Questions tagged [advice-and-nonuniformity]

Questions about advice and nonuniformity

1
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2answers
103 views

When is a problem specified on a TM contained in non-uniform classes such as P/poly? [closed]

In this paper by Gottesman and Irani: https://arxiv.org/abs/0905.2419 , they prove NEXP-hardness of tiling an $N\times N$ grid. They do so by encoding a TM in the tiles making up the grid. However, ...
9
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3answers
583 views

What are examples of how non-uniformity can be useful?

I'm curious about ways in which you have seen non-uniformity be useful in computation. One way is randomness, as in $BPP \subseteq P/poly$, and another is look-up tables which are used to show that ...
11
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1answer
206 views

Does the space hierarchy theorem generalize to non-uniform computation?

General Question Does the space hierarchy theorem generalize to non-uniform computation? Here are a few more specific questions: Is $L/poly \subsetneq PSPACE/poly$? For all space ...
14
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1answer
190 views

What are the consequences of $P \subseteq L/poly$?

A language is in $L/poly$ if there exists a logspace Turing machine that decides the language with polynomial amount of advice. See here for more info: https://en.wikipedia.org/wiki/L/poly ...
0
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1answer
189 views

What do stronger circuit lower bounds give in terms of derandomization?

We have $EXP\not\subseteq P/poly\implies BPP\subseteq io-DTIME(2^{n^\epsilon})$ at every $\epsilon>0$. This is essentially $DTIME(2^{O(n)})\not\subseteq P/poly\implies BPP\subseteq io-DTIME(2^{n^\...
2
votes
1answer
103 views

Non-Uniform Lower Bounds for NSPACE

If I'm not mistaken it is not known whether $E^{NP} \subseteq {\rm SIZE}(n)$ where $E^{NP}$ is the class of problems solvable by a TM which works in time $2^{O(n)}$ and is allowed to make queries of ...
2
votes
0answers
107 views

Complexity of $FP^{NP[O(n)]}$ with advice string?

$FP^{NP[O(n)]}$ is the functional complexity class with $O(n)$ queries to an $NP$ oracle. Are there any interesting classes $\mathcal C$ such that $$FP^{NP[O(n)]}\subseteq\mathcal C/\log$$ besides $...
3
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0answers
266 views

Oracle separating $coNP$ and $NP/poly$

I'd like to prove that, with respect to some adversarial oracle $O$, $coNP^O \not\subseteq NP/poly^O$. I was thinking of using $\textsf{UNSAT}$ for this and to build my oracle as follows: $O$ will "...
9
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1answer
254 views

Can a random oracle change which TFNP problems are strongly hard-on-average?

I've been thinking about the following question at various times since I saw this question on Cryptography. Question Let $R$ be a TFNP relation. ​ Can a random oracle help P/poly to break $R$ ...
5
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1answer
264 views

Smallest $f(n)$ such that $P/f(n) = BPP/f(n)$?

It is well-known that $\mathsf{P/poly}(n) = \mathsf{BPP/poly}(n)$. It is a major open problem to prove the conjecture $\mathsf{P} = \mathsf{BPP}$. $\mathsf{P} = \mathsf{BPP}$ implies $\mathsf{P}/f(n) ...
12
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1answer
699 views

Does P/poly $\neq$ NP/poly have any interesting implications?

$P/poly = NP/poly$ implies $NP \subseteq P/poly$, which in turn has interesting consequences like the collapse of the polynomial hierarchy. Are there interesting implications for $P/poly \neq NP/poly$...
3
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0answers
105 views

What is the strongest known lower bound against SIZE(n)?

What is the best known lower bound against (nonuniform) circuits of size $O(n)$? I understand that we don't know of any explicit functions that need circuits of size more than something like $5n$. But ...
17
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0answers
438 views

What if an $\mathsf L$-complete problem has $\mathsf{NC}^1$ circuits? More generally, what evidence is there against $\mathsf{NC}^1=\mathsf{L}$?

Edit: let me reformulate the question in a more specific way (and change the title accordingly). A slightly edited version of the original question follows. Is there a result comparable to the Karp-...
0
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0answers
165 views

ExpSpace problems whose configuration reachability problems are in P/poly?

Is anything known about ExpSpace problems whose configuration reachability problems are in P/poly? Let $M$ be an ExpSpace machine. Given two configurations $a$ and $b$ of $M$ (of max length), ...
0
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1answer
139 views

Assume that $\mathsf{NP} \subseteq \mathsf{P}/\text{log(n)}$, does it imply that $\mathsf{P} = \mathsf{NP}$? [closed]

I am trying to either prove or refute the claim mentioned in the title. Any ideas ?
8
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0answers
155 views

Is nonuniform $\mathsf{TC^0}$ equal to the composition closure of $\mathsf{AC^0}$ and Majority?

D.A.M. Barrington, N. Immerman and H. Straubing show in their 1990 paper "On Uniformity Within $\mathsf{NC^1}$" that the uniform $\mathsf{TC^0}$ is equal to $\mathsf{FOM}$ ($\mathsf{FO}$ plus ...
2
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0answers
108 views

Conditional Results on Bounded Depth Circuit Hierarchy

$\mathsf{AC,ACC,TC}$-hierarchy are basic bounded depth circuit hierarchies. $AC$-hierarchy is $\bigcup _{i =0}^{\infty} AC^{i} $ , where $AC^{i}$ is the $i$-th level of the hierarchy: a family of $\...
1
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0answers
135 views

Smallest Nonuniform Complexity Classes including uniform-P

As we know, studiyng differences between uniform complexity and nonuniform complexity class is crucial. For example, P/poly is defined as challenges to derive a separation between P and NP, because ...
5
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0answers
244 views

Is Solomonoff Induction in $\mathsf{P/poly}$?

