A Survey of Lower Bounds for Satisfiability and Related Problems

Melkebeek, Dieter van

doi:10.1561/0400000012

Cited by 29 publications

(8 citation statements)

References 39 publications

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“…The following theorem is a special case of Theorem 1.3 in the excellent survey article by van Melkebeek [31] Proof. Assume that the claim fails for some depth d; thus for every > 0, SAT has Dlogtime-uniform depth d TC 0 circuits with fewer than n 1+ wires.…”

Section: Corollary 22mentioning

confidence: 96%

“…There are several examples of nonlinear lower bounds for various models of computation. For example Håstad presents a nearly-cubic lower bound on the formula size for a certain function [16], lower bounds on branching program size have been presented [1,7], and the time-space tradeoff results that are surveyed by van Melkebeek [31] give run-time lower bounds of the form n c for small-space computations. None of these lower bounds has led to separations of complexity classes.…”

Section: What Are the Main Contributions?mentioning

confidence: 99%

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Amplifying lower bounds by means of self-reducibility

Allender

Koucký

2010

J. ACM

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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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Section: Corollary 22mentioning

confidence: 96%

Section: What Are the Main Contributions?mentioning

confidence: 99%

Amplifying lower bounds by means of self-reducibility

Allender

Koucký

2010

J. ACM

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“…Several of the known models only allow measurements at the end of the computation but not during the computation. As another example, the known time-space lower bounds for classical algorithms (see [37] for a survey) hold for models with random access to the input and memory. This makes the lower bounds more meaningful as they do not exploit artifacts due to sequential access.…”

Section: Theorem 12 (Quantum Simulation)mentioning

confidence: 99%

“…However, the space usage can be reduced to at most the running time with at most a polylogarithmic factor increase in the latter by compressing the data and using an appropriate data structure to store (old address, new address) pairs. (See Section 2.3.1 of [37] for a similar construction. )…”

Section: Complexity Measuresmentioning

confidence: 99%

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Melkebeek¹,

Watson²

2012

Theory of Comput.

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On the Size of Depth-Three Boolean Circuits for Computing Multilinear Functions

Goldreich

Wigderson

2020

Lecture Notes in Computer Science

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We propose that multi-linear functions of relatively low degree over GF(2) may be good candidates for obtaining exponential 1 lower bounds on the size of constant-depth Boolean circuits (computing explicit functions). Specifically, we propose to move gradually from linear functions to multilinear ones, and conjecture that, for any t ≥ 2, some explicit t-linear functions F : ({0, 1} n ) t → {0, 1} require depth-three circuits of size exp(Ω(tn t/(t+1) )).Towards studying this conjecture, we suggest to study two frameworks for the design of depththree Boolean circuits computing multilinear functions, yielding restricted models for which lower bounds may be easier to prove. Both correspond to constructing a circuit by expressing the target polynomial as a composition of simpler polynomials. The first framework corresponds to a direct composition, whereas the second (and stronger) framework corresponds to nested composition and yields depth-three Boolean circuits via a "guess-and-verify" paradigm. The corresponding restricted models of circuits are called D-canonical and ND-canonical, respectively.Our main results are (1) a generic upper bound on the size of depth-three D-canonical circuits for computing any t-linear function, and (2) a lower bound on the size of any depththree ND-canonical circuits for computing some (in fact, almost all) t-linear functions. These bounds match the foregoing conjecture (i.e., they have the form of exp(tn t/(t+1) )). Another important result is a separation of the two models: We prove that ND-canonical circuits can be super-polynomially smaller than their D-canonical counterparts. We also reduce proving lower bounds for the ND-model to Valiant's matrix rigidity problem (for parameters that were not the focus of previous works).The study of the foregoing (Boolean) models calls for an understanding of new types of arithmetic circuits, which we define in this paper and may be of independent interest. These circuits compute multilinear polynomials by using arbitrary multilinear gates of some limited arity. It turns out that a GF(2)-polynomial is computable by such circuits with at most s gates of arity at most s if and only if it can be computed by ND-canonical circuits of size exp(s). A similar characterization holds for D-canonical circuits if we further restrict the arithmetic circuits to have depth two. We note that the new arithmetic model makes sense over any field, and indeed all our results carry through to all fields. Moreover, it raises natural arithmetic complexity problems which are independent of our original motivation.

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A Survey of Lower Bounds for Satisfiability and Related Problems

Cited by 29 publications

References 39 publications

Amplifying lower bounds by means of self-reducibility

Amplifying lower bounds by means of self-reducibility

Untitled

On the Size of Depth-Three Boolean Circuits for Computing Multilinear Functions

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