Related Computer Research Articles

FINDING THE LIMIT OF INCOMPLETENESS I

AbstractIn this paper, we examine the limit of applicability of Gödel’s first incompleteness theorem ($\textsf {G1}$ for short). We first define the notion “$\textsf {G1}$ holds for the theory $T$”. This paper is motivated by the following question: can we find a theory with a minimal degree of interpretation for which $\textsf {G1}$ holds. To approach this question, we first examine the following question: is there a theory T such that Robinson’s $\mathbf {R}$ interprets T but T does not interpret $\mathbf {R}$ (i.e., T is weaker than $\mathbf {R}$ w.r.t. interpretation) and $\textsf {G1}$ holds for T? In this paper, we show that there are many such theories based on Jeřábek’s work using some model theory. We prove that for each recursively inseparable pair $\langle A,B\rangle $, we can construct a r.e. theory $U_{\langle A,B\rangle }$ such that $U_{\langle A,B\rangle }$ is weaker than $\mathbf {R}$ w.r.t. interpretation and $\textsf {G1}$ holds for $U_{\langle A,B\rangle }$. As a corollary, we answer a question from Albert Visser. Moreover, we prove that for any Turing degree $\mathbf {0}< \mathbf {d}<\mathbf {0}^{\prime }$, there is a theory T with Turing degree $\mathbf {d}$ such that $\textsf {G1}$ holds for T and T is weaker than $\mathbf {R}$ w.r.t. Turing reducibility. As a corollary, based on Shoenfield’s work using some recursion theory, we show that there is no theory with a minimal degree of Turing reducibility for which $\textsf {G1}$ holds.

The Non-hardness of Approximating Circuit Size

The Minimum Circuit Size Problem (\(\mathsf {MCSP}\)) has been the focus of intense study recently; \(\mathsf {MCSP}\) is hard for \(\mathsf {SZK}\) under rather powerful reductions [4], and is provably not hard under “local” reductions computable in \({\mathsf {TIME}}(n^{0.49})\) [22]. The question of whether \(\mathsf {MCSP}\) is \(\mathsf {NP}\)-hard (or indeed, hard even for small subclasses of \(\mathsf {P}\)) under some of the more familiar notions of reducibility (such as many-one or Turing reductions computable in polynomial time or in \(\mathsf {AC}^0\)) is closely related to many of the longstanding open questions in complexity theory [7, 8, 16, 17, 18, 20, 22].

NON-DENSITY OF GENERIC M-REDUCIBILITY FOR C.E. SETS

Generic-case approach to algorithmic problems was suggested by I. Kapovich, A. Myasnikov, V. Shpilrain and P. Schupp in 2003. This approach studies behavior of an al-gorithm on typical (almost all) inputs and ignores the rest of inputs. C. Jockusch and P. Schupp in 2012 began the study of generic computability in the context of classical computability theory. In particular, they defined a generic analog of Turing reducibility. A. Rybalov in 2018 introduced a generic analog of classical m-reducibility. In this paper we study the generic m-reducibility for c.e. sets and prove that unlike classical m-reducibility, generic m-reducibility does not have the density property for c.e. sets.

On Decidable Categoricity and Almost Prime Models

The complexity of isomorphisms for computable and decidable structures plays an important role in computable model theory. Goncharov [26] defined the degree of decidable categoricity of a decidable model $$\mathcal {M} $$ to be the least Turing degree, if it exists, which is capable of computing isomorphisms between arbitrary decidable copies of $$\mathcal {M} $$ . If this degree is $$\mathbf {0} $$ , we say that the structure $$\mathcal {M} $$ is decidably categorical. Goncharov established that every computably enumerable degree is the degree of categoricity of a prime model, and Bazhenov showed that there is a prime model with no degree of categoricity. Here we investigate the degrees of categoricity of various prime models with added constants, also called almost prime models. We relate the degree of decidable categoricity of an almost prime model $$\mathcal {M} $$ to the Turing degree of the set $$C(\mathcal {M}) $$ of complete formulas. We also investigate uniform decidable categoricity, characterizing it by primality of $$\mathcal {M} $$ and Turing reducibility of $$C(\mathcal {M}) $$ to the theory of $$\mathcal {M} $$ .

