Computably Extendible Order Types
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Order Types and Structure of Orders
ORDER TYPES AND STRUCTURE OF ORDERS BY ANDRE GLEYZALp) 1. Introduction. This paper is concerned with operations on order types or order properties a and the construction of order types related to a. The reference throughout is to simply or linearly ordered sets, and we shall speak of a as either property or type. Let a and ß be any two order types. An order A will be said to be of type aß if it is the sum of /3-orders (orders of type ß) over an a-order; i.e., if A permits of decomposition into nonoverlapping seg- ments each of order type ß, the segments themselves forming an order of type a. We have thus associated with every pair of order types a and ß the product order type aß. The definition of product for order types automatically associates with every order type a the order types aa = a2, aa2 = a3, ■ ■ ■ . We may further- more define, for all ordinals X, a Xth power of a, a\ and finally a limit order type a1. This order type has certain interesting properties. It has closure with respect to the product operation, for the sum of ar-orders over an a7-order is an a'-order, i.e., a'al = aI. For this reason we call a1 iterative. In general, we term an order type ß having the property that ßß = ß iterative, a1 has the following postulational identification: 1. a7 is a supertype of a; that is to say, all a-orders are a7-orders. 2. a1 is iterative. -
Handout from Today's Lecture
MA532 Lecture Timothy Kohl Boston University April 23, 2020 Timothy Kohl (Boston University) MA532 Lecture April 23, 2020 1 / 26 Cardinal Arithmetic Recall that one may define addition and multiplication of ordinals α = ot(A, A) β = ot(B, B ) α + β and α · β by constructing order relations on A ∪ B and B × A. For cardinal numbers the foundations are somewhat similar, but also somewhat simpler since one need not refer to orderings. Definition For sets A, B where |A| = α and |B| = β then α + β = |(A × {0}) ∪ (B × {1})|. Timothy Kohl (Boston University) MA532 Lecture April 23, 2020 2 / 26 The curious part of the definition is the two sets A × {0} and B × {1} which can be viewed as subsets of the direct product (A ∪ B) × {0, 1} which basically allows us to add |A| and |B|, in particular since, in the usual formula for the size of the union of two sets |A ∪ B| = |A| + |B| − |A ∩ B| which in this case is bypassed since, by construction, (A × {0}) ∩ (B × {1})= ∅ regardless of the nature of A ∩ B. Timothy Kohl (Boston University) MA532 Lecture April 23, 2020 3 / 26 Definition For sets A, B where |A| = α and |B| = β then α · β = |A × B|. One immediate consequence of these definitions is the following. Proposition If m, n are finite ordinals, then as cardinals one has |m| + |n| = |m + n|, (where the addition on the right is ordinal addition in ω) meaning that ordinal addition and cardinal addition agree. Proof. The simplest proof of this is to define a bijection f : (m × {0}) ∪ (n × {1}) → m + n by f (hr, 0i)= r for r ∈ m and f (hs, 1i)= m + s for s ∈ n. -
Intrinsic Bounds on Complexity and Definability at Limit Levels*
Intrinsic bounds on complexity and definability at limit levels∗ John Chisholm Ekaterina Fokina Department of Mathematics Department of Mathematics Western Illinois University Novosibirsk State University Macomb, IL 61455 630090 Novosibirsk, Russia [email protected] [email protected] Sergey S. Goncharov Academy of Sciences, Siberian Branch Mathematical Institute 630090 Novosibirsk, Russia [email protected] Valentina S. Harizanov Julia F. Knight Department of Mathematics Department of Mathematics George Washington University University of Notre Dame [email protected] [email protected] Sara Miller Department of Mathematics University of Notre Dame [email protected] May 31, 2007 Abstract We show that for every computable limit ordinal α,thereisacom- 0 0 putable structure that is ∆α categorical, but not relatively ∆α categor- A 0 ical, i.e., does not have a formally Σα Scott family. We also show that for every computable limit ordinal α, there is a computable structure with 0 A an additional relation R that is intrinsically Σα on , but not relatively 0 A intrinsically Σ on , i.e., not definable by a computable Σα formula with α A ∗The authors gratefully acknowledge support of the Charles H. Husking endowment at the University of Notre Dame. The last five authors were also partially supported by the NSF binational grant DMS-0554841, and the fourth author by the NSF grant DMS-0704256. In addition, the second author was supported by a grant of the Russian Federation as a 2006-07 research visitor to the University of Notre Dame. 1 finitely many parameters. Earlier results in [7], [10], [8] establish the same facts for computable successor ordinals α. -
Arxiv:2010.12452V1 [Math.LO]
