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### On Defining Integers in the Counting Hierarchy and Proving Arithmetic Circuit Lower Bounds [chapter]

Peter Bürgisser
STACS 2007
We prove that if there are arithmetic circuits for computing the permanent of n by n matrices having size polynomial in n, then τ (n!) is polynomially bounded in log n.  ...  The constant-free Valiant model An arithmetic circuit over the field Q is an acyclic finite digraph, where all nodes except the input nodes have fan-in 2 and are labelled by +, −, × or /. The circuit  ...  Integers definable in the counting hierarchy We consider sequences of integers a(n, k) defined for n, k ∈ N and 0 ≤ k ≤ q(n), where q is polynomially bounded, such that ∀n > 1 ∀k ≤ q(n) |a(n, k)| ≤ 2 n  ...

### Shallow Circuits with High-Powered Inputs [article]

Pascal Koiran
2010 arXiv   pre-print
In this third version of our paper we show that the same lower bound would follow even if one could only prove a slightly superpolynomial upper bound on the number of real roots.  ...  We also show that an even weaker bound on the number of real roots would suffice to obtain a lower bound on the size of depth 4 circuits computing the permanent.  ...  The counting hierarchy contains all the polynomial hierarchy PH and is contained in PSPACE. The arithmetic circuit classes defined in Section 2.1 are nonuniform.  ...

### Arithmetic Constant-Depth Circuit Complexity Classes [chapter]

Hubie Chen
2003 Lecture Notes in Computer Science
One reason for interest in these classes is that they contain the boundary marking the limits of current lower bound technology: such technology exists for AC 0 and some of the classes AC 0 [m], while  ...  Continuing a line of research originating from Valiant's work on the counting class ♯P , the arithmetic circuit complexity classes ♯AC 0 and ♯N C 1 have recently been studied.  ...  The author would like to thank Eric Allender for many interesting discussions. Riccardo Pucella deserves thanks for comments on a draft of this paper.  ...

### Permanent does not have succinct polynomial size arithmetic circuits of constant depth

Maurice Jansen, Rahul Santhanam
2013 Information and Computation
From this we obtain the lower bound by explicitly constructing a hitting set against arithmetic circuits in the polynomial hierarchy. 5 It is possible to give a uniform upper of E NP RP for L. 6 One of  ...  To obtain this result we develop a novel technique that further strengthens the connection between black-box derandomization of polynomial identity testing and lower bounds for arithmetic circuits.  ...  We thank Pavel Hrubeš for pointing out to us that without division gates a lower bound can be obtained for succinct circuits of constant depth by a reduction to the Razborov-Smolensky lower bound.  ...

### Permanent Does Not Have Succinct Polynomial Size Arithmetic Circuits of Constant Depth [chapter]

Maurice Jansen, Rahul Santhanam
2011 Lecture Notes in Computer Science
From this we obtain the lower bound by explicitly constructing a hitting set against arithmetic circuits in the polynomial hierarchy. 5 It is possible to give a uniform upper of E NP RP for L. 6 One of  ...  To obtain this result we develop a novel technique that further strengthens the connection between black-box derandomization of polynomial identity testing and lower bounds for arithmetic circuits.  ...  We thank Pavel Hrubeš for pointing out to us that without division gates a lower bound can be obtained for succinct circuits of constant depth by a reduction to the Razborov-Smolensky lower bound.  ...

### On Defining Integers And Proving Arithmetic Circuit Lower Bounds

Peter Bürgisser
2009 Computational Complexity
We prove that if there are arithmetic circuits of size polynomial in n for computing the permanent of n by n matrices, then τ (n!) is polynomially bounded in log n.  ...  X k and n k=1 1 k X k of exp and log, respectively, can be computed by arithmetic circuits of size polynomial in log n (allowing divisions).  ...  The author was partially supported by DFG grant BU 1371 and the Paderborn Institute for Scientific Computation (PaSCo).  ...

### The complexity of two problems on arithmetic circuits

Pascal Koiran, Sylvain Perifel
2007 Theoretical Computer Science
This gives a coNP Mod k P algorithm for deciding an upper bound on the degree of a polynomial given by a circuit in fields of characteristic k > 0.  ...  By using arithmetic circuits, encoding multivariate polynomials may be drastically more efficient than writing down the list of monomials.  ...  Acknowledgements The authors thank Erich Kaltofen for the suggestion of encoding d in unary in the language DEG b as well as the anonymous referees for useful remarks.  ...

