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Unconditional Lower Bounds against Advice [chapter]

Harry Buhrman, Lance Fortnow, Rahul Santhanam
2009 Lecture Notes in Computer Science  
We show several unconditional lower bounds for exponential time classes against polynomial time classes with advice, including: 1. For any constant c, NEXP ⊆ P NP[n c ] /n c 2.  ...  For the probabilistic classes, no lower bounds for uniform exponential time against advice were known before.  ...  In this paper, we show unconditional lower bounds for exponential-time classes such as NEXP, MAEXP, BPEXP and REXP against their polynomial-time versions with advice.  ... 
doi:10.1007/978-3-642-02927-1_18 fatcat:t44e2glkrzd37dw4sviqs72gwy

An Average-Case Lower Bound Against $$\mathsf {ACC}^0$$ ACC 0 [chapter]

Ruiwen Chen, Igor C. Oliveira, Rahul Santhanam
2018 Lecture Notes in Computer Science  
one-sided error) and ZPE/1 (zero-error randomized time 2 O(n) with 1 bit of advice) against polynomial size ACC 0 circuits.  ...  Williams' technique inherently gives a worst-case lower bound, and until now, no average-case version of his result was known.  ...  Motivation and Background Significant advances in unconditional lower bounds are few and far between, specially in non-monotone boolean circuit models.  ... 
doi:10.1007/978-3-319-77404-6_24 fatcat:wdjooay4yjhe3attx4mgssb2km

Natural Proofs Versus Derandomization [article]

Ryan Williams
2015 arXiv   pre-print
These characterizations are applied to yield several new results, including improved ACC^0 lower bounds and new unconditional derandomizations.  ...  We prove: ∙ Constructivity is unavoidable, even for NEXP lower bounds.  ...  Interestingly, the proof of Theorem 1.4 crucially uses the previous lower bound framework against NEXP, and builds on it, via Theorem 1.1 and a modification of the NEXP ⊂ ACC 0 lower bound.  ... 
arXiv:1212.1891v3 fatcat:pks6jxag7jhmvbjypvqxylqbwu

On Medium-Uniformity and Circuit Lower Bounds

Rahul Santhanam, Ryan Williams
2013 2013 IEEE Conference on Computational Complexity  
Lower Bounds Against Medium-Uniform Circuits.  ...  Using a new kind of indirect diagonalization argument, we prove several new unconditional lower bounds against medium uniform circuit classes, including: • For all k, P is not contained in P-uniform SIZE  ...  LOWER BOUNDS AGAINST MEDIUM UNIFORMITY We will use a folklore result about a time hierarchy for deterministic time, where the lower bound holds against sublinear advice.  ... 
doi:10.1109/ccc.2013.40 dblp:conf/coco/SanthanamW13 fatcat:mm4ls445rjd37mkuqv5sg2lqiy

Natural proofs versus derandomization

Ryan Williams
2013 Proceedings of the 45th annual ACM symposium on Symposium on theory of computing - STOC '13  
These characterizations are applied to yield several new results, including improved ACC lower bounds and new unconditional derandomizations. * A preliminary version of this paper appeared in  ...  We prove: • Constructivity is unavoidable, even for NEXP lower bounds.  ...  Interestingly, our proof of Theorem 1.3 crucially uses the previous lower bound framework against NEXP, and builds on it, via Theorem 1.1 and a modification of the NEXP ⊂ ACC lower bound.  ... 
doi:10.1145/2488608.2488612 dblp:conf/stoc/Williams13 fatcat:ecvxel2o2vdw3aafixugvf4xzu

Natural Proofs versus Derandomization

R. Ryan Williams
2016 SIAM journal on computing (Print)  
These characterizations are applied to yield several new results, including improved ACC lower bounds and new unconditional derandomizations. * A preliminary version of this paper appeared in  ...  We prove: • Constructivity is unavoidable, even for NEXP lower bounds.  ...  Interestingly, our proof of Theorem 1.3 crucially uses the previous lower bound framework against NEXP, and builds on it, via Theorem 1.1 and a modification of the NEXP ⊂ ACC lower bound.  ... 
doi:10.1137/130938219 fatcat:gxa2yzaxkfgo7pojgjz7bkkcui

Pseudodeterministic Algorithms and the Structure of Probabilistic Time [article]

Zhenjian Lu, Igor C. Oliveira, Rahul Santhanam
2021 arXiv   pre-print
stretch computable in pseudodeterministic polynomial time (with one bit of advice) that is secure infinitely often against polynomial-time computations.  ...  We build on techniques developed to prove hierarchy theorems for probabilistic time with advice (Fortnow and Santhanam, FOCS 2004) to construct the first unconditional pseudorandom generator of polynomial  ...  Theorem 25 (An unconditional complexity lower bound for estimating rK poly ). For any ε > 0 and d ≥ 1 there exists a constant k ≥ 1 for which the following holds.  ... 
arXiv:2103.08539v1 fatcat:t344sx34orgormjvstk3l3n2l4

Strong co-nondeterministic lower bounds for NP cannot be proved feasibly

Ján Pich, Rahul Santhanam
2021 Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing  
We also show similar unconditional unprovability results in APC 1 for the conjecture of Rudich about the existence of super-bits. * jan.pich@cs.ox.ac.uk  ...  Theorem 1 talks about average-case lower bounds for NP against co-nondeterministic circuits.  ...  In this paper, we rule out this pessimistic situation for certain kinds of strong complexity lower bounds, namely for lower bounds for NP against co-nondeterministic circuits.  ... 
doi:10.1145/3406325.3451117 fatcat:zwtwvpqcfbcvtlkxx75w7jkf5i

Why are Proof Complexity Lower Bounds Hard?

