Showing papers on "Recursively enumerable language published in 2012"

Lambek Calculus with Nonlogical Axioms

Abstract: We study Nonassociative Lambek Calculus and Associative Lambek Calculus enriched with finitely many nonlogical axioms. We prove that the nonassociative systems are decidable in polynomial time and generate context-free languages. In [Buszkowski 1982] it has been shown that finite axiomatic extensions of Associative Lambek Calculus generate all recursively enumerable languages; here we give a new proof of this fact. We also obtain similar results for systems with permutation and n−ary

Evolutionary Automata

Abstract: Expressiveness and convergence of evolutionary computation (EC) is studied using the evolutionary automata model. It turns out that all standard classes of evolutionary automata are equally expressive when they operate in the terminal mode, i.e. in the terminal mode, evolutionary finite automata (EFA) are as expressive as evolutionary pushdown automata, evolutionary linearly bounded automata, evolutionary Turing machines or evolutionary inductive Turing machines. For example, the simplest class of evolutionary automata, EFA, can accept all recursively enumerable languages (i.e. EFA have power of Turing machines) and even more—they can accept languages that are not recursively enumerable. Due to utilization of evolutionary automata, we obtain also very simple sufficient conditions for convergence of EC.

Language Equations with Symmetric Difference

Abstract: The paper investigates the expressive power of language equations with the operations of concatenation and symmetric difference. For equations over every finite alphabet Σ with |Σ| ≥ 1, it is demonstrated that the sets representable by unique solutions of such equations are exactly the recursive sets over Σ, and the sets representable by their least (greatest) solutions are exactly the recursively enumerable sets (their complements, respectively). If |_| ≥ 2, the same characterization holds already for equations using symmetric difference and linear concatenation with regular constants. In both cases, the solution existence problem is P01-complete, the existence of a unique, a least or a greatest solution is P02-complete, while the existence of finitely many solutions is P03-complete.

Domination, forcing, array nonrecursiveness and relative recursive enumerability

Abstract: We present some abstract theorems showing how domination properties equivalent to being $\overline{GL}_2$ or array nonrecursive can be used to construct sets generic for different notions of forcing. These theorems are then applied to give simple proofs of some known results. We also give a direct uniform proof of a recent result of Ambos-Spies, Ding, Wang and Yu [2009] that every degree above any in $\overline{GL}_2$ is recursively enumerable in a 1-generic degree strictly below it. Our major new result is that every array nonrecursive degree is r.e. in some degree strictly below it. Our analysis of array nonrecursiveness and construction of generic sequences below $\mathbf{ANR}$ degrees also reveal a new level of uniformity in these types of results.

On the Properties of Language Classes Defined by Bounded

Abstract: Reaction automata are a formal model that has been introduced to investigate the computing powers of interactive behaviors of biochemical reactions([14]). Reaction automata are language acceptors with multiset rewriting mechanism whose basic frameworks are based on reaction systems introduced in [4]. In this paper we continue the investigation of reaction automata with a focus on the formal language theoretic properties of subclasses of reaction automata, called linearbounded reaction automata (LRAs) and exponentially-bounded reaction automata (ERAs). Besides LRAs, we newly introduce an extended model (denoted by �-LRAs) by allowing �-moves in the accepting process of reaction, and investigate the closure properties of language classes accepted by both LRAs and �-LRAs. Further, we establish new relationships of language classes accepted by LRAs and by ERAs with the Chomsky hierarchy. The main results include the following : (i) the class of languages accepted by �-LRAs forms an AFL with additional closure properties, (ii) any recursively enumerable language can be expressed as a homomorphic image of a language accepted by an LRA, (iii) the class of languages accepted by ERAs coincides with the class of context-sensitive languages.

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.

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 t precisely when two given programs are equivalent on halting computations. We show that virtually any variant of propositional dynamic logic has a Π11-hard validity problem if it can express even just the equivalence of well-structured programs with the empty program 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 axiomatization for finitely valid propositions.

