Questions tagged [advice-and-nonuniformity]

Questions about advice and nonuniformity

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29
votes
3answers
957 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 $\...
20
votes
1answer
613 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 $\...
17
votes
0answers
450 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-...
17
votes
0answers
322 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 \...
15
votes
0answers
333 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}...
14
votes
1answer
205 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 ...
12
votes
1answer
816 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 $P/poly \neq NP/poly$...
11
votes
1answer
209 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 ...
10
votes
1answer
343 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) ...
9
votes
3answers
593 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 ...
9
votes
3answers
633 views

P/Poly vs Uniform Complexity Classes

It is not known whether NEXP is contained in P/poly. Indeed proving that NEXP is not in P/poly would have some applications in derandomization. What is the smallest uniform class C for which one can ...
9
votes
1answer
867 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 ...
9
votes
1answer
257 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 $R$ ...
9
votes
0answers
158 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 ...
9
votes
0answers
296 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, $M(\cdot)$ ...
8
votes
1answer
166 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 ...
8
votes
0answers
126 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))\...
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: $...
5
votes
1answer
386 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?
5
votes
1answer
268 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) ...
5
votes
1answer
271 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
votes
0answers
246 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 ...
5
votes
0answers
195 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 ...
3
votes
1answer
155 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 ...
3
votes
2answers
282 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
votes
0answers
287 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 "...
3
votes
0answers
106 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 ...
3
votes
0answers
150 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
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) = ...
2
votes
1answer
281 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 ...
2
votes
1answer
104 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
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 ...
2
votes
0answers
110 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 $...
2
votes
0answers
109 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
vote
2answers
106 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, ...
1
vote
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 ...
0
votes
1answer
190 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^\...
0
votes
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 ?
0
votes
0answers
167 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), ...
-2
votes
0answers
51 views

$BPP$ type algorithms with slightly more capability

A language $L$ is in $BPP$ if and only if there exists a polynomial $p$ and deterministic Turing machine $M$, such that $M$ runs for polynomial time on all inputs For all $x$ in $L$, the fraction of ...