Showing papers on "Recursively enumerable language published in 2005"
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02 Feb 2005TL;DR: It is found that set-conditional insertion-deletion systems with two axioms generate any recursively enumerable language, as well as that membrane systems with one membrane having context-free insertion- deleletion rules without conditional use of them generate all recursive enumerable languages.
Abstract: We consider a class of insertion-deletion systems which have not been investigated so far, those without any context controlling the insertion-deletion operations. Rather unexpectedly, we found that context-free insertion-deletion systems characterize the recursively enumerable languages. Moreover, this assertion is valid for systems with only one axiom, and also using inserted and deleted strings of a small length. As direct consequences of the main result we found that set-conditional insertion-deletion systems with two axioms generate any recursively enumerable language (this solves an open problem), as well as that membrane systems with one membrane having context-free insertion-deleletion rules without conditional use of them generate all recursively enumerable languages (this improves an earlier result). Some open problems are also formulated.
86 citations
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TL;DR: The existence of polynomial time algorithms to approximate the Julia sets of given hyperbolic rational functions is proved and strict computable error estimation is given w.r.t. the Hausdorff metric on the complex sphere.
48 citations
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TL;DR: It is proved that in the case of non-elementary networks, any recursively enumerable language over a common alphabet can be obtained with an HNEP whose underlying structure is a fixed graph depending on the common alphabet only.
Abstract: A hybrid network of evolutionary processors (an HNEP) consists of several language processors which are located in the nodes of a virtual graph and able to perform only one type of point mutations (insertion, deletion, substitution) on the words found in that node, according to some predefined rules. Each node is associated with an input and an output filter, defined by some random-context conditions. After applying in parallel a point mutation to all the words existing in every node, the new words which are able to pass the output filter of the respective node navigate simultaneously through the network and enter those nodes whose input filter they are able to pass. We show that even the so-called elementary HNEPs are computationally complete. In this case every node is able to perform only one instance of the specified operation: either an insertion, or a deletion, or a substitution of a certain symbol. We also prove that in the case of non-elementary networks, any recursively enumerable language over a common alphabet can be obtained with an HNEP whose underlying structure is a fixed graph depending on the common alphabet only.
44 citations
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02 Feb 2005TL;DR: It is proved that scattered context grammars having two context sensing productions and five nonterminals are sufficient to generate all recursively enumerable languages and it is shown that the same power can be reached by simple semi-conditional grammar having 10 conditional productions with conditions of the length two.
Abstract: We improve the upper bounds of certain descriptional complexity measures of two types of rewriting mechanisms regulated by context conditions. We prove that scattered context grammars having two context sensing productions and five nonterminals are sufficient to generate all recursively enumerable languages and we also show that the same power can be reached by simple semi-conditional grammars having 10 conditional productions with conditions of the length two or eight conditional productions with conditions of length three. The results are based on the common idea of using the so called Geffert normal forms for phrase structure grammars.
40 citations
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01 Sep 2005TL;DR: It is proved that four membranes suffice to a variant of P systems with symport/antiport with maximal parallelism to generate all recursively enumerable sets of numbers.
Abstract: It is proved that four membranes suffice to a variant of P systems with symport/antiport with maximal parallelism to generate all recursively enumerable sets of numbers. P systems with symport/antiport without maximal parallelism are also studied, considering two termination criteria.
34 citations
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18 Jul 2005TL;DR: It is shown that 8 rules suffice to recognise any recursively enumerable language if splicing tissue P systems are considered, and that this language cannot be described in terms of membranes or objects.
Abstract: In the last time several attempts to decrease different complexity parameters (number of membranes, size of rules, number of objects etc.) of universal P systems were done. In this article we consider another parameter which was not investigated yet: the number of rules. We show that 8 rules suffice to recognise any recursively enumerable language if splicing tissue P systems are considered.
31 citations
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01 Sep 2005TL;DR: In this article, a system with active membranes but without using electrical charges (polarization) is presented for generating recursively enumerable string languages, using only rules with membrane transitions as well as rules with membranes dissolving and elementary membrane division.
