Enumerable Research Articles

Degrees of d. c. e. reals

AbstractA real α is called a c. e. real if it is the halting probability of a prefix free Turing machine. Equivalently, α is c. e. if it is left computable in the sense that L(α) = {q ∈ ℚ : q ≤ α} is a computably enumerable set. The natural field formed by the c. e. reals turns out to be the field formed by the collection of the d. c. e. reals, which are of the form α—β, where α and β are c. e. reals. While c. e. reals can only be found in the c. e. degrees, Zheng has proven that there are Δ02 degrees that are not even n‐c. e. for any n and yet contain d. c. e. reals, where a degree is n‐c. e. if it contains an n‐c. e. set. In this paper we will prove that every ω‐c. e. degree contains a d. c. e. real, but there are ω + 1‐c. e. degrees and, hence Δ02 degrees, containing no d. c. e. real. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Characterising the Martin-Löf random sequences using computably enumerable sets of measure one

A sequence ω is Martin-Löf random if and only if it appears early in every Lebesgue measure one set of computably enumerable intervals.

Elimination of quantifiers from arithmetical formulas defining recursively enumerable sets

This is a short survey of known results about elimination of quantifiers over natural numbers, and some implications of these results on the power of computer algebra systems.

Mixed Variable Optimization of a Load-Bearing Thermal Insulation System Using a Filter Pattern Search Algorithm

This paper describes the optimization of a load-bearing thermal insulation system characterized by hot and cold surfaces with a series of heat intercepts and insulators between them. The optimization problem is represented as a mixed variable programming (MVP) problem with nonlinear constraints, in which the objective is to minimize the power required to maintain the heat intercepts at fixed temperatures so that one surface is kept sufficiently cold. MVP problems are more general than mixed integer nonlinear programming (MINLP) problems in that the discrete variables are categorical; i.e., they must always take on values from a predefined enumerable set or list. Thus, traditional approaches that use branch and bound techniques cannot be applied. In a previous paper, a linearly constrained version of this problem was solved numerically using the Audet-Dennis generalized pattern search (GPS) method for MVP problems. However, this algorithm may not work for problems with general nonlinear constraints. A new algorithm that extends that of Audet and Dennis by incorporating a filter to handle nonlinear constraints makes it possible to solve the more general problem. Additional nonlinear constraints on stress, mass, and thermal contraction are added to that of the previous work in an effort to find a more realistic feasible design. Several computational experiments show a substantial improvement in power required to maintain the system, as compared to the previous literature. The addition of the new constraints leads to a very different design without significantly changing the power required. The results demonstrate that the new algorithm can be applied to a very broad class of optimization problems, for which no previous algorithm with provable convergence results could be applied.

On the power of membrane division in P systems

First, we consider P systems with active membranes, hence with the possibility that the membranes can be divided, with non-cooperating evolution rules (the objects always evolve separately). These systems are known to be able to solve NP-complete problems in linear time. Here we give a normal form theorem for such systems: their computational universality is preserved even if only the elementary membranes are divided. The possibility of solving SAT in linear time is preserved only when non-elementary membranes may also be divided under the influence of objects in their region. Second, we consider a slight generalization, namely, we allow that a membrane can produce by division both a copy of itself and a copy of a membrane with a different label; again, only elementary membranes may be divided. In this case, we prove that the hierarchy on the maximal number of membranes present in the system collapses: three membranes at a time are sufficient in order to characterize the recursively enumerable sets of vectors of natural numbers. This result is optimal, two membranes are shown not to be sufficient. Third, we consider P systems with cooperating rules (several objects may evolve together). Making use of this powerful feature, we show that many NP-complete problems can be solved in linear time in a quite uniform way (by systems which are very similar to each other), using only elementary membranes division (and not further ingredients, such as electrical charges). The degree of cooperation is minimal: two objects at a time.

Lower bounds and the hardness of counting properties

Rice's Theorem states that all nontrivial language properties of recursively enumerable sets are undecidable. Borchert and Stephan (Math. Logic Quart. 46 (4) (2000) 489–504) started the search for complexity-theoretic analogs of Rice's Theorem, and proved that every nontrivial counting property of boolean circuits is UP-hard. Hemaspaandra and Rothe (Theoret. Comput. Sci. 244 (1–2) (2000) 205–217) improved the UP-hardness lower bound to UP O(1)-hardness. The present paper raises the lower bound for nontrivial counting properties from UP O(1)-hardness to FewP-hardness, i.e., from constant-ambiguity nondeterminism to polynomial-ambiguity nondeterminism. Furthermore, we prove that no relativizable technique can raise this lower bound to FewP-⩽ 1- tt p -hardness. We also prove a Rice-style theorem for NP, namely that every nontrivial language property of NP sets is NP-hard, and for a broad class of leaf-language classes we prove a sufficient condition for the natural analog of Rice's Theorem to hold.

