Every recursive linear ordering has a copy in DTIME-SPACE(n,log(n))

This paper is a contribution to the following natural problem in complexity theory:(*) Is there a complexity theory for isomorphism types of recursive countable relational structures? I.e. given a recursive relational structure ℛ over the set N of nonnegative integers, is there a nontrivial lower bound for the time-space complexity of recursive structures isomorphic (resp. recursively isomorphic) to ℛ?For unary recursive relations R, the answer is trivially negative: either R is finite or coinfinite or 〈N, R〉 is recursively isomorphic to 〈N, {x ϵ N: x is even}〉.The general problem for relations with arity 2 (or greater) is open.Related to this problem, a classical result (going back to S. C. Kleene [4], 1955) states that every recursive ordinal is in fact primitive recursive.In [3] Patrick Dehornoy, using methods relevant to computer science, improves this result, showing that every recursive ordinal can be represented by a recursive total ordering over N which has linear deterministic time complexity relative to the binary representation of integers. As he notices, his proof applies to every recursive total order type α such that the isomorphism type of α is not changed if points are replaced by arbitrary finite nonempty subsets of consecutive points.In this paper we extend Dehornoy's result to all recursive total orderings over N and get minimal complexity for both time and space simultaneously.

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On Binary Representation of Integers

PRIMUS ◽

10.1080/10511970.2018.1489923 ◽

2019 ◽

Vol 29 (5) ◽

pp. 474-486

Author(s):

Vardges Melkonian

Keyword(s):

Binary Representation ◽

Representation Of Integers

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On order-types of models

Journal of Symbolic Logic ◽

10.2307/2272546 ◽

1972 ◽

Vol 37 (1) ◽

pp. 69-70 ◽

Cited By ~ 2

Author(s):

Wilfrid Hodges

Keyword(s):

Order Type ◽

Linear Ordering ◽

Infinite Field ◽

Strong Limit ◽

First Order ◽

Order Types ◽

Order Language ◽

Language L ◽

New Feature

Let T be a theory in a first-order language L. Let L have a predicate ν0 ≺ ν1 such that in every model of T, the interpretation of ≺ is a linear ordering with infinite field. The order-type of this ordering will be called the order-type of the model .Several recent theorems have the following form: if T has a model of order-type ξ then T has a model of order-type ζ (see [1]). We shall add one to the list. The new feature of our result is that the order-type ζ may be in a sense “opposite” to ξ. Silver's Theorem 2.24 of [3] is a corollary of Theorem 1 below.Theorem 1. Let κ be a strong limit number (i.e. μ < κ implies 2μ < κ). Suppose λ < κ, and suppose that for every cardinal μ < κ, T has a model with where the order-type of contains no descending well-ordered sequences of length λ. Then for every cardinal μ ≥ the cardinality ∣L∣ of the language L, T has models and such that(a) the field of is the union of ≤ ∣L∣ well-ordered (inversely well-ordered) parts;(b) .The proof is by Ehrenfeucht-Mostowski models; we presuppose [2].

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An application of results by Hardy, Ramanujan and Karamata to Ackermannian functions

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.339 ◽

2003 ◽

Vol Vol. 6 no. 1 ◽

Author(s):

Andreas Weiermann

Keyword(s):

Fine Structure ◽

Structure Analysis ◽

Tauberian Theorem ◽

Classical Result ◽

Order Type ◽

Recursive Functions ◽

Fine Structure Analysis ◽

International Audience ◽

Primitive Recursive

International audience The Ackermann function is a fascinating and well studied paradigm for a function which eventually dominates all primitive recursive functions. By a classical result from the theory of recursive functions it is known that the Ackermann function can be defined by an unnested or descent recursion along the segment of ordinals below ω ^ω (or equivalently along the order type of the polynomials under eventual domination). In this article we give a fine structure analysis of such a Ackermann type descent recursion in the case that the ordinals below ω ^ω are represented via a Hardy Ramanujan style coding. This paper combines number-theoretic results by Hardy and Ramanujan, Karamata's celebrated Tauberian theorem and techniques from the theory of computability in a perhaps surprising way.

