Showing papers on "Recursively enumerable language published in 1997"
01 Jan 1997
50 citations
TL;DR: It is shown that if a is adegree such that a < 0', then there is a degree b such that 0 < b < 0' and a n b = 0, which follows at once from the following theorem.
Abstract: In their original paper on degrees [3], Kleene and Post showed that there is a degree between 0 and 0'. Later, Friedberg [1] and Muchnik [4] showed that there is a recursively enumerable degree between 0 and 0'. Since then, this phenomenon has been repeated several times: a result has been proved for degrees, and then, after considerable additional effort, it has been proved for recursively enumerable degrees. There are some obvious respects in which that set of all degrees differs from the set of recursively enumerable degrees; e.g., the former is uncountable and has no largest member. The resemblance between the set of degrees < 0' and the set of recursively enumerable degrees is much closer. It is not entirely trivial to prove that these sets are distinct; and it is quite difficult to find properties which are possessed by one of these sets and not by the other. The best known such property is the following: there are minimal degrees ? 0' (according to Sacks [6, p. 137] and Lacombe (unpublished)); but there are no minimal recursively enumerable degrees (according to Muchnik [5] and Friedberg [2]). We shall give another such property. Yates [7] has recently shown that there is a recursively enumerable degree a # 0' such that a n b # 0 for every recursively enumerable degree b # 0. We shall show that if a is a degree such that a < 0', then there is a degree b such that 0 < b < 0' and a n b = 0. (This answers (Q2) on page 171 of [6].) This follows at once from the following theorem. THEOREM. If a < 0', then there is a minimal degree b such that b < O' and b ffi a. It has been customary to present proofs of theorems on degrees in the symbolism of recursive function theory. While this facilitates complete proofs, it sometimes obscures the fundamental ideas. We shall attempt to avoid this symbolism, appealing to Church's thesis as a substitute when necessary. We need some preliminary definitions. By a sequence, we shall mean a finite (possibly empty) sequence of zeroes and ones; the number of elements in the sequence is called the length of the sequence. We use a and for sequences. If T is obtained from a by adding (possibly zero) elements at the end, we say that Xis an extension of v. We then designate a by T, where n is the length of a. If T is an extension of a different from a, we say that T is a proper extension of a. We say a and T are compatible if one
39 citations
01 Jan 1997
TL;DR: First, a result is improved about the so-called communicating distributed H systems (systems with seven components are able to characterize the recursively enumerable languages), then two new types of distributed H system are introduced: the separated two-level H systems and the periodically time-varying H systems, and it is proved that in all these cases one can design universal “DNA computers based on splicing”.
Abstract: Because splicing systems with a finite set of rules generate only regular languages, it is necessary to supplement such a system with a control mechanism on the use of rules. One fruitful idea is to use distributed architectures suggested by the grammar systems area. Three distributed computability (language generating) devices based on splicing are discussed here. First, we improve a result about the so-called communicating distributed H systems (systems with seven components are able to characterize the recursively enumerable languages — the best result known up to now is of ten components), then we introduce two new types of distributed H systems: the separated two-level H systems and the periodically time-varying H systems. In both cases we prove characterizations of recursively enumerable languages — which means that in all these cases we can design universal “DNA computers based on splicing”.
33 citations
TL;DR: The family of recursively enumerable languages is characterized by scattered context Grammars with four nonterminals if these grammars start their derivations from a word rather than a symbol.
Abstract: The family of recursively enumerable languages is characterized by scattered context grammars with four nonterminals. Moreover, this family is characterized by scattered context grammars with three nonterminals if these grammars start their derivations from a word rather than a symbol. Three open problem areas are suggested
14 citations
TL;DR: It is shown that inventories produced by SL Transactions are regular and every regular inventory can be generated by SL transactions and soundness and complet,eness for SL t,ransactions are decidahlt3.
13 citations
TL;DR: Several characterizations of recursively enumerable languages are given, using H systems with permitting contexts having splicing rules of small radius, and representation of context-free languages are obtained.