ORDINAL ANALYSIS OF PARTIAL COMBINATORY ALGEBRAS PAUL SHAFER AND SEBASTIAAN A. TERWIJN Abstract. For every partial combinatory algebra (pca), we define a hierarchy of extensionality rela- tions using ordinals. We investigate the closure ordinals of pca’s, i.e. the smallest ordinals where these CK relations become equal. We show that the closure ordinal of Kleene’s first model is ω1 and that the closure ordinal of Kleene’s second model is ω1. We calculate the exact complexities of the extensionality relations in Kleene’s first model, showing that they exhaust the hyperarithmetical hierarchy. We also discuss embeddings of pca’s. 1. Introduction Partial combinatory algebras (pca’s) were introduced by Feferman [9] in connection with the study of predicative systems of mathematics. Since then, they have been studied as abstract models of computation, in the same spirit as combinatory algebras (defined and studied long before, in the 1920’s, by Sch¨onfinkel and Curry) and the closely related lambda calculus, cf. Barendregt [2]. As such, pca’s figure prominently in the literature on constructive mathematics, see e.g. Beeson [5] and Troelstra and van Dalen [22]. This holds in particular for the theory of realizability, see van Oosten [24]. For example, pca’s serve as the basis of various models of constructive set theory, first defined by McCarty [14]. See Rathjen [18] for further developments using this construction. A pca is a set A equipped with a partial application operator · that has the same properties as a classical (total) combinatory algebra (see Section 2 below for precise definitions). In particular, it has the combinators K and S. -
2/3 Conjecture
PROGRESS ON THE 1=3 − 2=3 CONJECTURE By Emily Jean Olson A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of Mathematics | Doctor of Philosophy 2017 ABSTRACT PROGRESS ON THE 1=3 − 2=3 CONJECTURE By Emily Jean Olson Let (P; ≤) be a finite partially ordered set, also called a poset, and let n denote the cardinality of P . Fix a natural labeling on P so that the elements of P correspond to [n] = f1; 2; : : : ; ng. A linear extension is an order-preserving total order x1 ≺ x2 ≺ · · · ≺ xn on the elements of P , and more compactly, we can view this as the permutation x1x2 ··· xn in one-line notation. For distinct elements x; y 2 P , we define P(x ≺ y) to be the proportion 1 of linear extensions of P in which x comes before y. For 0 ≤ α ≤ 2, we say (x; y) is an α-balanced pair if α ≤ P(x ≺ y) ≤ 1 − α: The 1=3 − 2=3 Conjecture states that every finite partially ordered set that is not a chain has a 1=3-balanced pair. This dissertation focuses on showing the conjecture is true for certain types of partially ordered sets. We begin by discussing a special case, namely when a partial order is 1=2-balanced. For example, this happens when the poset has an automorphism with a cycle of length 2. We spend the remainder of the text proving the conjecture is true for some lattices, including Boolean, set partition, and subspace lattices; partial orders that arise from a Young diagram; and some partial orders of dimension 2. -
Computable Structure Theory on Ω 1 Using Admissibility
COMPUTABLE STRUCTURE THEORY USING ADMISSIBLE RECURSION THEORY ON !1 NOAM GREENBERG AND JULIA F. KNIGHT Abstract. We use the theory of recursion on admissible ordinals to develop an analogue of classical computable model theory and effective algebra for structures of size @1, which, under our assumptions, is equal to the contin- uum. We discuss both general concepts, such as computable categoricity, and particular classes of examples, such as fields and linear orderings. 1. Introduction Our aim is to develop computable structure theory for uncountable structures. In this paper we focus on structures of size @1. The fundamental decision to be made, when trying to formulate such a theory, is the choice of computability tools that we intend to use. To discover which structures are computable, we need to first describe which subsets of the domain are computable, and which functions are computable. In this paper, we use admissible recursion theory (also known as α-recursion theory) over the domain !1. We believe that this choice yields an interesting computable structure theory. It also illuminates the concepts and techniques of classical computable structure theory by observing similarities and differences between the countable and uncountable settings. In particular, it seems that as is the case for degree theory and for the study of the lattice of c.e. sets, the difference between true finiteness and its analogue in the generalised case, namely countability in our case, is fundamental to some constructions and reveals a deep gap between classical computability and attempts to generalise it to the realm of the uncountable. In this paper, we make a sweeping simplifying assumption about our set-theoretic universe. -
Types for Describing Coordinated Data Structures
