Showing papers on "Recursively enumerable language published in 2006"

Kolmogorov complexity and the recursion theorem

Abstract: We introduce the concepts of complex and autocomplex sets, where a set A is complex if there is a recursive, nondecreasing and unbounded lower bound on the Kolmogorov complexity of the prefixes (of the characteristic sequence) of A, and autocomplex is defined likewise with recursive replaced by A-recursive. We observe that exactly the autocomplex sets allow to compute words of given Kolmogorov complexity and demonstrate that a set computes a diagonally nonrecursive (DNR) function if and only if the set is autocomplex. The class of sets that compute DNR functions is intensively studied in recursion theory and is known to coincide with the class of sets that compute fixed-point free functions. Consequently, the Recursion Theorem fails relative to a set if and only if the set is autocomplex, that is, we have a characterization of a fundamental concept of theoretical computer science in terms of Kolmogorov complexity. Moreover, we obtain that recursively enumerable sets are autocomplex if and only if they are complete, which yields an alternate proof of the well-known completeness criterion for recursively enumerable sets in terms of computing DNR functions. All results on autocomplex sets mentioned in the last paragraph extend to complex sets if the oracle computations are restricted to truth-table or weak truth-table computations, for example, a set is complex if and only if it wtt-computes a DNR function. Moreover, we obtain a set that is complex but does not compute a Martin-Lof random set, which gives a partial answer to the open problem whether all sets of positive constructive Hausdorff dimension compute Martin-Lof random sets. Furthermore, the following questions are addressed: Given n, how difficult is it to find a word of length n that (a) has at least prefix-free Kolmogorov complexity n, (b) has at least plain Kolmogorov complexity n or (c) has the maximum possible prefix-free Kolmogorov complexity among all words of length n. All these questions are investigated with respect to the oracles needed to carry out this task and it is shown that (a) is easier than (b) and (b) is easier than (c). In particular, we argue that for plain Kolmogorov complexity exactly the PA-complete sets compute incompressible words, while the class of sets that compute words of maximum complexity depends on the choice of the universal Turing machine, whereas for prefix-free Kolmogorov complexity exactly the complete sets allow to compute words of maximum complexity.

Reasoning About Knowledge of Unawareness

Abstract: Awareness has been shown to be a useful addition to standard epistemic logic for many applications. However, standard propositional logics for knowledge and awareness cannot express the fact that an agent knows that there are facts of which he is unaware without there being an explicit fact that the agent knows he is unaware of. We propose a logic for reasoning about knowledge of unawareness, by extending Fagin and Halpern's \emph{Logic of General Awareness}. The logic allows quantification over variables, so that there is a formula in the language that can express the fact that ``an agent explicitly knows that there exists a fact of which he is unaware''. Moreover, that formula can be true without the agent explicitly knowing that he is unaware of any particular formula. We provide a sound and complete axiomatization of the logic, using standard axioms from the literature to capture the quantification operator. Finally, we show that the validity problem for the logic is recursively enumerable, but not decidable.

Modal Logics of Topological Relations

Abstract: Logical formalisms for reasoning about relations between spatial regions play a fundamental role in geographical information systems, spatial and constraint databases, and spatial reasoning in AI. In analogy with Halpern and Shoham's modal logic of time intervals based on the Allen relations, we introduce a family of modal logics equipped with eight modal operators that are interpreted by the Egenhofer-Franzosa (or RCC8) relations between regions in topological spaces such as the real plane. We investigate the expressive power and computational complexity of logics obtained in this way. It turns out that our modal logics have the same expressive power as the two-variable fragment of first-order logic, but are exponentially less succinct. The complexity ranges from (undecidable and) recursively enumerable to highly undecidable, where the recursively enumerable logics are obtained by considering substructures of structures induced by topological spaces. As our undecidability results also capture logics based on the real line, they improve upon undecidability results for interval temporal logics by Halpern and Shoham. We also analyze modal logics based on the five RCC5 relations, with similar results regarding the expressive power, but weaker results regarding the complexity.

Spiking Neural P Systems with Extended Rules

Abstract: We consider spiking neural P systems with spiking rules allowed to introduce zero, one, or more spikes at the same time. The computing power of the obtained systems is investigated, when considering them as number generating and as language generating devices. In the first case, a simpler proof of universality is obtained (universality is already known for the restricted rules), while in the latter case we find characterizations of finite and recursively enumerable languages (without using any squeezing mechanism, as it was necessary in the case of restricted rules). The relationships with regular languages are also investigated. In the end of the paper, a tool-kit for computing (some) operations with languages is provided.