Abstract: P systems with active membranes but without using electrical charges (polarizations) are shown to be complete for generating recursively enumerable string languages when working on string objects and using only rules with membrane transitions as well as rules with membrane dissolving and elementary membrane division, but also when using various other kinds of rules, even including a new type of rules allowing for membrane generation. In particular, allowing for changing membrane labels turns out to be a very powerful control feature.
26 citations
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TL;DR: This article studies the computational complexity of the agent design problem for tasks that are of the form “achieve this state of affairs” or “maintain thisstate of affairs,” and considers three general formulations of these problems (in both non-deterministic and deterministic environments).
Abstract: The agent design problem is as follows: given a specification of an environment, together with a specification of a task, is it possible to construct an agent that can be guaranteed to successfully accomplish the task in the environment? In this article, we study the computational complexity of the agent design problem for tasks that are of the form "achieve this state of affairs" or "maintain this state of affairs." We consider three general formulations of these problems (in both non-deterministic and deterministic environments) that differ in the nature of what is viewed as an "acceptable" solution: in the least restrictive formulation, no limit is placed on the number of actions an agent is allowed to perform in attempting to meet the requirements of its specified task. We show that the resulting decision problems are intractable, in the sense that these are non-recursive (but recursively enumerable) for achievement tasks, and non-recursively enumerable for maintenance tasks. In the second formulation, the decision problem addresses the existence of agents that have satisfied their specified task within some given number of actions. Even in this more restrictive setting the resulting decision problems are either pspace-complete or np-complete. Our final formulation requires the environment to be history independent and bounded. In these cases polynomial time algorithms exist: for deterministic environments the decision problems are nl-complete; in non-deterministic environments, p-complete.
25 citations
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22 Jul 2005TL;DR: In this article, the authors investigate the computational behavior of two-dimensional propositional temporal logics over (ℕ,<) (with and without the next-time operator O) that are capable of reasoning about states with transitive relations.
Abstract: We investigate the computational behaviour of ‘two-dimensional' propositional temporal logics over (ℕ,<) (with and without the next-time operator O) that are capable of reasoning about states with transitive relations. Such logics are known to be undecidable (even $\Pi^{\rm 1}_{\rm 1}$-complete) if the domains of states with those relations are assumed to be constant. Motivated by applications in the areas of temporal description logic and specification & verification of hybrid systems, in this paper we analyse the computational impact of allowing the domains of states to expand. We show that over finite expanding domains (with an arbitrary, tree-like, quasi-order, or linear transitive relation) the logics are recursively enumerable, but undecidable. If these finite domains eventually become constant then the resulting O-free logics are decidable (but not in primitive recursive time); on the other hand, when equipped with O they are not even recursively enumerable. Finally, we show that temporal logics over infinite expanding domains as above are undecidable even for the language with the sole temporal operator ‘eventually.' The proofs are based on Kruskal's tree theorem and reductions of reachability problems for lossy channel systems.
22 citations
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24 Feb 2005TL;DR: It is shown that one can construct a finite language L such that the largest language commuting with L is not recursively enumerable, giving a negative answer to the question raised by Conway in 1971 and disproves Conway's conjecture on context-freeness of maximal solutions of systems of semi-linear inequalities.
Abstract: We show that one can 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.
21 citations
01 Jan 2005
TL;DR: It follows that the decision problem, whether the Frechet distance of two given surfaces lies below some speci- fied value, is recursively enumerable.
Abstract: The Frechet distance is a distance measure for pa- rameterized curves or surfaces. Using a discrete ap- proximation, we show that for triangulated surfaces it is upper semi-computable, i.e., there is a non-halting Turing machine which produces a monotone decreas- ing sequence of rationals converging to the result. It follows that the decision problem, whether the Frechet distance of two given surfaces lies below some speci- fied value, is recursively enumerable.
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TL;DR: This work proposes a program for the separation of complexity classes and identifies three properties that are potential separators: auto-reducibility, robustness, and mitoticity.
