Context-free Components Research Articles

Parallel communicating grammar systems with context-free components are Turing complete for any communication model

Abstract Parallel Communicating Grammar Systems (PCGS) were introduced as a language-theoretic treatment of concurrent systems. A PCGS extends the concept of a grammar to a structure that consists of several grammars working in parallel, communicating with each other, and so contributing to the generation of strings. PCGS are usually more powerful than a single grammar of the same type; PCGS with context-free components (CF-PCGS) in particular were shown to be Turing complete. However, this result only holds when a specific type of communication (which we call broadcast communication, as opposed to one-step communication) is used. We expand the original construction that showed Turing completeness so that broadcast communication is eliminated at the expense of introducing a significant number of additional, helper component grammars. We thus show that CF-PCGS with one-step communication are also Turing complete. We introduce in the process several techniques that may be usable in other constructions and may be capable of removing broadcast communication in general.

On some derivation mechanisms and the complexity of their Szilard languages

A Szilard language is a language theoretical tool used to describe the derivation process in a formal grammar or grammar system. We investigate computational resources used by (alternating) Turing machines to accept Szilard languages of Chomsky grammars, regulated grammars and grammar systems. The results are related to the circuit complexity classes NC1 and NC2. The paper is a survey of the most important complexity results in the field, but it also brings several new insights into the parallel complexity of Szilard languages of grammar systems. We focus on parallel communicating grammar systems (PCGSs) with context-free components, and we prove that the class of Szilard languages of centralized (returning or non-returning) PCGSs is included in NC1.

Trade-Offs Between Time, Space, Cooperation, and Communication Complexity for CD Grammar Systems

In this paper we investigate the communication and cooperation phenomenon in Cooperating Distributed Grammar Systems (henceforth CDGSs). In this respect, we define several complexity structures and two complexity measures, the cooperation and communication complexity measures. These measures are studied with respect to the derivation modes and fairness conditions under which CDGSs may work. We deal with trade-offs between time, space, cooperation, and communication complexity of languages generated by CDGSs with regular, linear, and context-free components. Cooperation and communication processes in CDGSs with regular and linear components are of equal complexity. The two (equal) cooperation and communication complexity measures are either constant or linear, as functions of lengths of words in the generated language. The same result holds for the cooperation and communication complexity of q-fair languages generated by CDGSs with regular and linear components. For the case of non-constant communication (cooperation) the time and space used by a nondeterministic multitape Turing machine to recognize weakly q-fair languages are linear, as is the communication (cooperation) complexity. For CDGSs with context-free components the cooperation and communication complexity may be different. These measures are either linear or logarithmic functions, in terms of lengths of words in the generated language. In order to find trade-offs between time, space, cooperation, and communication complexity of languages generated by CDGSs with context-free components we study asymptotic behavior of growth functions characteristic to these measures. We prove that languages generated by CDGSs with context-free components are accepted by nondeterministic multitape Turing machines either in quadratic time, linear space, and with cooperation complexity that varies from logarithmic to linear, or in polynomial or exponential time and space, and linear cooperation complexity.

PC GRAMMAR SYSTEMS WITH CLUSTERS OF COMPONENTS

We introduce PC grammar systems where the components form clusters and the query symbols refer to clusters not individual grammars, i.e., the addressee of the query is not precisely identified. We prove that if the same component replies to all queries issued to a cluster in a rewriting step, then non-returning PC grammar systems with 3 clusters and 7 context-free components are able to generate any recursively enumerable language. We also provide open problems and directions for future research.

On the Size Complexity of Non-Returning Context-Free PC Grammar Systems

Improving the previously known best bound, we show that any recursively enumerable language can be generated with a non-returning parallel communicating (PC) grammar system having six context-free components. We also present a non-returning universal PC grammar system generating unary languages, that is, a system where not only the number of components, but also the number of productions and the number of nonterminals are limited by certain constants, and these size parameters do not depend on the generated language.

REMARKS ON CONTEXT-FREE PARALLEL COMMUNICATING GRAMMAR SYSTEMS GENERATING CROSSED AGREEMENTS

Parallel communicating grammar systems are language generating devices consisting of several grammars which derive synchronously their sentential forms and communicate with each other by sending their sentential forms to another component on request. Due to collaboration, grammar systems with context-free components are more powerful than a single context-free grammar; they even can generate all recursively enumerable languages. In natural languages, there occur constructions that cannot be modelled by context-free languages. Three important phenomena are the so-called multiple agreements, crossed agreements and replication which are represented by the three non-context-free languages K1 = { anbncn | n ≥ 1 }, K2 = { anbmcndm | m ≥ 1, n ≥ 1 }, and K3 = { ww | w ∈ { a, b }+}, respectively. In the present paper, we give parallel communicating grammar systems (PC grammar systems) that are context-free and that generate the language K2, working in different modes. In two cases, the results are optimal.

ON METALINEAR PARALLEL COMMUNICATING GRAMMAR SYSTEMS

A generalization of centralized and returning parallel communicating grammar systems with linear components (linear CPC grammar systems) is studied. It is known that linear CPC grammar systems are more powerful than regular CPC grammar systems and that CPC grammar systems with context-free components are more powerful than linear CPC grammar systems. Here, the intermediate model of metalinear CPC grammar systems is studied. This is a CPC grammar system where the master is allowed to use metalinear rules whereas the remaining components are restricted to use linear rules only. It turns out that metalinear CPC grammar systems are more powerful than linear CPC grammar systems and less powerful than CPC grammar systems with context-free components. Furthermore, it is shown that all languages generated by metalinear CPC grammar systems are semilinear.

