On computing the determinant in small parallel time using a small number of processors
Information Processing Letters
Uniform closure properties of P-computable functions
STOC '86 Proceedings of the eighteenth annual ACM symposium on Theory of computing
Feasible arithmetic computations: Valiant's hypothesis
Journal of Symbolic Computation
A new pebble game that characterizes parallel complexity classes
SIAM Journal on Computing
Properties that characterize LOGCFL
Journal of Computer and System Sciences
Circuit definitions of nondeterministic complexity classes
SIAM Journal on Computing
Why is Boolean complexity theory difficult?
Poceedings of the London Mathematical Society symposium on Boolean function complexity
The complexity of iterated multiplication
Information and Computation
Non-commutative arithmetic circuits: depth reduction and size lower bounds
Theoretical Computer Science
Some Exact Complexity Results for Straight-Line Computations over Semirings
Journal of the ACM (JACM)
Introduction to Circuit Complexity: A Uniform Approach
Introduction to Circuit Complexity: A Uniform Approach
Derandomizing polynomial identity tests means proving circuit lower bounds
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
The complexity of tensor calculus
Computational Complexity
Valiant's model and the cost of computing integers
Computational Complexity
On the Complexity of Numerical Analysis
CCC '06 Proceedings of the 21st Annual IEEE Conference on Computational Complexity
Expressing a fraction of two determinants as a determinant
Proceedings of the twenty-first international symposium on Symbolic and algebraic computation
Journal of Symbolic Computation
Characterizing arithmetic circuit classes by constraint satisfaction problems
ICALP'11 Proceedings of the 38th international colloquim conference on Automata, languages and programming - Volume Part I
Succinct algebraic branching programs characterizing non-uniform complexity classes
FCT'11 Proceedings of the 18th international conference on Fundamentals of computation theory
Balancing bounded treewidth circuits
CSR'10 Proceedings of the 5th international conference on Computer Science: theory and Applications
Separating multilinear branching programs and formulas
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
Arithmetic circuits: The chasm at depth four gets wider
Theoretical Computer Science
The complexity of weighted counting for acyclic conjunctive queries
Journal of Computer and System Sciences
Resource Trade-offs in Syntactically Multilinear Arithmetic Circuits
Computational Complexity
On Enumerating Monomials and Other Combinatorial Structures by Polynomial Interpolation
Theory of Computing Systems
Balancing Bounded Treewidth Circuits
Theory of Computing Systems
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Valiant introduced 20 years ago an algebraic complexity theory to study the complexity of polynomial families. The basic computation model used is the arithmetic circuit, which makes these classes very easy to define and open to combinatorial techniques. In this paper we gather known results and new techniques under a unifying theme, namely the restrictions imposed upon the gates of the circuit, building a hierarchy from formulas to circuits. As a consequence we get simpler proofs for known results such as the equality of the classes VNP and VNP"e or the completeness of the Determinant for VQP, and new results such as a characterization of the classes VQP and VP (which we can also apply to the Boolean class LOGCFL) or a full answer to a conjecture in Burgisser's book [Completeness and reduction in algebraic complexity theory, Algorithms and Computation in Mathematics, vol. 7, Springer, Berlin, 2000]. We also show that for circuits of polynomial depth and unbounded size these models all have the same expressive power and can be used to characterize a uniform version of VNP.