Showing papers on "Recursively enumerable language published in 2010"

A characterization of the entropies of multidimensional shifts of finite type

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

Computably Enumerable Equivalence Relations

Abstract: We study computably enumerable equivalence relations (ceers) on N and unravel a rich structural theory for a strong notion of reducibility among ceers.

A Proof of Completeness for Continuous First-Order Logic

Abstract: Continuous first-order logic has found interest among model theorists who wish to extend the classical analysis of “algebraic” structures (such as fields, group, and graphs) to various natural classes of complete metric structures (such as probability algebras, Hilbert spaces, and Banach spaces). With research in continuous first-order logic preoccupied with studying the model theory of this framework, we find a natural question calls for attention. Is there an interesting set of axioms yielding a completeness result? The primary purpose of this article is to show that a certain, interesting set of axioms does indeed yield a completeness result for continuous first-order logic. In particular, we show that in continuous first-order logic a set of formulae is (completely) satisfiable if (and only if) it is consistent. From this result it follows that continuous first-order logic also satisfies an approximated form of strong completeness, whereby Σ⊧ φ (if and) only if Σ⊢ φ ∸2 −n for all n . This approximated form of strong completeness asserts that if Σ⊧ φ , then proofs from Σ, being finite, can provide arbitrarily better approximations of the truth of φ . Additionally, we consider a different kind of question traditionally arising in model theory—that of decidability. When is the set of all consequences of a theory (in a countable, recursive language) recursive? Say that a complete theory T is decidable if for every sentence φ , the value φ T is a recursive real, and moreover, uniformly computable from φ . If T is incomplete, we say it is decidable if for every sentence φ the real number φ T o is uniformly recursive from φ , where φ T o is the maximal value of φ consistent with T . As in classical first-order logic, it follows from the completeness theorem of continuous first-order logic that if a complete theory admits a recursive (or even recursively enumerable) axiomatization then it is decidable.

An Axiomatization of a First-order Branching Time Temporal Logic

Abstract: We introduce a first-order temporal logic for reasoning about branching time. It is well known that the set of valid formulas is not recursively enumerable and there is no finitary axiomatization. We offer a sound and strongly complete axiomati- zation for the considered logic.

Presenting parabolic subgroups

Abstract: Consider a relatively hyperbolic group G. We prove that if G is finitely presented, so are its parabolic subgroups. Moreover, a presentation of the parabolic subgroups can be found algorithmically from a presentation of G, a solution of its word problem, and generating sets of the parabolic subgroups. We also give an algorithm that finds parabolic subgroups in a given recursively enumerable class of groups.

Accepting Hybrid Networks of Evolutionary Processors with Special Topologies and Small Communication

Abstract: Starting from the fact that complete Accepting Hybrid Networks of Evolutionary Processors allow much communication between the nodes and are far from network structures used in practice, we propose in this paper three network topologies that restrict the communication: star networks, ring networks, and grid networks. We show that ring-AHNEPs can simulate 2-tag systems, thus we deduce the existence of a universal ring-AHNEP. For star networks or grid networks, we show a more general result; that is, each recursively enumerable language can be accepted efficiently by a star- or grid-AHNEP. We also present bounds for the size of these star and grid networks. As a consequence we get that each recursively enumerable can be accepted by networks with at most 13 communication channels and by networks where each node communicates with at most three other nodes.

Diophantine sets of polynomials over number fields

Abstract: Let R be a number field or a recursive subring of a number field and consider the polynomial ring R[T]. We show that the set of polynomials with integer coefficients is diophantine over R[7]. Applying a result by Denef, this implies that every recursively enumerable subset of R[T](k) is diophantine over R[T].

Pure Nash equilibria: complete characterization of hard and easy graphical games

Abstract: We consider the computational complexity of pure Nash equilibria in graphical games. It is known that the problem is NP-complete in general, but tractable (i.e., in P) for special classes of graphs such as those with bounded treewidth. It is then natural to ask: is it possible to characterize all tractable classes of graphs for this problem? In this work, we provide such a characterization for the case of bounded in-degree graphs, thereby resolving the gap between existing hardness and tractability results. In particular, we analyze the complexity of PURE-GG(C, -), the problem of deciding the existence of pure Nash equilibria in graphical games whose underlying graphs are restricted to class C. We prove that, under reasonable complexity theoretic assumptions, for every recursively enumerable class C of directed graphs with bounded in-degree, PURE-GG(C, -) is in polynomial time if and only if the reduced graphs (the graphs resulting from iterated removal of sinks) of C have bounded treewidth. We also give a characterization for PURE-CHG(C, -), the problem of deciding the existence of pure Nash equilibria in colored hypergraphical games, a game representation that can express the additional structure that some of the players have identical local utility functions. We show that the tractable classes of bounded-arity colored hypergraphical games are precisely those whose reduced graphs have bounded treewidth modulo homomorphic equivalence. Our proofs make novel use of Grohe's characterization of the complexity of homomorphism problems.

