A Bounded Arithmetic Theory for Constant Depth Threshold Circuits

Gödel '96 ◽  
2017 ◽  
pp. 224-234
Author(s):  
Jan Johannsen
Keyword(s):  
Bounded Arithmetic ◽  
Constant Depth ◽  
Arithmetic Theory ◽  
Threshold 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).

Some Results on Uniform Arithmetic Circuit Complexity

1991 ◽  
Vol 20 (343) ◽  
Author(s):  
Gudmund Skovbjerg Frandsen ◽  
Mark Valence ◽  
David Mix Barrington
Keyword(s):  
Arithmetic Function ◽  
Circuit Complexity ◽  
Arithmetic Circuits ◽  
Arithmetic Functions ◽  
Constant Depth ◽  
Arithmetic Circuit ◽  
Arithmetic Complexity ◽  
Polynomial Size ◽  
Threshold Circuits ◽  
Class Of Functions

We introduce a natural set of arithmetic expressions and define the complexity class AE to consist of all those arithmetic functions (over the fields F_(2)n) that are described by these expressions. We show that AE coincides with the class of functions that are computable with constant depth and polynomial size unbounded fan-in arithmetic circuits satisfying a natural uniformity constraint (DLOGTIME-uniformity). A 1-input and 1-output arithmetic function over the fields F_(2)n may be identified with an <em>n</em>-input and an n-output Boolean function when field elements are represented as bit strings. We prove that if some such representation is X-uniform (where X is P or DLOGTIME) then the arithmetic complexity of a function (measured with X-uniform unbounded fan-in arithmetic circuits) is identical to the Boolean complexity of this function (measured with X-uniform threshold circuits). We show the existence of a P-uniform representation and we give partial results concerning the existence of representations with more restrictive uniformity properties.

scholarly journals Uniform constant-depth threshold circuits for division and iterated multiplication

2002 ◽  
Vol 65 (4) ◽  
pp. 695-716 ◽  
Author(s):  
William Hesse ◽  
Eric Allender ◽  
David A. Mix Barrington
Keyword(s):  
Constant Depth ◽  
Threshold Circuits

Implicit proofs

2004 ◽  
Vol 69 (2) ◽  
pp. 387-397 ◽  
Author(s):  
Jan Krajíček
Keyword(s):  
Proof System ◽  
Bounded Arithmetic ◽  
Polynomial Size ◽  
Arithmetic Theory ◽  
General Method

Abstract.We describe a general method how to construct from a prepositional proof system P a possibly much stronger proof system iP. The system iP operates with exponentially long P-proofs described “implicitly” by polynomial size circuits.As an example we prove that proof system iEF, implicit EF, corresponds to bounded arithmetic theory and hence, in particular, polynomially simulates the quantified prepositional calculus G and the -consequences of proved with one use of exponentiation. Furthermore, the soundness of iEF is not provable in . An iteration of the construction yields a proof system corresponding to T2 + Exp and, in principle, to much stronger theories.

scholarly journals Uniform, Integral, and Feasible Proofs for the Determinant Identities

2021 ◽  
Vol 68 (2) ◽  
pp. 1-80
Author(s):  
Iddo Tzameret ◽  
Stephen A. Cook
Keyword(s):  
Linear Algebra ◽  
Order Theory ◽  
Bounded Arithmetic ◽  
First Order ◽  
First Order Theory ◽  
Polynomial Size ◽  
Arithmetic Theory ◽  
Integral Version ◽  
Determinant Function ◽  
Basic Determinant

Aiming to provide weak as possible axiomatic assumptions in which one can develop basic linear algebra, we give a uniform and integral version of the short propositional proofs for the determinant identities demonstrated over GF (2) in Hrubeš-Tzameret [15]. Specifically, we show that the multiplicativity of the determinant function and the Cayley-Hamilton theorem over the integers are provable in the bounded arithmetic theory VNC 2 ; the latter is a first-order theory corresponding to the complexity class NC 2 consisting of problems solvable by uniform families of polynomial-size circuits and O (log 2 n )-depth. This also establishes the existence of uniform polynomial-size propositional proofs operating with NC 2 -circuits of the basic determinant identities over the integers (previous propositional proofs hold only over the two-element field).

scholarly journals Counting, fanout and the complexity of quantum ACC

2002 ◽  
Vol 2 (1) ◽  
pp. 35-65
Author(s):  
F. Green ◽  
S. Homer ◽  
C. Moore ◽  
C. Pollett
Keyword(s):  
Complexity Classes ◽  
Constant Depth ◽  
Classical Counterpart ◽  
Classical Class ◽  
Polynomial Size ◽  
Constant Depth Circuits ◽  
Language Classes ◽  
Threshold Circuits ◽  
Quantum Analog ◽  
Quantum Complexity

We propose definitions of QAC^0, the quantum analog of the classical class AC^0 of constant-depth circuits with AND and OR gates of arbitrary fan-in, and QACC[q], the analog of the class ACC[q] where Mod_q gates are also allowed. We prove that parity or fanout allows us to construct quantum MOD_q gates in constant depth for any q, so QACC[2] = QACC. More generally, we show that for any q,p > 1, MOD_q is equivalent to MOD_p (up to constant depth and polynomial size). This implies that QAC^0 with unbounded fanout gates, denoted QACwf^0, is the same as QACC[q] and QACC for all q. Since \ACC[p] \ne ACC[q] whenever p and q are distinct primes, QACC[q] is strictly more powerful than its classical counterpart, as is QAC^0 when fanout is allowed. This adds to the growing list of quantum complexity classes which are provably more powerful than their classical counterparts. We also develop techniques for proving upper bounds for QACC in terms of related language classes. We define classes of languages closely related to QACC[2] and show that restricted versions of them can be simulated by polynomial-size circuits. With further restrictions, language classes related to QACC[2] operators can be simulated by classical threshold circuits of polynomial size and constant depth.

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