Nonterminal complexity of one-sided random context grammars

Abstract: In the present paper, we study the nonterminal complexity of one-sided random context grammars. More specifically, we prove that every recursively enumerable language can be generated by a one-sided random context grammar with no more than ten nonterminals. An analogical result holds for thirteen nonterminals in terms of these grammars with the set of left random context rules coinciding with the set of right random context rules. Furthermore, we introduce the notion of a right random context nonterminal, defined as a nonterminal that appears on the left-hand side of a right random context rule. We demonstrate how to convert any one-sided random context grammar G to an equivalent one-sided random context grammar H with two right random context nonterminals. An analogical conversion is given in terms of (1) propagating one-sided random context grammars and (2) left random context nonterminals. In the conclusion, two open problems are stated.

Easy lambda-terms are not always simple

Abstract: A closed λ -term M is easy if, for any other closed term N , the lambda theory generated by M = N is consistent. Recently, it has been introduced a general technique to prove the easiness of λ -terms through the semantical notion of simple easiness. Simple easiness implies easiness and allows to prove consistency results via construction of suitable filter models of λ -calculus living in the category of complete partial orderings: given a simple easy term M and an arbitrary closed term N , it is possible to build (in a canonical way) a non-trivial filter model which equates the interpretation of M and N . The question whether easiness implies simple easiness constitutes Problem 19 in the TLCA list of open problems. In this paper we negatively answer the question providing a non-empty co-r.e. (complement of a recursively enumerable) set of easy, but not simple easy, λ -terms.

Grammars Controlled by Petri Nets

Abstract: Formal language theory, introduced by Noam Chomsky in the 1950s as a tool for a description of natural languages [8–10], has also been widely involved in modeling and investigating phenomena appearing in computer science, artificial intelligence and other related fields because the symbolic representation of a modeled system in the form of strings makes its processes by information processing tools very easy: coding theory, cryptography, computation theory, computational linguistics, natural computing, and many other fields directly use sets of strings for the description and analysis of modeled systems. In formal language theory a model for a phenomenon is usually constructed by representing it as a set of words, i.e., a language over a certain alphabet, and defining a generative mechanism, i.e., a grammar which identifies exactly the words of this set. With respect to the forms of their rules, grammars and their languages are divided into four classes of Chomsky hierarchy: recursively enumerable, context-sensitive, context-free and regular.

The n-r.E. Degrees: Undecidability and σ1substructures

Abstract: We study the global properties of , the Turing degrees of the n-r.e. sets. In Theorem 1.5, we show that the first order of is not decidable. In Theorem 1.6, we show that for any two n and m with n < m, is not a Σ1-substructure of .

Rule-restricted automaton-grammar transducers: power and linguistic applications

Abstract: This paper introduces the notion of a new transducer as a two-component system, which consists of a nite automaton and a context-free grammar. In essence, while the automaton reads its input string, the grammar produces its output string, and their cooperation is controlled by a set, which restricts the usage of their rules. From a theoretical viewpoint, the present paper discusses the power of this sys- tem working in an ordinary way as well as in a leftmost way. In addition, the paper introduces an appearance checking, which allows us to check whether some symbols are present in the rewritten string, and studies its eect on the power. It achieves the following three main results. First, the system generates and accepts languages dened by matrix grammars and partially blind multi-counter automata, respec- tively. Second, if we place a leftmost restriction on derivation in the context-free grammar, both accepting and generating power of the system is equal to generative power of context-free grammars. Third, the system with appearance checking can accept and generate all recursively enumerable languages. From more pragmatical viewpoint, this paper describes several linguistic applications. A special attention is paid to the Japanese-Czech translation.

Godel's Incompleteness Phenomenon - Computationally

Abstract: We argue that Godel's completeness theorem is equivalent to completability of consistent theories, and Godel's incompleteness theorem is equivalent to the fact that this completion is not constructive, in the sense that there are some consistent and recursively enumerable theories which cannot be extended to any complete and consistent and recursively enumerable theory. Though any consistent and decidable theory can be extended to a complete and consistent and decidable theory. Thus deduction and consistency are not decidable in logic, and an analogue of Rice's Theorem holds for recursively enumerable theories: all the non-trivial properties of such theories are undecidable.