### Interpolation in Valiant's theory [article]

Pascal Koiran
2007 arXiv   pre-print
Our proof method relies on Lagrange interpolation and on recent results connecting the (boolean) counting hierarchy to algebraic complexity classes.  ...  We investigate the following question: if a polynomial can be evaluated at rational points by a polynomial-time boolean algorithm, does it have a polynomial-size arithmetic circuit?  ...  Acknowledgments We would like to thank Erich Kaltofen and Christos Papadimitriou for sharing their thoughts on question (*).  ...

### Marginal hitting sets imply super-polynomial lower bounds for permanent

Maurice Jansen, Rahul Santhanam
2012 Proceedings of the 3rd Innovations in Theoretical Computer Science Conference on - ITCS '12
We prove that the hypothesis implies that Permanent does not have polynomial size constant-free arithmetic circuits.  ...  Suppose f is a univariate polynomial of degree r = r(n) that is computed by a size n arithmetic circuit.  ...  Assuming τ (per n ) = n O(1) , for a first compression step, one uses the relation between the counting hierarchy CH and TC 0 to get the coefficients of fn "weakly-definable" in CH.  ...

### Monomials in Arithmetic Circuits: Complete Problems in the Counting Hierarchy

Hervé Fournier, Guillaume Malod, Stefan Mengel
2014 Computational Complexity
We consider the complexity of two questions on polynomials given by arithmetic circuits: testing whether a monomial is present and counting the number of monomials.  ...  We show that these problems are complete for subclasses of the counting hierarchy which had few or no known natural complete problems before.  ...  The results of this paper were conceived while the third author was visiting the Équipe de Logique Mathématique at Université Paris Diderot Paris 7.  ...

### Interpolation in Valiant's Theory

Pascal Koiran, Sylvain Perifel
2011 Computational Complexity
Our proof method relies on Lagrange interpolation and on recent results connecting the (boolean) counting hierarchy to algebraic complexity classes.  ...  Answering it negatively would indeed imply that the constant-free versions of the algebraic complexity classes VP and VNP defined by Valiant are different.  ...  Acknowledgements We would like to thank Erich Kaltofen and Christos Papadimitriou for sharing their thoughts on question (*), and the anonymous referees for their useful remarks.  ...

### Bounded Arithmetic, Cryptography and Complexity

SAMUEL R. BUSS
2008 Theoria
2 can prove the polynomial time hierarchy collapses in a strong way [14, 6, 23] .  ...  This survey discusses theories of bounded arithmetic, growth rates of definable functions, natural proofs, interpolation theorems, connections to cryptography, and the difficulty of obtaining independence  ...  Impagliazzo for pointing out the construction of section 4.4 and J. Krajíček, P. Pudlák and C. Pollett for comments on a preliminary version of this paper.  ...

### Symmetric Computation (Invited Talk)

Anuj Dawar, Michael Wagner
2020 Annual Conference for Computer Science Logic
This is at once a rich class of problems and one for which we have methods for proving lower bounds.  ...  The mismatch between algorithms working on high-level data structures and complexity defined in terms of low-level machines is one of the central concerns of the field of descriptive complexity, which  ...  The lower bound on counting width established in Theorem 5 has interesting consequences for lift-and-project hierarchies.  ...

### Finite and Algorithmic Model Theory (Dagstuhl Seminar 17361)

Anuj Dawar, Erich Grädel, Phokion G. Kolaitis, Thomas Schwentick, Marc Herbstritt
2018 Dagstuhl Reports
This report documents the program and the outcomes of Dagstuhl Seminar 17361 "Finite and Algorithmic Model Theory".  ...  We formalize agent-based models as stochastic processes whose states are metafinite models, and we define a notion of abstraction.  ...  Our main results are conditions that imply an abstraction is sound, and further conditions that imply it preserves the Markov property.  ...

### Monomials in arithmetic circuits: Complete problems in the counting hierarchy [article]

Hervé Fournier and Guillaume Malod and Stefan Mengel
2012 arXiv   pre-print
We consider the complexity of two questions on polynomials given by arithmetic circuits: testing whether a monomial is present and counting the number of monomials.  ...  We show that these problems are complete for subclasses of the counting hierarchy which had few or no known natural complete problems.  ...  The results of this paper were conceived while the third author was visiting theÉquipe de Logique Mathématique at Université Paris Diderot Paris 7. He would like to thank Arnaud Durand  ...
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