ON COMPETENCE IN CD GRAMMAR SYSTEMS WITH PARALLEL REWRITING

We continue our investigation of the generative power of cooperating distributed grammar systems (CDGSs), using the previously introduced ≤k-, =k-, and ≥k-competence-based cooperation strategies and context-free components that rewrite the sentential form in a parallel manner. This leads to new characterizations of the languages generated by (random context) ET0L systems and recurrent programmed grammars.

A remark on evolutionary systems

Evolutionary systems have been introduced by Csuhaj-Varjú and Mitrana (Acta Inform. 36 (2000) 913) who proved that two context-sensitive or three context-free components are sufficient to obtain all recursively enumerable languages. We improve these results by showing that two context-free components are sufficient to generate all recursively enumerable languages. Furthermore, we study the power of systems with one component.

Hybrid modes in cooperating distributed grammar systems: combining the t-mode with the modes ⩽k and =k

We continue our research on hybrid derivation modes of cooperating distributed (CD) grammar systems started in (Fernau et al., Proc. 3rd Internat. Conf. Developments in Language Theory, Aristotle University of Thessaloniki, 1997, pp. 261–272; Theoret. Comput. Sci. 259 (2001) 405). Here, we focus on (hybrid) systems with context-free components working in a hybrid t- and Δk-mode, Δ∈{⩽,=}. We prove that CD grammar systems working in the mode (t∧=1), which means that each component cannot perform more than one derivation step in a row, characterize the context-free programmed languages with finite index. Moreover, we show that additional external hybridization of this mode with, e.g., the =1-mode (or =2-mode, respectively) leads to a characterization of the context-free ordered languages (or the context-free programmed languages with appearance checking, respectively). In particular, hybrid CD grammar systems with classical and internally hybrid modes have universal computational power, a result which is unknown for hybrid CD grammar systems whose components work only in classical modes, since such systems are computationally equivalent to recurrent programmed grammars with appearance checking, as shown in (Theoret. Computer. Sci. 259 (2001) 405). Furthermore, we obtain a characterization of ordered languages within the framework of CD grammar systems.

PC grammar systems with five context-free components generate all recursively enumerable languages

Parallel communicating grammar systems (PC grammar systems, in short) are language generating devices consisting of several context-free grammars which work synchronously on their own sentential forms and communicate the generated strings to each other by request. These systems with eleven components are known to have the power of the Turing machines. We considerably improve this result, proving that five components suffice in order to generate any recursively enumerable language.

On the computational power of context-free PC grammar systems

It is shown that non-returning parallel communicating grammar systems with λ-free context-free components can generate any recursively enumerable language. Since it was proven that such systems can be simulated by returning PC grammar systems with context-free components, the result extends to those.

On simulating non-returning PC grammar systems with returning systems

Non-returning PC grammar systems with n context free components are simulated in [3] by returning PC grammar systems with 4 n 2 − 3 n + 1 components. In this paper we reduce the number of components of the simulating system by using a different simulating construction. The number of simulating components can be further decreased if the queries appearing in some of the components satisfy a simple condition, which all queries in regular and linear grammars naturally do. This number can be as low as 2 n, while in general it is still ( n + 1) 2.

Remarks on the Relationship Between Cooperating Distributed Grammar Systems and Lindenmayer Systems

In [2] it is shown that the family of languages generated by cooperating distributed (CD) grammar systems with context-free components in the derivation modes = k, ≥ k, for k ≥ 2, and the family of E0L languages are incomparable. Because one of the languages used in the proof of the incomparability does not work, we fill here the gap, using another language. We also give an effective procedure for constructing an ET0L system from a given context-free cooperating/distributed (CD) grammar system working in the maximum mode of cooperation.

Pattern Languages Versus Parallel Communicating Grammar Systems

We compare the power of two (fairly different) recently investigated language identifying devices: patterns and parallel communicating (PC) grammar systems. The simulation of multi-patterns by context-free PC grammar systems is rather obvious, but, unexpectedly, this can be realized also by (non-centralized) PC grammar systems with right-linear components. Moreover, infinite multi-patterns forming a regular set can also be simulated by PC grammar systems with right-linear components, whereas PC grammar systems with context-free components can simulate context-free multi-patterns with context-free domains for variables.

Context-free attentional operators: The generalized symmetry transform

Active vision systems, and especially foveated vision systems, depend on efficient attentional mechanisms. We propose that machine visual attention should consist of both high-level, context-dependent components, and low-level, context free components. As a basis for the context-free component, we present an attention operator based on the intuitive notion of symmetry, which generalized many of the existing methods of detecting regions of interest. It is a low-level operator that can be applied successfully without a priori knowledge of the world. The resultingsymmetry edge map can be applied in various low, intermediate-and high- level tasks, such as extraction of interest points, grouping, and object recognition. In particular, we have implemented an algorithm that locates interest points in real time, and can be incorporated in active and purposive vision systems. The results agree with some psychophysical findings concerning symmetry as well as evidence concerning selection of fixation points. We demonstrate the performance of the transform on natural, cluttered images.