Left computably enumerable reals and initial segment complexity

Theoretical Computer Science, 2002

We study the relationship between a computably enumerable real and its presentations: ways of approximating the real by enumerating a prefix-free set of binary strings.

Domains and Processes, 2001

There have been many suggestions for what should be a computable real number or function. Some of them exhibited pathological properties. At present, research concentrates either on an application of Weihrauch's Type Two Theory of Effectivity or on domain-theoretic approaches, in which case the partial objects appearing during computations are made explicit. A further, more analysis-oriented line of research is based on Grzegorczyk's work. All these approaches are claimed to be equivalent, but not in all cases proofs have been given. In this paper it is shown that a real number as well as a real-valued function are computable in Weihrauch's sense if and only if they are definable in Escardó's functional language Real PCF, an extension of the language PCF by a new ground type for (total and partial) real numbers. This is exactly the case if the number is a computable element in the continuous domain of all compact real intervals and/or the function has a computable extension to this domain. For defining the semantics of the language Real PCF a full subcategory of the category of bounded-complete ω-continuous directed-complete partial orders is introduced and it is defined when a domain in this category is effectively given. The subcategory of effectively given domains contains the interval domain and is Cartesian closed. * The paper mainly contains results from the second author's diploma thesis [18] written under the supervision of the first author.

Journal of Complexity, 2003

A real number x is called h-monotonically computable (h-mc for short), for some function h : N-N; if there is a computable sequence ðx s Þ of rational numbers converging to x such that hðnÞjx À x n jXjx À x m j for all m4n: x is called o-monotonically computable (o-mc) if it is h-mc for some computable function h: Thus, the class of o-mc real numbers is an extension of the class of monotonically computable real numbers introduced in (Math. Logic Quart. 48(3) (2002) 459), where only constant functions h c are considered and the corresponding real numbers are called c-monotonically computable. In (Math. Logic Quart. 48(3) (2002) 459) it is shown that the classes of c-mc real numbers form a proper hierarchy inside the class of weakly computable real numbers which is the arithmetical closure of the 1-mc real numbers. In this paper, we show that this hierarchy is dense, i.e., for any real numbers c 2 4c 1 X1; there is a c 2mc real number which is not c 1-mc and there is also an o-mc real number which is not c-mc for any cAR: Furthermore, we show that the class of all o-mc real numbers is incomparable with the class of weakly computable real numbers.

Proc. of the Thirty-Second Annual ACM/IEEE Symposium on Logic in Computer Science (LICS 2017), 2017

—We investigate interrelationships among different notions from mathematical analysis, effective topology, and classical computability theory. Our main object of study is the class of computable functions defined over a bounded domain with the boundary being a left-c.e. number. We investigate necessary and sufficient conditions under which such function can be computably extended. It turns out that this depends on the behavior of the function near the boundary as well as on the class of left-c.e. numbers to which the boundary belongs, that is, how it can be constructed. Of particular interest a class of functions is investigated: sawtooth functions constructed from computable enumerations of c.e. sets.

Journal of Computer and System Sciences, 2011

In this paper we introduce the notion of ε-universal prefix-free Turing machine (ε is a computable real in (0, 1]) and study its halting probability. The main result is the extension of the representability theorem for left-computable random reals to the case of ε-random reals: a real is left-computable ε-random iff it is the halting probability of an ε-universal prefixfree Turing machine. We also show that left-computable ε-random reals are provable εrandom in the Peano Arithmetic. The theory developed here parallels to a large extent the classical theory, but not completely. For example, random reals are Borel normal (in any base), but for ε ∈ (0, 1), some ε-random reals do not contain even arbitrarily long runs of 0s.

Theoretical Computer Science, 2002

We consider for a real number the Kolmogorov complexities of its expansions with respect to di erent bases. In the paper it is shown that, for usual and self-delimiting Kolmogorov complexity, the complexity of the preÿxes of their expansions with respect to di erent bases r and b are related in a way that depends only on the relative information of one base with respect to the other.

Electronic Notes in Theoretical Computer Science, 2002

Solovay showed that there are noncomputable reals α such that H(α n) H(1 n) + O(1), where H is prefix-free Kolmogorov complexity. Such H-trivial reals are interesting due to the connection between algorithmic complexity and effective randomness. We give a new, easier construction of an H-trivial real. We also analyze various computability-theoretic properties of the H-trivial reals, showing for example that no H-trivial real can compute the halting problem. Therefore, our construction of an H-trivial computably enumerable set is an easy, injury-free construction of an incomplete computably enumerable set. Finally, we relate the H-trivials to other classes of "highly nonrandom" reals that have been previously studied. * Some of the material in this paper was presented by Downey in his talk Algorithmic Randomness and Computability at the 8th Asian Logic Meeting in Chongqing, China.

MLQ, 2004

Analogous to Ershov's hierarchy for ∆ 0 2-subsets of natural numbers we discuss the similar hierarchy for recursively approximable real numbers. Namely, we define the k-computability for natural number k and f-computability for function f. We will show that these notions are not equivalent for different representations of real numbers based on Cauchy sequence, Dedekind cut and binary expansion.

MLQ, 2002

We study the relationship between a computably enumerable real and its presentations. A set A presents a computably enumerable real α if A is a computably enumerable prefix-free set of strings such that α = σ∈A 2 −|σ|. Note that σ∈A 2 −|σ| is precisely the measure of the set of reals that have a string in A as an initial segment. So we will simply abbreviate σ∈A 2 −|σ| by µ(A). It is known that whenever A so presents α then A ≤wtt α, where ≤wtt denotes weak truth table reducibility, and that the wtt degrees of presentations form an ideal I(α) in the computably enumerable wtt degrees. We prove that any such ideal is Σ 0 3 , and conversely that if I is any Σ 0 3 ideal in the computably enumerable wtt degrees then there is a computable enumerable real α such that I = I(α).

Journal of Complexity, 2006

A real number x is f-bounded computable (f-bc, for short) for a function f if there is a computable sequence (x s) of rational numbers which converges to x f-bounded effectively in the sense that, for any natural number n, the sequence (x s) has at most f (n) non-overlapping jumps of size larger than 2 −n. f-bc reals are called divergence bounded computable if f is computable. In this paper we give a hierarchy theorem for Turing degrees of different classes of f-bc reals. More precisely, we will show that, for any computable functions f and g, if there exists a constant > 1 such that, for any constant c, f (n)+n+c g(n) holds for almost all n, then the classes of Turing degrees given by f-bc and g-bc reals are different. As a corollary this implies immediately the result of [R. Rettinger, X. Zheng, On the Turing degrees of the divergence bounded computable reals, in: