Enumerable Research Articles

On the computational power of random strings

There are two fundamental computably enumerable sets associated with any Kolmogorov complexity measure. These are the set of non-random strings and the overgraph. This paper investigates the computational power of these sets. It follows work done by Kummer, Muchnik and Positselsky, and Allender and co-authors. Muchnik and Positselsky asked whether there exists an optimal monotone machine whose overgraph is not t t -complete. This paper answers this question in the negative by proving that the overgraph of any optimal monotone machine, or any optimal process machine, is t t -complete. The monotone results are shown for both descriptional complexity K m and K M , the complexity measure derived from algorithmic probability. A distinction is drawn between two definitions of process machines that exist in the literature. For one class of process machines, designated strict process machines, it is shown that there is a universal machine whose set of non-random strings is not t t -complete.

Prime models of finite computable dimension

AbstractWe study the following open question in computable model theory: does there exist a structure of computable dimension two which is the prime model of its first-order theory? We construct an example of such a structure by coding a certain family of c.e. sets with exactly two one-to-one computable enumerations into a directed graph. We also show that there are examples of such structures in the classes of undirected graphs, partial orders, lattices, and integral domains.

Computability of entropy and information in classical Hamiltonian systems

We consider the computability of entropy and information in classical Hamiltonian systems. We define the information part and total information capacity part of entropy in classical Hamiltonian systems using relative information under a computable discrete partition. Using a recursively enumerable nonrecursive set it is shown that even though the initial probability distribution, entropy, Hamiltonian and its partial derivatives are computable under a computable partition, the time evolution of its information capacity under the original partition can grow faster than any recursive function. This implies that even though the probability measure and information are conserved in classical Hamiltonian time evolution we might not actually compute the information with respect to the original computable partition.

The Settling Time Reducibility Ordering and Formula Sets

The settling time reducibility ordering gives an ordering on computably enumerable sets based on their enumerations. The <st ordering is in fact an ordering on c.e. sets, since it is independent of the particular enumeration chosen. In this article, we show that it is not possible to extend this ordering in an approximation-independent way to Δ02 sets in general, or even to n-c.e. sets for any fixed n ≥ 3.

Non-cupping, measure and computably enumerable splittings

We show that there is a computably enumerable function f (that is, computably approximable from below) that dominates almost all functions, and f ⊕ W is incomplete for all incomplete computably enumerable sets W. Our main methodology is the LR equivalence relation on reals: A ≡LRB if and only if the notions of A-randomness and B-randomness coincide. We also show that there are c.e. sets that cannot be split into two c.e. sets of the same LR degree. Moreover, a c.e. set is low for random if and only if it computes no c.e. set with this property.

Prime models of computably enumerable degree

AbstractWe examine the computably enumerable (c.e.) degrees of prime models of complete atomic decidable (CAD) theories. A structure has degree d if d is the degree of its elementary diagram. We show that if a CAD theory T has a prime model of c.e. degree c, then T has a prime model of strictly lower c.e. degree b, where, in addition, b is low (b′ = 0′), This extends Csima's result that every CAD theory has a low prime model. We also prove a density result for c.e. degrees of prime models. In particular, if c and d are c.e. degrees with d &lt; c and c not low2 (c″ &gt; 0″), then for any CAD theory T, there exists a c.e. degree b with d &lt; b &lt; c such that T has a prime model of degree b, where b can be chosen so that b′ is any degree c.e. in c with d′ ≤ b′. As a corollary, we show that for any degree c with 0 &lt; c &lt; 0′, every CAD theory has a prime model of low c.e. degree incomparable with c. We show also that every CAD theory has prime models of low c.e. degree that form a minimal pair, extending another result of Csima. We then discuss how these results apply to homogeneous models.

Almost computably enumerable families of sets

An almost computably enumerable family that is not -computably enumerable is constructed. Moreover, it is established that for any computably enumerable (c.e.) set there exists a family that is -c.e. if and only if the set is not -computable. Bibliography: 5 titles.

Relative enumerability in Ershov’s hierarchy

Generalizations to various levels of Ershov’s hierarchy of the relationship between n-computable enumerability and relative enumerability are considered.

Large-scale computation of elementary flux modes with bit pattern trees

Elementary flux modes (EFMs)--non-decomposable minimal pathways--are commonly accepted tools for metabolic network analysis under steady state conditions. Valid states of the network are linear superpositions of elementary modes shaping a polyhedral cone (the flux cone), which is a well-studied convex set in computational geometry. Computing EFMs is thus basically equivalent to extreme ray enumeration of polyhedral cones. This is a combinatorial problem with poorly scaling algorithms, preventing the large-scale analysis of metabolic networks so far. Here, we introduce new algorithmic concepts that enable large-scale computation of EFMs. Distinguishing extreme rays from normal (composite) vectors is one critical aspect of the algorithm. We present a new recursive enumeration strategy with bit pattern trees for adjacent rays--the ancestors of extreme rays--that is roughly one order of magnitude faster than previous methods. Additionally, we introduce a rank updating method that is particularly well suited for parallel computation and a residue arithmetic method for matrix rank computations, which circumvents potential numerical instability problems. Multi-core architectures of modern CPUs can be exploited for further performance improvements. The methods are applied to a central metabolism network of Escherichia coli, resulting in approximately 26 Mio. EFMs. Within the top 2% modes considering biomass production, most of the gain in flux variability is achieved. In addition, we compute approximately 5 Mio. EFMs for the production of non-essential amino acids for a genome-scale metabolic network of Helicobacter pylori. Only large-scale EFM analysis reveals the >85% of modes that generate several amino acids simultaneously. An implementation in Java, with integration into MATLAB and support of various input formats, including SBML, is available at http://www.csb.ethz.ch in the tools section; sources are available from the authors upon request.

