## Arithmetic circuits and counting complexity classes

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Venue: | In Complexity of Computations and Proofs,J.Krajíček, Ed. Quaderni di Matematica |

Citations: | 18 - 2 self |

### BibTeX

@INPROCEEDINGS{Allender_arithmeticcircuits,

author = {Eric Allender},

title = {Arithmetic circuits and counting complexity classes},

booktitle = {In Complexity of Computations and Proofs,J.Krajíček, Ed. Quaderni di Matematica},

year = {}

}

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### Abstract

Arithmetic circuits are the focus of renewed attention in the complexity theory community. It is easy to list a few of the reasons for the increased interest: • Innovative work by Kabanets and Impagliazzo [KI03] shows that, in

### Citations

215 | Bounded-width polynomial-size branching programs recognize exactly those languages in NC1
- Barrington
- 1989
(Show Context)
Citation Context ...) ,DegreenO(1) Context-free Languages [Ruz80, Sud78, Ven91] NL Size nO(1) ,Skew2 [Ven92] Shortest paths, Transitive Closure, etc. NC1 Depth O(log n), Bounded fan-in Regular sets, Non-solvable monoids =-=[Bar89] B-=-oolean Formula Evaluation [Bus93] TC0 Depth O(1), Size nO(1) , Majority gates ×, ÷, sorting [BCH86, RT92, HAB02] ACC0 Depth O(1), Size nO(1) , Unbounded fan-in, Modm Solvable monoids [BT88] AC0 Dept... |

124 | On uniformity within NC1 - Barrington, Immerman, et al. - 1990 |

120 |
LOG Depth Circuit for Division and Related Problems
- Beame, Cook, et al.
- 1986
(Show Context)
Citation Context ...act that for nearly two decades it was known that there are Puniform circuits for division and related problems, having logarithmic depth, but it was not known how to improve the uniformity condition =-=[BCH86]-=-. The first significant step in improving the uniformity condition was reported in [CDL01] and the final solution was presented in [HAB02]. (See also [All01] for additional information.) (An unrelated... |

55 | Algebraic Complexity Theory. Grundlehren der mathematischen Wissenschaften 315 - Bűrgisser, Clausen, et al. - 1997 |

43 | An optimal parallel algorithm for formula evaluation
- Buss, Cook, et al.
- 1992
(Show Context)
Citation Context ...metic formulae of polynomial size. As such, it has been studied as a complexity class at least since [Val79a]. Evaluating arithmetic formulae over N (Z) is complete for #NC 1 (GapNC 1 , respectively) =-=[BCGR92]-=-. It follows from [CDL01] that every function in #NC 1 is computable in logspace. Probably the most important and fascinating open question regarding arithmetic NC 1 is the following: Open Question 8 ... |

43 | Structure and importance of logspace-MOD-classes
- Buntrock, Damm, et al.
- 1992
(Show Context)
Citation Context ...imilarity, equivalence, and diagonalizability. These results are summarized in Figure 2. Note that for computation over finite fields of characteristic p, all of these problems are complete for ModpL =-=[BDHM91]-=-. Additional examples of problems complete for L #L are presented in [AV04], including the problem of testing if two permutation groups are isomorphic, and computing the order of a permutation group. ... |

36 | The complexity of matrix rank and feasible systems of linear equations
- Allender, Beals, et al.
- 1999
(Show Context)
Citation Context ...ass of problems AC0-reducible to computing the determinant of integer matrices. The first two of these hierarchies collapse, and they coincide with NC 1 reducibility. • AC 0 (C=L) = L C=L =NC 1 (C=L=-=) [ABO99]. • -=-AC 0 (PL) = PL = NC 1 (PL) [Ogi98, BF00]. (These hierarchies are defined using “Ruzzo-Simon-Tompa” reducibility [RST84], which is the usual notion of oracle access for space-bounded nondeterminist... |

33 | Relationships among PL, #L and the determinant
- Allender, Ogihara
- 1996
(Show Context)
Citation Context ...he class of problems reducible to A under NC 1 reductions coincides with the problems reducible to A under AC 0 reductions. Is this also the case for the determinant? This question was first posed in =-=[AO96], whe-=-re the following hierarchies were defined: • The Exact Counting Logspace Hierarchy = C=LC=L.·.C=L =AC0 (C=L) =theclassofproblemsAC0-reducible to the set of singular integer matrices. 14s• The PL ... |

29 | Reducing the complexity of reductions - Agrawal, Allender, et al. |

29 | The permanent requires large uniform threshold circuits
- Allender
- 1996
(Show Context)
Citation Context ...2] would allow us to compute the value over Z in uniform TC 0 .) 22sHowever, it was shown in [CMTV98] that #P is not in DLOGTIME-uniform TC 0 . (Specific superpolynomial lower bounds are presented in =-=[All99].)-=- Interestingly (and frustratingly), for any field F of characteristic p �= 2, it is not known if the permanent over F requires superpolynomial-size uniform constant-depth arithmetic circuits (even t... |

28 |
P-uniform circuit complexity
- Allender
- 1989
(Show Context)
Citation Context ...smodel what can be computed efficiently by circuits that are feasible to construct, then polynomial-time would seem to be the right notion of uniformity to consider. This point of view is explored in =-=[All89]. Ho-=-wever, the real reason for most occurrences of the phrase “P-uniformity” in the literature stems from the fact that for nearly two decades it was known that there are Puniform circuits for divisio... |

27 | Non-commutative arithmetic circuits: depth reduction and size lower bounds
- Allender, Jiao, et al.
- 1998
(Show Context)
Citation Context ...th-reduction results for uniform circuits were proved for the Boolean ring in [Ruz81], and for N in [Vin91]. A general proof that works in the uniform setting over any commutative semiring appears in =-=[AJMV98]-=-. The time has come to distinguish the two approaches to circuit complexity that have been hinted to earlier. It is really quite simple. In the traditional approach to circuit complexity, the circuit ... |