Types for Describing Coordinated Data Structures Michael F. Ringenburg∗ Dan Grossman [email protected] [email protected] Dept. of Computer Science & Engineering University of Washington, Seattle, WA 98195 ABSTRACT the n-th function). This section motivates why such invari- Coordinated data structures are sets of (perhaps unbounded) ants are important, explores why the scope of type variables data structures where the nodes of each structure may share makes the problem appear daunting, and previews the rest abstract types with the corresponding nodes of the other of the paper. structures. For example, consider a list of arguments, and 1.1 Low-Level Type Systems a separate list of functions, where the n-th function of the Recent years have witnessed substantial work on powerful second list should be applied only to the n-th argument of type systems for safe, low-level languages. Standard moti- the first list. We can guarantee that this invariant is obeyed vation for such systems includes compiler debugging (gener- by coordinating the two lists, such that the type of the n-th ated code that does not type check implies a compiler error), argument is existentially quantified and identical to the ar- proof-carrying code (the type system encodes a safety prop- gument type of the n-th function. In this paper, we describe erty that the type-checker verifies), automated optimization a minimal set of features sufficient for a type system to sup- (an optimizer can exploit the type information), and manual port coordinated data structures. We also demonstrate that optimization (humans can use idioms unavailable in higher- two known type systems (Crary and Weirich’s LX [6] and level languages without sacrificing safety). -
Counting Linear Extensions in Practice: MCMC Versus Exponential Monte Carlo
The Thirty-Second AAAI Conference on Artificial Intelligence (AAAI-18) Counting Linear Extensions in Practice: MCMC versus Exponential Monte Carlo Topi Talvitie Kustaa Kangas Teppo Niinimaki¨ Mikko Koivisto Dept. of Computer Science Dept. of Computer Science Dept. of Computer Science Dept. of Computer Science University of Helsinki Aalto University Aalto University University of Helsinki topi.talvitie@helsinki.fi juho-kustaa.kangas@aalto.fi teppo.niinimaki@aalto.fi mikko.koivisto@helsinki.fi Abstract diagram (Kangas et al. 2016), or the treewidth of the incom- parability graph (Eiben et al. 2016). Counting the linear extensions of a given partial order is a If an approximation with given probability is sufficient, #P-complete problem that arises in numerous applications. For polynomial-time approximation, several Markov chain the problem admits a polynomial-time solution for all posets Monte Carlo schemes have been proposed; however, little is using Markov chain Monte Carlo (MCMC) methods. The known of their efficiency in practice. This work presents an first fully-polynomial time approximation scheme of this empirical evaluation of the state-of-the-art schemes and in- kind for was based on algorithms for approximating the vol- vestigates a number of ideas to enhance their performance. In umes of convex bodies (Dyer, Frieze, and Kannan 1991). addition, we introduce a novel approximation scheme, adap- The later improvements were based on rapidly mixing tive relaxation Monte Carlo (ARMC), that leverages exact Markov chains in the set of linear extensions (Karzanov exponential-time counting algorithms. We show that approx- and Khachiyan 1991; Bubley and Dyer 1999) combined imate counting is feasible up to a few hundred elements on with Monte Carlo counting schemes (Brightwell and Win- various classes of partial orders, and within this range ARMC kler 1991; Banks et al. -
Counting Linear Extensions of Restricted Posets
COUNTING LINEAR EXTENSIONS OF RESTRICTED POSETS SAMUEL DITTMER? AND IGOR PAK? Abstract. The classical 1991 result by Brightwell and Winkler [BW91] states that the number of linear extensions of a poset is #P-complete. We extend this result to posets with certain restrictions. First, we prove that the number of linear extension for posets of height two is #P-complete. Furthermore, we prove that this holds for incidence posets of graphs. Finally, we prove that the number of linear extensions for posets of dimension two is #P-complete. 1. Introduction Counting linear extensions (#LE) of a finite poset is a fundamental problem in both Combinatorics and Computer Science, with connections and applications ranging from Sta- tistics to Optimization, to Social Choice Theory. It is primarily motivated by the following basic question: given a partial information of preferences between various objects, what are the chances of other comparisons? In 1991, Brightwell and Winkler showed that #LE is #P-complete [BW91], but for var- ious restricted classes of posets the problem remains unresolved. Notably, they conjectured that the following problem is #P-complete: #H2LE (Number of linear extensions of height-2 posets) Input: A partially ordered set P of height 2. Output: The number e(P ) of linear extensions. Here height two means that P has two levels, i.e. no chains of length 3. This problem has been open for 27 years, most recently reiterated in [Hub14, LS17]. Its solution is the first result in this paper. Theorem 1.1. #H2LE is #P-complete. Our second result is an extension of Theorem 1.1. -
Dimensional Functions Over Partially Ordered Sets
