Showing papers on "Recursively enumerable language published in 2007"

On String Languages Generated by Spiking Neural P Systems

Abstract: We continue the study of spiking neural P systems by considering these computing devices as binary string generators: the set of spike trains of halting computations of a given system constitutes the language generated by that system. Although the "direct" generative capacity of spiking neural P systems is rather restricted (some very simple languages cannot be generated in this framework), regular languages are inverse-morphic images of languages of finite spiking neural P systems, and recursively enumerable languages are projections of inverse-morphic images of languages generated by spiking neural P systems.

A Characterization of the Entropies of Multidimensional Shifts of Finite Type

Abstract: We show that the values of entropies of multidimensional shifts of finite type (SFTs) are characterized by a certain computation-theoretic property: a real number $h\geq 0$ is the entropy of such an SFT if and only if it is right recursively enumerable, i.e. there is a computable sequence of rational numbers converging to $h$ from above. The same characterization holds for the entropies of sofic shifts. On the other hand, the entropy of an irreducible SFT is computable.

The Power of Commuting with Finite Sets of Words

Abstract: We construct a finite language L such that the largest language commuting with L is not recursively enumerable. This gives a negative answer to the question raised by Conway in 1971 and also strongly disproves Conway's conjecture on context-freeness of maximal solutions of systems of semi-linear inequalities.

An extension of the recursively enumerable Turing degrees

Abstract: Consider the countable semilattice RT consisting of the recursively enumerable Turing degrees. AlthoughRT is known to be structurally rich, a major source of frustration is that no specific, natural degrees inRT have been discovered, except the bottom and top degrees, 0 and 0′. In order to overcome this difficulty, we embed RT into a larger degree structure which is better behaved. Namely, consider the countable distributive lattice Pw consisting of the weak degrees (a.k.a., Muchnik degrees) of mass problems associated with nonempty Π01 subsets of 2 . It is known that Pw contains a bottom degree 0 and a top degree 1 and is structurally rich. Moreover, Pw contains many specific, natural degrees other than 0 and 1. In particular, we show that in Pw one has 0 < d < r1 < inf(r2, 1) < 1. Here d is the weak degree of the diagonally nonrecursive functions, and rn is the weak degree of the n-random reals. It is known that r1 can be characterized as the maximum weak degree of a Π01 subset of 2 ω of positive measure. We now show that inf(r2,1) can be characterized as the maximum weak degree of a Π01 subset of 2 ω whose Turing upward closure is of positive measure. We exhibit a natural embedding of RT into Pw which is one-toone, preserves the semilattice structure of RT , carries 0 to 0, and carries 0 ′ to 1. Identifying RT with its image in Pw, we show that all of the degrees in RT except 0 and 1 are incomparable with the specific degrees d, r1, and inf(r2,1) in Pw.

Can we Compute the Similarity Between Surfaces

Abstract: A suitable measure for the similarity of shapes represented by parameterized curves or surfaces is the Frechet distance. Whereas efficient algorithms are known for computing the Frechet distance of polygonal curves, the same problem for triangulated surfaces is NP-hard. Furthermore, it remained open whether it is computable at all. Here, using a discrete approximation we show that it is {\em upper semi-computable}, i.e., there is a non-halting Turing machine which produces a monotone decreasing sequence of rationals converging to the result. It follows that the decision problem, whether the Frechet distance of two given surfaces lies below some specified value, is recursively enumerable. Furthermore, we show that a relaxed version of the problem, the computation of the {\em weak Frechet distance} can be solved in polynomial time. For this, we give a computable characterization of the weak Frechet distance in a geometric data structure called the {\em free space diagram}.

On the size complexity of universal accepting hybrid networks of evolutionary processors

Abstract: In this paper we discuss the following interesting question about accepting hybrid networks of evolutionary processors (AHNEP), which are a recently introduced bio-inspired computing model. The question is: how many processors are required in such a network to recognise a given language L? Two answers are proposed for the most general case, when L is a recursively enumerable language, and both answers improve on the previously known bounds. In the first case the network has a number of processors that is linearly bounded by the cardinality of the tape alphabet of a Turing machine recognising the given language L. In the second case we show that an AHNEP with a fixed underlying structure can accept any recursively enumerable language. The second construction has another useful property from a practical point of view as it includes a universal AHNEP as a subnetwork, and hence only a limited number of its parameters depend on the given language.