Abstract: In 1944, E. Post proposed a program that would lead to the identification of separate degrees of recursively enumerable sets. Post proposed to identify structural properties that sets of different degrees would not share. Thus proving such a property for sets in one degree would imply that these sets are not in the other. We propose a similar program for the separation of complexity classes and identify three properties that are potential separators: auto-reducibility, robustness, and mitoticity. Some partial results that do separate complexity classes have already been established. Also, answering the question whether complete sets in certain classes do or do not have these properties either way gives an answer to separation problems of central interest.
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25 Sep 2005TL;DR: It is proved that 10 + m symbols are enough to generate any recursively enumerable language over m symbols, and it is shown that accepting can be done by deterministic systems.
Abstract: We improve, by using register machines, some existing universality results for specific models of P systems. P systems with membrane creation are known to generate all recursively enumerable sets of vectors of non-negative integers, even when no region (except the environment) contains more than one object of the same kind. We here show that they generate all recursively enumerable languages, and two membrane labels are sufficient (the same result holds for accepting all recursively enumerable vectors of non-negative integers). Moreover, at most two objects are present inside the system at any time in the generative case. Then we prove that 10 + m symbols are enough to generate any recursively enumerable language over m symbols. P systems with active membranes without polarizations are known to generate all recursively enumerable sets of vectors of non-negative integers. We show that they generate all recursively enumerable languages; four starting membranes with three labels or seven starting membranes with two labels are sufficient. P systems with active membranes and two polarizations are known to generate/accept all recursively enumerable sets of vectors of non-negative integers, only using rules of rewriting and sending objects out. We show that accepting can be done by deterministic systems. Finally, remarks and open questions are presented.
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TL;DR: It turns out that starting from a Turing machine M with alphabet A and finite set of states Q which generates a given recursively enumerable language L, you need around 2 × |I| + 2 rules in order to define an extended H system which generates L.
Abstract: In this paper, we look at extended splicing systems (i.e., H systems) in order to find how small such a system can be in order to generate a recursively enumerable language. It turns out that starting from a Turing machine M with alphabet A and finite set of states Q which generates a given recursively enumerable language L, we need around 2×|I|+2 rules in order to define an extended H system H which generates L, where I is the set of instructions of Turing machine M. Next, coding the states of Q and the non-terminal symbols of L, we obtain an extended H system H 1 which generates L using |A|+2 symbols. At last, by encoding the alphabet, we obtain a splicing system U which generates a universal recursively enumerable set using only two letters.
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TL;DR: The hairpin inverted repeat excision is formalized, which is known in ciliate genetics as an operation on words and languages by defining as the set of all words xαyRαRz, and the status of decidability of the membership problem, emptiness problem and finiteness problem is presented.
Abstract: We formalize the hairpin inverted repeat excision, which is known in ciliate genetics as an operation on words and languages by defining as the set of all words xαyRαRz where w = xαyαRz and the pointer α is in P. We extend this concept to language families which results in families . For and be the families of finite, regular, context-free, context-sensitive or recursively enumerable language, respectively, we determine the hierarchy of the families and compare these families with those of the Chomsky hierarchy. Furthermore, we present the status of decidability of the membership problem, emptiness problem and finiteness problem for the families .
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TL;DR: A positive answer to the question whether or not insertion grammars with weight at least 7 can characterize recursively enumerable languages can be improved is come up with by decreasing the weight of the insertion grammar used to 5.
Abstract: Insertion grammars have been introduced in [1] and their computational power has been studied in several places. In [7] it is proved that insertion grammars with weight at least 7 can characterize recursively enumerable languages (modulo a weak coding and an inverse morphism), and the question was formulated whether or not this result can be improved. In this paper, we come up with a positive answer to this question, by decreasing the weight of the insertion grammar used to 5. We also give a characterization of recursively enumerable languages in terms of right quotients of insertion languages.
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TL;DR: It is proved that the class of all representable ordered monoids is not finitely axiomatisable, and reducts of relation algebras are considered, by dropping some of the operators from the signature.
