Many-one Degrees Research Articles

Topological reducibilities for discontinuous functions and their structures

In this article, we give a full description of the topological many-one degree structure of real-valued functions, recently introduced by Day—Downey—Westrick. We also clarify the relationship between the Martin conjecture and Day—Downey—Westrick’s topological Turing-like reducibility, also known as parallelized continuous strong Weihrauch reducibility, for single-valued functions: Under the axiom of determinacy, we show that the continuous Weihrauch degrees of parallelizable single-valued functions are well-ordered; and moreover, if f has continuous Weihrauch rank α, then f′ has continuous Weihrauch rank α + 1, where f′(x) is defined as the Turing jump of f(x).

A CHARACTERIZATION OF THE STRONGLY -REPRESENTABLE MANY-ONE DEGREES

Abstract $\eta $ -representations are a way of coding sets in computable linear orders that were first introduced by Fellner in his thesis. Limitwise monotonic functions have been used to characterize the sets with $\eta $ -representations, and give characterizations for several variations of $\eta $ -representations. The one exception is the class of sets with strong $\eta $ -representations, the only class where the order type of the representation is unique.We introduce the notion of a connected approximation of a set, a variation on $\Sigma ^0_2$ approximations. We use connected approximations to give a characterization of the many-one degrees of sets with strong $\eta $ -representations as well new characterizations of the variations of $\eta $ -representations with known characterizations.

The uniform Martin’s conjecture for many-one degrees

We study functions from reals to reals which are uniformly degree-invariant from Turing-equivalence to many-one equivalence, and compare them a cone. We prove that they are in one-to-one correspondence with the Wadge degrees, which can be viewed as a refinement of the uniform Martin's conjecture for uniformly invariant functions from Turing- to Turing-equivalence. Our proof works in the general case of many-one degrees on $\mathcal{Q}^\omega$ and Wadge degrees of functions $\omega^\omega\to\mathcal{Q}$ for any better quasi ordering $\mathcal{Q}$.

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.

Many-one reductions and the category of multivalued functions

Multivalued functions are common in computable analysis (built upon the Type 2 Theory of Effectivity), and have made an appearance in complexity theory under the monikersearch problemsleading to complexity classes such as PPAD and PLS being studied. However, a systematic investigation of the resulting degree structures has only been initiated in the former situation so far (the Weihrauch-degrees).A more general understanding is possible, if the category-theoretic properties of multivalued functions are taken into account. In the present paper, the category-theoretic framework is established, and it is demonstrated that many-one degrees of multivalued functions form a distributive lattice under very general conditions, regardless of the actual reducibility notions used (e.g. Cook, Karp, Weihrauch).Beyond this, an abundance of open questions arises. Some classic results for reductions between functions carry over to multivalued functions, but others do not. The basic theme here again depends on category-theoretic differences between functions and multivalued functions.

Some new results on decidability for elementary algebra and geometry

We carry out a systematic study of decidability for theories (a) of real vector spaces, inner product spaces, and Hilbert spaces and (b) of normed spaces, Banach spaces and metric spaces, all formalized using a 2-sorted first-order language. The theories for list (a) turn out to be decidable while the theories for list (b) are not even arithmetical: the theory of 2-dimensional Banach spaces, for example, has the same many-one degree as the set of truths of second-order arithmetic.We find that the purely universal and purely existential fragments of the theory of normed spaces are decidable, as is the ∀∃ fragment of the theory of metric spaces. These results are sharp of their type: reductions of Hilbertʼs 10th problem show that the ∃∀ fragments for metric and normed spaces and the ∀∃ fragment for normed spaces are all undecidable.

Low sets without subsets of higher many-one degree

Given a reducibility ⩽r, we say that an infinite set A is r-introimmune if A is not r-reducible to any of its subsets B with |A\B| = ∞. We consider the many-one reducibility ⩽m and we prove the existence of a low1 m-introimmune set in Π01 and the existence of a low1 bi-m-introimmune set.