Turing degrees and automorphism groups of substructure lattices

The study of automorphisms of computable and other structures connects computability theory with classical group theory. Among the noncomputable 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 $\mathbf{d}$, we investigate the groups of $\mathbf{d}$ -computable automorphisms of the lattice of $\mathbf{d}$-computably enumerable vector spaces, of the interval Boolean algebra $\mathcal{B}_{\eta }$ of the ordered set of rationals, and of the lattice of $\mathbf{d}$ -computably enumerable subalgebras of $\mathcal{B}_{\eta }$. For these groups we show that Turing reducibility 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 $\mathbf{d}$ , $\mathbf{d^{\prime \prime }}$.

Detecting mixed-unitary quantum channels is NP-hard

A quantum channel is said to be amixed-unitarychannel if it can be expressed as a convex combination of unitary channels. We prove that, given the Choi representation of a quantum channelΦ, it is NP-hard with respect to polynomial-time Turing reductions to determine whether or notΦis a mixed-unitary channel. This hardness result holds even under the assumption thatΦis not within an inverse-polynomial distance (in the dimension of the space upon whichΦacts) of the boundary of the mixed-unitary channels.

Turing Degrees and Automorphism Groups of Substructure Lattices

The study of automorphisms of computable and other structures connects computability theory with classical group theory. Among the noncomputable countable structures, computably enumerable structures are one of the most important objects of investigation in computable model theory. Here 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 reducibility 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 d′′ of d.

On generic NP-completeness of the graph clustering problem

Generic-case approach to algorithmic problems was suggested by Miasnikov, Kapovich, Schupp and Shpilrain in 2003. This approach studies behavior of an algorithm on typical (almost all) inputs and ignores the rest of inputs. In this paper, we study the generic complexity of the problem of clustering graphs. In this problem the structure of relations of objects is presented as a graph: vertices correspond to objects, and edges connect similar objects. It is required to divide a set of objects into disjoint groups (clusters) to minimize the number of connections between clusters and the number of missing links within clusters. We prove that the graph clustering problem is NP-hard with respect to generic analog of polynomial Turing reduction. Supported by Russian Science Foundation, grant 18-71-10028.

Quasiminimal pairs for c.e. degrees of generic and coarse reducibilities

Generic approach to algorithmic problems in combinatorial group theory was suggested in 2003 by Kapovich, Myasnikov, Schupp and Shpilrain. This approach deals with algorithms, which solves algorithmic problems on ”most” of the inputs (i.e., on a generic set) instead of the entire domain and output undefined answer (do not halt) on the rest of inputs (a negligible set). Generic analog of Turing reducibility was introduced by Jockusch and Schupp in 2012. Later Hirschfeldt, Jockusch, Kuyper and Schupp defined coarse reducibility. In this paper we prove that there exist quasiminimal pairs for generic and coarse reducibilities of computably enumerable (c.e.) degrees. The work was supported by Russian Science Foundation, grant number 18-71-10028.

Turing reducibility in the fine hierarchy

In the late 1980s, Selivanov used typed Boolean combinations of arithmetical sets to extend the Ershov hierarchy beyond Δ20 sets. Similar to the Ershov hierarchy, Selivanov's fine hierarchy{Σγ}γ<ε0 proceeds through transfinite levels below ε0 to cover all arithmetical sets. In this paper we use a 0‴ construction to show that the Σ30 Turing degrees are properly contained in the Σωω+2 Turing degrees (to be defined); intuitively, the latter class consists of “non-uniformly Σ30 sets” in the sense that will be clarified in the introduction. The question whether the hierarchy was proper at this level with respect to Turing reducibility remained open for over 20 years.

The complexity of simulating local measurements on quantum systems

An important task in quantum physics is the estimation of local quantities for ground states of local Hamiltonians. Recently, [Ambainis, CCC 2014] defined the complexity classPQMA[log], and motivated its study by showing that the physical task of estimating the expectation value of a local observable against the ground state of a local Hamiltonian isPQMA[log]-complete. In this paper, we continue the study ofPQMA[log], obtaining the following lower and upper bounds.Lower bounds (hardness results): - ThePQMA[log]-completeness result of [Ambainis, CCC 2014] requiresO(log⁡n)-local observables and Hamiltonians. We show that simulating even asingle qubitmeasurement on ground states of5-local Hamiltonians isPQMA[log]-complete, resolving an open question of Ambainis.- We formalize the complexity theoretic study of estimating two-point correlation functions against ground states, and show that this task is similarlyPQMA[log]-complete. - We identify a flaw in [Ambainis, CCC 2014] regarding aPUQMA[log]-hardness proof for estimating spectral gaps of local Hamiltonians. By introducing a ``query validation'' technique, we build on [Ambainis, CCC 2014] to obtainPUQMA[log]-hardness for estimating spectral gaps under polynomial-time Turing reductions. Upper bounds (containment in complexity classes): -PQMA[log]is thought of as ``slightly harder'' than QMA. We justify this formally by exploiting the hierarchical voting technique of [Beigel, Hemachandra, Wechsung, SCT 1989] to showPQMA[log]⊆PP. This improves the containmentQMA⊆PP[Kitaev, Watrous, STOC 2000]. This work contributes a rigorous treatment of the subtlety involved in studying oracle classes in which the oracle solves apromiseproblem. This is particularly relevant for quantum complexity theory, where most natural classes such as BQP and QMA are defined as promise classes.

