Turing Degrees Research Articles

The Gamma question for many-one degrees

A set A is coarsely computable with density r∈[0,1] if there is an algorithm for deciding membership in A which always gives a (possibly incorrect) answer, and which gives a correct answer with density at least r. To any Turing degree a we can assign a value ΓT(a): the minimum, over all sets A in a, of the highest density at which A is coarsely computable. The closer ΓT(a) is to 1, the closer a is to being computable. Andrews, Cai, Diamondstone, Jockusch, and Lempp noted that ΓT can take on the values 0, 1/2, and 1, but not any values in strictly between 1/2 and 1. They asked whether the value of ΓT can be strictly between 0 and 1/2. This is the Gamma question.Replacing Turing degrees by many-one degrees, we get an analogous question, and the same arguments show that Γm can take on the values 0, 1/2, and 1, but not any values strictly between 1/2 and 1. We will show that for any r∈[0,1/2], there is an m-degree a with Γm(a)=r. Thus the range of Γm is [0,1/2]∪{1}.Benoit Monin has recently announced a solution to the Gamma question for Turing degrees. Interestingly, his solution gives the opposite answer: the only possible values of ΓT are 0, 1/2, and 1.

Spectrum of the Field of Computable Real Numbers

Necessary and sufficient conditions for a Turing degree to be an element of the spectrum of the classical field of computable real numbers are established.

Diagonally non-computable functions and fireworks

A set C of reals is said to be negligible if there is no probabilistic algorithm which generates a member of C with positive probability. Various classes have been proven to be negligible, for example the Turing upper-cone of a non-computable real, the class of coherent completions of Peano Arithmetic or the class of reals of minimal Turing degree. One class of particular interest in the study of negligibility is the class of diagonally non-computable (DNC) functions, proven by Kučera to be non-negligible in a strong sense: every Martin-Löf random real computes a DNC function. Ambos-Spies et al. showed that the converse does not hold: there are DNC functions which compute no Martin-Löf random real. In this paper, we show that the set of such DNC functions is in fact non-negligible using a technique we call ‘fireworks argument’. We also use this technique to prove further results on DNC functions.

Defying the Facts: Justice and Being in Speculative Materialism

ABSTRACT Quentin Meillassoux claims that physical resurrection of the dead is necessary and sufficient for justice to be realized in the world. This article presents two arguments against Meillassoux's claim: (1) the permanence argument, which suggests that events of injustice cannot be erased by what happens later, and (2) the priority argument, which suggests that death is not the chief injustice. The article also investigates the consequences of rejecting Meillassoux's view, asking whether a commitment to justice is still viable if it cannot be fully realized in the world.

Computing halting probabilities from other halting probabilities

The halting probability of a Turing machine is the probability that the machine will halt if it starts with a random stream written on its one-way input tape. When the machine is universal, this probability is referred to as Chaitin's omega number, and is the most well known example of a real which is random in the sense of Martin-Löf. Although omega numbers depend on the underlying universal Turing machine, they are robust in the sense that they all have the same Turing degree, namely the degree of the halting problem. This means that, given two universal prefix-free machines U,V, the halting probability ΩU of U computes the halting probability ΩV of V. If this computation uses at most the first n+g(n) bits of ΩU for the computation of the first n bits of ΩV, we say that ΩU computes ΩV with redundancy g.In this paper we give precise bounds on the redundancy growth rate that is generally required for the computation of an omega number from another omega number. We show that for each ϵ>1, any pair of omega numbers compute each other with redundancy ϵlog⁡n. On the other hand, this is not true for ϵ=1. In fact, we show that for each omega number ΩU there exists another omega number which is not computable from ΩU with redundancy log⁡n. This latter result improves an older result of Frank Stephan.

Elementary theories and structural properties of d-c.e. and n-c.e. degrees

This paper is a survey on the upper semilattices of Turing and enumeration degrees of n-c.e. sets. Questions on the structural properties of these semilattices, and some model-theoretic properties are considered.

