Showing papers on "Recursively enumerable language published in 2011"

Generalized planning: synthesizing plans that work for multiple environments

Abstract: We give a formal definition of generalized planning that is independent of any representation formalism. We assume that our generalized plans must work on a set of deterministic environments, which are essentially unrelated to each other. We prove that generalized planning for a finite set of environments is always decidable and EXPSPACE-complete. Our proof is constructive and gives us a sound, complete and complexity-wise optimal technique. We also consider infinite sets of environments, and show that generalized planning for the infinite "one-dimensional problems," known in the literature to be recursively enumerable when restricted to finite-state plans, is EXPSPACE-decidable without sequence functions, and solvable by generalized planning for finite sets.

Degrees of autostability relative to strong constructivizations

Abstract: The spectra of the Turing degrees of autostability of computable models are studied. For almost prime decidable models, it is shown that the autostability spectrum relative to strong constructivizations of such models always contains a certain recursively enumerable Turing degree; moreover, it is shown that for any recursively enumerable Turing degree, there exist prime models in which this degree is the least one in the autostability spectrum relative to strong constructivizations.

Levels of undecidability in rewriting

Abstract: Undecidability of various properties of first-order term rewriting systems is well-known. An undecidable property can be classified by the complexity of the formula defining it. This classification gives rise to a hierarchy of distinct levels of undecidability, starting from the arithmetical hierarchy classifying properties using first order arithmetical formulas, and continuing into the analytic hierarchy, where quantification over function variables is allowed. In this paper we give an overview of how the main properties of first order term rewriting systems are classified in these hierarchies. We consider properties related to normalization (strong normalization, weak normalization and dependency problems) and properties related to confluence (confluence, local confluence and the unique normal form property). For all of these we distinguish between the single term version and the uniform version. Where appropriate, we also distinguish between ground and open terms. Most uniform properties are @P"2^0-complete. The particular problem of local confluence turns out to be @P"2^0-complete for ground terms, but only @P"2^0-complete (and thereby recursively enumerable) for open terms. The most surprising result concerns dependency pair problems without minimality flag: we prove this problem to be @P"1^1-complete, hence not in the arithmetical hierarchy, but properly in the analytic hierarchy. Some of our results are new or have appeared in our earlier publications. Others are based on folklore constructions, and are included for completeness as their precise classifications have hardly been noticed previously.

One-sided random context grammars

Abstract: The notion of a one-sided random context grammar is defined as a context-free-based regulated grammar, in which a set of permitting symbols and a set of forbidding symbols are attached to every rule, and its set of rules is divided into the set of left random context rules and the set of right random context rules. A left random context rule can rewrite a nonterminal if each of its permitting symbols occurs to the left of the rewritten symbol in the current sentential form while each of its forbidding symbols does not occur there. A right random context rule is applied analogically except that the symbols are examined to the right of the rewritten symbol. The paper demonstrates that without erasing rules, one-sided random context grammars characterize the family of context-sensitive languages, and with erasing rules, these grammars characterize the family of recursively enumerable languages. In fact, these characterization results hold even if the set of left random context rules coincides with the set of right random context rules. Several special cases of these grammars are considered, and their generative power is established. In its conclusion, some important open problems are suggested to study in the future.

Well structured program equivalence is highly undecidable

Abstract: We show that strict deterministic propositional dynamic logic with intersection is highly undecidable, solving a problem in the Stanford Encyclopedia of Philosophy. In fact we show something quite a bit stronger. We introduce the construction of program equivalence, which returns the value $\mathsf{T}$ precisely when two given programs are equivalent on halting computations. We show that virtually any variant of propositional dynamic logic has $\Pi_1^1$-hard validity problem if it can express even just the equivalence of well-structured programs with the empty program \texttt{skip}. We also show, in these cases, that the set of propositional statements valid over finite models is not recursively enumerable, so there is not even an axiomatisation for finitely valid propositions.

