Cracks in the Defenses: Scouting Out Approaches on Circuit Lower Bounds

Allender, Eric

doi:10.1007/978-3-540-79709-8_2

Cited by 6 publications

(4 citation statements)

References 45 publications

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“…I try to be optimistic. I wrote a survey recently [All08], outlining some of the approaches that have been proposed, to try to overcome the barriers that seem to block progress toward proving circuit lower bounds (and let me repeat that I think that circuit lower bounds are really the most important goal to strive for, if we want to understand randomness). In the survey, I tried to make the case that, although there is certainly a great deal of pessimism about the prospects for quick resolution of any of these problems (such as the P vs. NP problem), there is nonetheless some reason for hope.…”

Section: What Are the Most Important Open Problems In The Field?mentioning

confidence: 99%

Randomness as Circuit Complexity (and the Connection to Pseudorandomness)

Allender

2011

Randomness Through Computation

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This material was generated for the book Randomness through Computation, edited by Hector Zenil. The format of the book calls for various contributors to give responses to five questions. Why were you initially drawn to the study of computation and randomness? I started out my undergraduate studies majoring in theater. However, I also knew that I enjoyed math and computers, thanks to a terrific high school math teacher (Ed Rolenc) who installed a computer in one of the classrooms in my high school in Mount Pleasant, Iowa in the 1970's. Thus, when I was an undergrad at the University of Iowa and I decided to pick a second major that might make me more employable in case theater didn't work out, I picked Computer Science. My initial computing courses didn't really inspire me very much, and I continued to think of computing as a "reserve" career, until two things happened at more-or-less the same time: (1) I worked in summer theaters for a couple of summers, and I noticed that there were incredibly talented people who were working at the same undistinguished summer theaters where I was working, leading me to wonder how much impact I was likely to have in the field of theater. (2) I took my first courses in theoretical computer science (with Ted Baker and Don Alton), which opened my eyes to some of the fascinating open questions in the field. They gave me some encouragement to consider grad school, and thus (after taking a year off to "see the world" by working as a bellhop in Germany) I entered the doctoral program at Georgia Tech. At that time, in the early-to-mid 1980's, Kolmogorov complexity was finding application in complexity theory. (I'm referring to the material that's now covered in the chapter called "The Incompressibility Method" in the standard text by Li and Vitányi [LV08].) Also, there were influential papers by Mike Sipser [Sip83] and Juris Hartmanis [Har83] that discussed resource-bounded Kolmogorov complexity. Also, the theory of pseudorandom generators was just getting underway, with the work of Yao [Yao82] and of Blum and Micali [BM84]. Thus Kolmogorov complexity and randomness were very much in the air. Please recall that, at the time, most people believed that BPP and RP were likely to contain problems that could not be solved in deterministic polynomial time. (BPP is the class of problems that can be solved efficiently if we have access to randomness, and RP is the natural way to define a probabilistic subclass of NP.) I was struck by the observation that there was some tension between the popular conjectures relating deterministic, probabilistic, and nondeterministic computation. For example, if nondeterministic exponential time has no efficient deterministic simulation, it means

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Section: What Are the Most Important Open Problems In The Field?mentioning

confidence: 99%

Randomness as Circuit Complexity (and the Connection to Pseudorandomness)

Allender

2011

Randomness Through Computation

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“…(This topic is also discussed in a recent survey [3].) However, a number of other questions naturally arise.…”

Section: Conclusion and Open Problemsmentioning

confidence: 99%

Amplifying Lower Bounds by Means of Self-Reducibility

Allender

Koucký

2008

2008 23rd Annual IEEE Conference on Computational Complexity

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We observe that many important computational problems in NC 1 share a simple self-reducibility property. We then show that, for any problem A having this self-reducibility property, A has polynomial size TC 0 circuits if and only if it has TC 0 circuits of size n 1+ for every > 0 (counting the number of wires in a circuit as the size of the circuit). As an example of what this observation yields, consider the Boolean Formula Evaluation problem (BFE), which is complete for NC 1 and has the self-reducibility property. It follows from a lower bound of Impagliazzo, Paturi, and Saks, that BFE requires depth d TC 0 circuits of size n 1+ d . If one were able to improve this lower bound to show that there is some constant > 0 such that every TC 0 circuit family recognizing BFE has size n 1+ , then it would follow that TC 0 = NC 1 . We show that proving lower bounds of the form n 1+ is not ruled out by the Natural Proof framework of Razborov and Rudich and hence there is currently no known barrier for separating classes such as ACC 0 , TC 0 and NC 1 via existing "natural" approaches to proving circuit lower bounds.We also show that problems with small uniform constant-depth circuits have algorithms that simultaneously have small space and time bounds. We then make use of known time-space tradeoff lower bounds to show that SAT requires uniform depth d TC 0 and AC 0 [6] circuits of size n 1+c for some constant c depending on d.

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“…2 [All96] was still "depressingly up to date" in 2008, according to [All08]. 3 The number of outputs of f n is typically 1, but occasionally is some other specified function of n. When it's 1, complexity theorists call the set of n-tuple inputs for which f n outputs 1 (unioned over all n) "the language recognized by f n " (or by a circuit family or algorithm that computes f n ); this lets them identify "languages" with 1-output function families, since they're 1-1.…”

Section: Introduction 1the Problem Of Circuit Complexitymentioning

confidence: 99%

The Pattern Basis Approach to Circuit Complexity

Smith

2016

Preprint

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We describe and motivate a proposed new approach to the problem of finding a "complexity formula" which lower-bounds the circuit complexity (over circuits in a given class) of an arbitrary boolean function, given its truth table, and which can be used to prove nontrivial lower bounds on the complexity of specific functions of interest. This requires addressing the "natural proofs barrier" [Razborov & Rudich 1997], which (roughly) rules out any formula which can be computed in time polynomial in the truth table size. The proposed kind of formula would take as long to compute in general as a brute-force search among all possible circuits, but has a form amenable to proving lower bounds on its value for well-understood explicit functions.This form is based on a new formalization of "patterns", as elements of a special basis of the vector space of all truth table properties. We prove that any "pattern basis" gives a nonnegative and subadditive complexity formula, which (after scaling to fit primitive circuit costs) lower-bounds circuit complexity (though often trivially). We give specific conditions on the pattern basis which would lead to formula values having a useful range (in which all small circuits have low values, but most functions have high values). We discuss some of the issues involved in satisfying those conditions, and the features a pattern basis would need to make proving explicit-function lower bounds practical.This paper is intended for all mathematicians interested in the general phenomenon of complexity, including those in other fields which might be relevant to realizing this new approach (such as linear algebra, combinatorics, and finite group representation theory).

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Cracks in the Defenses: Scouting Out Approaches on Circuit Lower Bounds

Cited by 6 publications

References 45 publications

Randomness as Circuit Complexity (and the Connection to Pseudorandomness)

Randomness as Circuit Complexity (and the Connection to Pseudorandomness)

Amplifying Lower Bounds by Means of Self-Reducibility

The Pattern Basis Approach to Circuit Complexity

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