Modal Logics of Topological Relations

Abstract: Logical formalisms for reasoning about relations between spatial regions play a fundamental role in geographical information systems, spatial and constraint databases, and spatial reasoning in AI. In analogy with Halpern and Shoham's modal logic of time intervals based on the Allen relations, we introduce a family of modal logics equipped with eight modal operators that are interpreted by the Egenhofer-Franzosa (or RCC8) relations between regions in topological spaces such as the real plane. We investigate the expressive power and computational complexity of logics obtained in this way. It turns out that our modal logics have the same expressive power as the two-variable fragment of first-order logic, but are exponentially less succinct. The complexity ranges from (undecidable and) recursively enumerable to highly undecidable, where the recursively enumerable logics are obtained by considering substructures of structures induced by topological spaces. As our undecidability results also capture logics based on the real line, they improve upon undecidability results for interval temporal logics by Halpern and Shoham. We also analyze modal logics based on the five RCC5 relations, with similar results regarding the expressive power, but weaker results regarding the complexity.

Dynamic topological logics over spaces with continuous functions

Abstract: Dynamic topological logics are combinations of topological and temporal modal logics that are used for reasoning about dynamical systems consisting of a topological space and a continuous function on it. Here we partially solve a major open problem in the field by showing (by reduction of the ω-reachability problem for lossy channel systems) that the dynamic topological logic over arbitrary topological spaces as well as those over Rn, for each n ≥ 1, are undecidable. Actually, we prove this result for the natural and expressive fragment of the full dynamic topological language where the topological operators cannot be applied to formulas containing the temporal eventuality. Using Kruskal’s tree theorem we also show that the formulas of this fragment that are valid in arbitrary topological spaces with continuous functions are recursively enumerable, which is not the case for spaces with homeomorphisms.

Learning rational stochastic languages

Abstract: Given a finite set of words w1,...,wn independently drawn according to a fixed unknown distribution law P called a stochastic language, an usual goal in Grammatical Inference is to infer an estimate of P in some class of probabilistic models, such as Probabilistic Automata (PA). Here, we study the class of rational stochastic languages, which consists in stochastic languages that can be generated by Multiplicity Automata (MA) and which strictly includes the class of stochastic languages generated by PA. Rational stochastic languages have minimal normal representation which may be very concise, and whose parameters can be efficiently estimated from stochastic samples. We design an efficient inference algorithm DEES which aims at building a minimal normal representation of the target. Despite the fact that no recursively enumerable class of MA computes exactly the set of rational stochastic languages over Q, we show that DEES strongly identifies tis set in the limit. We study the intermediary MA output by DEES and show that they compute rational series which converge absolutely to one and which can be used to provide stochastic languages which closely estimate the target.

All NP-problems can be solved in polynomial time by accepting networks of splicing processors of constant size

Abstract: In this paper, we present two new results regarding ANSPs. The first one states that every recursively enumerable language can be accepted by an ANSP of size 7 out of which 6 do not depend on the given language. Then we propose a method for constructing, given an NP-language, an ANSP of size 7 accepting that language in polynomial time. Unlike the previous case, all nodes of this ANSP depend on the given language. Since each ANSP may be viewed as a problem solver as shown in [6], the later result may be interpreted as a method for solving every NP-problem in polynomial time by an ANSP of size 7.

Abstract Geometrical Computation 1: Embedding Black Hole Computations with Rational Numbers

Abstract: The Black hole model of computation provides super-Turing computing power since it offers the possibility to decide in finite (observer's) time any recursively enumerable (R.E.) problem. In this paper, we provide a geometric model of computation, conservative abstract geometrical computation, that, although being based on rational numbers (and not real numbers), has the same property: it can simulate any Turing machine and can decide any R.E. problem through the creation of an accumulation. Finitely many signals can leave any accumulation, and it can be known whether anything leaves. This corresponds to a black hole effect.

On the number of nodes in universal networks of evolutionary processors

Abstract: We consider the networks of evolutionary processors (NEP) introduced by J. Castellanos, C. Marti n-Vide, V. Mitrana and J. Sempere recently. We show that every recursively enumerable (RE) language can be generated by an NEP with three nodes modulo a terminal alphabet and moreover, NEPs with four nodes can generate any RE language. Thus, we improve existing universality result from five nodes down to four nodes. For mNEPs (a variant of NEPs where operations of different kinds are allowed in the same node) we obtain optimal results: each RE language can be generated by an mNEP with one node modulo a terminal alphabet, and mNEPs with two nodes can generate any RE language; this is not possible for mNEPs with one node. Some open problems are formulated.