Abstract: An ordered monoid is a structure with an identity element (1′), a binary composition operator (;) and an antisymmetric partial order (≤), satisfying certain axioms. A representation of an ordered monoid is a 1-1 map which maps elements of an ordered monoid to binary relations in such a way that 1′ is mapped to the identity relation, ; corresponds to composition of binary relations and ≤ corresponds to inclusion of binary relations. We devize a two player game that tests the representability of an ordered monoid n times and show that these games characterise representability. From this we obtain a recursively enumerable, universal axiomatisation of the class of all representable ordered monoids. For each n < ω we construct an unrepresentable ordered monoid An and show that the second player has a winning strategy in a game of length n. Hence we prove that the class of all representable ordered monoids is not finitely axiomatisable. Relation Algebras are badly behaved in a number of ways. The class of representable relation algebras cannot be defined by finitely many axioms [Mon64], nor by any set of equations using a finite number of variables [Jón91], nor by any Sahlqvist theory [Ven97], the equational theory of relation algebras and the equational theory of representable relation algebras is undecidable [Tar41], the problem of determining whether a finite relation algebra is representable or not is itself undecidable [HH01]. An important line of research is to consider reducts of relation algebras, by dropping some of the operators from the signature. We aim to find out exactly what causes this “bad behaviour” and how it can be avoided. Mikulás has surveyed much of this research [Mik03]. In the current paper we consider algebras in the reduced signature {≤, 1′, ; }. Such an algebra is representable if its elements can be interpretted as binary relations over some domain in such a way that ≤ is represented as inclusion of
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TL;DR: It is proved that given a “program” of the operator one can obtain positive information on the spectrum as a compact set in the sense that a dense subset of the spectrum can be enumerated and a bound on the set can be computed.
Abstract: Self-adjoint operators and their spectra play a crucial role in analysis and physics. For instance, in quantum physics self-adjoint operators are used to describe measurements and the spectrum represents the set of possible measurement results. Therefore, it is a natural question whether the spectrum of a self-adjoint operator can be computed from a description of the operator. We prove that given a “program” of the operator one can obtain positive information on the spectrum as a compact set in the sense that a dense subset of the spectrum can be enumerated (or equivalently: its distance function can be computed from above) and a bound on the set can be computed. This generalizes some non-uniform results obtained by Pour-El and Richards, which imply that the spectrum of any computable self-adjoint operator is a recursively enumerable compact set. Additionally, we show that the spectrum of compact selfadjoint operators can even be computed in the sense that also negative information is available (i.e. the distance function can be fully computed). Finally, we also discuss computability properties of the resolvent map.
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04 Jul 2005TL;DR: It is shown that there exist finite languages K and P and star-free languages L, M and R such that the largest solutions of the systems XK⊆LX and XKsubseteq LX are not recursively enumerable.
Abstract: It is known that for a regular language L and an arbitrary language K the largest solution of the inequality XK⊆LX is regular. Here we show that there exist finite languages K and P and star-free languages L, M and R such that the largest solutions of the systems $\{XK\subseteq LX,\ X\subseteq M\}$ and $\{XK\subseteq LX,\ XP\subseteq RX\}$ are not recursively enumerable.
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TL;DR: It is shown how game semantics can be employed to prove that program equivalence in finitary Idealized Algol with active expressions is undecidable.
Abstract: We show how game semantics can be employed to prove that program equivalence in finitary Idealized Algol with active expressions is undecidable. We also investigate a notion of representability of languages by terms and show that finitary Idealized Algol terms of respectively second, third and higher orders define exactly regular, context-free and recursively enumerable languages.
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Metz1
TL;DR: It is shown that ETVDH systems with 2 components, i.e., having two sets of rules which act periodically, may generate all recursively enumerable languages by simulating type-0 grammars.
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TL;DR: It is proved that for every recursively enumerable language L, there exists a propagating scattered context grammar whose language consists of L's sentences followed by their parses.
Abstract: Propagating scattered context grammars are used to generate their sentences together with their parses--that is, the sequences of labels denoting productions whose use lead to the generation of the corresponding sentences. It is proved that for every recursively enumerable language L, there exists a propagating scattered context grammar whose language consists of L's sentences followed by their parses.
