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Uniform normal form for general time-bounded complexity classes

Bernard R. Hodgson, Clement F. Kent
1986 Journal of computer and system sciences (Print)  
Refining techniques of previous works, we obtain a normal form arithmetical representation for non-deterministic computability, in which the polynomial matrix does not involve the time-bounding function  ...  This permits arithmetization of Turing machine complexity classes determined by quite general time bounds. Applications are made to complexity hierarchies and to obtain a single, uniform, normal form.  ...  for set membership are of two prenex normal form types.  ... 
doi:10.1016/0022-0000(86)90034-6 fatcat:cnnhapneqvgddmrqvrdsaka4se

Tuples of Disjoint NP-Sets [chapter]

Olaf Beyersdorff
2006 Lecture Notes in Computer Science  
In this paper we introduce a natural generalization of the notion of disjoint NP-pairs to disjoint k-tuples of NP-sets for k ≥ 2.  ...  In our main result we show that complete disjoint NP-pairs exist if and only if complete disjoint k-tuples of NP-sets exist for all k ≥ 2.  ...  For helpful conversations and suggestions on this work I am very grateful to Johannes Köbler, Jan Krajíček, Pavel Pudlák, and Zenon Sadowski.  ... 
doi:10.1007/11753728_11 fatcat:7nfv4x3tvbcddbwnioiwanf3mq

On the Complexity of Hilbert's 17th Problem [chapter]

Nikhil R. Devanur, Richard J. Lipton, Nisheeth K. Vishnoi
2004 Lecture Notes in Computer Science  
We consider arithmetic circuits as a natural representation of rational functions.  ...  In this setting, the following question is a natural relaxation: What is the descriptive complexity of the sum of squares representation (as rational functions) of a non-negative, n-variate, degree d polynomial  ...  Acknowledgments We are grateful to Marie-Françoise Roy for valuable comments on an earlier draft of this paper. We also thank Bruce Reznick for pointing us to references [3, 7] .  ... 
doi:10.1007/978-3-540-30538-5_20 fatcat:ke4ckzcjrfae5f7266juxxaxli

A Characterisation of Definable NP Search Problems in Peano Arithmetic [chapter]

Arnold Beckmann
2009 Lecture Notes in Computer Science  
The complexity class of ≺-bounded local search problems with goals is introduced for well-orderings ≺, and is used to give a characterisation of definable NP search problems in Peano Arithmetic.  ...  This is the basis for the well-known representation of ordinals less than 0 : namely, write an ordinal α < 0 as a term in Cantor normal form, recursively writing the exponents of ω in the same form.  ...  We simultaneously and inductively define a set of expressions, called normal compact forms for ordinals less than 0 , and a binary relation ≺ 0 on normal compact forms, as follows, where "=" denotes identity  ... 
doi:10.1007/978-3-642-02261-6_1 fatcat:idsg2izjl5hjlprgjpxxm4xtbq

Tuples of Disjoint $\mathsf{NP}$ -Sets

Olaf Beyersdorff
2007 Theory of Computing Systems  
In this paper we introduce a natural generalization of the notion of disjoint NP-pairs to disjoint k-tuples of NP-sets for k ≥ 2.  ...  In our main result we show that complete disjoint NP-pairs exist if and only if complete disjoint k-tuples of NP-sets exist for all k ≥ 2.  ...  For helpful conversations and suggestions on this work I am very grateful to Johannes Köbler, Jan Krajíček, Pavel Pudlák, and Zenon Sadowski.  ... 
doi:10.1007/s00224-007-9023-8 fatcat:45azqnufdjc6xbywxv6nzmnozi

Page 4693 of Mathematical Reviews Vol. , Issue 2002G [page]

2002 Mathematical Reviews  
In partic- ular, we have such a Galois representation for a Siegel modular eigenform Y for the full Siegel modular group Sp(4, Z) of weight 28; Y is unique up to normalization and Galois conjugation and  ...  F,(k)q” is the Fourier expansion of a classical modular form of weight k and level Np.  ... 

Page 4449 of Mathematical Reviews Vol. , Issue 86j [page]

1986 Mathematical Reviews  
We assume that there exists a cusp form H(r) = }>°°, b(n)e[nz] of weight 2 satisfying (1) H(r) is a normalized primitive cusp form, (2) b(n) € Z for all n > 1, (3) for almost all primes p, a(p) = b(p)  ...  +cx+d, over Q of degree 3 whose splitting field Ky over Q belongs to Cp; C2= the set of all normalized cusp forms F(r) of weight 1 on '9(N) whose Mellin transform is the L-function L(s,x) = }5a(n)n-* with  ... 