Abstract: Several characterizations of recursively enumerable languages are given, using H systems with permitting contexts having splicing rules of small radius. Representations of context-free languages are also obtained in certain particular cases. These results improve previous related results which were recently published. Some open problems are also pointed out.
13 citations
10 citations
TL;DR: It is proved that each recursively enumerable language can be generated using a distributed splicing system with a fixed number of test tubes.
Abstract: In this paper we prove that each recursively enumerable language can be generated using a distributed splicing system with a fixed number of test tubes. This improves a recent result by Csuhaj-Varju, Kari, Paun, proving computational completeness only for a system with a number of tubes depending on the cardinality of the used alphabet.
9 citations
01 Jan 1997
TL;DR: This investigation proves that four-nonterminal matrix grammars working in this way characterize the family of recursively enumerable languages.
Abstract: The paper investigates the descriptional complexity of matrix grammars that always rewrites the leftmost possible occurrence of a non-terminal. Measuring this complexity by the number of nonterminals, this investigation proves that four-nonterminal matrix grammars working in this way characterize the family of recursively enumerable languages.
9 citations
TL;DR: The algebraic structure of the upper semi-lattice formed by the recursively enumerable Turing degrees is investigated, and Lachlan's Nonsplitting Theorem is proved, yielding an alternative proof of their result that the theory of R has infinitely many one-types.
Abstract: We investigate the algebraic structure of the upper semi-lattice formed by the recursively enumerable Turing degrees. The following strong generalization of Lachlan's Nonsplitting Theorem is proved: Given n ≥ 1, there exists an r.e. degree d such that the interval [d, 0′] ⊂ R admits an embedding of the n-atom Boolean algebra preserving (least and) greatest element, but also such that there is no (n + 1 )-tuple of pairwise incomparable r.e. degrees above d which pairwise join to 0′ (and hence, the interval [d, 0′] ⊂ R does not admit a greatest-element-preserving embedding of any lattice which has n + 1 co-atoms, including ). This theorem is the dual of a theorem of Ambos-Spies and Soare, and yields an alternative proof of their result that the theory of R has infinitely many one-types.
9 citations
TL;DR: Computation theory could be reconstructed without rewriting (and non-terminal symbols) and without any loss in power and some consequences for the future development of generative theory of natural language are obtained.
Abstract: Computation not only takes place in provoked contexts of scientific experimentation, but in natural circumstances too. We are going to approach computation in natural contexts. How the nature computes? Turing machines and Chomsky grammars are rewriting systems, and the same is true for Post, Thue, Markov, Lindenmayer and other classes of axiomatic systems. If, among the whole set of natural objects, we focus natural language description, we must say that major trends in contemporary linguistics look at syntax as a rewriting process. Is rewriting unavoidable in this case, does our mind work by rewriting, does the nature compute in this way? We shall attempt to defend that the answer could be negative. The arguments will come from computability theory as well as from linguistics. First we'll formally explain the former ones, then informally the latter ones. With regard to computability theory arguments, we will see that, using the operation of adjoining, a large generative capacity is obtained. This is the case with contextual grammars. It has recently been proved that each recursively enumerable language is the quotient by a regular language of a language generated by a contextual grammar of a particular form. Thus, adjoining (paste) and quotient (cut) lead to computational universality. Recursively enumerable languages can also be characterized as the quotient by a regular language of a language generated by an insertion grammar. The same result is obtained if we take the splicing operation, a formal model of the DNA recombination. This is again a cut-and-paste operation. On the basis of the proof of this result, several further characterizations of recursively enumerable languages have been obtained. Computability theory, then, could be reconstructed without rewriting (and non-terminal symbols) and without any loss in power. Our first aim will be to show some formal aspects of such reconstruction. Later, we'll try to obtain some consequences for the future development of generative theory of natural language.
TL;DR: This investigation demonstrates that the family of recursively enumerable languages is characterized by six-nonterminal multi-sequential grammars.