Bounded Number of Parallel Productions in Scattered Context Grammars with Three Nonterminals

Abstract: Scattered context grammars with three nonterminals are known to be computationally complete. So far, however, it was an open problem whether the number of parallel productions can be bounded along with three nonterminals. In this paper, we prove that every recursively enumerable language is generated by a scattered context grammar with three nonterminals and five parallel productions, each of which simultaneously rewrites no more than nine nonterminals.

Decomposability of low 2-computably enumerable degrees and turing jumps in the Ershov hierarchy

Abstract: In this paper we prove the following theorem: For every notation of a constructive ordinal there exists a low 2-computably enumerable degree that is not splittable into two lower 2-computably enumerable degrees whose jumps belong to the corresponding Δ-level of the Ershov hierarchy.

On Networks of Evolutionary Processors with State Limited Filters.

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 lan- guages that are accepted by deterministic finite automata with a fixed number of states. We show that if the number of states is bounded by three then every recursively enumerable language can be generated by such a network. If the number of states is bounded by two then every context-free language can be generated. If the number of states is bounded by one then not all regular languages but non-context-free languages can be generated.

Representation theorems for recursively enumerable sets and a conjecture related to poonen’s large subring of $\bf \mathbb{Q} $

Abstract: It is remarked that unsolvability results can often be extended to yield novel “representation” theorems for the set of all recursively enumerable sets. In particular, it is shown that an analysis of the proof of the unsolvability of Hilbert’s 10th problem over Poonen’s large subring of $ \mathbb{Q} $ can provide such a theorem. Moreover, applying that theorem to the case of a simple set leads to a conjecture whose truth would imply the unsolvability of Hilbert’s 10th problem over $ \mathbb{Q} $ .

Least and greatest solutions of equations over sets of integers

Abstract: Systems of equations with sets of integers as unknowns are considered, with the operations of union, intersection and addition of sets, S + T = {m + n | m ∈ S, n ∈ T}. These equations were recently studied by the authors ("On equations over sets of integers", STACS 2010), and it was shown that their unique solutions represent exactly the hyperarithmetical sets. In this paper it is demonstrated that greatest solutions of such equations represent exactly the Σ11 sets in the analytical hierarchy, and these sets can already be represented by systems in the resolved form Xi = ϕi(X1,...,Xn). Least solutions of such resolved systems represent exactly the recursively enumerable sets.

Accepting Hybrid Networks of Evolutionary Processors with Special Topologies and Small Communication

Abstract: Starting from the fact that complete Accepting Hybrid Networks of Evolutionary Processors allow much communication between the nodes and are far from network structures used in practice, we propose in this paper three network topologies that restrict the communication: star networks, ring networks, and grid networks. We show that ring-AHNEPs can simulate 2-tag systems, thus we deduce the existence of a universal ring-AHNEP. For star networks or grid networks, we show a more general result; that is, each recursively enumerable language can be accepted efficiently by a star- or grid-AHNEP. We also present bounds for the size of these star and grid networks. As a consequence we get that each recursively enumerable can be accepted by networks with at most 13 communication channels and by networks where each node communicates with at most three other nodes.

Slopes of tilings

Abstract: We study here slopes of periodicity of tilings. A tiling is of slope θ if it is periodic along direction θ but has no other direction of periodicity. We characterize in this paper the set of slopes we can achieve with tilings, and prove they coincide with recursively enumerable sets of rationals.

How Long Can a Sentence Be and Should Anyone Care

Abstract: It is commonly assumed that natural languages, construed as sets of sentences, contain denumerably many sentences. One argument for this claim is that the sentences of a language must be recursively enumerable by a grammar, if we are to understand how a speaker-hearer could exhibit unbounded competence in a language. The paper defends this reasoning by articulating and defending a principle that excludes the construction of a sentence non-denumerably many words long.