Observation of String-Rewriting Systems

Abstract: Models of computation in theoretical computer science very frequently consist of a device performing some type of process, like a Turing machine and its computation or a grammar and its derivation. After the process halts, only some final output is regarded as the result. In adding an observer to such a device, one can obtain a protocol of the entire process and then use this as the result of the computation. In several recent articles this approach has proved to often exceed greatly the power of the observed system. Here we apply this architecture to string-rewriting systems. They receive a string as input, and a combination of observer and decider then determines whether this string is accepted. This decision is based only on the rewriting process performed. First we determine the power of painter, context-free, and inverse context-free rewriting systems in terms of McNaughton languages. Then these are investigated as components of rewriting/observer systems, and we obtain characterizations of the classes of context-sensitive and recursively enumerable languages. Finally we investigate some limitations, i.e. characterize some systems, where observation does not increase power.

Equality of streams is a Π0 over 2-complete problem

Abstract: This paper gives a precise characterization for the complexity of the problem of proving equal two streams defined with a finite number of equations: Π0 over 2. Since the Π 0 over 2 class includes properly both the reursively enumerable and the corecursively enumerable classes, this result implies that neither the set of pairs of equal streams nor the set of pairs of unequal streams is recursively enumerable. Consequently, one can find no algorithm for determining equality of streams, as well as no algorithm for determining inequality of streams. In particular, there is no complete proof system for equality of streams and no complete system for inequality of streams.

On the rule complexity of universal tissue P systems

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.

Equality of streams is a Π02-complete problem

Abstract: This paper gives a precise characterization for the complexity of the problem of proving equal two streams defined with a finite number of equations: Π 0 2 . Since the Π 0 2 class includes properly both the recursively enumerable and the co-recursively enumerable classes, this result implies that neither the set of pairs of equal streams nor the set of pairs of unequal streams is recursively enumerable. Consequently, one can find no algorithm for determining equality of streams, as well as no algorithm for determining inequality of streams. In particular, there is no complete proof system for equality of streams and no complete system for inequality of streams.

Parallel communicating Watson-Crick automata systems

Abstract: Watson-Crick automata are finite state automata working on double-stranded tapes, introduced to investigate the potential of DNA molecules for computing. In this paper we introduce the concept of parallel communicating Watson-Crick automata systems. It consists of several Watson-Crick finite automata parsing independently the same input and exchanging information on request, by communicating states to each other. We investigate the computational power of these systems and prove that they are more powerful than classical Watson-Crick finite automata, but still accepting at most context-sensitive languages. Moreover, if the complementarity relation is injective, then we obtain that this inclusion is strict. For the general case, we also give some closure properties, as well as a characterization of recursively enumerable languages based on these systems.

Computational completeness of tissue p systems with conditional uniport

Abstract: The paper introduces (purely communicative) tissue P systems with conditional uniport. Conditional uniport means that rules move only one object at a time, but this may be with the help of another one acting as an activator which is left untouched in the place where it is. Tissue P systems with conditional uniport are shown to be computationally complete in the sense that they can recognize all recursively enumerable sets of natural numbers. This is achieved by simulating deterministic register machines.

On the complexity of limit sets of cellular automata associated with probability measures

Abstract: We study the notion of limit sets of cellular automata associated with probability measures (μ-limit sets). This notion was introduced by P. Kůrka and A. Maass in [1]. It is a refinement of the classical notion of ω-limit sets dealing with the typical long term behavior of cellular automata. It focuses on the words whose probability of appearance does not tend to 0 as time tends to infinity (the persistent words). In this paper, we give a characterization of the persistent language for non sensitive cellular automata associated with Bernoulli measures. We also study the computational complexity of these languages. We show that the persistent language can be non-recursive. But our main result is that the set of quasi-nilpotent cellular automata (those with a single configuration in their μ-limit set) is neither recursively enumerable nor co-recursively enumerable.

On the Complexity of Limit Sets of Cellular Automata Associated with Probability Measures

Abstract: We study the notion of limit sets of cellular automata associated with probability measures (mu-limit sets). This notion was introduced by P. Kurka and A. Maass. It is a refinement of the classical notion of omega-limit sets dealing with the typical long term behavior of cellular automata. It focuses on the words whose probability of appearance does not tend to 0 as time tends to infinity (the persistent words). In this paper, we give a characterisation of the persistent language for non sensible cellular automata associated with Bernouilli measures. We also study the computational complexity of these languages. We show that the persistent language can be non-recursive. But our main result is that the set of quasi-nilpotent cellular automata (those with a single configuration in their mu-limit set) is neither recursively enumerable nor co-recursively enumerable.

Membrane division, restricted membrane creation and object complexity in P 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 show here that they generate all recursively enumerable languages, and that 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. We then prove that 10+m symbols are sufficient 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 memb...

On Σ₁-structural differences among finite levels of the Ershov hierarchy

Abstract: We show that the structure M of recursively enumerable degrees is not a Ei-elementary substructure of 3f?, where 9f? (n > 1) is the structure of n-r.e. degrees in the Ershov hierarchy. ?

