scholarly journals Lower Bounds for Arithmetic Circuits via the Hankel Matrix

2021 ◽  
Vol 30 (2) ◽  
Author(s):  
Nathanaël Fijalkow ◽  
Guillaume Lagarde ◽  
Pierre Ohlmann ◽  
Olivier Serre
Keyword(s):  
Lower Bounds ◽  
Hankel Matrix ◽  
Arithmetic Circuits

scholarly journals Uniform derandomization from pathetic lower bounds

2012 ◽  
Vol 370 (1971) ◽  
pp. 3512-3535 ◽  
Author(s):  
Eric Allender ◽  
V. Arvind ◽  
Rahul Santhanam ◽  
Fengming Wang
Keyword(s):  
Word Problem ◽  
Lower Bounds ◽  
Black Box ◽  
Arithmetic Circuits ◽  
List Type ◽  
Probabilistic Algorithms ◽  
Constant Depth ◽  
Subexponential Time ◽  
Polynomial Size ◽  
Threshold Circuits

The notion of probabilistic computation dates back at least to Turing, who also wrestled with the practical problems of how to implement probabilistic algorithms on machines with, at best, very limited access to randomness. A more recent line of research, known as derandomization, studies the extent to which randomness is superfluous. A recurring theme in the literature on derandomization is that probabilistic algorithms can be simulated quickly by deterministic algorithms, if one can obtain impressive (i.e. superpolynomial, or even nearly exponential) circuit size lower bounds for certain problems. In contrast to what is needed for derandomization, existing lower bounds seem rather pathetic. Here, we present two instances where ‘pathetic’ lower bounds of the form n 1+ ϵ would suffice to derandomize interesting classes of probabilistic algorithms. We show the following: — If the word problem over S 5 requires constant-depth threshold circuits of size n 1+ ϵ for some ϵ >0, then any language accepted by uniform polynomial size probabilistic threshold circuits can be solved in subexponential time (and, more strongly, can be accepted by a uniform family of deterministic constant-depth threshold circuits of subexponential size). — If there are no constant-depth arithmetic circuits of size n 1+ ϵ for the problem of multiplying a sequence of n  3×3 matrices, then, for every constant d , black-box identity testing for depth- d arithmetic circuits with bounded individual degree can be performed in subexponential time (and even by a uniform family of deterministic constant-depth AC 0 circuits of subexponential size).

Lower bounds on arithmetic circuits via partial derivatives

1996 ◽  
Vol 6 (3) ◽  
pp. 217-234 ◽  
Author(s):  
Noam Nisan ◽  
Avi Wigderson
Keyword(s):  
Lower Bounds ◽  
Arithmetic Circuits ◽  
Partial Derivatives

scholarly journals Lower Bounds for Depth-Three Arithmetic Circuits with small bottom fanin

2016 ◽  
Vol 25 (2) ◽  
pp. 419-454 ◽  
Author(s):  
Neeraj Kayal ◽  
Chandan Saha
Keyword(s):  
Lower Bounds ◽  
Arithmetic Circuits

scholarly journals A Compilation of Succinctness Results for Arithmetic Circuits

2021 ◽  
Author(s):  
Alexis de Colnet ◽  
Stefan Mengel
Keyword(s):  
Lower Bounds ◽  
Probabilistic Models ◽  
Probabilistic Reasoning ◽  
Efficient Algorithms ◽  
Arithmetic Circuits ◽  
Boolean Circuits ◽  
Real Numbers ◽  
The Real ◽  
Indicator Variables ◽  
Input Variables

Arithmetic circuits (AC) are circuits over the real numbers with 0/1-valued input variables whose gates compute the sum or the product of their inputs. Positive AC – that is, AC representing non-negative functions – subsume many interesting probabilistic models such as probabilistic sentential decision diagram (PSDD) or sum-product network (SPN) on indicator variables. Efficient algorithms for many operations useful in probabilistic reasoning on these models critically depend on imposing structural restrictions to the underlying AC. Generally, adding structural restrictions yields new tractable operations but increases the size of the AC. In this paper we study the relative succinctness of classes of AC with different combinations of common restrictions. Building on existing results for Boolean circuits, we derive an unconditional succinctness map for classes of monotone AC – that is, AC whose constant labels are non-negative reals – respecting relevant combinations of the restrictions we consider. We extend a small part of the map to classes of positive AC. Those are known to generally be exponentially more succinct than their monotone counterparts, but we observe here that for so-called deterministic circuits there is no difference between the monotone and the positive setting which allows us to lift some of our results. We end the paper with some insights on the relative succinctness of positive AC by showing exponential lower bounds on the representations of certain functions in positive AC respecting structured decomposability.

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