Proof system representations of degrees of disjoint NP-pairs

Let D be a set of many-one degrees of disjoint NP-pairs. We define a proof system representation of D to be a set of propositional proof systems P such that each degree in D contains the canonical NP-pair of a corresponding proof system in P and the degree structure of D is reflected by the simulation order among the corresponding proof systems in P . We also define a nesting representation of D to be a set of NP-pairs S such that each degree in D contains a representative NP-pair in S and the degree structure of D is reflected by the inclusion relations among their representative NP-pairs in S . We show that proof system and nesting representations both exist for D if the lower span of each degree in D overlaps with D on a finite set only. In particular, a linear chain of many-one degrees of NP-pairs has both a proof system representation and a nesting representation. This extends a result by Glaßer et al. (2009). We also show that in general D has a proof system representation if it has a nesting representation where all representative NP-pairs share the same set as their first components.

On the complexity of kings

A king in a directed graph is a vertex from which each vertex in the graph can be reached through paths of length at most two. There is a broad literature on tournaments (completely oriented digraphs), and it has been known for more than half a century that all tournaments have at least one king. Recently, kings have proven useful in theoretical computer science, in particular in the study of the complexity of reachability problems and semifeasible sets. In this article, we study the complexity of recognizing kings. For each succinctly specified family of tournaments, the king problem is already known to belong to Π 2 p . We prove that the complexity of kingship problems is a rich enough vocabulary to pinpoint every nontrivial many-one degree in Π 2 p . That is, we show that every set in Π 2 p other than 0̸ and Σ ∗ is equivalent to a king problem under ≤ m p -reductions. Indeed, we show that the equivalence can even be realized by relatively simple padding, and holds even if the notion of kings is redefined to refer to k -kings (for any fixed k ≥ 2 )—vertices from which all vertices can be reached through paths of length at most k . In contrast, we prove that for each succinctly specified family of tournaments the source problem (the problem of deciding whether a given vertex v has the property that there exists a k such that v is a k -king) also falls within Π 2 p , yet cannot be Π 2 p -complete—or even NP -hard—unless P = NP . Using these and related techniques, we obtain a broad range of additional results about the complexity of king problems, diameter problems, and radius problems. It follows easily from our proof approach that the problem of testing kingship in succinctly specified graphs (which need not be tournaments) is Π 2 p -complete. We show that the radius problem for arbitrary succinctly represented graphs is Σ 3 p -complete, but that in contrast the diameter problem for arbitrary succinctly represented graphs (or even tournaments) is Π 2 p -complete.

Indifferent Sets

We define the notion of indifferent set with respect to a given class of {0,1}-sequences. Roughly, for a set A in the class, a set of natural numbers I is indifferent for A with respect to the class if it does not matter how we change A at the positions in I: the new sequence continues to be in the given class. We are especially interested in studying those sets that are indifferent with respect to classes containing different types of stochastic sequences. For the class of Martin-Löf random sequences, we show that every random sequence has an infinite indifferent set and that there is no universal indifferent set. We show that indifferent sets must be sparse, in fact sparse enough to decide the halting problem. We prove the existence of co-c.e. indifferent sets, including a co-c.e. set that is indifferent for every 2-random sequence with respect to the class of random sequences. For the class of absolutely normal numbers, we show that there are computable indifferent sets with respect to that class and we conclude that there is an absolutely normal real number in every non-trivial many-one degree.

Effective Strong Dimension in Algorithmic Information and Computational Complexity