Consider any language $L$. Define $s(L) \in {\lbrace 0, 1 \rbrace}^\omega$ (an infinite sequence of bits) by the recursive formula $$s(L)_n=\chi_L(s(L)_{<n})$$ Here $\chi_L$ is the characteristic ...
17
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0answers
319 views

$\mathsf{P} \ne \mathsf{P/poly} \cap \mathsf{NP}$? [duplicate]

Assuming $\mathsf{P} \ne \mathsf{NP}$ can we show $\mathsf{P} \ne \mathsf{P/poly} \cap \mathsf{NP}$? Obviously this would be the case if $\mathsf{P} \ne \mathsf{NP}$ and $\mathsf{P/poly} \supset \...
5
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0answers
192 views

Non-uniform average-case complexity of NP

It is conjectured that NP-complete problems are hard not only in the worst case but also in the typical case. Formally, given a language $S \in \lbrace 0,1 \rbrace^*$ and for each $n$ a probability ...
29
votes
3answers
914 views

Is NPI contained in P/poly?

It is conjectured that $\mathsf{NP} \nsubseteq \mathsf{P}/\text{poly}$ since the converse would imply $\mathsf{PH} = \Sigma_2$. Ladner's theorem establishes that if $\mathsf{P} \ne \mathsf{NP}$ then $\...
8
votes
0answers
120 views

Can polynomial-sized circuits use garbage?

This is a non-uniform (and simplified) version of my previous question about Cook reductions. Let $R\subseteq \{0,1\}^*\times\{0,1\}$. A function $r\colon \{0,1\}^*\to\{0,1\}$ solves $R$ if $(x,r(x))\...
3
votes
1answer
153 views

Speed-up of universal computation by caching

A universal computer is a program that can execute any other program. It is interesting to ask whether there are "booster" computers that execute programs faster than they execute "on their own". In ...
6
votes
2answers
2k views

Complexity of the halting problem

One of the most celebrated results in computer science is that the halting problem is undecidable. However there are still notions of complexity that are applicable. Here are 3 that I have in mind: $...
3
votes
2answers
279 views

Separation of space complexity classes: differeces between uniform class and nonuniform one as an analogy of circuit lower bounds project

Boolean circuit is used to measure time in a nonuniform way, which Pippenger showed the relation between a time complexity of uniform model (Turing Machines) and size complexity of boolean circuits. ...
3
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0answers
148 views

Survey on the Power of Non-Uniformity

I know that BPP is in P/poly. I know that if NP is in P/poly then $PH = \Sigma^2_p$ Question: Is there a good survey on the power of non-uniformity? I'm basically looking for a list of known ...
2
votes
1answer
235 views

Recommendation for a good book on first order logic w.r.t inductive logic programming

I have had 10 days to read up on Computational Logic but the books I am following are only succeeding in confusing me. I find most of text's ( Niehuys-Cheng & de Wolf 1997, De Raedt 2008, Lloyd ...
15
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0answers
332 views

Intersecting Complexity Classes with Advice

In on hiding information from an oracle, the authors (Abadi, Feigenbaum, and Kilian) wrote: $(\mathsf{NP/poly} \cap \mathsf{co\text-NP}{/poly})$ ... is not known to be equal to $(\mathsf{NP}...
9
votes
1answer
820 views

Non-Uniform vs. Uniform Adversaries

This question arose in the context of cryptography, but below I will present it in terms of complexity theory, since people here are more acquainted with the latter. This question is related to ...
20
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1answer
582 views

Problems in NP but not in Average-P/poly

The Karp–Lipton Theoem states that if $\mathsf{NP} \subset \mathsf{P/poly}$, then $\mathsf{PH}$ collapses to $\mathsf{\Sigma^P_2}$. Therefore, assuming separations between $\mathsf{\Sigma^P_2}$ and $\...
9
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0answers
285 views

Beating Nonuniformity by Oracle Access

Informally, we say that a Turing machine $M(\cdot)$ approximates a function $f(\cdot)$ if their outputs on a series of inputs are indistinguishable. More formally, let $L$ be a language, $M(\cdot)$ ...
5
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1answer
266 views

The Complexity of Advice in Computational Indistinguishability

One of the cornerstones of the modern cryptography is the definition of computational indistinguishability: It is used in definition of cryptosystems, pseudorandom generators, zero-knowledge, etc. ...
5
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1answer
383 views

Logarithmic advice language in P?

Is something like DTIME(poly(n))/log(n) in P? Can the log-length advice be somehow hardwired into a DTM for P?
2
votes
2answers
252 views

Nonuniform circuit families - don't have to specify for arbitrarily large, but finite, input lengths?

This is a question about nonuniform circuit families that's kind of bothering me. Let $\lbrace C_n \rbrace$ be a family of circuits for a language $L$ such that for inputs $x$ of length $n$, $C_n(x) = ...
8
votes
1answer
162 views

Separation of classes with different amounts of advice?

The time hierarchy theorem lets one show that, for example, there are problems in P that cannot be solved in time less than const*n^2 by a Turing machine. But give the Turing machine some advice and ...
2
votes
1answer
276 views

Complexity of advice language?

Let $L$ be a language in P/poly. There is then a deterministic polynomial-time Turing machine $M$ with polynomial-sized advice that decides $L$. Consider the language $A(M)$ of all advice strings ...
10
votes
1answer
334 views

Has the derandomization of slightly non-uniform classes, e.g BPP/linear, been studied?

By BPP/linear I refer to BPP machines with linear advice, which fulfills the promise when given the "correct" advice, and the derandomization should give us, say, a P/linear or (SUBEXP/linear) ...