Abstract: The present paper investigates the descriptional complexity of multi-sequential grammars with respect to the number of nonterminals. This investigation demonstrates that the family of recursively enumerable languages is characterized by six-nonterminal multi-sequential grammars.
01 Jan 1997
TL;DR: It is proved that any recursively enumerable language L can be homomorphically represented by twisting a linear context-free language using the combination of twist and product.
Abstract: We show that in conjunction with the usual trio operations the combination of twist and product can simulate any combination of intersection, reversal and 1/2. It is proved that any recursively enumerable language L can be homomorphically represented by twisting a linear context-free language. Indeed, the recursively enumerable sets form the least twist-closed full trio generated by dMIR:=wcw rev ¦ w e a,b *.
TL;DR: A partial answer to the problem of identification of programs for computable functions from their graphs and identification of grammars for recursively enumerable languages from positive data is provided.
06 Oct 1997
TL;DR: The present paper investigates identification of indexed families of recursively enumerable languages from good examples and shows that the notions of finite and limit identification coincide in the context of class preserving learning from good text examples.
Abstract: The present paper investigates identification of indexed families of recursively enumerable languages from good examples. In the context of class preserving learning from good text examples, it is shown that the notions of finite and limit identification coincide. On the other hand, these two criteria are different in the context of class comprising learning from good text examples. In the context of learning from good informant examples, finite and limit identification criteria differ for both class preserving and class comprising cases. The above results resolve an open question posed by Lange, Nessel and Wiehagen in a similar study about indexed families of recursive languages.
01 Jan 1997
TL;DR: It is proved that: 1) any linear language can be generated by a splicing grammar system with two regular components; 2) any context-free language can been generated byA splicing grammars system with threeRegular components; 3) any recursively enumerable language can being generated by an splicing language system with four right linear components.
Abstract: The generative capacity of splicing grammar systems is investigated in this paper. It is proved that: 1) any linear language can be generated by a splicing grammar system with two regular components; 2) any context-free language can be generated by a splicing grammar system with three regular components; 3) any recursively enumerable language can be generated by a splicing grammar system with four right linear components. The first two results answer a problem left open in [18], the last result improves results in the same paper.
TL;DR: It is proved that the notions of effectively subcreative set, strongly effectively acceleratable set, andsQ-complete recursively enumerable set are equivalent.
Abstract: The notions of effectively subcreative set and strongly effectively acceleratable set are introduced. It is proved that the notions of effectively subcreative set, strongly effectively acceleratable set, andsQ-complete recursively enumerable set are equivalent.
01 Jan 1997
01 Jul 1997
TL;DR: Borchert and Stephan as discussed by the authors showed that every P-constructibly bi-infinite counting property of circuits is SPP-hard, and showed that this conclusion cannot be strengthened to UP-hardness unless unlikely complexity class containments hold.
Abstract: Rice''s theorem states that every nontrivial language property of the recursively enumerable sets is undecidable. Borchert and Stephan initiated a search for complexity-theoretic analogs of Rice''s Theorem. In particular, they proved that every nontrivial counting property of circuits is UP-hard. We extend their result by proving that every nontrivial counting property of circuits is UP_{O(1)}-hard; that is, we raise the lower bound from unambiguous nondeterminism to constant-ambiguity nondeterminism. We show that this conclusion cannot be strengthened to SPP-hardness unless unlikely complexity class containments hold. Nonetheless, we prove that every P-constructibly bi-infinite counting property of circuits is SPP-hard.
Book•
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01 Jan 1997
TL;DR: The central definition is that of a computable sequence of recursively enumerable open subsets of real numbers, which is equivalent to that of Calude and Jurgensen by means of random positional representations.
Abstract: In 1966 Martin-Lof [ML66] has defined random sequences of symbols
as a recursion theoretical concept. A sequence of symbols is
non-random, iff it is an element of the intersection of a
computable sequence of recursively enumerable open sets the
measures of which converge to zero with 2⁻ⁿ, a computable
null set for short. We use the same concept to introduce random
real numbers. The central definition is that of a computable
sequence of recursively enumerable open subsets of real numbers.
