Enumerable Research Articles

Cut, paste and filter

We study the combined and separate effects of three parts of finite multi-test-tube cut and paste DNA computing. First, we reformulate the ideas of Csuhaj-Varjú et al. (Comput. AI 15(2–3) (1996) 211), Freund (Pacific Symposium on Biocomputing, World Scientific Publishing Co., Singapore, 1997), and Priese et al. (in: L. Hunter, R.B. Altman, A. Kieth Dunker, T. Klein (Eds.), Pacific Symposium on Biocomputing, World Scientific Publishing Co., Singapore, 1998, pp. 547–558) about multi-test-tube splicing DNA computing in terms of cutting and pasting as in Pixton's work (Theoret. Comput. Sci. 234 (2000) 135). As Pixton shows (Discrete Appl. Math. 69 (1996) 101; Theoret. Comput. Sci. 234 (2000) 135) with just finite cutting and pasting only regular sets can be obtained from a finite set of initial molecules. The others cited above show that if filtering is allowed between a finite number of test-tubes in which finite splicing occurs then any recursively enumerable set can be obtained using finite sets of initial test-tube contents (Comput. AI 15(2–3) (1996) 211; in: Developments in Language Theory, Aachen, July 6–9, 1999, pp. 275–286; in: L. Hunter, R.B. Altman, A. Kieth Dunker, T. Klein (Eds.), Pacific Symposium on Biocomputing, World Scientific Publishing Co., Singapore, 1998, pp. 547–558). We confirm this result for cutting and pasting. Second, we show that when only finite pasting and filtering between tubes with finite initial contents are allowed then the result must be context free and that any context free language can be so obtained. Finally, we consider several forms of filtering and several ways of combining filtering with cutting, pasting or splicing and show that all are equivalent.

A survey and new results on computer enumeration of polyhex and fusene hydrocarbons.

After a short historic review, we briefly describe a new algorithm for constructive enumeration of polyhex and fusene hydrocarbons. In this process our algorithm also enumerates isomers and symmetry groups of molecules (which implies enumeration of enantiomers). Contrary to previous methods often based on the boundary code or its variants (which records orientation of edges along the boundary) or on the DAST code, which uses a rigid dualist graph (whose vertices are associated with faces and edges with adjacency between them), the proposed algorithm proceeds in two phases. First inner dual graphs are enumerated; then molecules obtained from each of them by specifying angles between adjacent edges are obtained. Favorable computational results are reported. The new algorithm is so fast that output of the structures is by far the most time-consuming part of the process. It thus contributes to enumeration in chemistry, a topic studied for over a century, and is useful in library making, QSAR/QSPR, and synthesis studies.

Density of the Medvedev lattice of ?01 classes

The partial ordering of Medvedev reducibility restricted to the family of Π0 1 classes is shown to be dense. For two disjoint computably enumerable sets, the class of separating sets is an important example of a Π0 1 class, which we call a ``c.e. separating class''. We show that there are no non-trivial meets for c.e. separating classes, but that the density theorem holds in the sublattice generated by the c.e. separating classes.

A decidable multi-modal logic of context

We give a logic for formulas φ−∘ ψ, with the informal reading “ ψ is true in the context described by φ”. These are interpreted as binary modalities, by quantification over an enumerable set of unary modalities c−∘ ψ, meaning “ ψ is true in context c”. The logic allows arbitrary nesting of contexts. A corresponding axiomatic presentation is given, and proven to be decidable, sound, and complete. Previously, quantificational logic of context restricted the nesting of contexts, and was only known to be decidable in very special cases.

Necessary Isomorphism Conditions for Rogers Semilattices of Finite Partially Ordered Sets

We establish a condition that is necessary for Rogers semilattices of computable numberings of finite families of computably enumerable sets to be isomorphic.

A Proof-theoretical Analysis of Semiconstructive Intermediate Theories

In the 80's Pierangelo Miglioli, starting from motivations in the framework of Abstract Data Types and Program Synthesis, introduced semiconstructive theories, a family of “large subsystems” of classical theories that guarantee the computability of functions and predicates represented by suitable formulas. In general, the above computability results are guaranteed by algorithms based on a recursive enumeration of the theorems of the whole system. In this paper we present a family of semiconstructive systems, we call uniformly semiconstructive, that provide computational procedures only involving formulas with bounded complexity. We present several examples of uniformly semiconstructive systems containing Harrop theories, induction principles and some well-known predicate intermediate principles. Among these, we give an account of semiconstructive and uniformly semiconstructive systems which lie between Intuitionistic and Classical Arithmetic and we discuss their constructive incompatibility.

