Logic and Degrees
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
A Hierarchy of Computably Enumerable Degrees, Ii: Some Recent Developments and New Directions
A HIERARCHY OF COMPUTABLY ENUMERABLE DEGREES, II: SOME RECENT DEVELOPMENTS AND NEW DIRECTIONS ROD DOWNEY, NOAM GREENBERG, AND ELLEN HAMMATT Abstract. A transfinite hierarchy of Turing degrees of c.e. sets has been used to calibrate the dynamics of families of constructions in computability theory, and yields natural definability results. We review the main results of the area, and discuss splittings of c.e. degrees, and finding maximal degrees in upper cones. 1. Vaughan Jones This paper is part of for the Vaughan Jones memorial issue of the New Zealand Journal of Mathematics. Vaughan had a massive influence on World and New Zealand mathematics, both through his work and his personal commitment. Downey first met Vaughan in the early 1990's, around the time he won the Fields Medal. Vaughan had the vision of improving mathematics in New Zealand, and hoped his Fields Medal could be leveraged to this effect. It is fair to say that at the time, maths in New Zealand was not as well-connected internationally as it might have been. It was fortunate that not only was there a lot of press coverage, at the same time another visionary was operating in New Zealand: Sir Ian Axford. Ian pioneered the Marsden Fund, and a group of mathematicians gathered by Vaughan, Marston Conder, David Gauld, Gaven Martin, and Rod Downey, put forth a proposal for summer meetings to the Marsden Fund. The huge influence of Vaughan saw to it that this was approved, and this group set about bringing in external experts to enrich the NZ scene. -
Computability and Incompleteness Fact Sheets
Computability and Incompleteness Fact Sheets Computability Definition. A Turing machine is given by: A finite set of symbols, s1; : : : ; sm (including a \blank" symbol) • A finite set of states, q1; : : : ; qn (including a special \start" state) • A finite set of instructions, each of the form • If in state qi scanning symbol sj, perform act A and go to state qk where A is either \move right," \move left," or \write symbol sl." The notion of a \computation" of a Turing machine can be described in terms of the data above. From now on, when I write \let f be a function from strings to strings," I mean that there is a finite set of symbols Σ such that f is a function from strings of symbols in Σ to strings of symbols in Σ. I will also adopt the analogous convention for sets. Definition. Let f be a function from strings to strings. Then f is computable (or recursive) if there is a Turing machine M that works as follows: when M is started with its input head at the beginning of the string x (on an otherwise blank tape), it eventually halts with its head at the beginning of the string f(x). Definition. Let S be a set of strings. Then S is computable (or decidable, or recursive) if there is a Turing machine M that works as follows: when M is started with its input head at the beginning of the string x, then if x is in S, then M eventually halts, with its head on a special \yes" • symbol; and if x is not in S, then M eventually halts, with its head on a special • \no" symbol. -
Weakly Precomplete Computably Enumerable Equivalence Relations
Math. Log. Quart. 62, No. 1–2, 111–127 (2016) / DOI 10.1002/malq.201500057 Weakly precomplete computably enumerable equivalence relations Serikzhan Badaev1 ∗ and Andrea Sorbi2∗∗ 1 Department of Mechanics and Mathematics, Al-Farabi Kazakh National University, Almaty, 050038, Kazakhstan 2 Dipartimento di Ingegneria dell’Informazione e Scienze Matematiche, Universita` Degli Studi di Siena, 53100, Siena, Italy Received 28 July 2015, accepted 4 November 2015 Published online 17 February 2016 Using computable reducibility ≤ on equivalence relations, we investigate weakly precomplete ceers (a “ceer” is a computably enumerable equivalence relation on the natural numbers), and we compare their class with the more restricted class of precomplete ceers. We show that there are infinitely many isomorphism types of universal (in fact uniformly finitely precomplete) weakly precomplete ceers , that are not precomplete; and there are infinitely many isomorphism types of non-universal weakly precomplete ceers. Whereas the Visser space of a precomplete ceer always contains an infinite effectively discrete subset, there exist weakly precomplete ceers whose Visser spaces do not contain infinite effectively discrete subsets. As a consequence, contrary to precomplete ceers which always yield partitions into effectively inseparable sets, we show that although weakly precomplete ceers always yield partitions into computably inseparable sets, nevertheless there are weakly precomplete ceers for which no equivalence class is creative. Finally, we show that the index set of the precomplete ceers is 3-complete, whereas the index set of the weakly precomplete ceers is 3-complete. As a consequence of these results, we also show that the index sets of the uniformly precomplete ceers and of the e-complete ceers are 3-complete. -
