The Complexity of Computing
The Design and Analysis of Computer Algorithms
The Design and Analysis of Computer Algorithms
Some NP-complete geometric problems
STOC '76 Proceedings of the eighth annual ACM symposium on Theory of computing
Some simplified NP-complete problems
STOC '74 Proceedings of the sixth annual ACM symposium on Theory of computing
On the parallel evaluation of multivariate polynomials
STOC '78 Proceedings of the tenth annual ACM symposium on Theory of computing
The complexity of theorem-proving procedures
STOC '71 Proceedings of the third annual ACM symposium on Theory of computing
A complexity theory based on Boolean algebra
Journal of the ACM (JACM)
Feasible arithmetic computations: Valiant's hypothesis
Journal of Symbolic Computation
Computing algebraic formulas with a constant number of registers
STOC '88 Proceedings of the twentieth annual ACM symposium on Theory of computing
Towards optimal simulations of formulas by bounded-width programs
STOC '90 Proceedings of the twenty-second annual ACM symposium on Theory of computing
On Datalog vs. polynomial time (extended abstract)
PODS '91 Proceedings of the tenth ACM SIGACT-SIGMOD-SIGART symposium on Principles of database systems
Depth reduction for noncommutative arithmetic circuits
STOC '93 Proceedings of the twenty-fifth annual ACM symposium on Theory of computing
The complexity of matrix rank and feasible systems of linear equations (extended abstract)
STOC '96 Proceedings of the twenty-eighth annual ACM symposium on Theory of computing
Is there an algebraic proof for P ≠ NC? (extended abstract)
STOC '97 Proceedings of the twenty-ninth annual ACM symposium on Theory of computing
An exponential lower bound for depth 3 arithmetic circuits
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
Some Exact Complexity Results for Straight-Line Computations over Semirings
Journal of the ACM (JACM)
Probabilistic Algorithms for Deciding Equivalence of Straight-Line Programs
Journal of the ACM (JACM)
Quantum computers that can be simulated classically in polynomial time
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
Affine projections of symmetric polynomials
Journal of Computer and System Sciences - Complexity 2001
On Implications between P-NP-Hypotheses: Decision versus Computation in Algebraic Complexity
MFCS '01 Proceedings of the 26th International Symposium on Mathematical Foundations of Computer Science
Lower Bounds Are Not Easier over the Reals: Inside PH
ICALP '00 Proceedings of the 27th International Colloquium on Automata, Languages and Programming
Complete Problems for Valiant's Class of qp-Computable Families of Polynomials
COCOON '01 Proceedings of the 7th Annual International Conference on Computing and Combinatorics
Efficient Zero-Knowledge Authentication Based on a Linear Algebra Problem MinRank
ASIACRYPT '01 Proceedings of the 7th International Conference on the Theory and Application of Cryptology and Information Security: Advances in Cryptology
Rook polynomials to and from permanents
Discrete Mathematics - The 2000 Com2MaC conference on association schemes, codes and designs
Derandomizing polynomial identity tests means proving circuit lower bounds
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
On TC/sup 0/, AC/sup 0/, and Arithmetic Circuits
CCC '97 Proceedings of the 12th Annual IEEE Conference on Computational Complexity
Negation can be exponentially powerful
STOC '79 Proceedings of the eleventh annual ACM symposium on Theory of computing
On monotone formulae with restricted depth
STOC '84 Proceedings of the sixteenth annual ACM symposium on Theory of computing
Exponential lower bounds for restricted monotone circuits
STOC '83 Proceedings of the fifteenth annual ACM symposium on Theory of computing
Computing the sign or the value of the determinant of an integer matrix, a complexity survey
Journal of Computational and Applied Mathematics - Special issue: Proceedings of the international conference on linear algebra and arithmetic, Rabat, Morocco, 28-31 May 2001
On the hardness of approximating the permanent of structured matrices
Computational Complexity
Counting complexity classes for numeric computations II: algebraic and semialgebraic sets
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Classical complexity and quantum entanglement
Journal of Computer and System Sciences - Special issue: STOC 2003
Derandomizing polynomial identity tests means proving circuit lower bounds
Computational Complexity
The complexity of matrix completion
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Improved construction for universality of determinant and permanent
Information Processing Letters
Circuit lower bounds for Merlin-Arthur classes
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
The complexity of two problems on arithmetic circuits
Theoretical Computer Science
Elusive functions and lower bounds for arithmetic circuits
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Expressing a fraction of two determinants as a determinant
Proceedings of the twenty-first international symposium on Symbolic and algebraic computation