Jan Pich, Rahul Santhanam
2019 2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)  
Circuit Lower Bounds Program: Separate NP and P by proving super-polynomial lower bounds for functions in NP against more and more expressive classes of Boolean circuits.  ...  It is well-known that P can be simulated by polynomialsize Boolean circuits, and therefore super-polynomial lower bounds on Boolean circuit size imply lower bounds against P.  ...  Formalizing Lower Bounds in the Propositional Setting Circuit lower bounds.  ... 
doi:10.1109/focs.2019.00080 dblp:conf/focs/PichS19 fatcat:4of45zsirzd6ldkebom27aeaqy

Pseudodeterministic algorithms and the structure of probabilistic time

Zhenjian Lu, Igor C. Oliveira, Rahul Santhanam
2021 Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing  
polynomial time (with one bit of advice) that is secure infinitely often against polynomial-time computations.  ...  We build on techniques developed to prove hierarchy theorems for probabilistic time with advice (Fortnow and Santhanam [FS04]) to construct an unconditional pseudorandom generator computable in pseudodeterministic  ...  We can also prove the following unconditional complexity lower bound. Theorem 25. For any ε > 0 and d ≥ 1 there exists a constant k ≥ 1 for which the following holds.  ... 
doi:10.1145/3406325.3451085 fatcat:yqh6zf42y5akrflgxvwvianpmm

On Uniformity and Circuit Lower Bounds

Rahul Santhanam, Ryan Williams
2014 Computational Complexity  
Lower bounds against medium-uniform circuits.  ...  Using a new kind of indirect diagonalization argument, we prove several new unconditional lower bounds against medium-uniform circuit classes, including: • For all k, P is not contained in P-uniform SIZE  ...  R.S. was supported in part by the 2013 ERC Consolidator Grant ALUnif: "Algorithms and Lower Bounds: A Unified Approach."  ... 
doi:10.1007/s00037-014-0087-y fatcat:4fy5xs7jrfbtzm37apd7wq2qza

Marginal hitting sets imply super-polynomial lower bounds for permanent

Maurice Jansen, Rahul Santhanam
2012 Proceedings of the 3rd Innovations in Theoretical Computer Science Conference on - ITCS '12  
hypothesis for any field of characteristic zero: There is a fixed depth d and some function s(n) = O(n), such that for arbitrarily small > 0, there exists a hitting set Hn ⊂ Z of size at most 2 s(n ) against  ...  Unconditional Lower Bounds Using Theorem 1 we will derive the following unconditional hardness result for the permanent: Theorem 2.  ...  How then do we obtain unconditional lower bounds? The key idea is to use a win-win argument.  ... 
doi:10.1145/2090236.2090275 dblp:conf/innovations/JansenS12 fatcat:nyq3xjitdvhmlmujyxrlojz22e

On the Advice Complexity of the Set Cover Problem [chapter]

Dennis Komm, Richard Královič, Tobias Mömke
2012 Lecture Notes in Computer Science  
We further show lower and upper bounds for achieving c-competitiveness also in m.  ...  set, we obtain bounds in both n and m.  ...  To do so, we again consider an adversary Adv that plays against some online algorithm A with advice, i. e., against 2 b deterministic algorithms at once (see Observation 1). Theorem 4.  ... 
doi:10.1007/978-3-642-30642-6_23 fatcat:qluinziykvbmfinqboqp3k5xpi

Pseudodeterministic constructions in subexponential time

Igor C. Oliveira, Rahul Santhanam
2017 Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing - STOC 2017  
Perhaps intriguingly, while our main results are unconditional, they have a non-constructive element, arising from a sequence of applications of the hardness versus randomness paradigm.  ...  It turns out that strong enough circuit lower bounds for BPE imply efficient bounded-error pseudodeterministic generation, and strong enough circuit lower bounds for ZPE imply efficient zero-error pseudodeterministic  ...  However, we appear to be very far from proving such strong circuit lower bounds.  ... 
doi:10.1145/3055399.3055500 dblp:conf/stoc/OliveiraS17 fatcat:p5uafhcwvrh5jfmcfo3lpru7ty

Algorithms versus Circuit Lower Bounds [article]

Igor C. Oliveira
2013 arXiv   pre-print
Different techniques have been used to prove several transference theorems of the form "nontrivial algorithms for a circuit class C yield circuit lower bounds against C".  ...  We discuss how circuit lower bounds can be obtained from derandomization, compression, learning, and satisfiability algorithms.  ...  Finally, Clément Canonne provided me valuable advice to improve the presentation.  ... 
arXiv:1309.0249v1 fatcat:t6hkghh4zrdcxgnntdo3zkdd5y
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