Conjectures for the first perimeter moment of directed animals

Using computer enumerations and a rational approximant method of series analysis, we conjecture an expression for the first perimeter moment of directed animals on the square lattice which are confined in a strip of a given width with open boundary conditions. When the width tends to infinity, the conjecture leads to an algebraic series for the first perimeter moment of directed animals on one-half of the square lattice, similar to Conway's earlier conjecture for the whole square lattice.

On Lachlan’s major sub-degree problem

The Major Sub-degree Problem of A. H. Lachlan (first posed in 1967) has become a long-standing open question concerning the structure of the computably enumerable (c.e.) degrees. Its solution has important implications for Turing definability and for the ongoing programme of fully characterising the theory of the c.e. Turing degrees. A c.e. degree a is a major subdegree of a c.e. degree b > a if for any c.e. degree x, \({{\bf 0' = b \lor x}}\) if and only if \({{\bf 0' = a \lor x}}\) . In this paper, we show that every c.e. degree b ≠ 0 or 0′ has a major sub-degree, answering Lachlan’s question affirmatively.

Accurate and efficient expression evaluation and linear algebra

We survey and unify recent results on the existence of accurate algorithms for evaluating multivariate polynomials, and more generally for accurate numerical linear algebra with structured matrices. By ‘accurate’ we mean that the computed answer has relative error less than 1,i.e., has some correct leading digits. We also address efficiency, by which we mean algorithms that run in polynomial time in the size of the input. Our results will depend strongly on the model of arithmetic: most of our results will use the so-calledtraditional model(TM), where the computed result of op(a,b), a binary operation likea+b, is given by op(a,b) * (1+δ) where all we know is that |δ| ≤ ε ≪ 1. Here ε is a constant also known as machine epsilon.We will see a common reason for the following disparate problems to permit accurate and efficient algorithms using only the four basic arithmetic operations: finding the eigenvalues of a suitably discretized scalar elliptic PDE, finding eigenvalues of arbitrary products, inverses, or Schur complements of totally non-negative matrices (such as Cauchy and Vandermonde), and evaluating the Motzkin polynomial. Furthermore, in all these cases the high accuracy is ‘deserved’,i.e., the answer is determined much more accurately by the data than the conventional condition number would suggest.In contrast, we will see that evaluating even the simple polynomialx+y+zaccurately is impossible in the TM, using only the basic arithmetic operations. We give a set of necessary and sufficient conditions to decide whether a high accuracy algorithm exists in the TM, and describe progress toward a decision procedure that will take any problem and provide either a high-accuracy algorithm or a proof that none exists.When no accurate algorithm exists in the TM, it is natural to extend the set of available accurate operations by a library of additional operations, such asx+y+z, dot products, or indeed any enumerable set which could then be used to build further accurate algorithms. We show how our accurate algorithms and decision procedure for finding them extend to this case.Finally, we address other models of arithmetic, and the relationship between (im)possibility in the TM and (in)efficient algorithms operating on numbers represented as bit strings.

On the orbits of computably enumerable sets

The goal of this paper is to show there is a single orbit of the c.e. sets with inclusion, E \mathcal {E} , such that the question of membership in this orbit is Σ 1 1 \Sigma ^1_1 -complete. This result and proof have a number of nice corollaries: the Scott rank of E \mathcal {E} is ω 1 C K + 1 \omega _1^{ {CK}}+1 ; not all orbits are elementarily definable; there is no arithmetic description of all orbits of E \mathcal {E} ; for all finite α ≥ 9 \alpha \geq 9 , there is a properly Δ α 0 \Delta ^0_\alpha orbit (from the proof).

Generalized communicating P systems

This paper considers a generalization of various communication models based on the P system paradigm where two objects synchronously move across components. More precisely, the model uses blocks of four cells such that pairs of objects from two input cells travel together to target output cells. It is shown that the model introduced, based on interactions between blocks, is complete, being able to generate all recursively enumerable sets of natural numbers. It is also proven that completeness is achievable by using a minimal interaction between blocks, i.e. every pair of cells is the input or output for at most one block. It is also shown that the concepts introduced in this paper to define the model may be simulated by more particular communication primitives, including symport, antiport and uniport rules. This enables us to automatically translate a system using interaction rules in any of minimal symport, minimal antiport or conditional uniport P systems.