P Transducers

Abstract: We introduce in this paper four classes of P transducers: arbitrary, initial, isolated arbitrary, isolated and initial. The first two classes are universal, they can compute the same word functions as Turing machines, the latter two are incomparable with finite state sequential transducers, generalized or not. We study the effect of the composition, and show that iteration increases the power of these latter classes, also leading to a new characterization of recursively enumerable languages. The "Sevilla carpet" of a computation is defined for P transducers, giving a representation of the control part for these P transducers.

Equality of streams is a Π0 over 2-complete problem

Abstract: This paper gives a precise characterization for the complexity of the problem of proving equal two streams defined with a finite number of equations: 0. Since the 0 class includes properly both the recursively enumerable and the co-recursively enumerable classes, this result implies that one can find no mechanical procedure to say when two streams are equal, as well as no procedure to say when two streams are not equal. In particular, there is no complete proof system for equality of streams and no complete system for dis-equality of streams.

Using Pseudo-Stochastic Rational Languages in Probabilistic Grammatical Inference

Abstract: In probabilistic grammatical inference, a usual goal is to infer a good approximation of an unknown distribution P called a stochastic language. The estimate of P stands in some class of probabilistic models such as probabilistic automata (PA). In this paper, we focus on probabilistic models based on multiplicity automata (MA). The stochastic languages generated by MA are called rational stochastic languages; they strictly include stochastic languages generated by PA; they also admit a very concise canonical representation. Despite the fact that this class is not recursively enumerable, it is efficiently identifiable in the limit by using the algorithm DEES, introduced by the authors in a previous paper. However, the identification is not proper and before the convergence of the algorithm, DEES can produce MA that do not define stochastic languages. Nevertheless, it is possible to use these MA to define stochastic languages. We show that they belong to a broader class of rational series, that we call pseudo-stochastic rational languages. The aim of this paper is twofold. First we provide a theoretical study of pseudo-stochastic rational languages, the languages output by DEES, showing for example that this class is decidable within polynomial time. Second, we have carried out a lot of experiments in order to compare DEES to classical inference algorithms such as ALERGIA and MDI. They show that DEES outperforms them in most cases.

The Role of True Finiteness in the Admissible Recursively Enumerable Degrees

Abstract: Introduction Coding into the R. E. degrees Coding effective successor models A negative result concerning effective successor models A nonembedding result Embedding the 1-3-1 lattice Appendix A. Basics Appendix B. The jump Appendix C. The projectum Appendix D. The admissible collapse Appendix E. Prompt permission Appendix. Bibliography.

Membrane systems with limited parallelism

Abstract: Membrane computing is an emerging research field that belongs to the more general area of molecular computing, which deals with computational models inspired from bio-molecular processes. Membrane computing aims at defining models, called membrane systems or P systems, which abstract the functioning and structure of the cell. A membrane system consists of a hierarchical arrangement of membranes delimiting regions, which represent various compartments of a cell, and with each region containing bio-chemical elements of various types and having associated evolution rules, which represent bio-chemical processes taking place inside the cell. This work is a continuation of the investigations aiming to bridge membrane computing (where in a compartmental cell-like structure the chemicals to evolve are placed in compartments defined by membranes) and brane calculi (where one considers again a compartmental cell-like structure with the chemicals/proteins placed on the membranes themselves). We use objects both in compartments and on membranes (the latter are called proteins), with the objects from membranes evolving under the control of the proteins. Several possibilities are considered (objects only moved across membranes or also changed during this operation, with the proteins only assisting the move/change or also changing themselves). Somewhat expected, computational universality is obtained for several combinations of such possibilities. We also present a method for solving the NP-complete SAT problem using P systems with proteins on membranes. The SAT problem is solved in O(nm) time, where n is the number of boolean variables and m is the number of clauses for an instance written in conjunctive normal form. Thus, we can say that the solution for each given instance is obtained in linear time. We succeeded in solving SAT by a uniform construction of a deterministic P system which uses rules involving objects in regions, proteins on membranes, and membrane division. Then, we investigate the computational power of P systems with proteins on membranes in some particular cases: when only one protein is placed on a membrane, when the systems have a minimal number of rules, when the computation evolves in accepting or computing mode, etc. This dissertation introduces also another new variant of membrane systems that uses context-free rewriting rules for the evolution of objects placed inside compartments of a cell, and symport rules for communication between membranes. The strings circulate across membranes depending on their membership to regular languages given by means of regular expressions. We prove that these rewriting-symport P systems generate all recursively enumerable languages. We investigate the computational power of these newly introduced P systems for three particular forms of the regular expressions that are used by the symport rules. A characterization of ET0L languages is obtained in this context.