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TL;DR: The main result proved shows that the natural embedding of any recursively enumerable one-dimensional array language in the two-dimensional space can be characterized by the projection of a two- dimensional array language generated by a contextual array grammar working in the t-mode and with norm one.
Abstract: The main result proved in this paper shows that the natural embedding of any recursively enumerable one-dimensional array language in the two-dimensional space can be characterized by the projection of a two-dimensional array language generated by a contextual array grammar working in the t-mode and with norm one Moreover, we show that any recursively enumerable one - dimensional array language can even be characterized by the projection of a two-dimensional array language generated by a contextual array grammar working in the t-mode where in the selectors of the contextual array productions only the ability to distinguish between blank and non-blank positions is necessary; in that case, the norm of the two-dimensional contextual array grammar working in the -mode cannot be bounded
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01 Jan 2005
TL;DR: It is shown that the mechanism underlying Logic Programming can be extended to handle the situation where the atoms are interpreted as subsets of a given space X by composing the old one-step consequence operator with a monotonic idempotent operator (miop) in the space of all subset of X, 2X.
Abstract: We propose a set of desiderata for extensions of Answer Set Programming to capture domains where the objects of interest are infinite sets and yet we can still process ASP programs effectively. We propose two different schemes to do this. One is to extend cardinality type constraints to set constraints which involve codes for finite, recursive and recursively enumerable sets. A second scheme to modify logic programming to reason about sets directly. In this setting, we can also augment logic programming with certain
monotone inductive operators so that we can reason about families of sets which have structure such a closed sets of a topological space or
subspaces of a vector space. We observe that under such conditions, the classic Gelfond-Lifschitz construction generalizes to at least two different notions of stable models.
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27 Jun 2005TL;DR: This work shows that the membership problem for deterministic catalytic systems is decidable and shows that for a deterministic 1-membrane catalytic system using only rules of type Ca →Cv, the set of reachable configurations from a given initial configuration is an effective semilinear set.
Abstract: We look at a 1-membrane catalytic P system with evolution rules of the form Ca →Cv or a →v, where C is a catalyst, a is a noncatalyst symbol, and v is a (possibly null) string representing a multiset of noncatalyst symbols. (Note that we are only interested in the multiplicities of the symbols.) A catalytic system can be regarded as a language acceptor in the following sense. Given an input alphabet Σ consisting of noncatalyst symbols, the system starts with an initial configuration wz, where w is a fixed string of catalysts and noncatalysts not containing any symbol in z, and $z = a_1^{n_1} \ldots a_k^{n_k}$ for some nonnegative integers n1, ..., nk, with { a1 ...ak } ⊆∑. At each step, a maximal multiset of rules is nondeterministically selected and applied in parallel to the current configuration to derive the next configuration (note that the next configuration is not unique, in general). The string z is accepted if the system eventually halts.
It is known that a 1-membrane catalytic system is universal in the sense that any unary recursively enumerable language can be accepted by a 1-membrane catalytic system (even by purely catalytic systems, i.e., when all rules are of the form Ca →Cv ). A catalytic system is said to be deterministic if at each step, there is a unique maximally parallel multiset of rules applicable. It has been an open problem whether deterministic systems of this kind are universal. We answer this question negatively: We show that the membership problem for deterministic catalytic systems is decidable. In fact, we show that the Parikh map of the language ( $\subseteq a_1^* \ldots a_k^*$ ) accepted by any deterministic catalytic system is a simple semilinear set which can be effectively constructed. Since nondeterministic 1-membrane catalytic system acceptors (with 2 catalysts) are universal, our result gives the first example of a variant of P systems for which the nondeterministic version is universal, but the deterministic version is not.
We also show that for a deterministic 1-membrane catalytic system using only rules of type Ca →Cv, the set of reachable configurations from a given initial configuration is an effective semilinear set. The application of rules of type a →v, however, is sufficient to render the reachability set non-semilinear. Our results generalize to multi-membrane deterministic catalytic systems. We also consider deterministic catalytic systems which allow rules to be prioritized and investigate three classes of such systems, depending on how priority in the application of the rules is interpreted. For these three prioritized systems, we obtain contrasting results: two are universal and one only accepts semilinear sets.