One-reversal counter machines and multihead automata: revisited

Abstract: Among the many models of language acceptors that have been studied in the literature are multihead finite automata (finite automata with multiple one-way input heads) and 1-reversal counter machines (finite automata with multiple counters, where each counter can only "reverse" once, i.e., once a counter decrements, it can no longer increment). The devices can be deterministic or nondeterministic and can be augmented with a pushdown stack. We investigate the relative computational power of these machines. Our results (where C1 and C2 are classes of machines) are of the following types: 1. Machines in C1 and C2 are incomparable. 2. Machines in C1 are strictly weaker than machines in C2. In obtaining results of these types, we use counting and "cut-and-paste" arguments as well as an interesting technique that shows that if a language were accepted by a device in a given class, then all recursively enumerable languages would be decidable.

Morphic characterizations of languages in Chomsky hierarchy with insertion and locality

Abstract: This paper concerns new characterizations of regular, context-free, and recursively enumerable languages, using insertion systems with lower complexity. This is achieved by using both strictly locally testable languages and morphisms. The representation is in a similar way to the Chomsky-Schutzenberger representation of context-free languages. Specifically, each recursively enumerable language L can be represented in the form L=h(L(γ)∩R), where γ is an insertion system of weight (3,3), R is a strictly 2-testable language, and h is a projection. A similar representation can be obtained for context-free languages, using insertion systems of weight (2,0) and strictly 2-testable languages, as well as for regular languages, using insertion systems of weight (1,0) and strictly 2-testable languages.

A Compact [0,1]-valued First-order Łukasiewicz Logic with Identity on Hilbert Space

Abstract: By an MV-set, we understand a pair (E,X) where X is a set of unit vectors in a Hilbert space E such that the linear span of X is dense in E, and 〈v,w〉 ≥ 0 for all v,w ∈ X. The scalar product 〈v,w〉 ∈ [0,1] is the identity degree of v and w. Building on MV-sets and continuous functions and relations defined on them, we construct a compact [0,1]-valued first-order Łukasiewicz logic, whose set of unsatisfiable formulas is recursively enumerable. In the particular case when X is an orthonormal basis of E we recover classical Skolem first-order logic with identity, constants, functions and relations. Our main tools are the Kolmogorov dilation theorem for positive semidefinite kernels, and the Tarski–Seidenberg decision method for elementary algebra and geometry.

On Networks of Evolutionary Processors with Filters Accepted by Two-State-Automata

Abstract: In this paper, we study networks of evolutionary processors where the filters are chosen as special regular sets. We consider networks where all the filters belong to a set of languages that are accepted by deterministic finite automata with a fixed number of states. We show that if the number of states is bounded by two, then every recursively enumerable language can be generated by such a network. If the number of states is bounded by one, then not all regular languages but non-context-free languages can be generated.

Morphic characterizations of language families in terms of insertion systems and star languages

Abstract: Insertion systems have a unique feature in that only string insertions are allowed, which is in marked contrast to a variety of the conventional computing devices based on string rewriting. This paper will mainly focus on those systems whose insertion operations are performed in a context-free fashion, called context-free insertion systems, and obtain several characterizations of language families with the help of other primitive languages (like star languages) as well as simple operations (like projections, weak-codings). For each k ≥ 1, a language L is a k-star language if L = F+ for some finite set F with the length of each string in F is no more than k. The results of this kind have already been presented in [10] by Paun et al., while the purpose of this paper is to prove enhanced versions of them. Specifically, we show that each context-free language L can be represented in the form L = h(L(γ)∩F+), where γ is an insertion system of weight (3, 0) (at most three symbols are inserted in a context-free manner), h is a projection, and F+ is a 2-star language. A similar characterization can be obtained for recursively enumerable languages, where insertion systems of weight (3, 3) and 2-star languages are involved.

Accepting Networks of Evolutionary Processors with Subregular Filters.

