On the Range Avoidance Problem for Circuits

Ren, Hanlin; Santhanam, Rahul; Wang, Zhikun

doi:10.1109/focs54457.2022.00067

Cited by 10 publications

(9 citation statements)

References 51 publications

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“…In fact, we reduce the problem of constructing explicit rigid matrices to a problem that we call degree-2-Avoid, where each output computes a degree-2 polynomial of the inputs. Following Ren, Santhanam, and Wang's approach [RSW22], this problem can be reduced to NC 0 3 -Avoid using the perfect encoding scheme of Applebaum, Ishai, and Kushilevitz [AIK06]. 4 On the algorithmic side, we make partial progress towards resolving Open Problem 2.…”

Section: Our Resultsmentioning

confidence: 99%

“…A solution to the degree-2-Avoid problem for the function g would give us a rigid matrix. Following [RSW22], we can then apply a perfect encoding scheme [IK00, IK02, AIK06] to g, and obtain an NC 0 3 function f , as required (see Lemma 2.7). Effectively, this reduces solving Avoid on g to solving Avoid on f .…”

Section: Proof Overviewmentioning

confidence: 99%

“…Composing these two encodings provides an encoding of NC 1 functions in NC 0 4 . Using this encoding, [RSW22] provides a polynomial time reduction from NC 1 -Avoid to NC 0 4 -Avoid. We use only one part of the result from [AIK06]: there is a polynomial time reduction from degree-d-Avoid to NC 0 d+1 -Avoid.…”

Section: Low Degree and Low Localitymentioning

confidence: 99%

“…For a class of circuits C, the C-Avoid problem is a special case of Avoid where each output is computed by a circuit from C. Recent works by Ren, Santhanam, and Wang [RSW22] and Guruswami, Lyu, and Wang [GLW22] proved that efficient algorithms for C-Avoid even for certain simple circuit classes C would be sufficient for getting various explicit constructions. Later, Chen, Huang, Li, and Ren [CHLR23] re-derived the best known lower bounds against ACC 0 circuits from an efficient algorithm for a certain C-Avoid problem.…”

Section: Introductionmentioning

confidence: 99%

“…Unlike the case of the general Avoid problem, NC 0 k -Avoid may be much easier for large values of the stretch m. Indeed, on one hand, NC 0 k -Avoid for small stretch m = n + o(n) is capable of encoding hard explicit construction problems [RSW22,GLW22]. On the other hand, NC 0 k -Avoid for m = Ω(n k ) is easily solvable in polynomial time: since the number of distinct functions depending on at most k out of n inputs is O(n k ), every such instance of the problem must have two outputs computing identical functions.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Range Avoidance for Constant-Depth Circuits: Hardness and Algorithms

Gajulapalli¹,

Golovnev²,

Nagargoje³

et al. 2023

Preprint

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Range Avoidance (Avoid) is a total search problem where, given a Boolean circuit C : {0, 1} n → {0, 1} m , m > n, the task is to find a y ∈ {0, 1} m outside the range of C. For an integer k ≥ 2, NC 0 k -Avoid is a special case of Avoid where each output bit of C depends on at most k input bits. Ren, Santhanam, and Wang (FOCS 2022) and Guruswami, Lyu, and Wang (RANDOM 2022) proved that explicit constructions of functions of high circuit complexity, rigid matrices, optimal linear codes, Ramsey graphs, and other combinatorial objects reduce to NC 0 4 -Avoid, thus establishing conditional hardness of the NC 0 4 -Avoid problem. On the other hand, NC 0 2 -Avoid admits polynomial-time algorithms, leaving the question about the complexity of NC 03 -Avoid open. We give the first reduction of an explicit construction question to NC 0 3 -Avoid. Specifically, we prove that a polynomial-time algorithm (with an NP oracle) for NC 0 3 -Avoid for the case of m = n + n 2/3 would imply an explicit construction of a rigid matrix, and, thus, a super-linear lower bound on the size of log-depth circuits.We also give deterministic polynomial-time algorithms for all NC 0 k -Avoid problems for m ≥ n k−1 / log(n). Prior work required an NP oracle, and required larger stretch, m ≥ n k−1 .

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Section: Our Resultsmentioning

confidence: 99%

Section: Proof Overviewmentioning

confidence: 99%

Section: Low Degree and Low Localitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Range Avoidance for Constant-Depth Circuits: Hardness and Algorithms

Gajulapalli¹,

Golovnev²,

Nagargoje³

et al. 2023

Preprint

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On the Existence of Strong Proof Complexity Generators

KRAJÍČEK

2023

Bull. symb. log

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Cook and Reckhow [5] pointed out that N P = coN P iff there is no propositional proof system that admits polynomial size proofs of all tautologies. The theory of proof complexity generators aims at constructing sets of tautologies hard for strong and possibly for all proof systems. We focus on a conjecture from [16] in foundations of the theory that there is a proof complexity generator hard for all proof systems. This can be equivalently formulated (for p-time generators) without a reference to proof complexity notions as follows:• There exists a p-time function g stretching each input by one bit such that its range rng(g) intersects all infinite N P sets.We consider several facets of this conjecture, including its links to bounded arithmetic (witnessing and independence results), to time-bounded Kolmogorov complexity, to feasible disjunction property of propositional proof systems and to complexity of proof search. We argue that a specific gadget generator from [18] is a good candidate for g. We define a new hardness property of generators, the -hardness, and show that one specific gadget generator is the -hardest (w.r.t. any sufficiently strong proof system). We define the class of feasibly infinite N P sets and show, assuming a hypothesis from circuit complexity, that the conjecture holds for all feasibly infinite N P sets.

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