Computing by observing insertion

Abstract: Computing by Observing is a theoretical model for computation that tries to formalize the standard setup of experiments in natural sciences. We establish that insertion systems with empty contexts and only one inserted letter suffice in this architecture to accept all recursively enumerable languages. While so far in most cases context-free power was needed, here a sub-regular system leads to computational completeness in this context. Further, we investigate more complicated insertion systems in a model with less powerful observer called Observing Change.

The Power of Centralized PC Systems of Pushdown Automata

Abstract: Parallel communicating systems of pushdown automata (PCPA) were introduced in (Csuhaj-Varj{\'u} et. al. 2000) and in their centralized variants shown to be able to simulate nondeterministic one-way multi-head pushdown automata. A claimed converse simulation for returning mode (Balan 2009) turned out to be incomplete (Otto 2012) and a language was suggested for separating these PCPA of degree two (number of pushdown automata) from nondeterministic one-way two-head pushdown automata. We show that the suggested language can be accepted by the latter computational model. We present a different example over a single letter alphabet indeed ruling out the possibility of a simulation between the models. The open question about the power of centralized PCPA working in returning mode is then settled by showing them to be universal. Since the construction is possible using systems of degree two, this also improves the previous bound three for generating all recursively enumerable languages. Finally PCPAs are restricted in such a way that a simulation by multi-head automata is possible.

Satisfaction relations for proper classes: Applications in logic and set theory

Abstract: We develop the theory of partial satisfaction relations for structures that may be proper classes and define a satisfaction predicate appropriate to such structures. We indicate the utility of this theory as a framework for the development of the metatheory of first-order predicate logic and set theory, and we use it to prove that for any recursively enumerable extension T of ZF (Zermelo-Fraenkel set theory) there is a finitely axiomatizable extension T' of GB (von Neumann-Bernays-G\"odel class theory) that is a conservative extension of T. We also prove a conservative extension result that justifies the use of this predicate to characterize ground models for forcing constructions.

Extended Watson---Crick L systems with regular trigger languages and restricted derivation modes

Abstract: Watson---Crick Lindenmayer systems add a control mechanism to ordinary Lindenmayer (L) system derivations. The mechanism is inspired by the complementarity relation in DNA strings, and it is formally defined in terms of a trigger language (trigger, for short). It is known that Watson---Crick E0L systems employed with the standard trigger (which is a context-free language) are computationally universal. Here we show that all recursively enumerable languages can be generated already by a Uni-Transitional Watson---Crick E0L system with a regular trigger. A system is Uni-Transitional if any derivation of a terminal word can apply the Watson---Crick morphism at most once. We introduce a weak derivation mode where, for sentential forms in the trigger language, the derivation chooses nondeterministically whether or not to apply the Watson---Crick morphism. We show that Watson---Crick E0L systems employing a regular trigger and the weak derivation mode remain computationally universal but, on the other hand, the corresponding Uni-Transitional systems generate only a subclass of the ET0L languages. We consider also the computational power of Watson---Crick (deterministic) ET0L systems.

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.

On the amount of nonconstructivity in learning formal languages from positive data

Abstract: Nonconstructive computations by various types of machines and automata have been considered by e.g., Karp and Lipton [18] and Freivalds [9, 10]. They allow to regard more complicated algorithms from the viewpoint of more primitive computational devices. The amount of nonconstructivity is a quantitative characterization of the distance between types of computational devices with respect to solving a specific problem. This paper studies the amount of nonconstructivity needed to learn classes of formal languages from positive data. Different learning types are compared with respect to the amount of nonconstructivity needed to learn indexable classes and recursively enumerable classes, respectively, of formal languages from positive data. Matching upper and lower bounds for the amount of nonconstructivity needed are shown.

On the Properties of Language Classes Defined by Bounded Reaction Automata

Abstract: Reaction automata are a formal model that has been introduced to investigate the computing powers of interactive behaviors of biochemical reactions([14]). Reaction automata are language acceptors with multiset rewriting mechanism whose basic frameworks are based on reaction systems introduced in [4]. In this paper we continue the investigation of reaction automata with a focus on the formal language theoretic properties of subclasses of reaction automata, called linearbounded reaction automata (LRAs) and exponentially-bounded reaction automata (ERAs). Besides LRAs, we newly introduce an extended model (denoted by lambda-LRAs) by allowing lambda-moves in the accepting process of reaction, and investigate the closure properties of language classes accepted by both LRAs and lambda-LRAs. Further, we establish new relationships of language classes accepted by LRAs and by ERAs with the Chomsky hierarchy. The main results include the following : (i) the class of languages accepted by lambda-LRAs forms an AFL with additional closure properties, (ii) any recursively enumerable language can be expressed as a homomorphic image of a language accepted by an LRA, (iii) the class of languages accepted by ERAs coincides with the class of context-sensitive languages.