Chern-Simons theory, analytic continuation and arithmetic [article]

Stavros Garoufalidis
2008 arXiv   pre-print
By the latter term, we mean analytic continuation as a multivalued analytic function in the complex numbers minus a discrete set of points, with restricted singularities, local and global monodromy.  ...  More precisely, we package the perturbative and nonperturbative invariants of knots and 3-manifolds into two power series of type P and NP, convergent in a neighborhood of zero, and we postulate their  ...  n, colored by the n-dimensional irreducible representation of sl 2 , and normalized by J unknot,n (q) = 1.  ... 
arXiv:0711.1716v3 fatcat:tp2ktkpvhja2rixbu4epy3xeea

On Frobenius semisimplicity in Hida families [article]

Jyoti Prakash Saha
2018 arXiv   pre-print
If k_f denotes the weight of a cusp form f(z)= ∑_n≥ 1 a_ℓ(f) e^2π i n z, then the inequality |a_ℓ(f) | ≤ 2 ℓ^(k_f-1)/2, predicted by the Ramanujan conjecture, is a strict inequality for almost all members  ...  We prove that the actions of the Frobenius element at ℓ on the Galois representations attached to almost all arithmetic specializations are semisimple and non-scalar.  ...  of Prof.  ... 
arXiv:1801.03070v2 fatcat:il5v2jwhcjer3oli4stp4ukwnq

Approaching Arithmetic Theories with Finite-State Automata [chapter]

Christoph Haase
2020 Lecture Notes in Computer Science  
In this article, which accompanies an invited talk, we give a high-level exposition of the NP upper bound for existential Büchi arithmetic, obtain some derived results, and further discuss some open problems  ...  This approach has recently been instrumental for settling long-standing open problems about the complexity of deciding the existential fragments of Büchi arithmetic and linear arithmetic over p-adic fields  ...  Hence we can guess in NP a clause of the disjunctive normal form of Φ, which is of the form (5) , and in turn check in NP a series of guessed reachability queries in p-automata induced by the guessed  ... 
doi:10.1007/978-3-030-40608-0_3 fatcat:y4yvocbpvzdcba4jlsttgv2nwq

Page 350 of Mathematical Reviews Vol. 24, Issue 4A [page]

1962 Mathematical Reviews  
The author obtains asymptotic formulae for the number of representations of a large number by a quadratic form (definite) in 5 or more variables when the variables lie in arithmetic progressions with the  ...  Mat. 19 (1955), 361-380; MR 17, 590] constructed a set of real numbers « such that, for large P, Np(M) = P|M| + O(4/ P), where M is a subinterval of [0, 1] of length || and Np(M) is the number of fractional  ... 

Data exchange with arithmetic operations

Balder ten Cate, Phokion G. Kolaitis, Walied Othman
2013 Proceedings of the 16th International Conference on Extending Database Technology - EDBT '13  
In the case of arithmetic schema mappings with a weakly acyclic set of target dependencies, a universal solution may not exist, but a finite universal basis exists (if a solution exists) and can be computed  ...  We obtain analogous complexity results for the data complexity of computing the certain answers of arithmetic conjunctive queries and linear arithmetic conjunctive queries.  ...  We thank Nimrod Megiddo and Marcus Schaefer for helpful discussions and pointers. References  ... 
doi:10.1145/2452376.2452439 dblp:conf/edbt/CateKO13 fatcat:rt5ah3lhrbfylonbwhjcupjsfm

Transcendence of Hecke operators in the big Hecke algebra

Haruzo Hida
2014 Duke mathematical journal  
Let S k (Np r , ; C) denote the space of (parallel) weight k adelic Hilbert cusp forms of level Np r with Neben character modulo Np r .  ...  We work with cusp forms in S k (Np r P, ; C) for increasing p-power level (but fixed prime-to-p part N).  ...  Valuations v with v(l) = 1 are said "normalized". Let V be the set of normalized valuations v of K, which is a finite set.  ... 
doi:10.1215/00127094-2690478 fatcat:pbyuvhc33bhubddfvxeloqmzim

Decidable Membership Problems for Finite Recurrent Systems over Sets of Naturals [chapter]

Daniel Meister
2005 Lecture Notes in Computer Science  
A finite recurrent system over the power set of the natural numbers of dimension n is a pair composed of n n-ary functions over the power set of the natural numbers and an n-tuple of singleton sets of  ...  We will show completeness results for the complexity classes NL, NP and PSPACE. M. Liśkiewicz and R. Reischuk (Eds.): FCT 2005, LNCS 3623, pp. 80-91, 2005.  ...  The idea to study finite recurrent systems was the result of a discussion with Klaus Wagner when a lot of people at Würzburg were studying membership problems for arithmetic circuits.  ... 
doi:10.1007/11537311_8 fatcat:5wdqt35wrzer5bzkstst4ska4m

Derandomizing Polynomial Identity over Finite Fields Implies Super-Polynomial Circuit Lower Bounds for NEXP [article]

Bin Fu
2013 arXiv   pre-print
In other words, for any finite field F(q) of size q, PIT_q∈ NSUBEXP NEXP⊆ P/poly, where PIT_q is the polynomial identity testing problem over F(q), and NSUBEXP is the nondeterministic subexpoential time  ...  class of languages.  ...  Acknowledegments: The author is grateful to Bohan Fan, Cynthia Fu, and Feng Li for their proofreading and suggestions for an earlier version of this paper.  ... 
arXiv:1311.2358v1 fatcat:i7iqcjn2v5bnjkmhxrcncogpsq
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