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Diophantine Definability and Decidability in Large Subrings of Totally Real Number Fields and Their Totally Complex Extensions of Degree 2

A Shlapentokh
2002 Journal of Number Theory  
Let O M;W ¼ fx 2 M j ord p x50 8peW g: The author continues her investigation of Diophantine definability and decidability in rings O M;W where W is infinite.  ...  In particular, the following results are proved: (1) Let M be a totally real field or a totally complex extension of degree 2 of a totally real field.  ...  INTRODUCTION The interest in the questions of Diophantine definability and decidability goes back to a question which was posed by Hilbert: given an arbitrary polynomial equation in several variables over  ... 
doi:10.1016/s0022-314x(01)92759-3 fatcat:nfau4nyfura25j7tsftc3xgeaa

Diophantine Definability and Decidability in Large Subrings of Totally Real Number Fields and Their Totally Complex Extensions of Degree 2

Alexandra Shlapentokh
2002 Journal of Number Theory  
Let O M;W ¼ fx 2 M j ord p x50 8peW g: The author continues her investigation of Diophantine definability and decidability in rings O M;W where W is infinite.  ...  In particular, the following results are proved: (1) Let M be a totally real field or a totally complex extension of degree 2 of a totally real field.  ...  INTRODUCTION The interest in the questions of Diophantine definability and decidability goes back to a question which was posed by Hilbert: given an arbitrary polynomial equation in several variables over  ... 
doi:10.1006/jnth.2001.2759 fatcat:zexf47cwojfdxgcqtyqcis6s6y

Uniform Diagonalization Theorem for Complexity Classes of Promise Problems including Randomized and Quantum Classes [article]

Friederike Anna Dziemba
2019 arXiv   pre-print
we will introduce as "total decidability" of promise problems.  ...  The theorem requires from the underlying computing model not only the decidability of its acceptance and rejection behaviour but also of its promise-contradicting indifferent behaviour - a property that  ...  Hence, the gap language G[r] is well-defined. And since it is decidable, B is totally decidable.  ... 
arXiv:1712.07276v3 fatcat:oizuqnerfbesri63uhvcbaymui

Exotic Quantifiers, Complexity Classes, and Complete Problems

Peter Bürgisser, Felipe Cucker
2007 Foundations of Computational Mathematics  
We define new complexity classes in the Blum-Shub-Smale theory of computation over the reals, in the spirit of the polynomial hierarchy, with the help of infinitesimal and generic quantifiers.  ...  All attempts to classify the complexity of these problems in terms of the previously studied complexity classes have failed.  ...  We formally define EDense R as follows: EDense R (Euclidean Denseness) Given a decision circuit C with n input gates, decide whether S C = R n .  ... 
doi:10.1007/s10208-007-9006-9 fatcat:as2mnecmdzeo7ncj5qq2eij74a

Pseudo-natural algorithms for finitely generated presentations of monoids and groups

Klaus Madlener, Friedrich Otto
1988 Journal of symbolic computation  
-decidable for some n ~> 1.  ...  On the other hand, the strong derivational complexity induces an upper bound for the intrinsic complexity of the word problem provided that the set of defining relators is easily decidable (Lemma 3.2).  ...  in the Grzegorczyk class E,, and (ii) the set L of defining relators is easily decidable?  ... 
doi:10.1016/s0747-7171(88)80034-8 fatcat:ujwmpjagzfeqnpqclh75iaypjm

A Rice-like theorem for primitive recursive functions [article]

Mathieu Hoyrup
2015 arXiv   pre-print
We provide an explicit characterization of the properties of primitive recursive functions that are decidable or semi-decidable, given a primitive recursive index for the function.  ...  In order to identify the properties that are decidable or semi-decidable, given a C -index of the input function, we introduce a notion of Kolmogorov complexity adapted to the class C .  ...  Here we take the simplest definition of complexity following directly from the enumeration of C , to avoid technicality. First we give a class of decidable properties.  ... 
arXiv:1503.05025v1 fatcat:2qzzyjbpyrhghk5bpi3frebkta

The Legacy of Turing in Numerical Analysis [chapter]

Felipe Cucker
2012 Lecture Notes in Computer Science  
Using these operators we may define many new complexity classes. Notations such as ∃ * ∀, H∀, or ∃ * H denote some of the newly created complexity classes in an obvious manner.  ...  Also, the class coNP R is defined to be ∀P R and we will denote it by ∀. Definition Let C be a complexity class of decision problems.  ... 
doi:10.1007/978-3-642-27660-6_1 fatcat:khf4ruc435bx5ib2rqijpanlfi

The word problem for finitely presented monoids and finite canonical rewriting systems [chapter]

Craig Squier, Friedrich Otto
1987 Lecture Notes in Computer Science  
groups) with decidable word problem that cannot be presented by finite canonical rewriting systems.  ...  The main purpose of this paper is to describe a negative answer to the following question: Does every finitely presented monoid with a decidable word problem have a presentation (~;R) where R is a finite  ...  Although the word problem is defined using a particular presentation, its decidability and intrinsic complexity are independent of the actually chosen presentation as long as this presentation is finitely  ... 
doi:10.1007/3-540-17220-3_7 fatcat:7chzcveea5a5feprziyku25vyu

