Unbalancing Sets and An Almost Quadratic Lower Bound for Syntactically Multilinear Arithmetic Circuits

Alon, Noga; Kumar, Mrinal; Volk, Ben Lee

doi:10.1007/s00493-019-4009-0

Cited by 8 publications

(12 citation statements)

References 32 publications

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“…D-AC p sD-AC p : there only is sD-AC p ≤ D-AC p to prove. Let C ∈ D-AC p , if g ∈ C is a +-node such that x ∈ var(g r ) and x ∈ var(g l ), then add a ×-node between g l and g whose successors are g l and Note that completing the map for positive AC might be very hard: in fact, it would in particular require showing strong lower bounds for D-AC p , a well-known open problem in complexity theory for which the best current result is a recent nearly quadratic lower bound (Alon, Kumar, and Volk 2020). Another question is the relations between the map of monotone AC and that of positive AC: when imposing determinism, the expressive power of positive AC is exactly that of monotone AC, while for unrestricted circuits it is known that positive AC are more succinct than monotone AC (Valiant 1980).…”

Section: Beginning the Map For Positive Acmentioning

confidence: 99%

A Compilation of Succinctness Results for Arithmetic Circuits

Colnet

Mengel

2021

Proceedings of the Eighteenth International Conference on Principles of Knowledge Representation and Reasoning

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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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Section: Beginning the Map For Positive Acmentioning

confidence: 99%

A Compilation of Succinctness Results for Arithmetic Circuits

Colnet

Mengel

2021

Proceedings of the Eighteenth International Conference on Principles of Knowledge Representation and Reasoning

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“…My proof used a combination of Gröbner basis methods and linear algebra. Srinivasan gave a simpler proof which combined Fermat's little Theorem with linear algebra (see [5]). Alon found a third proof based on the Combinatorial Nullstellensatz (see [3]).…”

Section: Introductionmentioning

confidence: 99%

$L$-balancing families

Hegedüs¹

2021

Preprint

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Let p be a prime and let f ∈ F p [x 1 , . . . , x 2p ] be a polynomial. Suppose that f (v F ) = 0 for each F ⊆ [2p], where |F | = p and that f (0) = 0. Then deg(f ) ≥ p.We prove here the following generalization of their result.Let p be a prime and q = p α > 1, α ≥ 1. Let n > 0 be a positive integer and q − 1 ≤ d ≤ n − q + 1 be an integer. Let F be a field of characteristic p. Suppose that f (v F ) = 0 for each F ⊆ [n], whereLet t = 2d be an even number and L ⊆ [d − 1] be a given subset. We say that F ⊆ 2 [t] is an L-balancing family if for each F ⊆ [t], whereWe give a general upper bound for the size of an L-balancing family.

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“…A seminal work of Raz [13] showed that multilinear formulas computing det n or perm n must have size n Ω(log n) . Although we know strong lower bounds for multilinear formulas, the best known lower bound against syntactic multilinear circuits is almost quadratic in the number of variables [1]. Note that any multilinear ABP of n O (1) size computing f on n variables can be converted to a multilinear formula of size n O(log n) computing f .…”

Section: Introductionmentioning

confidence: 99%

“…Although we know strong lower bounds for multilinear formulas, the best known lower bound against syntactic multilinear circuits is almost quadratic in the number of variables [1]. Note that any multilinear ABP of n O (1) size computing f on n variables can be converted to a multilinear formula of size n O(log n) computing f . In order to prove super-polynomial lower bounds for ABPs, it is enough to obtain a multilinear formula computing f of size n o(log n) or prove a lower bound of n ω(log n) for multilinear formulas, both of which are not known.…”

Section: Introductionmentioning

confidence: 99%