The Computational Power of Extended Watson-Crick L Systems

Abstract: Lindenmayer (L) systems form a class of interesting computational formalisms due to their parallel nature, the various circumstances under which they operate, the restrictions imposed on language acceptance, and other attributes. These systems have been extensively studied in the Formal Languages literature. In the past decade a new type of Lindenmayer system had been proposed: Watson-Crick Lindenmayer Systems. These systems are essentially a marriage between Developmental systems and DNA Computing. At their heart they are Lindenmayer systems augmented with a complementary relation amongst elements in the system just as the base pairs of DNA strands can be complementary with respect to one another. When conditions and a mechanism for ’switching’ the state of a computation to its complementary version are provided then these systems can become surprisingly more powerful than the L systems which form their backbone. This dissertation explores the computational power of new variants of Watson-Crick L systems. It is found that many of these systems are Computationally-Complete. These investigations differ from prior ones in that the systems under consideration have extended alphabets and usually Regular Triggers for complementation are considered as opposed to Context-Free Triggers investigated in previous works.

On eco-foraging systems

Abstract: In this paper we present a formal language theoretic approach to Internet crawlers seeking novel information on the World Wide Web, based on a modified version of eco–grammar systems, called eco–foraging systems. In our model, the grammars correspond to very simple autonomous agents and the generated language to the behaviour of the system. In fact, the agents are represented by regulated rewriting devices, which impose some constraint on the search strategy of the agent. The letters of the generated strings symbolize the web pages. We prove that if we ignore the aging of the web environment in the model, then through the simulation of certain normal form grammars, the eco–foraging systems determine the class of recursively enumerable languages. If the web pages may become obsolete, then the language family generated by unordered scattered context grammars of finite index can be obtained. The ignorance of lifetime implies that the crawlers communicating only through the environment are able to identify any computable set of the environmental states. The lifetime constraint, however, considerably decreases the efficiency of the cooperation of the agents. Key-words: crawlers, information harvest on the World Wide Web, eco– foraging systems, unordered scattered context grammar of finite index, recursively enumerable languages. 280 K. A. Lázár et al.

A formal language theoretic approach to distributed computing on dynamic networks

Abstract: In this paper, we present a formal language theoretic approach to the behavior of complex systems of cooperating and communicating agents performing distributed computation on dynamic networks. In particular, we model peer-to-peer networks and the information harvest of Internet crawlers on the World Wide Web, employing grammar systems theoretical constructions. In grammar systems theory, the grammars can be interpreted as agents, whilst the generated language describes the behavior of the system. To characterize the various phenomena that may arise in peer-to-peer networks, we apply networks of parallel multiset string processors. The multiset string processors form teams, send and receive information through collective and individual filters. We deal with the dynamics of the string collections. To describe the information harvest of the crawlers, we employ certain regulated rewriting devices in eco-grammar systems. We illustrate the wide range of applicability of the regulated rewriting devices in the field of web crawling techniques. We demonstrate that these eco-grammar systems with rather simple component grammars suffice to identify any recursively enumerable language.

On optimal heuristic randomized semidecision procedures, with applications to proof complexity and cryptography.

Abstract: The existence of an optimal propositional proof system is a major open question in proof complexity; many people conjecture that such systems do not exist. Krajicek and Pudlak (J. Symbol. Logic 54(3):1063, 1989) show that this question is equivalent to the existence of an algorithm that is optimal on all propositional tautologies. Monroe (Theor. Comput. Sci. 412(4–5):478, 2011) recently presented a conjecture implying that such an algorithm does not exist. We show that if one allows errors, then such optimal algorithms do exist. The concept is motivated by the notion of heuristic algorithms. Namely, we allow an algorithm, called a heuristic acceptor, to claim a small number of false “theorems” and err with bounded probability on other inputs. The amount of false “theorems” is measured according to a polynomial-time samplable distribution on non-tautologies. Our result remains valid for all recursively enumerable languages and can also be viewed as the existence of an optimal weakly automatizable heuristic proof system. The notion of a heuristic acceptor extends the notion of a classical acceptor; in particular, an optimal heuristic acceptor for any distribution simulates every classical acceptor for the same language. We also note that the existence of a co-NP-language L with a polynomial-time samplable distribution on $\overline{L}$ that has no polynomial-time heuristic acceptors is equivalent to the existence of an infinitely-often one-way function.

Constructions in the Recursively Enumerable Degrees

Abstract: In these lectures we survey some of the most important results and the fundamental methods concerning degrees of recursively enumerable (r. e.) sets. We begin §1 with Post's simple sets and a recent elegant generalization of the recursion theorem. In § 2 we give the finite injury priority method, the solution of Post's problem, and the Sacks splitting theorem, In § 3 the infinite injury method is introduced and applied to prove the thickness lemma and the Sacks density theorem. In §4 and §5 we develop the minimal pair method for embedding distributive lattices in the r. e. degrees by maps preserving infimums as well as Superscript>remums. In §6 we present the non-diamond theorem which asserts that such embeddings cannot always preserve greatest and least elements. For background reading we suggest Rogers [17], Shoenfield [23], and Soare [25].