A uniform version of non-low2-ness

We introduce a property of Turing degrees: being uniformly non-low2. We prove that, in the c.e. Turing degrees, there is an incomplete uniformly non-low2 degree, and not every non-low2 degree is uniformly non-low2. We also build some connection between (uniform) non-low2-ness and computable Lipschitz reducibility (≤cl), as a strengthening of weak truth table reducibility:(1) If a c.e. Turing degree d is uniformly non-low2, then for any non-computable Δ20 real there is a c.e. real in d such that both of them have no common upper bound in c.e. reals under cl-reducibility.(2) A c.e. Turing degree d is non-low2 if and only if for any Δ20 real there is a real in d which is not cl-reducible to it.

Randomness for computable measures and initial segment complexity

We study the possible growth rates of the Kolmogorov complexity of initial segments of sequences that are random with respect to some computable measure on 2ω, the so-called proper sequences. Our main results are as follows: (1) We show that the initial segment complexity of a proper sequence X is bounded from below by a computable function (that is, X is complex) if and only if X is random with respect to some computable, continuous measure. (2) We prove that a uniform version of the previous result fails to hold: there is a family of complex sequences that are random with respect to a single computable measure such that for every computable, continuous measure μ, some sequence in this family fails to be random with respect to μ. (3) We show that there are proper sequences with extremely slow-growing initial segment complexity, that is, there is a proper sequence the initial segment complexity of which is infinitely often below every computable function, and even a proper sequence the initial segment complexity of which is dominated by all computable functions. (4) We prove various facts about the Turing degrees of such sequences and show that they are useful in the study of certain classes of pathological measures on 2ω, namely diminutive measures and trivial measures.

TRIAL AND ERROR MATHEMATICS II: DIALECTICAL SETS AND QUASIDIALECTICAL SETS, THEIR DEGREES, AND THEIR DISTRIBUTION WITHIN THE CLASS OF LIMIT SETS

AbstractThis paper is a continuation of Amidei, Pianigiani, San Mauro, Simi, & Sorbi (2016), where we have introduced the quasidialectical systems, which are abstract deductive systems designed to provide, in line with Lakatos’ views, a formalization of trial and error mathematics more adherent to the real mathematical practice of revision than Magari’s original dialectical systems. In this paper we prove that the two models of deductive systems (dialectical systems and quasidialectical systems) have in some sense the same information content, in that they represent two classes of sets (the dialectical sets and the quasidialectical sets, respectively), which have the same Turing degrees (namely, the computably enumerable Turing degrees), and the same enumeration degrees (namely, the ${\rm{\Pi }}_1^0$ enumeration degrees). Nonetheless, dialectical sets and quasidialectical sets do not coincide. Even restricting our attention to the so-called loopless quasidialectical sets, we show that the quasidialectical sets properly extend the dialectical sets. As both classes consist of ${\rm{\Delta }}_2^0$ sets, the extent to which the two classes differ is conveniently measured using the Ershov hierarchy: indeed, the dialectical sets are ω-computably enumerable (close inspection also shows that there are dialectical sets which do not lie in any finite level; and in every finite level n ≥ 2 of the Ershov hierarchy there is a dialectical set which does not lie in the previous level); on the other hand, the quasidialectical sets spread out throughout all classes of the hierarchy (close inspection shows that for every ordinal notation a of a nonzero computable ordinal, there is a quasidialectical set lying in ${\rm{\Sigma }}_a^{ - 1}$, but in none of the preceding levels).

Autostability spectra for decidable structures

We study autostability spectra relative to strong constructivizations (SC-autostability spectra). For a decidable structure$\mathcal{S}$, the SC-autostability spectrum of$\mathcal{S}$is the set of all Turing degrees capable of computing isomorphisms among arbitrary decidable copies of$\mathcal{S}$. The degree of SC-autostability for$\mathcal{S}$is the least degree in the spectrum (if such a degree exists).We prove that for a computable successor ordinal α, every Turing degree c.e. in and above0(α)is the degree of SC-autostability for some decidable structure. We show that for an infinite computable ordinal β, every Turing degree c.e. in and above0(2β+1)is the degree of SC-autostability for some discrete linear order. We prove that the set of all PA-degrees is an SC-autostability spectrum. We also obtain similar results for autostability spectra relative ton-constructivizations.