Abstract: We propose a new variant of Accepting Networks of Evolutionary Processors, in which the operations are applied at arbitrary positions to the processed words (rather than at the ends of words only) and where the filters are languages from several special classes of regular sets. More precisely, we show that the use of filters from the class of non-counting, ordered, power-separating, suffix-closed regular, union-free, definite and combinational languages is as powerful as the use of arbitrary regular languages and yields networks that can accept all the recursively enumerable languages. On the other hand, by using filters that are only finite languages, monoids, nilpotent languages, commutative regular languages, or circular regular languages, one cannot generate all recursively enumerable languages. These results seem interesting as they provide both upper and lower bounds on the classes of languages that one can use as filters in an accepting network of evolutionary processors in order to obtain a complete computational model.

On the Power of Small Size Insertion P Systems

Abstract: In this article we investigate insertion systems of small size in the framework of P systems. We consider P systems with insertion rules having one symbol context and we show that they have the computational power of context-free matrix grammars. If contexts of length two are permitted, then any recursively enumerable language can be generated. In both cases a squeezing mechanism, an inverse morphism, and a weak coding are applied to the output of the corresponding P systems. We also show that if no membranes are used then corresponding family is equal to the family of context-free languages.

On normal forms for networks of evolutionary processors

Abstract: In this paper we show that some aspects of networks of evolutionary processors can be normalized or simplified without loosing generative power. More precisely, we show that one can use very small finite automata for the control of the communication. We first prove that the networks with evolutionary processors remain computationally complete if one restricts the control automata to have only one state, but underlying graphs of the networks have no fixed structure and the rules are applied in three different modes. Moreover, we show that networks where the rules are applied arbitrary, and all the automata for control have one state, cannot generate all recursively enumerable languages. Finally, we show that one can generate all recursively enumerable languages by complete networks, where the rules are applied arbitrary, but the automata for control have at most two states.

A Computational Approach to the Ordinal Numbers

Abstract: This paper expands Kleene’s notations for recursive ordinals to larger countable ordinals by defining notations for limit ordinals using total recursive functions on nonrecursively enumerable domains such as Kleene’s O. This leads to a hierarchy related to that developed with Turing Machine oracles or relative recursion. The recursive functions that define notations for limit ordinals form a typed hierarchy. They are encoded as Turing Machines that identify the type of parameters (labeled by ordinal notations) they accept as inputs and the type of input that can be constructed from them. It is practical to partially implement these recursive functional hierarchies and perform computer experiments as an aid to understanding and intuition. This approach is both based on and compliments an ordinal calculator research tool. 3.1 Objective mathematics Kleene’s O is part of what I call objective mathematics. O is a set of recursive ordinal notations defined using finite structures and recursive functions. ω 1 is the set of all recursive ordinals. The recursive ordinal notations in Kleene’s O are constructed using computer programs whose actions mirror the structure of the ordinal. Thus the ordinals represented in O have an objective interpretation in an always finite but possibly potentially infinite universe. Part of the motivation for generalizing Kleene’s O to notations for larger countable ordinals is to help to draw the boundaries of unambiguous mathematics that is physically definable and physically meaningful in an always finite but potentially infinite universe. Many others have attempted to draw related lines in various ways. Section 3.1.2 discusses the objectivity of creative divergent processes with important properties that can only be defined by quantification over the reals. Even so these properties are objective and important in am always finite but potentially infinite universe. The Lowenheim-Skolem theorem proves that the uncountable is ambiguous in any finite or r. e. (recursively enumerable) formalization of mathematics. Any effective formal system, that has a model, must have a countable model. Thus, in contrast to much of mathematics[5], cardinality of the integers. This means that if the reals can be mapped onto a set (with a unique real for every object in the set) and the the integers cannot then that set can be mapped onto the reals with a unique object for every real.

Language classes generated by tree controlled grammars with bounded nonterminal complexity

Abstract: A tree controlled grammar is specified as a pair (G,G^') where G is a context-free grammar and G^' is a regular grammar. Its language consists of all terminal words with a derivation in G such that all levels of the corresponding derivation tree-except the last level-belong to L(G^'). We define the nonterminal complexity V ar(H) of H=(G,G^') as the sum of the numbers of nonterminals of G and G^'. In Turaev et al. (2011) [23] it is shown that tree controlled grammars H with V ar(H)@?9 are sufficient to generate all recursively enumerable languages. In this paper, we improve the bound to seven. Moreover, we show that all linear and regular simple matrix languages can be generated by tree controlled grammars with a nonterminal complexity bounded by three, and we prove that this bound is optimal for the mentioned language families. Furthermore, we show that any context-free language can be generated by a tree controlled grammar (G,G^') where the number of nonterminals of G and G^' is at most four.