The two most important notions of fractal dimension are Hausdorff dimension, developed by Hausdorff [Math. Ann., 79 (1919), pp. 157-179], and packing dimension, developed independently by Tricot [Math. Proc. Cambridge Philos. Soc., 91 (1982), pp. 57-74] and Sullivan [Acta Math., 153 (1984), pp. 259-277]. Both dimensions have the mathematical advantage of being defined from measures, and both have yielded extensive applications in fractal geometry and dynamical systems. Lutz [Proceedings of the 15th IEEE Conference on Computational Complexity, Florence, Italy, 2000, IEEE Computer Society Press, Piscataway, NJ, 2000, pp. 158-169] has recently proven a simple characterization of Hausdorff dimension in terms of gales, which are betting strategies that generalize martingales. Imposing various computability and complexity constraints on these gales produces a spectrum of effective versions of Hausdorff dimension, including constructive, computable, polynomial-space, polynomial-time, and finite-state dimensions. Work by several investigators has already used these effective dimensions to shed significant new light on a variety of topics in theoretical computer science. In this paper we show that packing dimension can also be characterized in terms of gales. Moreover, even though the usual definition of packing dimension is considerably more complex than that of Hausdorff dimension, our gale characterization of packing dimension is an exact dual of—and every bit as simple as—the gale characterization of Hausdorff dimension. Effectivizing our gale characterization of packing dimension produces a variety of effective strong dimensions, which are exact duals of the effective dimensions mentioned above. In general (and in analogy with the classical fractal dimensions), the effective strong dimension of a set or sequence is at least as great as its effective dimension, with equality for sets or sequences that are sufficiently regular. We develop the basic properties of effective strong dimensions and prove a number of results relating them to fundamental aspects of randomness, Kolmogorov complexity, prediction, Boolean circuit-size complexity, polynomial-time degrees, and data compression. Aside from the above characterization of packing dimension, our two main theorems are the following. 1. If $\vec{\beta} = (\beta_0,\beta_1,\ldots)$ is a computable sequence of biases that are bounded away from 0 and $R$ is random with respect to $\vec{\beta}$, then the dimension and strong dimension of $R$ are the lower and upper average entropies, respectively, of $\vec{\beta}$. 2. For each pair of $\Delta^0_2$-computable real numbers $0 < \alpha \le \beta \le 1$, there exists $A \in {\rm E}$ such that the polynomial-time many-one degree of $A$ has dimension $\alpha$ in E and strong dimension $\beta$ in E. Our proofs of these theorems use a new large deviation theorem for self-information with respect to a bias sequence $\vec{\beta}$ that need not be convergent.

Sets without Subsets of Higher Many-One Degree

Previously, both Soare and Simpson considered sets without subsets of higher $\leq_T$, define the concept of a $\leq_r$-introimmune set. For the most common reducibilities $\leq_r$, a set does not contain subsets of higher $\leq_r$-degree if and only if it is $\leq_r$-introimmune. In this paper we consider $\leq_m$-introimmune and $\leq^P_m$-introimmune sets and examine how structurally easy such sets can be. In other words we ask, What is the smallest class of the Kleene's Hierarchy containing $\leq_r$-introimmune sets for $\leq_r\in\{\leq_m,\leq^P_m\}$? We answer the question by proving the existence of $\leq_m$-introimmune sets in the class $\Pi^0_1$, bi-$\leq_m$-introimmune sets in $\Delta^0_2$, and bi-$\leq^P_m$-introimmune sets in $\Delta^0_1$.

On the structures inside truth-table degrees

AbstractThe following theorems on the structure inside nonrecursive truth-table degrees are established: Dëgtev's result that the number of bounded truth-table degrees inside a truth-table degree is at least two is improved by showing that this number is infinite. There are even infinite chains and antichains of bounded truth-table degrees inside every truth-table degree. The latter implies an affirmative answer to the following question of Jockusch: does every truth-table degree contain an infinite antichain of many-one degrees? Some but not all truth-table degrees have a least bounded truth-table degree. The technique to construct such a degree is used to solve an open problem of Beigel, Gasarch and Owings: there are Turing degrees (constructed as hyperimmune-free truth-table degrees) which consist only of 2-subjective sets and therefore do not contain any objective set. Furthermore, a truth-table degree consisting of three positive degrees is constructed where one positive degree consists of enumerable semirecursive sets, one of coenumerable semirecursive sets and one of sets, which are neither enumerable nor coenumerable nor semirecursive. So Jockusch's result that there are at least three positive degrees inside a truth-table degree is optimal. The number of positive degrees inside a truth-table degree can also be some other odd integer as for example nineteen, but it is never an even finite number.