On a generic Turing reducibility of computably enumerable sets

Kapovich, Myasnikov, Schupp and Shpilrain in 2003 developed generic approach to algorithmic problems, which considers an algorithmic problem on “most” of the inputs (i.e., on a generic set) instead of the entire domain and ignores it on the rest of inputs (a negligible set). Jockusch and Schupp in 2012 defined a generic analog of Turing reducibility. In this paper we investigate a generic reducibility of computably enumerable (c.e.) sets. We prove that there exists a pair of incomparable c.e. sets, that there is a complete c.e. set, that there are no minimal and maximal c.e. sets. Also we prove some analog of classical Sacks density theorem. Supported by Russian Science Foundation, grant 18-71-10028.

Domination in some subclasses of bipartite graphs

A set D⊆V is called a dominating set of G=(V,E) if |NG[v]∩D|≥1 for all v∈V. The Minimum Domination problem is to find a dominating set of minimum cardinality of the input graph. In this paper, we study the Minimum Domination problem for star-convex bipartite graphs, circular-convex bipartite graphs and triad-convex bipartite graphs. It is known that the Minimum Domination Problem for a graph with n vertices can be approximated with an approximation ratio of lnn+1. However, we show that for any ϵ>0, the Minimum Domination problem does not admit a (1−ϵ)lnn-approximation algorithm even for star-convex bipartite graphs with n vertices unless NP ⊆ DTIME(nO(loglogn)). On the positive side, we propose polynomial time algorithms for computing a minimum dominating set of circular-convex bipartite graphs and triad-convex bipartite graphs, by making polynomial time Turing reductions from the Minimum Domination problem for these graph classes to the Minimum Domination problem for convex bipartite graphs.

The Conjugacy Problem in Free Solvable Groups and Wreath Products of Abelian Groups is in TC0

We show that the conjugacy problem in a wreath product A ≀ B is uniform-TC0-Turing-reducible to the conjugacy problem in the factors A and B and the power problem in B. If B is torsion free, the power problem in B can be replaced by the slightly weaker cyclic submonoid membership problem in B. Moreover, if A is abelian, the cyclic subgroup membership problem suffices, which itself is uniform-AC0-many-one-reducible to the conjugacy problem in A ≀ B. Furthermore, under certain natural conditions, we give a uniform TC0 Turing reduction from the power problem in A ≀ B to the power problems of A and B. Together with our first result, this yields a uniform TC0 solution to the conjugacy problem in iterated wreath products of abelian groups – and, by the Magnus embedding, also in free solvable groups.

Jumps of computably enumerable equivalence relations

We study computably enumerable equivalence relations (or, ceers), under computable reducibility ≤, and the halting jump operation on ceers. We show that every jump is uniform join-irreducible, and thus join-irreducible. Therefore, the uniform join of two incomparable ceers is not equivalent to any jump. On the other hand there exist ceers that are not equivalent to jumps, but are uniform join-irreducible: in fact above any non-universal ceer there is a ceer which is not equivalent to a jump, and is uniform join-irreducible. We also study transfinite iterations of the jump operation. If a is an ordinal notation, and E is a ceer, then let E(a) denote the ceer obtained by transfinitely iterating the jump on E along the path of ordinal notations up to a. In contrast with what happens for the Turing jump and Turing reducibility, where if a set X is an upper bound for the A-arithmetical sets then X(2) computes A(ω), we show that there is a ceer R such that R≥Id(n), for every finite ordinal n, but, for all k, R(k)≱Id(ω) (here Id is the identity equivalence relation). We show that if a,b are notations of the same ordinal less than ω2, then E(a)≡E(b), but there are notations a,b of ω2 such that Id(a) and Id(b) are incomparable. Moreover, there is no non-universal ceer which is an upper bound for all the ceers of the form Id(a) where a is a notation for ω2.