Primitive Recursive Functions Are Recursively Enumerable
AN ENUMERATION OF THE PRIMITIVE RECURSIVE FUNCTIONS WITHOUT REPETITION SHIH-CHAO LIU (Received April 15,1900) In a theorem and its corollary [1] Friedberg gave an enumeration of all the recursively enumerable sets without repetition and an enumeration of all the partial recursive functions without repetition. This note is to prove a similar theorem for the primitive recursive functions. The proof is only a classical one. We shall show that the theorem is intuitionistically unprovable in the sense of Kleene [2]. For similar reason the theorem by Friedberg is also intuitionistical- ly unprovable, which is not stated in his paper. THEOREM. There is a general recursive function ψ(n, a) such that the sequence ψ(0, a), ψ(l, α), is an enumeration of all the primitive recursive functions of one variable without repetition. PROOF. Let φ(n9 a) be an enumerating function of all the primitive recursive functions of one variable, (See [3].) We define a general recursive function v(a) as follows. v(0) = 0, v(n + 1) = μy, where μy is the least y such that for each j < n + 1, φ(y, a) =[= φ(v(j), a) for some a < n + 1. It is noted that the value v(n + 1) can be found by a constructive method, for obviously there exists some number y such that the primitive recursive function <p(y> a) takes a value greater than all the numbers φ(v(0), 0), φ(y(Ϋ), 0), , φ(v(n\ 0) f or a = 0 Put ψ(n, a) = φ{v{n), a). -
Polya Enumeration Theorem
Polya Enumeration Theorem Sasha Patotski Cornell University [email protected] December 11, 2015 Sasha Patotski (Cornell University) Polya Enumeration Theorem December 11, 2015 1 / 10 Cosets A left coset of H in G is gH where g 2 G (H is on the right). A right coset of H in G is Hg where g 2 G (H is on the left). Theorem If two left cosets of H in G intersect, then they coincide, and similarly for right cosets. Thus, G is a disjoint union of left cosets of H and also a disjoint union of right cosets of H. Corollary(Lagrange's theorem) If G is a finite group and H is a subgroup of G, then the order of H divides the order of G. In particular, the order of every element of G divides the order of G. Sasha Patotski (Cornell University) Polya Enumeration Theorem December 11, 2015 2 / 10 Applications of Lagrange's Theorem Theorem n! For any integers n ≥ 0 and 0 ≤ m ≤ n, the number m!(n−m)! is an integer. Theorem (ab)! (ab)! For any positive integers a; b the ratios (a!)b and (a!)bb! are integers. Theorem For an integer m > 1 let '(m) be the number of invertible numbers modulo m. For m ≥ 3 the number '(m) is even. Sasha Patotski (Cornell University) Polya Enumeration Theorem December 11, 2015 3 / 10 Polya's Enumeration Theorem Theorem Suppose that a finite group G acts on a finite set X . Then the number of colorings of X in n colors inequivalent under the action of G is 1 X N(n) = nc(g) jGj g2G where c(g) is the number of cycles of g as a permutation of X . -
Enumeration of Finite Automata 1 Z(A) = 1
INFOI~MATION AND CONTROL 10, 499-508 (1967) Enumeration of Finite Automata 1 FRANK HARARY AND ED PALMER Department of Mathematics, University of Michigan, Ann Arbor, Michigan Harary ( 1960, 1964), in a survey of 27 unsolved problems in graphical enumeration, asked for the number of different finite automata. Re- cently, Harrison (1965) solved this problem, but without considering automata with initial and final states. With the aid of the Power Group Enumeration Theorem (Harary and Palmer, 1965, 1966) the entire problem can be handled routinely. The method involves a confrontation of several different operations on permutation groups. To set the stage, we enumerate ordered pairs of functions with respect to the product of two power groups. Finite automata are then concisely defined as certain ordered pah's of functions. We review the enumeration of automata in the natural setting of the power group, and then extend this result to enumerate automata with initial and terminal states. I. ENUMERATION THEOREM For completeness we require a number of definitions, which are now given. Let A be a permutation group of order m = ]A I and degree d acting on the set X = Ix1, x~, -.. , xa}. The cycle index of A, denoted Z(A), is defined as follows. Let jk(a) be the number of cycles of length k in the disjoint cycle decomposition of any permutation a in A. Let al, a2, ... , aa be variables. Then the cycle index, which is a poly- nomial in the variables a~, is given by d Z(A) = 1_ ~ H ~,~(°~ . (1) ~$ a EA k=l We sometimes write Z(A; al, as, .. -
The Axiom of Choice and Its Implications