Semidefinite programming and arithmetic circuit evaluation
Discrete Applied Mathematics
On the Shortest Linear Straight-Line Program for Computing Linear Forms
MFCS '08 Proceedings of the 33rd international symposium on Mathematical Foundations of Computer Science
Arithmetic Circuits, Syntactic Multilinearity, and the Limitations of Skew Formulae
MFCS '08 Proceedings of the 33rd international symposium on Mathematical Foundations of Computer Science
Proceedings of the forty-first annual ACM symposium on Theory of computing
VPSPACE and a transfer theorem over the complex field
Theoretical Computer Science
On defining integers in the counting hierarchy and proving arithmetic circuit lower bounds
STACS'07 Proceedings of the 24th annual conference on Theoretical aspects of computer science
VPSPACE and a transfer theorem over the reals
STACS'07 Proceedings of the 24th annual conference on Theoretical aspects of computer science
Decision versus evaluation in algebraic complexity
MCU'07 Proceedings of the 5th international conference on Machines, computations, and universality
On the expressive power of planar perfect matching and permanents of bounded treewidth matrices
ISAAC'07 Proceedings of the 18th international conference on Algorithms and computation
Small-space analogues of Valiant's classes
FCT'09 Proceedings of the 17th international conference on Fundamentals of computation theory
Non-commutative circuits and the sum-of-squares problem
Proceedings of the forty-second ACM symposium on Theory of computing
CSR'08 Proceedings of the 3rd international conference on Computer science: theory and applications
Journal of Symbolic Computation
Arithmetic Circuits: A survey of recent results and open questions
Foundations and Trends® in Theoretical Computer Science
On the arithmetic complexity of euler function
CSR'11 Proceedings of the 6th international conference on Computer science: theory and applications
An extended tree-width notion for directed graphs related to the computation of permanents
CSR'11 Proceedings of the 6th international conference on Computer science: theory and applications
On the power of algebraic branching programs of width two
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
Deterministic Polynomial Time Algorithms for Matrix Completion Problems
SIAM Journal on Computing
From a zoo to a zoology: descriptive complexity for graph polynomials
CiE'06 Proceedings of the Second conference on Computability in Europe: logical Approaches to Computational Barriers
Valiant's model: from exponential sums to exponential products
MFCS'06 Proceedings of the 31st international conference on Mathematical Foundations of Computer Science
Short proofs for the determinant identities
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
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
FAW-AAIM'12 Proceedings of the 6th international Frontiers in Algorithmics, and Proceedings of the 8th international conference on Algorithmic Aspects in Information and Management
A dichotomy theorem for homomorphism polynomials
MFCS'12 Proceedings of the 37th international conference on Mathematical Foundations of Computer Science
VPSPACE and a transfer theorem over the complex field
MFCS'07 Proceedings of the 32nd international conference on Mathematical Foundations of Computer Science
Permanent does not have succinct polynomial size arithmetic circuits of constant depth
Information and Computation
Explicit lower bounds via geometric complexity theory
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
Quasi-polynomial hitting-set for set-depth-Δ formulas
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
Most Tensor Problems Are NP-Hard
Journal of the ACM (JACM)
The complexity of weighted counting for acyclic conjunctive queries
Journal of Computer and System Sciences
Tensor-Rank and Lower Bounds for Arithmetic Formulas
Journal of the ACM (JACM)
Non-commutative arithmetic circuits with division
Proceedings of the 5th conference on Innovations in theoretical computer science
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In the theory of recursive functions and computational complexity it has been demonstrated repeatedly that the natural problems tend to cluster together in “completeness classes”. These are families of problems that (A) are computationally interreducible and (B) are the hardest members of some computationally defined class. The aim of this paper is to demonstrate that for both algebraic and combinatorial problems this phenomenon exists in a form that is purely algebraic in both of the respects (A) and (B). Such computational consequences as NP-completeness are particular manifestations of something more fundamental. The core of the paper is self-contained, consisting as it does essentially of the two notions of “p-definability” and the five algebraic relations that are proved as theorems. In the remainder our aim is to elucidate the computational consequences of these basic results. Hence in the auxiliary propositions and discussion for convenience we do assume familiarity with algebraic and Boolean complexity theory.