Model theory of the computably enumerable many-one degrees

Model theory of the computably enumerable many-one degrees

On MODkP Counting Degrees

For a prime k, the embeddability of finite lattices are discussed for various kind of the MODkP degrees of recursive sets. In particular, all finite lattices are embeddable into the MODkP Turing degrees, whereas the non distributive lattice M3 is embeddable into the MOD2P many-one degrees but N5 is not.

Automorphisms in the PTIME-Turing degrees of recursive sets

We consider questions related to the rigidity of the structure R, the PTIME-Turing degrees of recursive sets of strings together with PTIME-Turing reducibility, ⩽ pT , and related structures; do these structures have nontrivial automorphisms? We prove that there is a nontrivial automorphism of an ideal of R. This can be rephrased in terms of partial relativizations. We consider the sets which are PTIME-Turing computable from a set A, and call this class PTIME A . Our result can be stated as follows: There is an oracle, A, relative to which the PTIME-Turing degrees are not rigid (i.e. there is a nontrivial automorphism of the structure 〈 PTIME A , ⩽ pT 〉). Furthermore, the automorphism can be made to preserve the complexity classes DTIME A ( n k ) (the collection of sets computable from A by a deterministic computation with time bound of order n k ) for all k ⩾ 1, or to move any DTIME A ( n k ) for n ⩾ 2. From the existence of such an automorphism we conclude as a corollary that there is an oracle A relative to which the classes DTIME( n k ) are not definable over R. We carry out the corresponding partial relativization for the many-one degrees to construct an oracle, A, relative to which the PTIMA-many-one degrees relative to A have a nontrivial automorphism, and one relative to which the lattice of sets in PTIME A under inclusion have a nontrivial automorphism. The proof is phrased as a forcing argument; we construct the set A to meet a particular collection of dense sets in our notion of forcing. Roughly, the dense sets will guarantee that if A meets these sets and we split A into two pieces, A 0 and A 1, in a simple way, and then switching the roles of A 0 and A 1 in all computations from A will produce an automorphism of the ideal of PTIMA-degrees below A. We force A 0 and A 1 to have different PTIME-Turing degree; this will then make the automorphism nontrivial. An appropriately generic set A is constructed using a priority argument.

Decidability of the two-quantifier theory of the recursively enumerable weak truth-table degrees and other distributive upper semi-lattices

AbstractWe give a decision procedure for the ∀∃-theory of the weak truth-table (wtt) degrees of the recursively enumerable sets. The key to this decision procedure is a characterization of the finite lattices which can be embedded into the r.e.wtt-degrees by a map which preserves the least and greatest elements: a finite lattice has such an embedding if and only if it is distributive and the ideal generated by its cappable elements and the filter generated by its cuppable elements are disjoint.We formulate general criteria that allow one to conclude that a distributive upper semi-lattice has a decidable two-quantifier theory. These criteria are applied not only to the weak truth-table degrees of the recursively enumerable sets but also to various substructures of the polynomial many-one (pm) degrees of the recursive sets. These applications to thepmdegrees require no new complexity-theoretic results. The fact that thepm-degrees of the recursive sets have a decidable two-quantifier theory answers a question raised by Shore and Slaman in [21].

Undecidable fragments of elementary theories

We introduce a general framework to prove undecidability of fragments. This is applied to fragments of theories arising in algebra and recursion theory. For instance, the V3V-theories of the class of finite distributive lattices and of the p.o. of recursively enumerable many-one degrees are shown to be undecidable.

On the topological size of p-m-complete degrees

All polynomial many-one degrees are shown to be of second Baire category in the superset topology when witness functions are allowed to run in 2 log h n time, for any h . Any improvement of this result for the complete p-m-degrees of RE, EXP or NP implies P ≠ NP or the nonisomorphism of the NP-complete sets.

The degrees of conditional problems

AbstractIn this paper we define and study conditional problems and their degrees. The main result is that the class of conditional degrees is a lattice extending the ordinary Turing degrees and it is dense. These properties are not shared by ordinary Turing degrees. We show that the class of conditional many-one degrees is a distributive lattice. We also consider properties of semidecidable problems and their degrees, which are analogous to r.e. sets and degrees.