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Uniform normal form for general time-bounded complexity classes

1986
*
Journal of computer and system sciences (Print)
*

Refining techniques

doi:10.1016/0022-0000(86)90034-6
fatcat:cnnhapneqvgddmrqvrdsaka4se
*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. ...##
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Tuples of Disjoint NP-Sets
[chapter]

2006
*
Lecture Notes in Computer Science
*

In this paper we introduce

doi:10.1007/11753728_11
fatcat:7nfv4x3tvbcddbwnioiwanf3mq
*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. ...##
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On the Complexity of Hilbert's 17th Problem
[chapter]

2004
*
Lecture Notes in Computer Science
*

We consider

doi:10.1007/978-3-540-30538-5_20
fatcat:ke4ckzcjrfae5f7266juxxaxli
*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] . ...##
###
A Characterisation of Definable NP Search Problems in Peano Arithmetic
[chapter]

2009
*
Lecture Notes in Computer Science
*

The complexity class

doi:10.1007/978-3-642-02261-6_1
fatcat:idsg2izjl5hjlprgjpxxm4xtbq
*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 ...##
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Tuples of Disjoint $\mathsf{NP}$ -Sets

2007
*
Theory of Computing Systems
*

In this paper we introduce

doi:10.1007/s00224-007-9023-8
fatcat:45azqnufdjc6xbywxv6nzmnozi
*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. ...##
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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]

2008
*
arXiv
*
pre-print

By the latter term, we mean analytic continuation as

arXiv:0711.1716v3
fatcat:tp2ktkpvhja2rixbu4epy3xeea
*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. ...##
###
On Frobenius semisimplicity in Hida families
[article]

2018
*
arXiv
*
pre-print

If k_f denotes the weight

arXiv:1801.03070v2
fatcat:il5v2jwhcjer3oli4stp4ukwnq
*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. ...##
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Approaching Arithmetic Theories with Finite-State Automata
[chapter]

2020
*
Lecture Notes in Computer Science
*

In this article, which accompanies an invited talk, we give

doi:10.1007/978-3-030-40608-0_3
fatcat:y4yvocbpvzdcba4jlsttgv2nwq
*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 ...##
###
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 ...##
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Data exchange with arithmetic operations

2013
*
Proceedings of the 16th International Conference on Extending Database Technology - EDBT '13
*

In the case

doi:10.1145/2452376.2452439
dblp:conf/edbt/CateKO13
fatcat:rt5ah3lhrbfylonbwhjcupjsfm
*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 ...##
###
Transcendence of Hecke operators in the big Hecke algebra

2014
*
Duke mathematical journal
*

Let S k (

doi:10.1215/00127094-2690478
fatcat:pbyuvhc33bhubddfvxeloqmzim
*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*. ...##
###
Decidable Membership Problems for Finite Recurrent Systems over Sets of Naturals
[chapter]

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. ...

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

2013
*
arXiv
*
pre-print

In other words,

arXiv:1311.2358v1
fatcat:i7iqcjn2v5bnjkmhxrcncogpsq
*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. ...
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