On Kurtz randomness

Kurtz randomness is a notion of algorithmic randomness for real numbers. In particular a real α is called Kurtz random (or weakly random) iff it is contained in every computably enumerable set U of (Lebesgue) measure 1. We prove a number of characterizations of this notion, relating it to other notions of randomness such as the well-known notions of computable randomness, Martin-Löf randomness and Schnorr randomness. For the first time we give machine characterizations of Kurtz randomness. Whereas the Turing degree of every Martin-Löf random c.e. real is the complete degree, and the degrees of Schnorr random c.e. reals are all high, we show that Kurtz random c.e. reals occur in every non-zero c.e. degree. Additionally, we show that the sets that are low for Kurtz randomness are all hyperimmune and include those that are low for Schnorr randomness, characterized previously by Terwijn and Zambella.

Splittings of effectively speedable sets and effectively levelable sets

AbstractWe prove that a computably enumerable set A is effectively speedable (effectively levelable) if and only if there exists a splitting (A0, A1) of A such that both A0 and A1 are effectively speedable (effectively levelable). These results answer two questions raised by J. B. Remmel.

A quantitative risk assessment of waterborne cryptosporidiosis in France using second-order Monte Carlo simulation.

A pragmatic quantitative risk assessment (QRA) of the risks of waterborne Cryptosporidium parvum infection and cryptosporidiosis in immunocompetent and immunodeficient French populations is proposed. The model takes into account French specificities such as the French technique for oocyst enumeration performance and tap water consumption. The proportion of infective oocysts is based on literature review and expert knowledge. The probability of infection for a given number of ingested viable oocysts is modeled using the exponential dose-response model applied on published data from experimental infections in immunocompetent human volunteers challenged with the IOWA strain. Second-order Monte Carlo simulations are used to characterize the uncertainty and variability of the risk estimates. Daily risk of infection and illness for the immunocompetent and the immunodeficient populations are estimated according to the number of oocysts observed in a single storage reservoir water sample. As an example, the mean daily risk of infection in the immunocompetent population is estimated to be 1.08 x 10(-4) (95% confidence interval: [0.20 x 10(-4); 6.83 x 10(-4)]) when five oocysts are observed in a 100 L storage reservoir water sample. Annual risks of infection and disease are estimated from a set of oocyst enumeration results from distributed water samples, assuming a negative binomial distribution of day-to-day contamination variation. The model and various assumptions used in the model are fully explained and discussed. While caveats of this model are well recognized, this pragmatic QRA could represent a useful tool for the French Food Safety Agency (AFSSA) to define recommendations in case of water resource contamination by C. parvum whose infectivity is comparable to the IOWA strain.

Plus cupping degrees do not form an ideal

A computably enumerable (c.e., for short) degree a is called plus cupping , if every c.e. degree x with 0 x ≤ a is cuppable. Let PC be the set of all plus cupping degrees. In the present paper, we show that PC is not closed under the join operation ∨ by constructing two plus cupping degrees which join to a high degree. So by the Harrington's noncupping theorem, PC is not an ideal of e .

Boolean Hierarchies of Partitions over a Reducible Base

The Boolean hierarchy of partitions was introduced and studied by Kosub and Wagner, primarily over the lattice of NP-sets. Here, this hierarchy is treated over lattices with the reduction property, showing that it has a much simpler structure in this instance. A complete characterization is given for the hierarchy over some important lattices, in particular, over the lattices of recursively enumerable sets and of open sets in the Baire space.

Catalytic P systems, semilinear sets, and vector addition systems

We look at 1-region membrane computing systems which only use rules of the form Ca→ Cv, where C is a catalyst, a is a noncatalyst, and v is a (possibly null) string of noncatalysts. There are no rules of the form a→ v. Thus, we can think of these systems as “purely” catalytic. We consider two types: (1) when the initial configuration contains only one catalyst, and (2) when the initial configuration contains multiple catalysts. We show that systems of the first type are equivalent to communication-free Petri nets, which are also equivalent to commutative context-free grammars. They define precisely the semilinear sets. This partially answers an open question (in: WMC-CdeA’02, Lecture Notes in Computer Science, vol. 2597, Springer, Berlin, 2003, pp. 400–409; Computationally universal P systems without priorities: two catalysts are sufficient, available at http://psystems.disco.unimib.it, 2003). Systems of the second type define exactly the recursively enumerable sets of tuples (i.e., Turing machine computable). We also study an extended model where the rules are of the form q : (p,Ca→Cv) (where q and p are states), i.e., the application of the rules is guided by a finite-state control. For this generalized model, type (1) as well as type (2) with some restriction correspond to vector addition systems. Finally, we briefly investigate the closure properties of catalytic systems.