Generalized derivations with synchronized context-free grammars

Abstract: Synchronized context-free grammars are special context-free grammars together with a relation which must be satisfied between every pair of paths from root to leaf in a derivation tree, in order to contribute towards the generated language. In the past, only the equality relation and the prefix relation have been studied, with both methods generating exactly the ET0L languages. In this paper, we study arbitrary relations, and in particular, those defined by a transducer. We show that if we use arbitrary a-transducers, we can generate all recursively enumerable languages, and moreover, there exists a single fixed transducer, even over a two letter alphabet, which allows to generate all recursively enumerable languages. We also study the problem over unary transducers. Although it is left open whether or not we can generate all recursively enumerable languages with unary transducers, we are able to demonstrate that we can generate all ET0L languages as well as a language that is not an indexed language. Only by varying the transducer used to define the relation, this generalization is natural, and can give each of the following language families: context-free languages, a family between the E0L and ET0L languages, ET0L languages, and recursively enumerable languages.

Reflection in the Chomsky Hierarchy

Abstract: The class of regular languages can be generated from the regular expressions. These regular expressions, however, do not themselves form a regular language, as can be seen using the pumping lemma. On the other hand, the class of enumerable languages can be enumerated by a universal language that is one of its elements. We say that the enumerable languages are reflexive. In this paper we investigate what other classes of the Chomsky Hierarchy are reflexive in this sense. To make this precise we require that the decoding function is itself specified by a member of the same class. Could it be that the regular languages are reflexive, by using a different collection of codes? It turns out that this is impossible: the collection of regular languages is not reflexive. Similarly the collections of the context-free, context-sensitive, and computable languages are not reflexive. Therefore the class of enumerable languages is the only reflexive one in the Chomsky Hierarchy.

What's Decidable About Asynchronous Multi-Tape Automata?

Abstract: When their reading heads are allowed to move completely asynchronously, finite-state automata with multiple tapes achieve a significant expressive power, but also lose useful closure properties—closure under intersection, in particular. This paper investigates to what extent it is still feasible to use multi-tape automata as recognizer of polyadic predicates on words. On the negative side, determining whether the intersection of asynchronous multi-tape automata is expressible is not even semidecidable. On the positive side, we present an algorithm that computes under-approximations of the intersection; and discuss conditions under which it can construct complete intersections. A prototype implementation and a few non-trivial examples demonstrate the algorithm in practice. 1 Automata As Decision Procedures Software verification needs expressive logics and powerful decision procedures. Since these requirements are contrasting—with great expressive power comes great undecidability—the chief research challenge is finding new combinations of formalisms that achieve an advantageous trade-off between expressiveness and complexity. In this paper, we investigate using multi-tape finite automata to build decision procedures for fragments of first-order theories with interpreted functions that are germane to program verification. Standard finite-state automata are simple computing devices widely used in computer science. They define a robust class of language acceptors, as each automaton instance A identifies a set L(A) of words that it accepts as input. The connection between finite-state automata and predicate logic has been well-known since the work of Buchi [4,5] and others [30,7], and is widely used in applications such as modelchecking: each automaton AP can be seen as implementing a monadic (that is, unary) predicate P (x), in the sense that the set L(AP ) of words accepted by the automaton corresponds to the set {x | x |= P (x)} of models of the predicate. Logic connectives (negation ¬, conjunction ∧, etc.) translate into composition operations on automata (complement, intersection ∩, etc.), so that finite-state automata can capture the semantics of arbitrary first-order monadic formulas whose interpreted atomic predicates are implementable. This gives a very efficient way to decide the satisfiability of monadic logic formulas representable by finite-state automata: unsatisfiability of a formula corresponds to emptiness of its automaton, which is testable efficiently in time linear in the automaton size. ar X iv :1 20 6. 48 60 v4 [ cs .L O ] 2 D ec 2 01 3 It is natural to extend this framework [1,29] to represent n-ary predicates, for n > 1, by means of multi-tape finite-state automata. An n-tape automaton AR is a device that accepts n-tuples of words, corresponding to the set of models of a predicate R(x1, . . . , xn) over n variables. Section 2 defines multi-tape automata and summarizes some of their fundamental properties. It turns out that the class of multi-tape automata (in their most expressive asynchronous variant) is not as robust as one-tape automata. In particular, multi-tape automata1 are not closed under intersection [11], and hence the conjunction of n-ary predicates is not implementable in general. This paper investigates to what extent this hurdle can be bypassed in practice. On the negative side, we prove that determining whether the intersection of two multi-tape automata A,B is expressible as an automaton is neither decidable nor semi-decidable; and that closure under intersection with equality is tantamount to the simpler class of synchronous automata. On the positive side, we provide an algorithm I(A,B, d) that computes an under-approximation of the intersection A ∩B of A and B, bounded by a given maximum delay d between heads on different tapes. The algorithm has the property that, if the intersection is expressible, then there exists a finite delay d such that I(A,B, d) returns the complete intersection. We also detail simple sufficient syntactic conditions on A and B for the algorithm to return complete intersections. Based on these, we implemented the algorithm and tried it on a number of natural examples inspired by the verification conditions of programs operating on sequences. While the examples are preliminary, they suggest that the framework based on multi-tape automata can supply new ways to reason automatically about expressive theories, as automata make for succinct implementations of atomic predicates.