Refining the nonterminal complexity of graph-controlled, programmed, and matrix grammars

Abstract: We refine the classical notion of the nonterminal complexity of graph-controlled grammars, programmed grammars, and matrix grammars by also counting, in addition, the number of nonterminal symbols that are actually used in the appearance checking mode. We prove that every recursively enumerable language can be generated by a graph-controlled grammar with only two nonterminal symbols when both symbols are used in the appearance checking mode. This result immediately implies that programmed grammars with three nonterminal symbols where two of them are used in the appearance checking mode as well as matrix grammars with three nonterminal symbols all of them used in the appearance checking mode are computationally complete. Moreover, we prove that matrix grammars with four nonterminal symbols with only two of them being used in the appearance checking mode are computationally complete, too. On the other hand, every language is recursive if it is generated by a graph-controlled grammar with an arbitrary number of nonterminal symbols but only one of the nonterminal symbols being allowed to be used in the appearance checking mode. This implies, in particular, that the result proving the computational completeness of graph-controlled grammars with two nonterminal symbols and both of them being used in the appearance checking mode is already optimal with respect to the overall number of nonterminal symbols as well as with respect to the number of nonterminal symbols used in the appearance checking mode, too. Finally, we also investigate in more detail the computational power of several language families which are generated by graph-controlled, programmed grammars or matrix grammars, respectively, with a very small number of nonterminal symbols and therefore are proper subfamilies of the family of recursively enumerable languages.

On finitely recursive programs

Abstract: Finitary programs are a class of logic programs admitting functions symbols and hence infinite domains. In this paper we prove that a larger class of programs, called finitely recursive programs, preserves most of the good properties of finitary programs under the stable model semantics, namely: (i) finitely recursive programs enjoy a compactness property; (ii) inconsistency check and skeptical reasoning are semidecidable; (iii) skeptical resolution is complete. Moreover, we show how to check inconsistency and answer skeptical queries using finite subsets of the ground program instantiation.

On the power of networks of evolutionary processors

Abstract: We discuss the power of networks of evolutionary processors where only two types of nodes are allowed. We prove that (up to an intersection with a monoid) every recursively enumerable language can be generated by a network with one deletion and two insertion nodes. Networks with an arbitrary number of deletion and substitution nodes only produce finite languages, and for each finite language one deletion node or one substitution node is sufficient. Networks with an arbitrary number of insertion and substitution nodes only generate contextsensitive languages, and (up to an intersection with a monoid) every contextsensitive language can be generated by a network with one substitution node and one insertion node.

Post's Programme for the Ershov Hierarchy

Abstract: 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.

Lambda theories of effective lambda models

Abstract: A longstanding open problem is whether there exists a nonsyntactical model of the untyped λ-calculus whose theory is exactly the least λ-theory λβ. In this paper we investigate the more general question of whether the equational/order theory of a model of the untyped λ-calculus can be recursively enumerable (r.e. for brevity). We introduce a notion of effective model of λ-calculus, which covers in particular all the models individually introduced in the literature. We prove that the order theory of an effective model is never r.e.; from this it follows that its equational theory cannot be λβ, λβ. We then show that no effective model living in the stable or strongly stable semantics has an r.e. equational theory. Concerning Scott's semantics, we investigate the class of graph models and prove that no order theory of a graph model can be r.e., and that there exists an effective graph model whose equational/ order theory is the minimum one. Finally, we show that the class of graph models enjoys a kind of downwards Lowenheim-Skolem theorem.

Minimal cooperation in symport/antiport tissue p systems

Abstract: We investigate tissue P systems with symport/antiport with minimal cooperation, i.e., when only 2 objects may interact. We show that 2 cells are enough in order to generate all recursively enumerable sets of numbers. Moreover, constructed systems simulate register machines and have purely deterministic behavior. We also investigate systems with one cell and we show that they may generate only finite sets of numbers.