EMBEDDING FINITELY GENERATED ABELIAN LATTICE-ORDERED GROUPS: HIGMAN'S THEOREM AND A REALISATION OF $\pi$

Graham Higman proved that a finitely generated group can be embedded in a finitely presented group if and only if it has a recursively enumerable set of defining relations. The analogue for lattice-ordered groups is considered here. Clearly, the finitely generated lattice-ordered groups that can be $\ell$ -embedded in finitely presented lattice-ordered groups must have recursively enumerable sets of defining relations. The converse direction is proved for a special class of lattice-ordered groups. T HEOREM . Every finitely generated Abelian lattice-ordered group that has finite rank and a recursively enumerable set of defining relations can be $\ell$ - embedded in a finitely presented lattice-ordered group . If $\xi$ is a real number, let $D(\xi)$ be the Abelian rank 2 group $\Z^2$ with order $(m,n)>0$ if and only if $m+n\xi>0$ . C OROLLARY . $D(\xi)$ can be $\ell$ - embedded in a finitely presented lattice-ordered group if and only if $\xi$ is a recursive real number . Thus an algebraic characterisation of recursive real numbers is obtained. In particular, $\pi$ is ‘ $\ell$ -algebraic’ in that it can be captured by finitely many relations in this language.

On existence of complete sets for bounded reducibilities

AbstractClassical reducibilities have complete sets U that any recursively enumerable set can be reduced to U. This paper investigates existence of complete sets for reducibilities with limited oracle access. Three characteristics of classical complete sets are selected and a natural hierarchy of the bounds on oracle access is built. As the bounds become stricter, complete sets lose certain characteristics and eventually vanish. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Isomorphisms of splits of computably enumerable sets

AbstractWe show that if A and are automorphic via Φ then the structures (A) and () are Δ30-isomorphic via an isomorphism Ψ induced by Φ. Then we use this result to classify completely the orbits of hhsimple sets.

On a class of P automata

In this paper, we propose a class of P automata in which each membrane has a state, like in tissue P systems [5], and the computation starts at some initial state and ends in a final state. Unlike the automaton considered in [2], where rules are used in sequential manner, here we consider a variant such that the rules can be applied in maximal mode (as defined in tissue P systems). We show that P automata characterize the recursively enumerable sets of vectors of natural numbers.

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.

User-friendly programs for easy calculations in paternity testing and kinship determinations

Four computer programs are developed for easy calculations of likelihood ratios (LRs) for paternity testing and kinship determinations. The programs are constructed based on the concepts of Bayes Theorem, conditional probability and pedigree analysis. Computer enumeration is used for handling the calculations. The programs have wide applicability, and users can save and check the results easily. The programs employ the distinctive pull-down manual, making them very easy to use. In this paper, we explain the theory and describe various features of the software.

Recent Advances in Σ-definability over Continuous Data Types

The purpose of this paper is to survey our recent research in computability and definability over continuous data types such as the real numbers, real-valued functions and functionals. We investigate the expressive power and algorithmic properties of the language of Sigma-formulas intended to represent computability over the real numbers. In order to adequately represent computability we extend the reals by the structure of hereditarily finite sets. In this setting it is crucial to consider the real numbers without equality since the equality test is undecidable over the reals. We prove Engeler's Lemma for Sigma-definability over the reals without the equality test which relates Sigma-definability with definability in the constructive infinitary language L_{omega_1 omega}. Thus, a relation over the real numbers is Sigma-definable if and only if it is definable by a disjunction of a recursively enumerable set of quantifier free formulas. This result reveals computational aspects of Sigma-definability and also gives topological characterisation of Sigma-definable relations over the reals without the equality test. We also illustrate how computability over the real numbers can be expressed in the language of Sigma-formulas.

On the number of hexagonal polyominoes

A combination of the refined finite lattice method and transfer matrices allows a radical increase in the computer enumeration of polyominoes on the hexagonal lattice (equivalently, site clusters on the triangular lattice), p n with n hexagons. We obtain p n for n⩽35. We prove that p n = τ n+o( n) , obtain the bounds 4.8049⩽ τ⩽5.9047, and estimate that τ=5.1831478(17). Finally, we provide compelling numerical evidence that the generating function ∑ p n z n ≈ A( z)log(1− τz), for z→(1/ τ) − with A( z) holomorphic in a cut plane, estimate A(1/ τ) and predict the sub-leading asymptotic behaviour, identifying a non-analytic correction-to-scaling term with exponent Δ=3/2. On the basis of universality and previous numerical work we argue that the mean-square radius of gyration 〈 R g 2〉 n of polyominoes of size n grows as n 2 ν , with ν=0.64115(5).