Symport/antiport P systems with look-ahead mode

Abstract: The look-ahead is a forbidding condition formalized by a set of forbidden rules that are checked after all assignment of objects to rules are done. The look-ahead mode can decrease the inherent non-determinism of P systems and helps to the practical implementation of P systems on computers. In this work, the computational power of P systems with symport/antiport rules working in the look-ahead mode are investigated. Communication P systems with 3 membranes and the weight of symport and antiport rules being 2 and 1, respectively, working in the look-ahead mode, can recognize any recursively enumerable languages; a characterization of context-sensitive languages is obtained by communication P systems with 2 membranes working in the look-ahead mode.

Variants of competence-based derivations in cd grammar systems

Abstract: In this paper we introduce and study some new cooperation protocols for cooperating distributed (CD) grammar systems. These derivation modes depend on the number of different nonterminals present in the sentential form obtained when a component finished a derivation phase. This measure describes the competence of the grammar on the string (the competence is high if the number of the different nonterminals is small). It is also a measure of the efficiency of the grammar on the given string (a component is more efficient than another one if it is able to decrease the number of nonterminals in the string to a greater extent). We prove that if the underlying derivation mode is the t-mode derivation, then some variants of these systems determine the class of random context ET0L languages. If these CD grammar systems use the k step limited derivations as underlying derivation mode, then they are able to generate any recursively enumerable language.

Co-Recursively Enumerable Triods with Computable Endpoints

Abstract: Recursive sets in the Euclidean space are those sets which can be effectively approximated by finitely many points for an arbitrary given precision. On the other hand, co-recursively enumerable sets are those sets whose complements can be effectively covered by open balls. If a set is recursive, then it is corecursively enumerable, however the converse is not true in general. In this paper we investigate the subsets of the Euclidean space called triods and we prove that each co-r.e. triod with computable endpoints is recursive. Keywords-recursive set, co-recursively enumerable set, triod

Pure Nash Equilibria: Complete Characterization of Hard and Easy Graphical Games

Abstract: We consider the computational complexity of pure Nash equilibria in graphical games. It is known that the problem is NP-complete in general, but tractable (i.e., in P) for special classes of graphs such as those with bounded treewidth. It is then natural to ask: is it possible to characterize all tractable classes of graphs for this problem? In this work, we provide such a characterization for the case of bounded in-degree graphs, thereby resolving the gap between existing hardness and tractability results. In particular, we analyze the complexity of PUREGG(C, -), the problem of deciding the existence of pure Nash equilibria in graphical games whose underlying graphs are restricted to class C. We prove that, under reasonable complexity theoretic assumptions, for every recursively enumerable class C of directed graphs with bounded in-degree, PUREGG(C, -) is in polynomial time if and only if the reduced graphs (the graphs resulting from iterated removal of sinks) of C have bounded treewidth. We also give a characterization for PURECHG(C,-), the problem of deciding the existence of pure Nash equilibria in colored hypergraphical games, a game representation that can express the additional structure that some of the players have identical local utility functions. We show that the tractable classes of bounded-arity colored hypergraphical games are precisely those whose reduced graphs have bounded treewidth modulo homomorphic equivalence. Our proofs make novel use of Grohe's characterization of the complexity of homomorphism problems.

Slopes of Tilings

Abstract: We study here slopes of periodicity of tilings. A tiling is of slope if it is periodic along direction but has no other direction of periodicity. We characterize in this paper the set of slopes we can achieve with tilings, and prove they coincide with recursively enumerable sets of rationals.

Priority Arguments in Higher Recursion Theory

Abstract: This is the first of six lectures on priority arguments in higher recursion theory. The term “priority argument” refers to a mode of combinatorial reasoning introduced independently by Friedberg [1] and Muchnik [2] to obtain a positive solution to Post's problem, the existence of two recursively enumerable sets such that neither is recursive in the other. I will try to show how their idea lifts to three generalizations of ordinary recursion theory. The three are: α-recursion, where α is a ∑1 admissible ordinal; β-recursion, where β is little more than a limit ordinal; and Kleene recursion in F, a normal object of type 3. Some of the results to be described are new and, in the case of Kleene recursion, not hitherto announced. Others are old but viewed, it is hoped, in the light of a new day.