We give several arguments which show that our definition is
natural. We prove that, as in the case of random sequences, there
is a universal randomness test. Finally we prove that a real
number is random, iff its natural positional representation with
basis Q (Q ≥ 2) is a random sequence. This shows that our
definition of random real numbers is equivalent to that of Calude
and Jurgensen [Cal94, CJ94] by means of random positional
representations.
TL;DR: An up-to-dated survey of results about the power of controlled extended H systems of various types, and disprove a conjecture that such systems of radius one and using only one-sided context conditions can generate all context-free languages.
Abstract: This paper is an up-to-dated survey of results about the power of controlled extended H systems of various types. Such systems are finite generative devices based on the splicing operation, which, in turn, is a model of the DNA recombination. When no control is imposed on the splicing operation, we get a characterization of regular languages. When a control is considered, we obtain characterizations of recursively enumerable languages. We briefly discuss here H systems with control mechanisms similar to those in regulated rewriting area in formal language theory. We do not present proofs, but only definitions, results, and bibliographical information. Looking for characterizations of families in Chomsky hierarchy other than those of regular and of recursively enumerable languages, we closely investigate the H systems with permitting context conditions. We disprove a conjecture of Chakaravarthy and Krithivasan: such systems of radius one and using only one-sided context conditions can generate all context-free languages (the conjecture was that a characterization of linear languages is obtained). Some open problems are also mentioned.
TL;DR: It is shown that if an infinite regressive isol has comparability of summands, then every predecessor of the isol has a weak comparabilityof summands property.
Abstract: In this paper we present a collection of results related to the comparability of summands property of regressive isols. We show that if an infinite regressive isol has comparability of summands, then every predecessor of the isol has a weak comparability of summands property. Recently R. Downey proved that there exist regressive isols that are both hyper-torre and cosimple. There is a surprisingly close connection between non-recursive recursively enumerable sets and particular retraceable sets and regressive isols. We apply the theorem of Downey to show that among the regressive isols that are related to recursively enumerable sets there are some with a new property.
01 Jan 1997
TL;DR: This book focuses on theorem proving in the context of first-order logic, a rich language by which algebraic theories, computational problems, and substantial knowledge representation in Artificial Intelligence can be expressed.
Abstract: Throughout the whole book we will focus on theorem proving in the context of first-order logic. First-order logic plays an important role in mathematical logic and computer science. First of all it is a rich language, by which algebraic theories, computational problems, and substantial knowledge representation in Artificial Intelligence can be expressed; due to its ability to represent undecidable problems (like the halting problem for Turing machines) first-order logic (or more precisely, the validity problem for first-order logic) is undecidable. On the other hand, the valid formulas of first-order logic can be obtained by logical calculi and thus are recursively enumerable. In this sense, first-order logic is mechanizable (we can find a proof for every valid sentence, but there is no decision procedure for validity).
TL;DR: It is shown that for every nontrivial r.
Abstract: We show that for every nontrivial r.e. wtt-degree a, there are r.e. wtt-degrees b and c incomparable to a such that the infimum of a and b exists but the infimum of a and c fails to exist. This shows in particular that there are no strongly noncappable r.e. wtt-degrees, in contrast to the situation in the r.e. Turing degrees.
01 Jan 1997
TL;DR: It is proved that each recursively enumerable language can be generated by a cooperating distributed grammar system with two Q+ registers and right-linear rules.
Abstract: We prove that each recursively enumerable language can be generated by a cooperating distributed grammar system with two Q+ registers and right-linear rules.
01 Jan 1997
TL;DR: This work provides purely homomorphic characterizations of the regular sets and the recursively enumerable sets, and of the classes MULTI-RESET and BNP, and provides extensive use of notions, techniques and results introduced by Professor Ron Book.