Genericity for Mathias forcing over general Turing ideals

In Mathias forcing, conditions are pairs (D, S) of sets of natural numbers, in which D is finite, S is infinite, and max D < min S. The Turing degrees and computational characteristics of generics for this forcing in the special (but important) case where the infinite sets S are computable were thoroughly explored by Cholak, Dzhafarov, Hirst and Slaman [2]. In this paper, we undertake a similar investigation for the case where the sets S are members of general countable Turing ideals, and give conditions under which generics for Mathias forcing over one ideal compute generics for Mathias forcing over another. It turns out that if I does not contain only the computable sets, then non-trivial information can be encoded into the generics for Mathias forcing over I. We give a classification of this information in terms of computability-theoretic properties of the ideal, using coding techniques that also yield new results about introreducibility. In particular, we extend a result of Slaman and Groszek and show that there is an infinite Δ 3 0 set with no introreducible subset of the same degree.

Generic muchnik reducibility and presentations of fields

We prove that if I is a countable ideal in the Turing degrees, then the field RI of real numbers in I is computable from exactly the degrees that list the functions (i.e., elements of ωω) in I. This implies, for example, that the degree spectrum of the field of computable real numbers consists exactly of the high degrees. We also prove that if I is a countable Scott ideal, then it is strictly easier to list the sets (i.e., elements of 2ω) in I than it is to list the functions in I. This allows us to answer a question of Knight, Montalban, and Schweber. They introduced generic Muchnik reducibility to extend the idea of Muchnik reducibility between countable structures to arbitrary structures. They asked if R is generically Muchnik reducible to the structure that consists of all sets of natural numbers. Our result for Scott ideals shows that this is not the case.

The weakness of being cohesive, thin or free in reverse mathematics

Informally, a mathematical statement is robust if its strength is left unchanged under variations of the statement. In this paper, we investigate the lack of robustness of Ramsey’s theorem and its consequence under the frameworks of reverse mathematics and computable reducibility. To this end, we study the degrees of unsolvability of cohesive sets for different uniformly computable sequence of sets and identify different layers of unsolvability. This analysis enables us to answer some questions of Wang about how typical sets help computing cohesive sets. We also study the impact of the number of colors in the computable reducibility between coloring statements. In particular, we strengthen the proof by Dzhafarov that cohesiveness does not strongly reduce to stable Ramsey’s theorem for pairs, revealing the combinatorial nature of this nonreducibility and prove that whenever k is greater than l, stable Ramsey’s theorem for n-tuples and k colors is not computably reducible to Ramsey’s theorem for n-tuples and l colors. In this sense, Ramsey’s theorem is not robust with respect to his number of colors over computable reducibility. Finally, we separate the thin set and free set theorem from Ramsey’s theorem for pairs and identify an infinite decreasing hierarchy of thin set theorems in reverse mathematics. This shows that in reverse mathematics, the strength of Ramsey’s theorem is very sensitive to the number of colors in the output set. In particular, it enables us to answer several related questions asked by Cholak, Giusto, Hirst and Jockusch.

ON REALS WITH -BOUNDED COMPLEXITY AND COMPRESSIVE POWER

AbstractThe (prefix-free) Kolmogorov complexity of a finite binary string is the length of the shortest description of the string. This gives rise to some ‘standard’ lowness notions for reals: A isK-trivial if its initial segments have the lowest possible complexity and A is low forKif using A as an oracle does not decrease the complexity of strings by more than a constant factor. We weaken these notions by requiring the defining inequalities to hold only up to all${\rm{\Delta }}_2^0$orders, and call the new notions${\rm{\Delta }}_2^0$-bounded K-trivialand${\rm{\Delta }}_2^0$-bounded low for K. Several of the ‘nice’ properties ofK-triviality are lost with this weakening. For instance, the new weaker definitions both give uncountable set of reals. In this paper we show that the weaker definitions are no longer equivalent, and that the${\rm{\Delta }}_2^0$-boundedK-trivials are cofinal in the Turing degrees. We then compare them to other previously studied weakenings, namelyinfinitely-often K-trivialityandweak lowness for K(in each, the defining inequality must hold up to a constant, but only for infinitely many inputs). We show that${\rm{\Delta }}_2^0$-boundedK-trivial implies infinitely-oftenK-trivial, but no implication holds between${\rm{\Delta }}_2^0$-bounded low forKand weakly low forK.