24 | Isolation, matching, and counting: Uniform and nonuniform upper bounds
- Allender, Reinhardt, et al.
- 1999
(Show Context)
Citation Context ... least as powerful as the Boolean circuits. Analogous results hold for #SAC 1 . Furthermore, there is reason to believe that this simulation should also be possible in the uniform setting; results of =-=[ARZ99] -=-and [KvM02] (using the Nisan-Wigderson generators [NW94]) show that if there is any problem in DSPACE(n) that requires circuits of size 2 ɛn ,thenNL=UL, and every Boolean function computed by polynom... |

23 | A uniform circuit lower bound for the permanent
- Allender, Gore
- 1994
(Show Context)
Citation Context ...ant-depth arithmetic circuits (even though uniform arithmetic circuits over finite fields characterize uniform ACC 0 , and the permanent is known to require exponential size on uniform ACC 0 circuits =-=[AG94]).-=- In contrast, for fields F of characteristic 2 it is known that the determinant (and hence the permanent) is complete for ⊕L, which contains functions that are not in (non-uniform) AC 0 [2] [Raz87, ... |

19 |
On TC0, AC0, and arithmetic circuits
- Agrawal, Allender, et al.
(Show Context)
Citation Context ...ssed above are at least as powerful as Boolean NC 1 , and thus we do not know if there is any problem in NP that does not have small arithmetic circuits of that sort. On the other hand it is shown in =-=[AAD00]-=- that the zero-one-valued functions in GapAC 0 are exactly the languages in AC 0 [2] (that is, the languages accepted by constant-depth polynomial-size circuits of AND, OR, and PARITY gates). The resu... |

12 |
A Very Hard Log Space Counting Class
- Àlvarez, Jenner
(Show Context)
Citation Context ...xity class is #P, the class of functions of the form #accM (x), counting the number of accepting paths of an NP machine M on input x [Val79b]. The class #L is defined similarly, but for NL machines M =-=[AJ93]-=-. Using the characterizations of NL and NP in terms of Boolean circuits as presented in Figure 1, we obtain a circuit-based characterization of the classes #P and #L, as follows. Start with a uniform ... |

12 |
Making computation count: Arithmetic circuits
- Allender
- 1997
(Show Context)
Citation Context ...e is essentially no difference between the two settings. This is discussed in more detail in later sections. Much of the material contained herein appeared earlier in a survey written for SIGACT News =-=[All97]-=-. In the intervening years, many open questions listed in [All97] have been solved, and many new developments have come about. Hence it was felt that an updated survey, with an updated list of open qu... |

11 | Circuits over PP and PL - Beigel, Fu - 1997 |

10 | Bounded depth arithmetic circuits: Counting and closure - Allender, Ambainis, et al. - 1999 |

10 | The complexity of policy evaluation for finite-horizon partially-observable Markov decision processes - Mundhenk, Goldsmith, et al. - 1997 |

7 | Arithmetic complexity, Kleene closure, and formal power series
- Allender, Arvind, et al.
(Show Context)
Citation Context ...the #L hierarchy is the following “iterated determinant” problem: Given as input n 2 matrices Mi,j, compute the determinant of the matrix M whose (i, j)th entry is DET(Mi,j). However, it was shown=-= in [AAM03]-=- that this function actually lies in GapL. The main reason to be interested in PL, C=L and related classes is this: They characterize the complexity of some important and natural problems. For instanc... |

7 | The first-order isomorphism theorem
- AGRAWAL
(Show Context)
Citation Context ... was presented in [HAB02]. (See also [All01] for additional information.) (An unrelated problem for which P-uniformity was needed was presented in [AAI + 01], but it was subsequently shown by Agrawal =-=[Agr01]-=- that, in this instance also, it was possible to state the theorems in terms of DLOGTIME-uniformity.) The circuits for division and iterated product presented in [HAB02] are for arithmetic over the in... |

7 |
The division breakthroughs
- Allender
(Show Context)
Citation Context ...wn how to improve the uniformity condition [BCH86]. The first significant step in improving the uniformity condition was reported in [CDL01] and the final solution was presented in [HAB02]. (See also =-=[All01]-=- for additional information.) (An unrelated problem for which P-uniformity was needed was presented in [AAI + 01], but it was subsequently shown by Agrawal [Agr01] that, in this instance also, it was ... |

3 |
On counting ac0 circuits with negated constants
- Ambainis, Barrington, et al.
- 1998
(Show Context)
Citation Context ...btain several other “gap” classes: GapL, GapAC 0 ,GapNC 1 , and GapSAC 1 .Itrequiresa clever argument to show that every function in GapSAC 1 can be expressed as the difference of two #AC 0 functi=-=ons [ABL98]-=-, but for all of the other GapC classes, it is straightforward to show that they can be expressed as the difference of two #C functions. GapP was originally introduced in [FFK94] as a tool for studyin... |

2 |
Abelian permutation group problems and logspace counting classes
- Arvind, Vijayaraghavan
- 2004
(Show Context)
Citation Context ... for C=L, and a variety of other problems regarding computation of the rank and determining if a system of linear equations is feasible are complete for L C=L [ABO99]. (Interestingly, it was shown in =-=[AV04]-=- that determining feasibility of a system of linear equations modulo a given small prime p is complete for the seemingly larger class L #L [ABO99].) Some other problems in linear algebra and problems ... |

1 | Complete problems for Valiant’s class of qpcomputable families of polynomials - Blaeser |

1 | The complexity of solving linear equations over a finite ring - Arvind, Vijayaraghavan - 2005 |