Diophantine Sets over Polynomial Rings and Hilbert's Tenth Problem for Function Fields

Abstract: In 1900, the German mathematician David Hilbert proposed a list of 23 unsolved mathematical problems. In his Tenth Problem, he asked to find an algorithm to decide whether or not a given diophantine equation has a solution (in integers). Hilbert's Tenth Problem has a negative solution, in the sense that such an algorithm does not exist. This was proven in 1970 by Y. Matiyasevich, building on earlier work by M. Davis, H. Putnam and J. Robinson. Actually, this result was the consequence of something much stronger: the equivalence of recursively enumerable and diophantine sets (we will refer to this result as "DPRM"). The first new result in the thesis is about Hilbert's Tenth Problem for function fields of curves over valued fields in characteristic zero. Under some conditions on the curve and the valuation, we have undecidability for diophantine equations over the function field of the curve. One interesting new case are function fields of curves over formal Laurent series. The proof relies on the method with two elliptic curves as developed by K. H. Kim and F. Roush and generalised by K. Eisentrager. Additionally, the proof uses the theory quadratic forms and valuations. And especially for non-rational function fields there is some algebraic geometry coming in. The second type of results establishes the equivalence of recursively enumerable and diophantine sets in certain polynomial rings. The most important is the one-variable polynomial ring over a finite field. This is the first generalisation of DPRM in positive characteristic. My proof uses the structure of finite fields and in particular the properties of cyclotomic polynomials. In the last chapter, this result for polynomials over finite fields is generalised to polynomials over recursive algebraic extensions of a finite field. For these rings we don't have a good definition of "recursively enumerable" set, therefore we consider sets which are recursively enumerable for every recursive presentation. We show that these are exactly the diophantine sets. In addition to infinite extensions of finite fields, we also show the analogous result for polynomials over a ring of integers in a recursive totally real algebraic extension of the rationals. This generalises results by J. Denef and K. Zahidi.

Prescribed Learning of R.E. Classes

Abstract: This work extends studies of Angluin, Lange and Zeugmann on the dependence of learning on the hypotheses space chosen for the class. In subsequent investigations, uniformly recursively enumerable hypotheses spaces have been considered. In the present work, the following four types of learning are distinguished: class-comprising (where the learner can choose a uniformly recursively enumerable superclass as hypotheses space), class-preserving (where the learner has to choose a uniformly recursively enumerable hypotheses space of the same class), prescribed (where there must be a learner for every uniformly recursively enumerable hypotheses space of the same class) and uniform (like prescribed, but the learner has to be synthesized effectively from an index of the hypothesis space). While for explanatory learning, these four types of learnability coincide, some or all are different for other learning criteria. For example, for conservative learning, all four types are different. Several results are obtained for vacillatory and behaviourally correct learning; three of the four types can be separated, however the relation between prescribed and uniform learning remains open. It is also shown that every (not necessarily uniformly recursively enumerable) behaviourally correct learnable class has a prudent learner, that is, a learner using a hypotheses space such that it learns every set in the hypotheses space. Moreover the prudent learner can be effectively built from any learner for the class.

A general comparison of language learning from examples and from queries

Abstract: In language learning, strong relationships between Gold-style models and query models have recently been observed: in some quite general setting Gold-style learners can be replaced by query learners and vice versa, without loss of learning capabilities. These 'equalities' hold in the context of learning indexable classes of recursive languages. Former studies on Gold-style learning of such indexable classes have shown that, in many settings, the enumerability of the target class and the recursiveness of its languages are crucial for learnability. Moreover, studying query learning, non-indexable classes have been mainly neglected up to now. So it is conceivable that the recently observed relations between Gold-style and query learning are not due to common structures in the learning processes in both models, but rather to the enumerability of the target classes or the recursiveness of their languages. In this paper, the analysis is lifted onto the context of learning arbitrary classes of recursively enumerable languages. Still, strong relationships between the approaches of Gold-style and query learning are proven, but there are significant changes to the former results. Though in many cases learners of one type can still be replaced by learners of the other type, in general this does not remain valid vice versa. All results hold even for learning classes of recursive languages, which indicates that the recursiveness of the languages is not crucial for the former 'equality' results. Thus we analyze how constraints on the algorithmic structure of the target class affect the relations between two approaches to language learning.