THE AXIOM OF CHOICE AND ITS IMPLICATIONS KEVIN BARNUM Abstract. In this paper we will look at the Axiom of Choice and some of the various implications it has. These implications include a number of equivalent statements, and also some less accepted ideas. The proofs discussed will give us an idea of why the Axiom of Choice is so powerful, but also so controversial. Contents 1. Introduction 1 2. The Axiom of Choice and Its Equivalents 1 2.1. The Axiom of Choice and its Well-known Equivalents 1 2.2. Some Other Less Well-known Equivalents of the Axiom of Choice 3 3. Applications of the Axiom of Choice 5 3.1. Equivalence Between The Axiom of Choice and the Claim that Every Vector Space has a Basis 5 3.2. Some More Applications of the Axiom of Choice 6 4. Controversial Results 10 Acknowledgments 11 References 11 1. Introduction The Axiom of Choice states that for any family of nonempty disjoint sets, there exists a set that consists of exactly one element from each element of the family. It seems strange at first that such an innocuous sounding idea can be so powerful and controversial, but it certainly is both. To understand why, we will start by looking at some statements that are equivalent to the axiom of choice. Many of these equivalences are very useful, and we devote much time to one, namely, that every vector space has a basis. We go on from there to see a few more applications of the Axiom of Choice and its equivalents, and finish by looking at some of the reasons why the Axiom of Choice is so controversial. -
17 Axiom of Choice
Math 361 Axiom of Choice 17 Axiom of Choice De¯nition 17.1. Let be a nonempty set of nonempty sets. Then a choice function for is a function f sucFh that f(S) S for all S . F 2 2 F Example 17.2. Let = (N)r . Then we can de¯ne a choice function f by F P f;g f(S) = the least element of S: Example 17.3. Let = (Z)r . Then we can de¯ne a choice function f by F P f;g f(S) = ²n where n = min z z S and, if n = 0, ² = min z= z z = n; z S . fj j j 2 g 6 f j j j j j 2 g Example 17.4. Let = (Q)r . Then we can de¯ne a choice function f as follows. F P f;g Let g : Q N be an injection. Then ! f(S) = q where g(q) = min g(r) r S . f j 2 g Example 17.5. Let = (R)r . Then it is impossible to explicitly de¯ne a choice function for . F P f;g F Axiom 17.6 (Axiom of Choice (AC)). For every set of nonempty sets, there exists a function f such that f(S) S for all S . F 2 2 F We say that f is a choice function for . F Theorem 17.7 (AC). If A; B are non-empty sets, then the following are equivalent: (a) A B ¹ (b) There exists a surjection g : B A. ! Proof. (a) (b) Suppose that A B. -
Turing Degrees and Automorphism Groups of Substructure Lattices 3
TURING DEGREES AND AUTOMORPHISM GROUPS OF SUBSTRUCTURE LATTICES RUMEN DIMITROV, VALENTINA HARIZANOV, AND ANDREY MOROZOV Abstract. The study of automorphisms of computable and other structures connects computability theory with classical group theory. Among the non- computable countable structures, computably enumerable structures are one of the most important objects of investigation in computable model theory. In this paper, we focus on the lattice structure of computably enumerable substructures of a given canonical computable structure. In particular, for a Turing degree d, we investigate the groups of d-computable automorphisms of the lattice of d-computably enumerable vector spaces, of the interval Boolean algebra Bη of the ordered set of rationals, and of the lattice of d-computably enumerable subalgebras of Bη. For these groups we show that Turing reducibil- ity can be used to substitute the group-theoretic embedding. We also prove that the Turing degree of the isomorphism types for these groups is the second Turing jump of d, d′′. 1. Automorphisms of effective structures Computable model theory investigates algorithmic content (effectiveness) of notions, objects, and constructions in classical mathematics. In algebra this in- vestigation goes back to van der Waerden who in his Modern Algebra introduced an explicitly given field as one the elements of which are uniquely represented by distinguishable symbols with which we can perform the field operations algo- rithmically. The formalization of an explicitly given field led to the notion of a computable structure, one of the main objects of study in computable model the- ory. A structure is computable if its domain is computable and its relations and arXiv:1811.01224v1 [math.LO] 3 Nov 2018 functions are uniformly computable. -
Enumeration of Structure-Sensitive Graphical Subsets: Calculations (Graph Theory/Combinatorics) R