Intro Dimensional functions over posets Application I Application II Dimensional functions over partially ordered sets V.N.Remeslennikov, E. Frenkel May 30, 2013 1 / 38 Intro Dimensional functions over posets Application I Application II Plan The notion of a dimensional function over a partially ordered set was introduced by V. N. Remeslennikov in 2012. Outline of the talk: Part I. Definition and fundamental results on dimensional functions, (based on the paper of V. N. Remeslennikov and A. N. Rybalov “Dimensional functions over posets”); Part II. 1st application: Definition of dimension for arbitrary algebraic systems; Part III. 2nd application: Definition of dimension for regular subsets of free groups (L. Frenkel and V. N. Remeslennikov “Dimensional functions for regular subsets of free groups”, work in progress). 2 / 38 Intro Dimensional functions over posets Application I Application II Partially ordered sets Definition A partial order is a binary relation ≤ over a set M such that ∀a ∈ M a ≤ a (reflexivity); ∀a, b ∈ M a ≤ b and b ≤ a implies a = b (antisymmetry); ∀a, b, c ∈ M a ≤ b and b ≤ c implies a ≤ c (transitivity). Definition A set M with a partial order is called a partially ordered set (poset). 3 / 38 Intro Dimensional functions over posets Application I Application II Linearly ordered abelian groups Definition A set A equipped with addition + and a linear order ≤ is called linearly ordered abelian group if 1. A, + is an abelian group; 2. A, ≤ is a linearly ordered set; 3. ∀a, b, c ∈ A a ≤ b implies a + c ≤ b + c. Definition The semigroup A+ of all nonnegative elements of A is defined by A+ = {a ∈ A | 0 ≤ a}. -
Generic Haskell: Applications
Generic Haskell: applications Ralf Hinze1 and Johan Jeuring2,3 1 Institut f¨urInformatik III, Universit¨atBonn R¨omerstraße 164, 53117 Bonn, Germany [email protected] http://www.informatik.uni-bonn.de/~ralf/ 2 Institute of Information and Computing Sciences, Utrecht University P.O.Box 80.089, 3508 TB Utrecht, The Netherlands [email protected] http://www.cs.uu.nl/~johanj/ 3 Open University, Heerlen, The Netherlands Abstract. 1 Generic Haskell is an extension of Haskell that supports the construc- tion of generic programs. This article describes generic programming in practice. It discusses three advanced generic programming applications: generic dictionaries, compressing XML documents, and the zipper. When describing and implementing these examples, we will encounter some advanced features of Generic Haskell, such as type-indexed data types, dependencies between and generic abstractions of generic functions, ad- justing a generic function using a default case, and generic functions with a special case for a particular constructor. 1 Introduction A polytypic (or generic, type-indexed) function is a function that can be instan- tiated on many data types to obtain data type specific functionality. Examples of polytypic functions are the functions that can be derived in Haskell [50], such as show, read, and ‘ ’. In [23] we have introduced type-indexed functions, and we have shown how to implement them in Generic Haskell [7]. For an older introduction to generic programming, see Backhouse et al [4]. Why is generic programming important? Generic programming makes pro- grams easier to write: – Programs that could only be written in an untyped style can now be written in a language with types. -
Effective Descriptive Set Theory
Effective Descriptive Set Theory Andrew Marks December 14, 2019 1 1 These notes introduce the effective (lightface) Borel, Σ1 and Π1 sets. This study uses ideas and tools from descriptive set theory and computability theory. Our central motivation is in applications of the effective theory to theorems of classical (boldface) descriptive set theory, especially techniques which have no classical analogues. These notes have many errors and are very incomplete. Some important topics not covered include: • The Harrington-Shore-Slaman theorem [HSS] which implies many of the theorems of Section 3. • Steel forcing (see [BD, N, Mo, St78]) • Nonstandard model arguments • Barwise compactness, Jensen's model existence theorem • α-recursion theory • Recent beautiful work of the \French School": Debs, Saint-Raymond, Lecompte, Louveau, etc. These notes are from a class I taught in spring 2019. Thanks to Adam Day, Thomas Gilton, Kirill Gura, Alexander Kastner, Alexander Kechris, Derek Levinson, Antonio Montalb´an,Dean Menezes and Riley Thornton, for helpful conversations and comments on earlier versions of these notes. 1 Contents 1 1 1 1 Characterizing Σ1, ∆1, and Π1 sets 4 1 1.1 Σn formulas, closure properties, and universal sets . .4 1.2 Boldface vs lightface sets and relativization . .5 1 1.3 Normal forms for Σ1 formulas . .5 1.4 Ranking trees and Spector boundedness . .7 1 1.5 ∆1 = effectively Borel . .9 1.6 Computable ordinals, hyperarithmetic sets . 11 1 1.7 ∆1 = hyperarithmetic . 14 x 1 1.8 The hyperjump, !1 , and the analogy between c.e. and Π1 .... 15 2 Basic tools 18 2.1 Existence proofs via completeness results .