Networks of evolutionary processors: computationally complete normal forms

Abstract: Networks of evolutionary processors (NEPs, for short) form a bio-inspired language generating computational model that was shown to be equivalent to the model of phrase-structure grammars. In this paper, we analyse different restricted variants of NEPs that preserve the computational power of the general model. We prove that any recursively enumerable language can be generated by a NEP where the derivation rules can be applied at arbitrarily chosen positions, the control of the communication is done by finite automata with at most three states, and either the rule sets are singletons or the underlying graph is a complete graph. If one uses networks with arbitrary underlying graphs and allows the additional application of insertions and deletions only to the right-most or the to left-most position of the derived words for some nodes, then we only need automata with only one state to control the communication in the network. Clearly, this result is optimal; moreover, finite automata with two states are necessary and sufficient in order to generate all the recursively enumerable languages when the derivation rules can be applied only at arbitrarily chosen positions.

On the Nonterminal Complexity of Tree Controlled Grammars

Abstract: A tree controlled grammar is a regulated rewriting device which can be given as a pair (G,R) where G is a context-free grammar and R is a regular set over the terminal and nonterminal alphabets of G. The language generated by the tree controlled grammar contains those words of L(G) which have a derivation tree where all the words obtained by reading the symbols labeling the nodes belonging to the different levels of the tree, from left to right, belong to the language R. The nonterminal complexity of tree controlled grammars can be given as the number of nonterminals of the context-free grammar G, and the number of nonterminals that a regular grammar needs to generate the control language R. Here we improve the currently known best upper bound on the nonterminal complexity of tree controlled grammars from seven to six, that is, we show that a context-free grammar with five nonterminals and a control language which can be generated by a grammar with one nonterminal is sufficient to generate any recursively enumerable language.

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.

Derivation "Trees" and Parallelism in Chomsky-Type Grammars

Abstract: In this paper we discuss parallel derivations for context-free, contextsensitive and phrase-structure grammars. For regular and linear grammars only sequential derivation can be applied, but a kind of parallelism is present in linear grammars. We show that nite languages can be generated by a recursion-free rule-set. It is well-known that in context-free grammars the derivation can be in maximal (independent) parallel way. We show that in cases of context-sensitive and recursively enumerable languages the parallel branches of the derivation have some synchronization points. In the case of context-sensitive grammars this synchronization can only be local, but in a derivation of an arbitrary grammar we cannot make this restriction. We present a framework to show how the concept of parallelism can be t to the derivations in formal language theory using tokens.