Initial Segments in Rogers Semilattices of \(\Sigma _n^0 \)-Computable Numberings

S. Goncharov and S. Badaev showed that for \(n \geqslant 2\), there exist infinite families whose Rogers semilattices contain ideals without minimal elements. In this connection, the question was posed as to whether there are examples of families that lack this property. We answer this question in the negative. It is proved that independently of a family chosen, the class of semilattices that are principal ideals of the Rogers semilattice of that family is rather wide: it includes both a factor lattice of the lattice of recursively enumerable sets modulo finite sets and a family of initial segments in the semilattice of \(m\)-degrees generated by immune sets.

Contiguity and distributivity in the enumerable Turing degrees — Corrigendum

A computably enumerable Turing degree a is called contiguous iff it contains only a single computably enumerable weak truth table degree (Ladner and Sasso [2]). In [1], the authors proved that a nonzero computably enumerable degree a is contiguous iff it is locally distributive, that is, for all a1, a2, c with a1 ∪a2 = a and c ≤ a, there exist ci, ≤ ai with c1 ∪ c2 = c.To do this we supposed that W was a computably enumerable set and ∪ a computably set with a Turing functional Φ such that ΦW = U. Then we constructed computably enumerable sets A0, A1 and B together with functionals Γ0, Γ1, Γ, and Δ so thatand so as to satisfy all the requirements below.That is, we built a degree-theoretical splitting A0, A1 of W and a set B ≤TW such that if we cannot beat all possible degree-theoretical splittings V0, V1 of B then we were able to witness the fact that U ≤WW (via Λ).After the proof it was observed that the set U of the proof (page 1222, paragraph 4) needed only to be Δ20. It was then claimed that a consequence to the proof was that every contiguous computably enumerable degree was, in fact, strongly contiguous, in the sense that all (not necessarily computably enumerable) sets of the degree had the same weak truth table degree.

ON THE DEFINABILITY OF THE DOUBLE JUMP IN THE COMPUTABLY ENUMERABLE SETS

We show that the double jump is definable in the computably enumerable sets. Our main result is as follows: let [Formula: see text] is the Turing degree of a [Formula: see text] set J ≥T0″}. Let [Formula: see text] such that [Formula: see text] is upward closed in [Formula: see text]. Then there is an ℒ(A) property [Formula: see text] such that [Formula: see text] if and only if there is an A where A ≡T F and [Formula: see text]. A corollary of this is that, for all n ≥ 2, the high n ([Formula: see text]) computably enumerable degrees are invariant in the computably enumerable sets. Our work resolves Martin's Invariance Conjecture.

Computably Enumerable Reals and Uniformly Presentable Ideals

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 Σ3, and conversely that if I is any Σ3 ideal in the computably enumerable wtt degrees then there is a computable enumerable real α such that I = I(α).

Incompleteness and Inconsistency

Graham Priest's In Contradiction (Dordrecht: Martinus Nijhoff Publishers, 1987, chapter 3) contains an argument concerning the intuitive, or 'naive' notion of (arithmetic) proof, or provability. He argues that the intuitively provable arithmetic sentences constitute a recursively enumerable set, which has a Godel sentence which is itself intuitively provable. The incompleteness theorem does not apply, since the set of provable arithmetic sentences is not consistent.

Membrane systems with promoters/inhibitors

The computational model of membrane computing (formalized through membrane systems, also called P systems) is based on the way that biological membranes define compartments, each having its set of molecules and (enzymes enhancing) reactions, with compartments communicating through the transport of molecules through membranes. In this paper we augment the basic model of membrane systems with promoters and inhibitors, which formalize the reaction enhancing and reaction prohibiting roles of various substances (molecules) present in cells. We formalize such membrane systems with promoters/inhibitors and investigate their basic properties. In particular we establish universality results, i.e., we provide characterizations of recursively enumerable sets (of vectors of natural numbers) using these systems. It turns out that systems with promoters/inhibitors achieve universal computations without using the standard “auxiliary” features of membrane systems, for instance, without using catalysts.

Chaitin [formula omitted] numbers, Solovay machines, and Gödel incompleteness

Computably enumerable (c.e.) reals can be coded by Chaitin machines through their halting probabilities. Tuning Solovay's construction of a Chaitin universal machine for which ZFC (if arithmetically sound) cannot determine any single bit of the binary expansion of its halting probability, we show that every c.e.~random real is the halting probability of a universal Chaitin machine for which ZFC cannot determine more than its initial block of 1 bits—as soon as you get a 0, it is all over. Finally, a constructive version of Chaitin information-theoretic incompleteness theorem is proven.

