Douglas Cenzer

Polynomial-time versus recursive models

Abstract The central problem considered in this paper is whether a given recursive structure is recursively isomorphic to a polynomial-time (p-time) structure. Positive results are obtained for all relational structures, for all Boolean algebras and for the natural numbers with addition, multiplication and the unary function 2x. Counterexamples are constructed for recursive structures with one unary function and for Abelian groups and also for relational structures when the universe of the structure is fixed. Results are also given which distinguish primitive recursive structures, exponential-time structures and structures with honest witnesses.

Algorithmic Randomness of Closed Sets

We investigate notions of randomness in the space C[2N] of non-empty closed subsets of {0,1}N. A probability measure is given and a version of the Martin-Lof test for randomness is defined. Π20 random closed sets exist but there are no random Π10 closed sets. It is shown that any random closed set is perfect, has measure 0, and has box dimension log2(4/3). A random closed set has no n-c.e. elements. A closed subset of 2N may be defined as the set of infinite paths through a tree and so the problem of compressibility of trees is explored. If Tn = T∩{0,1}n, then for any random closed set [T] where T has no dead ends, K(Tn)≥n-O(1) but for any k, K(Tn) ≤ 2n − k + O(1), where K(σ) is the prefix-free complexity of σ∈{0,1}*.

Recursively presented games and strategies

Abstract The complexity of an infinite game is determined in part by the complexity of the set of winning strategies for the game. In this paper we consider effectively closed, recursively bounded Gale–Stewart games. We give a coding for the winning strategies of such games so that any two different strategies are winning for different sets of games. We show that under our coding of winning strategies, the set of winning strategies can be effectively isomorphic to an arbitrary recursively bounded Π 0 1 -class. This strengthens the well-known result that recursively presented games need not have recursive winning strategies. We also show that for every recursively presented game, there is a polynomial time presented game which has the same set of winning strategies. Thus even feasibly presented games need not have recursive solutions. Finally we develop some criteria which ensures that a polynomial time presented game has a polynomial time winning strategy.

Chapter 2 – ∏10 Classes in Computability Theory

This chapter presents a survey of recent results on π 0 1 classes and their applications. It includes some new results that show the existence of an r.b. π 0 1 class with no member of high degree. The focus is on the study of recursively bounded (r.b.) π 0 1 classes as a branch of recursion theory. In particular the notion of a Π 0 1 classes may be viewed as the generalization of the notion of a π 0 1 set, which is simply the complement of a ∑ 0 1 or recursively enumerable set. Of particular interest for recursion theory is the connection between a retraceable π 0 1 set subset A of ω and the π 0 1 class P(A) of initial subsets of A. The notion of a co-maximal π 0 1 set plays an important role and has a natural version for π 0 1 classes that of a minimal π 0 1 class P. π 0 1 classes are also important in the application of recursion theory to numerous branches of mathematics, including combinatorics, algebra, and analysis. The chapter discusses basis and anti-basis results and develops the notion of the Cantor–Bendixson (C–B) rank to give a finer analysis of the complexity of the members of π 0 1 class. The minimal and thin classes and applications of π 0 1 class are discussed as well.

Index sets for Π01 classes

Abstract A Π01 class is an effectively closed set of reals. We study properties of these classes determined by cardinality, measure and category as well as by the complexity of the members of a class P. Given an effective enumeration {Pe:e

Countable thin Π01 classes

Abstract Cenzer, D., R. Downey, C. Jockusch and R.A. Shore, Countable thin Π01 classes, Annals of Pure and Applied Logic 59 (1993) 79–139. A Π01 class P ⊂ {0, 1}ω is thin if every Π01 subclass Q of P is the intersection of P with some clopen set. Countable thin Π01 classes are constructed having arbitrary recursive Cantor- Bendixson rank. A thin Π01 class P is constructed with a unique nonisolated point A and furthermore A is of degree 0’. It is shown that no set of degree ≥0” can be a member of any thin Π01 class. An r.e. degree d is constructed such that no set of degree d can be a member of any thin Π01 class. It is also shown that between any two distinct comparable r.e. degrees, there is a degree (not necessarily r.e.) that contains a set which is of rank one in some thin Π01 class. It is shown that no maximal set can have rank one in any Π01 class, while there exist maximal sets of rank 2. The connection between Π01 classes, propositional theories and recursive Boolean algebras is explored, producing several corollaries to the results on Π01 classes. For example, call a recursive Boolean algebra thin if it has no proper nonprincipal recursive ideals. Then no thin recursive Boolean algebra can have a maximal ideal of degree ≥0”.

Inductive definability: Measure and category

The purpose of this paper is threefold. Our first purpose is to exposit a major concept from descriptive set theory, inductive definability, and to present some of the major results concerning inductive operators. The classical version of this theory is carried out in the first two sections and the effective version in the fifth section. Our second purpose is to demonstrate a powerful unity of viewpoint provided by inductive definitions. This is shown by deriving several wellknown results from this viewpoint which had been previously proved by diverse methods. This point is demonstrated in the first section by deriving several well-known examples of analytic (and coanalytic) sets which are not Bore1 sets; in the third and fourth sections by the proofs of some “faithful extension” and reflection theorems. Again, this point is demonstrated in the fifth section in the presentation of several results from effective descriptive set theory. Our third purpose is to present some new results. Our first new result is given in the third section where it is shown (Theorem 3.3) that several classical “definability” results may be unified with the use of inductive definitions. New results are given in Theorems 4.1 and 4.2 where we demonstrate a definability and reflection principle with respect to conditional probability distributions. In the fifth section, we present new proofs of several known results and give a new characterization of p(x), the least ordinal not recursive in x. 55 OOOl-8708/80/100055-36

Computable symbolic dynamics

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Chapter 13 Π10 classes in mathematics

We investigate computable subshifts and the connection with effective symbolic dynamics. It is shown that a decidable Π01 class P is a subshift if and only if there exists a computable function F mapping 2ℕ to 2ℕ such that P is the set of itineraries of elements of 2ℕ. Π01 subshifts are constructed in 2ℕ and in 2ℤ which have no computable elements. We also consider the symbolic dynamics of maps on the unit interval. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Algorithmic randomness of continuous functions

Publisher Summary This chapter presents the applications of the ∏ 0 1 classes in a wide variety of fields including logic, nonmonotonic logic, algebra, combinatorics, orderings, and game theory. ∏ 0 1 Classes occur in many areas of recursive mathematic.. In each case, it shows that the set of solutions to a given recursive instance of the problem may be represented as an arbitrary (bounded, or recursively bounded) ∏ 0 1 class. For example, Bean observed that the set of κ-colorings of a recursive graph “G” is a recursively bounded ∏ 0 1 class. The chapter presents results on ∏ 0 1 classes and their members that can be applied to the mathematical problems. Through these results on various recursive problems in mathematics, the solution sets are shown to be represented by “r.b.” (Recursively bound) ∏ 0 1 classes and, in some cases, to represent all possible r.b. ∏ 0 1 classes (Certain problems are represented by bounded or unbounded represent all possible r.b. represent∏ 0 1 classes). The results are then applied to derive corollaries such as the existence of recursive instances of each such problem with no recursive solutions.