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TL;DR: In this article, it was shown that some natural games introduced by Lachlan in 1970 as a model of recursion theoretic constructions are undecidable, contrary to what was previously conjectured.
Abstract: We show that some natural games introduced by Lachlan in 1970 as a model of recursion theoretic constructions are undecidable, contrary to what was previously conjectured. Several consequences are pointed out; for instance, the set of all Π 2 -sentences that are uniformly valid in the lattice of recursively enumerable sets is undecidable. Furthermore we show that these games are equivalent to natural subclasses of effectively presented Borel games.
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25 Sep 2005TL;DR: It is proved that each recursively enumerable tree language can be obtained by this P systems with membrane creation working with symbol objects.
Abstract: In this paper, we consider P systems with membrane creation working with symbol objects. As a result of a halting computation we do not take the set of numbers generated in a designated output membrane, instead we take the resulting tree representing the membrane structure of the final configuration. We prove that each recursively enumerable tree language can be obtained by this system.
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TL;DR: In this article, it was shown that a weak version of Hilber's metamathematics is compatible with Godel's Incompleteness Theorems by employing only what are clearly natural provability predicates.
Abstract: In this paper, we attempt to show that a weak version of Hilberťs metamathematics is compatible with Godel's Incompleteness Theorems by employing only what are clearly natural provability predicates. Defining first 4T proves the consistency of a theory S indirectly in one step", we subsequently prove (i) "PA proves its own consistency indirectly in one step" and sketch the proof for (ii) "If S is a recursively enumerable extension of (QF-IA), S proves its own consistency indirectly in one step". The formalizations of the metatheoretical consistency assertions that occur in these theorems are clearly the natural ones. We conclude the paper with reflections on indirect consistency proofs and soundness proofs. 1. Goders Incompleteness Theorems and Consistency Proofs The main goal of Hilberťs foundational project was to vindicate all of classical mathematics by means of a finitist metamathematical consistency "proof'. Hilbert considered classical mathematics to be the paradigm of unassailable truth and believed that finitist means, as conceived by him, were absolutely reliable.1 For decades it has been widely held that Godel's Second Incompleteness Theorem put an end to Hilberťs original proof-theoretic programme. On the face of it, this view seems plausible: if we succeeded in carrying out a consistency proof for all of mathematics in metamathematics, mathematics would prove its own consistency, given that metamathematics is only a small fragment of mathematics in its entirety. Yet the very possibility that mathematics proves its own consistency is ruled out by Godel's Second
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TL;DR: Algebras of operations defined on recursively enumerable sets of different kinds are considered and fuzzy recursivity sets can be treated as fuzzy counterparts of sets representable by formulas of Presburger’s arithmetic system.
Abstract: Algebras of operations defined on recursively enumerable sets of different kinds are considered. Every such algebra is specified by a list of operations involved and a list of basic elements. An element of an algebra is said to be representable in this algebra if it can be obtained from given basic elements by operations of the algebra. Two kinds of recursively enumerable sets are considered: recursively enumerable sets in the usual sense and fuzzy recursively enumerable sets. On binary, i.e., two-dimensional recursively enumerable sets of these kinds, algebras of operations are introduced. An algebra θ is constructed in which all binary recursively enumerable sets are representable. A subalgebra θ0 of θ is constructed in which all binary recursively enumerable sets are representable if and only if they are described by formulas of Presburger’s arithmetic system. An algebra Ω is constructed in which all binary recursively enumerable fuzzy sets are representable. A subalgebra Ω0 of the algebra Ω is constructed such that fuzzy recursively enumerable sets representable in Ω0 can be treated as fuzzy counterparts of sets representable by formulas of Presburger’s system. Bibliography: 16 titles.
01 Jan 2005
TL;DR: It is proved 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.
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. On the other hand, every unary 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.