COARSE REDUCIBILITY AND ALGORITHMIC RANDOMNESS

AbstractA coarse description of a set A ⊆ ω is a set D ⊆ ω such that the symmetric difference of A and D has asymptotic density 0. We study the extent to which noncomputable information can be effectively recovered from all coarse descriptions of a given set A, especially when A is effectively random in some sense. We show that if A is 1-random and B is computable from every coarse description D of A, then B is K-trivial, which implies that if A is in fact weakly 2-random then B is computable. Our main tool is a kind of compactness theorem for cone-avoiding descriptions, which also allows us to prove the same result for 1-genericity in place of weak 2-randomness. In the other direction, we show that if $A \le _{{\rm{T}}} \emptyset {\rm{'}}$ is a 1-random set, then there is a noncomputable c.e. set computable from every coarse description of A, but that not all K-trivial sets are computable from every coarse description of some 1-random set. We study both uniform and nonuniform notions of coarse reducibility. A set Y is uniformly coarsely reducible to X if there is a Turing functional Φ such that if D is a coarse description of X, then ΦD is a coarse description of Y. A set B is nonuniformly coarsely reducible to A if every coarse description of A computes a coarse description of B. We show that a certain natural embedding of the Turing degrees into the coarse degrees (both uniform and nonuniform) is not surjective. We also show that if two sets are mutually weakly 3-random, then their coarse degrees form a minimal pair, in both the uniform and nonuniform cases, but that the same is not true of every pair of relatively 2-random sets, at least in the nonuniform coarse degrees.

The universality of polynomial time Turing equivalence

We show that polynomial time Turing equivalence and a large class of other equivalence relations from computational complexity theory are universal countable Borel equivalence relations. We then discuss ultrafilters on the invariant Borel sets of these equivalence relations which are related to Martin's ultrafilter on the Turing degrees.

The automorphism group of the enumeration degrees

We investigate the extent to which Slaman and Woodin's framework for the analysis of the automorphism group of the structure of the Turing degrees can be transferred to analyze the automorphism group of the structure of the enumeration degrees.

Degrees of Autostability Relative to Strong Constructivizations for Boolean Algebras

It is proved that for every computable ordinal α, the Turing degree 0 (α) is a degree of autostability of some computable Boolean algebra and is also a degree of autostability relative to strong constructivizations for some decidable Boolean algebra. It is shown that a Harrison Boolean algebra has no degree of autostability relative to strong constructivizations. It is stated that the index set of decidable Boolean algebras having degree of autostability relative to strong constuctivizations is ∏ 1 1 -complete.

Abelian p-groups and the Halting problem

We investigate which effectively presented abelian p-groups are isomorphic relative to the halting problem. The standard approach to this and similar questions uses the notion of Δ20-categoricity (to be defined). We partially reduce the description of Δ20-categorical p-groups of Ulm type 1 to the analogous problem for equivalence structures. Using this reduction, we solve a problem left open in [5]. For the sake of the reduction mentioned above, we introduce a new notion of effective Δ20-categoricity that lies strictly in-between plain Δ20-categoricity and relative Δ20-categoricity (to be defined). We then reduce the problem of classifying effective Δ20-categoricity to a question stated in terms of Σ20-sets. Among other results, we show that for c.e. Turing degrees bounding such sets is equal to being complete.

Mass problems and intuitionistic higher-order logic

In this paper we study a model of intuitionistic higher-order logic which we call the Muchnik topos. The Muchnik topos may be defined briefly as the category of sheaves of sets over the topological space consisting of the Turing degrees, where the Turing cones form a base for the topology. We note that our Muchnik topos interpretation of intuitionistic mathematics is an extension of the well known Kolmogorov/Muchnik interpretation of intuitionistic propositional calculus via Muchnik degrees, i.e., mass problems under weak reducibility. We introduce a new sheaf representation of the intuitionistic real numbers, the Muchnik reals, which are different from the Cauchy reals and the Dedekind reals. Within the Muchnik topos we obtain a choice principle (8x9y A(x,y)) ) 9w 8xA(x,wx) and a bound