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ALGEBRAIC LINEAR ORDERINGS

International Journal of Foundations of Computer Science ◽

10.1142/s0129054111008155 ◽

2011 ◽

Vol 22 (02) ◽

pp. 491-515 ◽

Cited By ~ 10

Author(s):

S. L. BLOOM ◽

Z. ÉSIK

Keyword(s):

Order Type ◽

Linear Ordering ◽

Type Result ◽

First Order ◽

Linear Orderings ◽

Context Free Language ◽

Categorical Algebra ◽

Context Free ◽

Free Language

An algebraic linear ordering is a component of the initial solution of a first-order recursion scheme over the continuous categorical algebra of countable linear orderings equipped with the sum operation and the constant 1. Due to a general Mezei-Wright type result, algebraic linear orderings are exactly those isomorphic to the linear ordering of the leaves of an algebraic tree. Using Courcelle's characterization of algebraic trees, we obtain the fact that a linear ordering is algebraic if and only if it can be represented as the lexicographic ordering of a deterministic context-free language. When the algebraic linear ordering is a well-ordering, its order type is an algebraic ordinal. We prove that the Hausdorff rank of any scattered algebraic linear ordering is less than ωω. It follows that the algebraic ordinals are exactly those less than ωωω.

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Primitive Recursion with Existential Types1

Fundamenta Informaticae ◽

10.3233/fi-1993-191-209 ◽

1993 ◽

Vol 19 (1-2) ◽

pp. 201-222

Author(s):

Pawel Urzyczyn

Keyword(s):

Standard Approach ◽

Halting Problem ◽

Primitive Recursion ◽

Representation Of Integers ◽

Existential Quantification ◽

Primitive Recursive ◽

Recursive Definitions ◽

Recursive Set

We consider computability over abstract structures with help of primitive recursive definitions (an appropriate modification of Gödel’s system T). Unlike the standard approach, we do not assume any fixed representation of integers, but instead we allow primitive recursion to be polymorphic, so that iteration is performed with help of counters viewed as objects of an abstract type Int of arbitrary (hidden) implementation. This approach involves the use of existential quantification in types, following the ideas of Mitchell and Plotkin. We show that the halting problem over finite interpretations is primitive recursive for each program involving primitive recursive definitions. Conversely, each primitive recursive set of interpretations is defined by the termination property of some program.

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An introduction to well-ordering proofs in Martin-Löf’s type theory

Twenty Five Years of Constructive Type Theory ◽

10.1093/oso/9780198501275.003.0015 ◽

1998 ◽

Author(s):

Anton Setzer

Keyword(s):

Lower Bound ◽

Classical Theory ◽

Type Theory ◽

Direct Method ◽

Order Type ◽

Transfinite Induction ◽

Linear Orderings ◽

Heyting Arithmetic ◽

Primitive Recursive Arithmetic ◽

Primitive Recursive

The proof-theoretic strength α of a theory is the supremum of all ordinals up to which we can prove transfmite induction in that theory. Whereas for classical theories the main problem is to show that α is an upper bound for the strength—this usually means to reduce the theory to a weak theory like primitive recursive arithmetic or Heyting arithmetic extended by transfmite induction up to α, which can be considered to be more constructive than the classical theory itself—for constructive theories this is in most cases not difficult, since we can easily build a term model in a classical theory of known strength. For constructive theories in general the main problem is to show that α is a lower bound: that despite the restricted principles available one has a proof-theoretically strong theory. In this article we will concentrate on the direct method for showing that α is a lower bound, namely well-ordering proofs: to carry out in the theory a sequence of proofs of the well-foundedness of linear orderings of order type αn, such that supn∈ω αn = α. Such proofs can be considered to be the logically most complex proofs which one can carry out in the theory; in most cases, in addition to transfinite induction up to αn for each n, only primitive recursive arithmetic is needed in order to analyze the theory proof-theoretically and in order to prove the same Π02-sentences. Griffor and Rathjen (1994) have used the more indirect method of interpreting theories of known strength in type theory for obtaining lower bounds for the strength of it. Apart from the fact that in the case of one universe and W-type Griffor and Rathjens’ approach did not yield sharp bounds, we believe that the direct method has the advantage of giving a deeper insight into the theory, since one examines the principles of the theory directly without referring to the analysis of another theory, and that the programs obtained by it are of independent interest. In Setzer (1995) and Setzer (1996) we have carried out well-ordering proofs for Martin-Löf’s type theory with W-type and one universe and for the Mahlo universe.

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Countable homogeneous relational structures and ℵ0-categorical theories

Journal of Symbolic Logic ◽

10.2307/2272734 ◽

1972 ◽

Vol 37 (3) ◽

pp. 494-500 ◽

Cited By ~ 41

Author(s):

C. Ward Henson

Keyword(s):

Order Theory ◽

Linear Ordering ◽

Categorical Theory ◽

Symmetric Relation ◽

First Order ◽

Relational Structures ◽

First Order Theory ◽

Partial Orderings ◽

Homogeneous Structures ◽

Binary Relation Symbol

A relational structure of cardinality ℵ0 is called homogeneous by Fraissé [1] if each isomorphism between finite substructures of can be extended to an automorphism of . In §1 of this paper it is shown that there are isomorphism types of such structures for the first order language L0 with a single (binary) relation symbol, answering a question raised by Fraissé. In fact, as is shown in §2, a family of nonisomorphic homogeneous structures for L0 can be constructed, each member of which satisfies the following conditions (where U is the homogeneous, ℵ0-universal graph, the structure of which is considered in [4]):(i) The relation R of is asymmetric (R ∩ R−1 = ∅);(ii) If A is the domain of and S is the symmetric relation R ∪ R−1, then (A, S) is isomorphic to U. That is, each may be regarded as the result of assigning a unique direction to each edge of the graph U.Let T0 be the first order theory of all homogeneous structures for L0 which have cardinality ℵ0. In §3 (which can be read independently of §2) it is shown that T0 has complete extensions (in L0), each of which is ℵ0-categorical. Moreover, among the complete extensions of T0 are theories of arbitrary (preassigned) degree of unsolvability. In particular, there exists an undecidable, ℵ0-categorieal theory in L0, which answers a question raised by Grzegorczyk [2], [3].It follows from Theorem 6 of [3] that there are ℵ0-categorical theories of partial orderings which have arbitrarily high degrees of unsolvability. This is in sharp contrast to the situation for linear orderings, which were the motivation for Fraissé's early work. Indeed, as is shown in [10], every ℵ0-categorical theory of a linear ordering is finitely axiomatizable. (W. Glassmire [12] has independently shown the existence of theories in L0 which are all ℵ0-categorical, and C. Ash [13] has independently shown that such theories exist with arbitrary degree of unsolvability.)

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On Minimum Weight Binary Representation of Integers and Continued Fractions with Application to Computer Arithmetic

DAIMI Report Series ◽

10.7146/dpb.v10i130.7406 ◽

1981 ◽

Vol 10 (130) ◽

Author(s):

David W. Matula ◽

Peter Kornerup

Keyword(s):

Rational Number ◽

Continued Fractions ◽

Continued Fraction ◽

Minimum Weight ◽

Computer Arithmetic ◽

Binary Representation ◽

Representation Of Integers

We develop the concept of minimum weight binary continued fraction representation of a rational number as an extension of minimum weight binary radix representation of an integer. The relation of these representations to the attainment of optimum efficiency in the shift and add or subtract model of binary computer arithmetic is discussed.

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Indicators, Chains, Antichains, Ramsey Property

Canadian Mathematical Bulletin ◽

10.4153/cmb-2013-028-0 ◽

2014 ◽

Vol 57 (3) ◽

pp. 631-639

Author(s):

Miodrag Sokić

Keyword(s):

Partial Ordering ◽

Linear Ordering ◽

Ramsey Property ◽

Relational Structures ◽

Weak Ordering ◽

Finite Structures ◽

Maximal Chains

AbstractWe introduce two Ramsey classes of finite relational structures. The first class contains finite structures of the form , where ≤ is a total ordering on A and ⪯ is a linear ordering on the set fa 2 A : Ii (a)g. The second class contains structures of the form a , where (A,≤) is a weak ordering and ⪯ is a linear ordering on A such that A is partitioned by into maximal chains in the partial ordering ≤ and each is an interval with respect to .

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