Learning in Friedberg Numberings

Abstract: In this paper we consider learnability in some special numberings, such as Friedberg numberings, which contain all the recursively enumerable languages, but have simpler grammar equivalence problem compared to acceptable numberings. We show that every explanatorily learnable class can be learnt in some Friedberg numbering. However, such a result does not hold for behaviourally correct learning or finite learning. One can also show that some Friedberg numberings are so restrictive that all classes which can be explanatorily learnt in such Friedberg numberings have only finitely many infinite languages. We also study similar questions for several properties of learners such as consistency, conservativeness, prudence, iterativeness and non U-shaped learning. Besides Friedberg numberings, we also consider the above problems for programming systems with K-recursive grammar equivalence problem.

Non-preserving splicing with delay

Abstract: In this paper we investigate H systems with strongly non-preserving splicing that exhibit a new feature, namely delay, and introduce a variant of the H system that lies between H systems with strongly non-preserving splicing and H systems with non-reflexively evolving splicing. Informally, the new splicing system behaves as follows: (1) each splicing step is exactly a splicing step in a system with non-reflexively evolving splicing; and (2) the generated language is obtained exactly as in a system with strongly non-preserving splicing. For both H systems with non-reflexively evolving and non-preserving splicing we have a remarkable jump in power between systems with a finite but arbitrarily large delay, and those with infinite delay. The first can generate non-context-free languages, whereas the second do not get beyond the regular limit. Moreover, H systems with null delay generate all recursively enumerable languages.

Automata with two-sided pushdowns defined over free groups generated by reduced alphabets

Abstract: This paper introduces and discusses a modification of pushdown automata. This modification is based on two-sided pushdowns into which symbols are pushed from both ends. These pushdowns are defined over free groups, not free monoids, and they can be shortened only by the standard group reduction. We demonstrate that these automata characterize the family of recursively enumerable languages even if the free groups are generated by no more than four symbols.

Skin output in P systems with minimal symport/antiport and two membranes

Abstract: It is known that symport/antiport P systems with two membranes and minimal cooperation can generate any recursively enumerable sets of natural numbers using exactly one superfluous object in the output membrane, where the output membrane is an elementary membrane. In this paper we consider symport/antiport P systems where the output membrane is the skin membrane. In this case we prove an unexpected characterization: symport/antiport P systems (and purely symport P systems) with two membranes and minimal cooperation generate exactly the recursively enumerable sets of natural numbers.

Non-returning PC grammar systems generate any recursively enumerable language with eight context-free components

Abstract: We show how to generate any recursively enumerable language with a nonreturning PC grammar system having eight context-free components. This is an improvement of descriptional complexity compared to the previously known construction where the number of component grammars depends on the generated language and can be arbitrarily high.

Mitotic classes

Abstract: For the natural notion of splitting classes into two disjoint subclasses via a recursive classifier working on texts, the question is addressed how these splittings can look in the case of learnable classes. Here the strength of the classes is compared using the strong and weak reducibility from intrinsic complexity. It is shown that, for explanatorily learnable classes, the complete classes are also mitotic with respect to weak and strong reducibility, respectively. But there is a weak complete class which cannot be split into two classes which are of the same complexity with respect to strong reducibility. It is shown that for complete classes for behaviourally correct learning, one half of each splitting is complete for this learning notion as well. Furthermore, it is shown that explanatorily learnable and recursively enumerable classes always have a splitting into two incomparable classes; this gives an inductive inference counterpart of Sacks Splitting Theorem from recursion theory.

On the recognition of context-free languages using accepting hybrid networks of evolutionary processors

Abstract: In this paper we approach the problem of constructing an accepting device for the class of context-free languages, based on a newly defined computational model: Accepting Hybrid Network of Evolutionary Processors (AHNEPs). Although it is known that AHNEPs are Turing complete, and, consequently, for every context-free language, seen as a recursively enumerable language, we can construct an AHNEP that simulates the Turing machine accepting it, we choose a direct approach: the AHNEPs we design simulate the computation done by a non-deterministic push-down automata accepting the language. This approach leads to a more economic AHNEP since the number of processors we use depends linearly on the number of states and the number of working symbols of the automaton, while for the network obtained in the general case of recursively enumerable languages, the number of processors is linear in the number of states, but quadratic in the number of working symbols of a Turing machine accepting the given language. Finally, we particularize the AHNEP architecture we proposed in order to recognize regular languages. We obtain an upper bound for the number of processors needed close to that known for generating hybrid networks of evolutionary processors.