On the number of components and clusters of non-returning parallel communicating grammar systems

Abstract: In this paper, we study the size complexity of nonreturning parallel communicating grammar systems First we consider the problem of determining the minimal number of components necessary to generate all recursively enumerable languages We present a construction which improves the currently known best bounds of seven (with three predefined clusters) and six (in the non-clustered case) to five, in both cases (having four clusters in the clustered variant) We also show that in the case of unary languages four components are sufficient Then, by defining systems with dynamical clusters, we investigate the minimal number of different query symbols necessary to obtain computational completeness We prove that for this purpose three dynamical clusters (which means two different query symbols) are sufficient in general, which (although the number of components is higher) can also be interpreted as an improvement in the number of necessary clusters when compared to the case of predefined clusters

Universal computably enumerable sets and initial segment prefix-free complexity

Abstract: We show that there are Turing complete computably enumerable sets of arbitrarily low non-trivial initial segment prefix-free complexity. In particular, given any computably enumerable set $A$ with non-trivial prefix-free initial segment complexity, there exists a Turing complete computably enumerable set $B$ with complexity strictly less than the complexity of $A$. On the other hand it is known that sets with trivial initial segment prefix-free complexity are not Turing complete. Moreover we give a generalization of this result for any finite collection of computably enumerable sets $A_i, i

Implementation of Recursively Enumerable Languages in Universal Turing Machine

Abstract: This paper presents the design and working of a Universal Turing Machine (UTM) for the JFLAP platform. Automata play a major role in compiler design and parsing. The class of formal languages that work for the most complex problems belong to the set of Recursively Enumerable Languages (REL). RELs are accepted by the type of automata known as Turing Machines. Turing Machines are the most powerful computational machines and are the theoretical basis for modern computers. Still it is a tedious task to create and maintain Turing Machines for all the problems. To solve this Universal Turing Machine (UTM) is designed in this paper. The UTM works for all classes of languages including regular languages, Context Free Languages as well as Recursively Enumerable Languages. A UTM simulates any other TM, thus providing a single model and solution for all the computational problems. The creation of UTM is very tedious because of the underlying complexities. Also many of the existing tools do not support the creation of UTM which makes the task very difficult to accomplish. Hence a Universal Turing Machine is developed for the JFLAP platform. JFLAP is most successful and widely used tool for visualizing and simulating all types of automata.

Turing machines based on unsharp quantum logic

Abstract: In this paper, we consider Turing machines based on unsharp quantum logic. For a lattice-ordered quantum multiple-valued (MV) algebra E, we introduce E-valued non-deterministic Turing machines (ENTMs) and E-valued deterministic Turing machines (EDTMs). We discuss different E-valued recursively enumerable languages from width-first and depth-first recognition. We find that width-first recognition is equal to or less than depth-first recognition in general. The equivalence requires an underlying E value lattice to degenerate into an MV algebra. We also study variants of ENTMs. ENTMs with a classical initial state and ENTMs with a classical final state have the same power as ENTMs with quantum initial and final states. In particular, the latter can be simulated by ENTMs with classical transitions under a certain condition. Using these findings, we prove that ENTMs are not equivalent to EDTMs and that ENTMs are more powerful than EDTMs. This is a notable difference from the classical Turing machines.

Further Results on Languages of Membrane Structures

Abstract: In [3], P systems with active membranes were used to generate languages, in the sense of languages associated with the structure of membrane systems. Here, we analyze the power of P systems with membrane creation and dissolution restricted to elementary membranes, P systems without membrane dissolution operating according to certain output modes. This leads us to characterizations of recursively enumerable languages.