Trivial Reals

Solovay showed that there are noncomputable reals α such that H(α ∥ n) ≤ H(1n)+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 (which means that our construction of an H-trivial computably enumerable set is a particularly easy, injury-free construction of an incomplete c.e. set). Finally, we relate the H-trivials to other classes of “highly nonrandom” reals that have been previously studied.

On orbits, of prompt and low computably enumerable sets

AbstractThis paper concerns automorphisms of the computably enumerable sets. We prove two results relating semilow sets and prompt degrees via automorphisms, one of which is complementary to a recent result of Downey and Harrington. We also show that the property of effective simplicity is not invariant under automorphism, and that in fact every promptly simple set is automorphic to an effectively simple set. A major technique used in these proofs is a modification of the Harrington-Soare version of the method of Harrington-Soare and Cholak for constructing Δ30 automorphisms; this modification takes advantage of a recent result of Soare on the extension of “restricted” automorphisms to full automorphisms.

Definable incompleteness and Friedberg splittings

AbstractWe define a property R(A0, A1) in the partial order of computably enumerable sets under inclusion, and prove that R implies that A0 is noncomputable and incomplete. Moreover, the property is nonvacuous. and the A0 and A1 which we build satisfying R form a Friedberg splitting of their union A, with A1 prompt and A promptly simple. We conclude that A0 and A1 lie in distinct orbits under automorphisms of , yielding a strong answer to a question previously explored by Downey, Stob, and Soare about whether halves of Friedberg splittings must lie in the same orbit.

Congruence relations on lattices of recursively enumerable sets

Let {We}e∈ω be a standard enumeration of the recursively enumerable (r. e.) subsets of ω = {0, 1, 2, …}. The lattice of recursively enumerable sets, is the structure ({We}e∈ω, ∪, ∩). is the sublattice of consisting of the recursive sets.Suppose is a lattice of subsets of ω. ≡ is said to be a congruence relation on if ≡ is an equivalence relation on and if for all U, U′ ∈ and V, V ∈ , if U ≡ U′ and V ≡ V′ then U ∪ U′ ≡ V ∪ V′ and U ∩ U′ ≡ V ∩ V′. [U] = {V ∈ | V ≡ U} is the equivalence class of U. If ≡ is a congruence relation on , the elements of the quotient lattice / ≡ are the equivalence classes of ≡. [U] ∪ [V] is defined as [U ∪ V], and [U] ∩ [V] is defined as [U ∩ V].The quotient lattices of (or of some sublattice ) correspond naturally with the congruence relations which give rise to them, and in turn the congruence relations of sublattices of can be characterized in part by their computational complexity. The aim of the present paper is to characterize congruence relations in some of the most important complexity classes.

Isolation and the high/low hierarchy

Say that a d.c.e. degree d is isolated by a c.e. degree b, if b<d and any c.e. degree c below d is also below b, namely, b is the largest c.e. degree below d. Wu proved that there is a high d.c.e. degree d isolated by a low2 c.e. degree b. In this paper, we improve this result by making b low, not merely low2. RID=""ID="" Mathematics Subject Classification (2000): 03D25, 03D30, 03D35 RID=""ID="" Key words or phrases: Computably enumerable set – d.c.e. degree – Isolation – High/low hierarchy RID=""ID="" Ishmukhametov's research is supported by RFBR grant 01-01-00733, and Wu's research is supported by the Marsden Fund of New Zealand. Wu would like to thank his supervisor, Prof. Rod Downey, for his many helpful suggestions and comments.

On the number of benzenoid hydrocarbons.

We present a new algorithm which allows a radical increase in the computer enumeration of benzenoids b(h) with h hexagons. We obtain b(h) up to h = 35. We prove that b(h) approximately const.kappa(h), prove the rigorous bounds 4.789 < or = kappa < or = 5.905, and estimate that kappa = 5.16193016(8). Finally, we provide strong numerical evidence that the generating function summation operator b(h)z(h) approximately A(z) log(1 - kappa z), estimate A(1/kappa) and predict the subleading asymptotic behavior. We also provide compelling arguments that the mean-square radius of gyration <R(g)(2)>(h) of benzenoids of size h grows as h(2 nu), with nu = 0.64115(5).

On Some Properties of 𝑟-Maximal Sets and 𝑄1–𝑁-Reducibility

Abstract It is shown that if 𝑀1, 𝑀2 are 𝑟-maximal sets and 𝑀1 ≡ 𝑄1–𝑁𝑀2, then 𝑀1 ≡ 𝑚𝑀2. In addition, we prove that there exists a simultaneously 𝑄1–𝑁- and 𝑊-complete recursively enumerable set which is not 𝑠𝑄-complete.