The Complexity of Orbits of Computably Enumerable Sets

AbstractThe goal of this paper is to announce there is a single orbit of the c.e. sets with inclusion, ε, such that the question of membership in this orbit is complete. This result and proof have a number of nice corollaries: the Scott rank of ε is + 1; not all orbits are elementarily definable; there is no arithmetic description of all orbits of ε; for all finite α ≥ 9, there is a properly orbit (from the proof).

On the computing power of fuzzy Turing machines

We work with fuzzy Turing machines ( FTMs) and we study the relationship between this computational model and classical recursion concepts such as computable functions, recursively enumerable (r.e.) sets and universality. FTMs are first regarded as acceptors. It has recently been shown by J. Wiedermann that these machines have more computational power than classical Turing machines. Still, the context in which this formulation is valid has an unnatural implicit assumption. We settle necessary and sufficient conditions for a language to be r.e., by embedding it in a fuzzy language recognized by a FTM. We do the same thing for n-r.e. set. It is shown that there is no universal fuzzy machine, and “universality” is analyzed for smaller classes of FTMs. We argue for a definition of computable fuzzy function, when FTMs are understood as transducers. It is shown that, in this case, our notion of computable fuzzy function coincides with the classical one.

Extension theorems, orbits, and automorphisms of the computably enumerable sets

We prove an algebraic extension theorem for the computably enumerable sets, E \mathcal {E} . Using this extension theorem and other work we then show if A A and A ^ \widehat {A} are automorphic via Ψ \Psi , then they are automorphic via Λ \Lambda where Λ ↾ L ∗ ( A ) = Ψ \Lambda \restriction \mathcal {L}^*(A) = \Psi and Λ ↾ E ∗ ( A ) \Lambda \restriction \mathcal {E}^*(A) is Δ 3 0 \Delta ^0_3 . We give an algebraic description of when an arbitrary set A ^ \widehat {A} is in the orbit of a computably enumerable set A A . We construct the first example of a definable orbit which is not a Δ 3 0 \Delta ^0_3 orbit. We conclude with some results which restrict the ways one can increase the complexity of orbits. For example, we show that if A A is simple and A ^ \widehat {A} is in the same orbit as A A , then they are in the same Δ 6 0 \Delta ^0_6 -orbit and, furthermore, we provide a classification of when two simple sets are in the same orbit.

Q-degrees of $n$-c.e. sets

In this paper we study Q-degrees of $n$-computably enumerable ($n$-c.e.) sets. It is proved that $n$-c.e. sets form a true hierarchy in terms of Q-degrees, and that for any $n\ge 1$ there exists a $2n$-c.e. Q-degree which bounds no noncomputable c.e. Q-degree, but any $(2n+1)$-c.e. non $2n$-c.e. Q-degree bounds a c.e. noncomputable Q-degree. Studying weak density properties of $n$-c.e. Q-degrees, we prove that for any $n\ge 1$, properly $n$-c.e. Q-degrees are dense in the ordering of c.e. Q-degrees, but there exist c.e. sets $A$ and $B$ such that $A-B<_QA\equiv_Q\emptyset'$, and there are no c.e. sets for which the Q-degrees are strongly between $A-B$ and $A$.

Recursively enumerable sets of polynomials over a finite field are Diophantine

We construct a Diophantine interpretation of $\mathbb{F}_q[W,Z]$ over $\mathbb{F}_q[Z]$ . Using this together with a previous result that every recursively enumerable (r.e.) relation over $\mathbb{F}_q[Z]$ is Diophantine over $\mathbb{F}_q[W,Z]$ , we will prove that every r.e. relation over $\mathbb{F}_q[Z]$ is Diophantine over $\mathbb{F}_q[Z]$ . We will also look at recursive infinite base fields $\mathbb{F}$ , algebraic over $\mathbb{F}_p$ . It turns out that the Diophantine relations over $\mathbb{F}[Z]$ are exactly the relations which are r.e. for every recursive presentation.

Post's Programme for the Ershov Hierarchy

This article extends Post's; programme to finite levels of the Ershov hierarchy of Δ2 sets. Our initial characterization, in the spirit of Post (1994, Bulletin of the American Mathematical Society, 50, 284–316), of the degrees of the immune and hyperimmune n-enumerable sets leads to a number of results setting other immunity properties in the context of the Turing and wtt-degrees derived from the Ershov hierarchy. For instance, we show that any n-enumerable hyperhyperimmune set must be co-enumerable, for each n ≥ 2. The situation with regard to the wtt-degrees is particularly interesting, as demonstrated by a range of results concerning the wtt-predecessors of hypersimple sets. Finally, we give a number of results directed at characterizing basic classes of n-enumerable degrees in terms of natural information content. For example, a 2-enumerable degree contains a 2-enumerable dense immune set iff it contains a 2-enumerable r-cohesive set iff it bounds a high enumerable set. This result is extended to a characterization of n-enumerable degrees which bound high enumerable degrees. Furthermore, a characterization for n-enumerable degrees bounding only low2 enumerable degrees is given.