Bounded Turing Reductions and Data Processing Inequalities for Sequences

A data processing inequality states that the quantity of shared information between two entities (e.g. signals, strings) cannot be significantly increased when one of the entities is processed by certain kinds of transformations. In this paper, we prove several data processing inequalities for sequences, where the transformations are bounded Turing functionals and the shared information is measured by the lower and upper mutual dimensions between sequences. We show that, for all sequences X, Y, and Z, if Z is computable Lipschitz reducible to X, then $$mdim(Z:Y) \leq mdim(X:Y) \text{ and} Mdim(Z:Y) \leq Mdim(X:Y). $$ We also show how to derive different data processing inequalities by making adjustments to the computable bounds of the use of a Turing functional. The yield of a Turing functional Φ S with access to at most n bits of the oracle S is the smallest input $m \in \mathbb {N}$ such that ${\Phi }^{S \upharpoonright n}(m)\uparrow $ . We show how to derive reverse data processing inequalities (i.e., data processing inequalities where the transformation may significantly increase the shared information between two entities) for sequences by applying computable bounds to the yield of a Turing functional.

Where join preservation fails in the bounded Turing degrees of c.e. sets

We look at the question of join preservation for bounded Turing reducibilities r and r′ such that r is stronger than r′. We say that join preservation holds for two reducibilities r and r′ if every join in the computably enumerable (c.e.) r-degrees is also a join in the c.e. r′-degrees. We consider the class of monotone admissible (uniformly) bounded Turing reducibilities, i.e. the reflexive and transitive Turing reducibilities with use bounded by a function that is contained in a (uniformly computable) family of strictly increasing computable functions. This class contains for example identity bounded Turing (ibT-) and computable Lipschitz (cl-) reducibility. Our main result is that join preservation fails for cl and any strictly weaker monotone admissible uniformly bounded Turing reducibility (Theorem 10). We also look at the dual question of meet preservation and show that for all monotone admissible bounded Turing reducibilities r and r′ such that r is stronger than r′, meet preservation holds. Finally, we completely solve the question of join and meet preservation in the classical reducibilities 1, m, tt, wtt, and T. This is an extended version of our article [18].

Continuous higher randomness

We investigate the role of continuous reductions and continuous relativization in the context of higher randomness. We define a higher analogue of Turing reducibility and show that it interacts well with higher randomness, for example with respect to van Lambalgen’s theorem and the Miller–Yu/Levin theorem. We study lowness for continuous relativization of randomness, and show the equivalence of the higher analogues of the different characterizations of lowness for Martin-Löf randomness. We also characterize computing higher [Formula: see text]-trivial sets by higher random sequences. We give a separation between higher notions of randomness, in particular between higher weak 2-randomness and [Formula: see text]-randomness. To do so we investigate classes of functions computable from Kleene’s [Formula: see text] based on strong forms of the higher limit lemma.

Bounded low and high sets

Anderson and Csima (Notre Dame J Form Log 55(2):245–264, 2014) defined a jump operator, the bounded jump, with respect to bounded Turing (or weak truth table) reducibility. They showed that the bounded jump is closely related to the Ershov hierarchy and that it satisfies an analogue of Shoenfield jump inversion. We show that there are high bounded low sets and low bounded high sets. Thus, the information coded in the bounded jump is quite different from that of the standard jump. We also consider whether the analogue of the Jump Theorem holds for the bounded jump: do we have \(A \le _{bT}B\) if and only if \(A^b \le _1 B^b\)? We show the forward direction holds but not the reverse.

Kobayashi compressibility

Kobayashi [21] introduced a uniform notion of compressibility of infinite binary sequences X in terms of relative Turing computations with sub-identity use of the oracle. Given f:N→N we say that X is f-compressible if there exists Y such that for each n we compute X↾n using at most the first f(n) bits of the oracle Y. Kobayashi compressibility has remained a relatively obscure notion, with the exception of some work on resource bounded Kolmogorov complexity. The main goal of this note is to show that it is relevant to a number of topics in current research on algorithmic randomness.We prove that Kobayashi compressibility can be used in order to define Martin-Löf randomness, a strong version of finite randomness and Kurtz randomness, strictly in terms of Turing reductions. Moreover these randomness notions naturally correspond to Turing reducibility, weak truth-table reducibility and truth-table reducibility respectively. Finally we discuss Kobayashi's main result from [21] regarding the compressibility of computably enumerable sets, and provide additional related original results.