Proc. Nati Acad. Sci. USA Vol. 78, No. 3, pp. 1329-1332, March 1981 Mathematics Enumeration of structure-sensitive graphical subsets: Calculations (graph theory/combinatorics) R. E. MERRIFIELD AND H. E. SIMMONS Central Research and Development Department, E. I. du Pont de Nemours and Company, Experimental Station, Wilmington, Delaware 19898 Contributed by H. E. Simmons, October 6, 1980 ABSTRACT Numerical calculations are presented, for all Table 2. Qualitative properties of subset counts connected graphs on six and fewer vertices, of the -lumbers of hidepndent sets, connected sets;point and line covers, externally Quantity Branching Cyclization stable sets, kernels, and irredundant sets. With the exception of v, xIncreases Decreases the number of kernels, the numbers of such sets are al highly p Increases Increases structure-sensitive quantities. E Variable Increases K Insensitive Insensitive The necessary mathematical machinery has recently been de- Variable Variable veloped (1) for enumeration of the following types of special cr Decreases Increases subsets of the vertices or edges of a general graph: (i) indepen- p Increases Increases dent sets, (ii) connected sets, (iii) point and line covers, (iv;' ex- X Decreases Increases ternally stable sets, (v) kernels, and (vi) irredundant sets. In e Increases Increases order to explore the sensitivity of these quantities to graphical K Insensitive Insensitive structure we have employed this machinery to calculate some Decreases Increases explicit subset counts. (Since no more efficient algorithm is known for irredundant sets, they have been enumerated by the brute-force method of testing every subset for irredundance.) 1. e and E are always odd. The 10 quantities considered in ref. -
Algebraic Aspects of the Computably Enumerable Degrees
Proc. Natl. Acad. Sci. USA Vol. 92, pp. 617-621, January 1995 Mathematics Algebraic aspects of the computably enumerable degrees (computability theory/recursive function theory/computably enumerable sets/Turing computability/degrees of unsolvability) THEODORE A. SLAMAN AND ROBERT I. SOARE Department of Mathematics, University of Chicago, Chicago, IL 60637 Communicated by Robert M. Solovay, University of California, Berkeley, CA, July 19, 1994 ABSTRACT A set A of nonnegative integers is computably equivalently represented by the set of Diophantine equations enumerable (c.e.), also called recursively enumerable (r.e.), if with integer coefficients and integer solutions, the word there is a computable method to list its elements. The class of problem for finitely presented groups, and a variety of other sets B which contain the same information as A under Turing unsolvable problems in mathematics and computer science. computability (<T) is the (Turing) degree ofA, and a degree is Friedberg (5) and independently Mucnik (6) solved Post's c.e. if it contains a c.e. set. The extension ofembedding problem problem by producing a noncomputable incomplete c.e. set. for the c.e. degrees Qk = (R, <, 0, 0') asks, given finite partially Their method is now known as the finite injury priority ordered sets P C Q with least and greatest elements, whether method. Restated now for c.e. degrees, the Friedberg-Mucnik every embedding ofP into Rk can be extended to an embedding theorem is the first and easiest extension of embedding result of Q into Rt. Many of the most significant theorems giving an for the well-known partial ordering of the c.e. -
Recursion Theory Notes, Fall 2011 0.1 Introduction
Recursion Theory Notes, Fall 2011 Lecturer: Lou van den Dries 0.1 Introduction Recursion theory (or: theory of computability) is a branch of mathematical logic studying the notion of computability from a rather theoretical point of view. This includes giving a lot of attention to what is not computable, or what is computable relative to any given, not necessarily computable, function. The subject is interesting on philosophical-scientific grounds because of the Church- Turing Thesis and its role in computer science, and because of the intriguing concept of Kolmogorov complexity. This course tries to keep touch with how recursion theory actually impacts mathematics and computer science. This impact is small, but it does exist. Accordingly, the first chapter of the course covers the basics: primitive recur- sion, partial recursive functions and the Church-Turing Thesis, arithmetization and the theorems of Kleene, the halting problem and Rice's theorem, recur- sively enumerable sets, selections and reductions, recursive inseparability, and index systems. (Turing machines are briefly discussed, but the arithmetization is based on a numerical coding of combinators.) The second chapter is devoted to the remarkable negative solution (but with positive aspects) of Hilbert's 10th Problem. This uses only the most basic notions of recursion theory, plus some elementary number theory that is worth knowing in any case; the coding aspects of natural numbers are combined here ingeniously with the arithmetic of the semiring of natural numbers. The last chapter is on Kolmogorov complexity, where concepts of recursion theory are used to define and study notions of randomness and information content.