Abstract: The use of homomorphisms for simple representations of certain classes of languages is investigated. The homomorphisms are combined to tuples which then define a homomorphic replication. This operation plays an important role in the characterization of classes of languages like BNP and MULTI-RESET. If the homomophisms of a tuple coincide on a string, then their homomorphic equality is defined. This operation is closely related to equality sets of homomorphisms and is extremely powerful. We provide purely homomorphic characterizations of the regular sets and the recursively enumerable sets, and of the classes MULTI-RESET and BNP. Starting from {$}, three homomorphisms suffice to represent every regular set and four homomorphisms suffice to represent every recursively enumerable set. For these statements we make extensive use of notions, techniques and results introduced by Professor Ron Book.
01 Jan 1997
TL;DR: Of primary interest are those questions related to possible applications of contextual grammars as models of natural or artificial languages syntax and those problems related to characterizations of recursively enumerable languages.
Abstract: Although the theory of contextual grammars is about thirty years old and, as one can see in the previous chapters, this theory is pretty well developed, many problems still wait for further research efforts. Some of them are “local” technical problems, others are research topics of a larger interest. We list here a number of problems stated already in this book; in most cases, we reformulate them in a more general form. Of course, not all of them have the same importance (hence not all of them deserve the same interest). In our opinion, of primary interest are those questions related to possible applications of contextual grammars as models of natural or artificial languages syntax and those problems related to characterizations of recursively enumerable languages. The interest for the first research direction is obvious, the second one can shed a new light on the “structure” of computability: Chomsky grammars, Turing machines, Markov normal algorithms, Thue and Post systems, etc, are basically rewriting mechanisms; contextual grammars (as well as computing devices appearing in the DNA computing area, such as splicing systems) are not using rewriting, but adjoining operations; still, they characterize RE as soon as some erasing possibilities are provided. And, it seems, the nature uses mainly such adjoining, cut-and-paste (splicing), insertion, and deletion operations when “computing” (see the genetic area).
15 Oct 1997
TL;DR: It is found that characterizations of recursively enumerable languages starting from languages generated by (non-returning non-centralized) parallel communicating grammar systems with right-linear rules are similar.
Abstract: We give characterizations of recursively enumerable languages starting from copy languages, that is languages of the form $\{x\bar x\mid x\in L\}$, where $L$ is a regular language and $\bar x$ is the barred version of $x$. One characterization uses an intersection of morphic images of two copy languages, the other one uses a quotient of morphic images of two copy languages. As a consequence, we find similar characterizations of recursively enumerable languages starting from languages generated by (non-returning non-centralized) parallel communicating grammar systems with right-linear rules.
TL;DR: In this article, it was shown that for any A, B C w, (EA*) is isomorphic to (EB*) if and only if A' = T B', where A' is the Turing jump of A.
Abstract: Let w = {0,1,2,...}. , and for A C w, let EA be the lattice of subsets of w which are recursively enumerable relative to the "oracle" A. Let (EA)* be EA/_, where I is the ideal of finite subsets of w. It is established that for any A, B C w, (EA)* is effectively isomorphic to (EB)* if and only if A' = T B', where A' is the Turing jump of A. A consequence is that if A' =T B', then EA EB. A second consequence is that (EA)* can be effectively embedded into (EB)* preserving least and greatest elements if and only if A'
Journal Article•
TL;DR: The degree- theoretic structure of set-theoretical splittings of recursively enumerable (r.e.) sets into differences of r.e. sets is studied to deduce that the ordering of wtt–degrees of unsolvability of differences of R.E. sets are not a distributive semilattice and is not elementarily equivalent to the ordering.
Abstract: We study here the degree-theoretic structure of set-theoretical splittings of recursively enumerable (re) sets into differences of re sets As a corollary we deduce that the ordering of wtt–degrees of unsolvability of differences of re sets is not a distributive semilattice and is not elementarily equivalent to the ordering of re wtt–degrees of unsolvability