On Promoters/Inhibitors and Symport/Antiport with Traces in P Systems

Abstract: This article brings together some rather powerful results on P systems in which the computation is performed by the communication of objects through symport and antiport rules considering the trace of an object through membranes, on the one hand, and by P systems with object-rewriting non-cooperative rules, promoters/inhibitors at the level of rules and only one catalyst, on the other. It is recalled here that computational universality can be reached whit these formalisms and that some of the proofs can be sketched. Three ideas are also put forward to brake the direct relationship (infinite hierarchy) induced by the size of the considered alphabet and the number of the membranes needed in a P system (with traces) to generate recursively enumerable languages on the chosen alphabet.

Double Cross - Over Circular Array Splicing

Abstract: Splicing system is proposed to model the recombinant behaviour of DNA molecules. This paper introduces the concept of double cross-over splicing in circular arrays. The theoretical result as persistence in circular array splicing give rise to important properties. we obtained recursively enumerable language L for any double cross-over circular array splicing system S.

Kolmogorov complexity and computably enumerable sets

Abstract: We study the computably enumerable sets in terms of the: (a) Kolmogorov complexity of their initial segments; (b) Kolmogorov complexity of finite programs when they are used as oracles. We present an extended discussion of the existing research on this topic, along with recent developments and open problems. Besides this survey, our main original result is the following characterization of the computably enumerable sets with trivial initial segment prefix-free complexity. A computably enumerable set $A$ is $K$-trivial if and only if the family of sets with complexity bounded by the complexity of $A$ is uniformly computable from the halting problem.

Generating Networks of Splicing Processors.

Abstract: In this paper, we introduce generating networks of splicing processors (GNSP for short), a formal languages generating model related to networks of evolutionary processors and to accepting networks of splicing processors. We show that all recursively enumerable languages can be generated by GNSPs with only nine processors. We also show, by direct simulation, that two other variants of this computing model, where the communication between processors is conducted in different ways, have the same computational power.

On the Power of P Systems with Valuations

Abstract: WE CONSIDER SOME SLIGTH VARIANTS OF P SYSTEMS WITH VALUATION AS INTRODUCED IN MARTIN-VIDE AND MITRANA (2000) AND WE STUDY THEIR GENERATIVE POWER AN D COMPUTACIONAL EFFICIENCY. WHEN REWRITING TAKES PLACE ANLY IN THE ENDS OF THE STRINGS, EACH RECURSIVELY ENUMERABLE LANGUAGE CON BE GENERATED BY A SYSTEM WITH ONLY TWO MENBRANES. IF THE STRING REPLICATION IS ALLOWED (WHEN THE VALUATION OF A STRING ALLOWS THE STRING TO GO TO SEVERAL MEMBRANES, THEN COPIES OF THE STRING ARE SENT TO ALL THESE MEMBRANES), THEN NP-COMPLETE PROBLEMS CAN BE SOLVED IN LINEAR TIME. WE PROVE THIS FOR HPP (THE EXISTENCE OF A HAMILTONIAN PATH IN A DIRECTED GRAPH) AND GIVE SOME INFORMAL AXPLANATIONS ON A SIMILAR SOLUTION FOR SAT.

Recursively enumerable hierarchy through Complexity classes

Abstract: This paper divide some complexity class by using fixpoint and fixpointless area of Decidable Universal Turing Machine (UTM). Decidable Deterministic Turing Machine (DTM) have fixpointless combinator that add no extra resources (like Negation), but UTM makes some fixpoint in the combinator. This means that we can jump out of the fixpointless combinator system by making more complex problem from diagonalisation argument of UTM. As a concrete example, we proof L is not P . We can make Polynomial time UTM that emulate all Logarithm space DTM (LDTM). LDTM set close under Negation, therefore UTM does not close under LDTM set. (We can proof this theorem like halting problem and time/space hierarchy theorem, and also we can extend this proof to divide time/space limited DTM set.) In the same way, we proof P is not NP. These are new hierarchy that use UTM and Negation.