A New Look at Some Classical Results in Computational Complexity [article]

Arnaldo Moura, Igor Carboni Oliveira
2009 Electronic colloquium on computational complexity  
We propose a generalization of the traditional algorithmic space and time complexities.  ...  This opens the possibility for the unification and generalization of other results that apply to both the time and space complexities.  ...  Next, we define a general complexity measure based on the usual time and space complexities, which we call f -complexity. Definition 3.  ... 
dblp:journals/eccc/MouraO09 fatcat:yjopypquerfl7ke5gvmbdjvs2a

The Decidable Properties of Subrecursive Functions

Mathieu Hoyrup, Marc Herbstritt
2016 International Colloquium on Automata, Languages and Programming  
This characterization uses a variant of Kolmogorov complexity where only programs in a subrecursive programming language are allowed.  ...  More precisely, we prove that all the decidable and semidecidable properties can be obtained as combinations of two classes of basic decidable properties: (i) the function takes some particular values  ...  We borrow the terminology from [3] , where the notion of anti-complex set is defined in terms of usual Kolmogorov complexity, and is studied from a computability-theoretic perspective.  ... 
doi:10.4230/lipics.icalp.2016.108 dblp:conf/icalp/Hoyrup16 fatcat:k2xljz6l6zhptdslbue7nfsc7a

Constitution of a Plant Complex with Heat Recovery

Tetsuo FUCHINO, Masaaki MURAKI
1997 Journal of the Japan Petroleum Institute  
Deciding the constitution of a complex is defined by optimizing the relative production capacities from their characteristics.  ...  In early planning phase of a plant complex, its constitution (combination of plants and their production capacities) is decided.  ...  Thus, the material balance of these products among the plants is one of the key factors in deciding the constitution of a complex.  ... 
doi:10.1627/jpi1958.40.474 fatcat:rk4eo4fuyng33pjkierq64xkbm

A note on parallel and alternating time

Felipe Cucker, Irénée Briquel
2007 Journal of Complexity  
In this note we consider some complexity classes defined through alternation of mixed digital and unrestricted quantifiers in different patterns.  ...  We show that the class of sets decided in parallel polynomial time is sandwiched between two such classes for different patterns.  ...  Given a complexity class C (defined in terms of a class of resource-bounded machines) and a set A as the above one denotes by C A the class of sets decidable by machines in C which query the oracle A.  ... 
doi:10.1016/j.jco.2007.02.005 fatcat:ik3fywqtrbfo7kpqygzxgpwcnm

Locality and Checkability in Wait-Free Computing [chapter]

Pierre Fraigniaud, Sergio Rajsbaum, Corentin Travers
2011 Lecture Notes in Computer Science  
tasks T = (I, O, ∆) for which there exists a wait-free distributed algorithm enabling, given a pair (s, t), s ∈ I, t ∈ O, to check whether t ∈ ∆(s), i.e., to decide whether t is a valid output for s.  ...  First, we define a task to be projection-closed if π(∆(s)) ⊆ ∆(π(s)), and prove the perhaps surprising fact that projection-closed tasks are precisely those tasks that are wait-free checkable, that is  ...  S n consists of all simplexes where processes decide values in {0, 1}. Now, to define ∆ c , let S J be the sub-complex of S n induced by the processes in J, for J ⊆ [n].  ... 
doi:10.1007/978-3-642-24100-0_34 fatcat:jkcgmmiefzfbfdw5rxd27yj6zm

Curiouser and Curiouser: The Link between Incompressibility and Complexity [chapter]

Eric Allender
2012 Lecture Notes in Computer Science  
R is not a decidable set, and thus it is absurd to suggest that the class of problems reducible to it constitutes a complexity class.  ...  The absurdity fades if, for example, we interpret "NP R " to be "the class of problems that are NP-Turing reducible to R, no matter which universal machine we use in defining Kolmogorov complexity".  ...  Complexity theory is supposed to deal with decidable sets (and preferably with sets that are very decidable -primitive recursive at least, and ideally much lower in the complexity hierarchy than that).  ... 
doi:10.1007/978-3-642-30870-3_2 fatcat:ofl5kpdavrfqxfar5gkla6erdi

Universal equivalence and majority of probabilistic programs over finite fields

Gilles Barthe, Charlie Jacomme, Steve Kremer
2020 Proceedings of the 35th Annual ACM/IEEE Symposium on Logic in Computer Science  
While the first problem is obviously decidable, we establish its exact complexity which lies in the counting hierarchy.  ...  We study decidability problems for equivalence of probabilistic programs, for a core probabilistic programming language over finite fields of fixed characteristic.  ...  We first study the complexity of deciding if the distributions of two programs are equal on a specific point.  ... 
doi:10.1145/3373718.3394746 dblp:conf/lics/BartheJK20 fatcat:l6pnwohxlfgxvamsflunlczqg4
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