correcting codes in theory besides error correction itself? I am aware of three applications: Goldreich-Levin theorem about hard core bit, Trevisan's construction of ...
584 views

### Lower bounds for constant-depth formulae?

We know a lot about the limitations of (polynomial size) constant-depth circuits. Since (polynomial size) constant-depth formulae are an even more restricted model of computation, all problems known ...
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540 views

### Are recursive forms of Godel's statement possible?

The self-referentiality of the P/NP problem has sometimes been highlighted as a barrier to its resolution, see, for instance, Scott Aaronson's paper, is P vs. NP formally independent? One of the many ...
2k views

### Complexity of testing for a value versus computing a function

In general we know that the complexity of testing whether a function takes a particular value at a given input is easier than evaluating the function at that input. For example: Evaluating the ...
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### Is there an official name for a notion of "reusably universal"?

There are several different (probably inequivalent) notions of computational universality (see for example the last couple pages of http://www.dna.caltech.edu/~woods/download/WoodsNearyTCS07-DRAFT.pdf)...
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### A more intuitive proof of the Zone theorem ?

The Zone theorem says that if we stab an arrangement of n lines with another line, the total complexity of its zone, the set of all 0-, 1-, and 2-faces adjacent to it, is O(n). The actual constant is ...
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### What are good references on understanding online learning?

Specifically, I'm asking for resources to learn about machine learning systems that can update their respective belief networks (or equivalent) during operation. I've even run across a few, though I ...