Matrix Grammars Research Articles

On the computational completeness of generalized forbidding matrix grammars

Matrix grammars are one of the first approaches ever proposed in regulated rewriting, prescribing that rules have to be applied in a certain order. In traditional regulated rewriting, the most interesting case shows up when all rules are context-free. Typical descriptional complexity measures incorporate the number of nonterminals or the length, i.e., the number of rules per matrix. When viewing matrices as program fragments, it becomes natural to consider additional applicability conditions for such matrices. Here, we focus on forbidding sets, i.e., a matrix is applicable to a sentential form w only if none of the words in its forbidding set occurs as a subword in w. This gives rise to further natural descriptional complexity measures: How long could words in forbidding sets be? How many words could be in any forbidding set? How many matrices contain non-empty forbidding contexts? As context-free grammars with forbidding sets are known as generalized forbidding grammars, we call this variant of matrix grammars also generalized forbidding. In this paper, we attempt to answer the four questions above while studying the computational completeness of generalized forbidding matrix grammars. In the course of our studies, we also define several new normal forms for type-0 grammars that might be of independent interest.

Parallel contextual array insertion deletion P systems and Siromoney matrix grammars

James et al. have introduced a variant of P systems called parallel contextual array insertion deletion P system and studied some of its properties. The family of array languages generated by this variant of PCAIDPS includes families of array languages like recognisable picture languages (REC) and context-sensitive matrix languages . In this paper we show that another interesting family of Siromoney matrix languages is included in the family of array languages generated by PCAIDPS with two membranes.

On the computational completeness of matrix simple semi-conditional grammars

In matrix grammars, context-free rules have to be applied in a certain order. In simple semi-conditional (SSC) grammars, the derivations are controlled either by a permitting string or by a forbidden string associated to each rule. In SSC grammars, the maximum length i of permitting strings and the maximum length j of forbidden strings, the numbers of conditional rules and of nonterminals serve as measures of descriptional complexity and the pair (i,j) is called the degree of such SSC grammars. Matrix grammars with appearance checking with three nonterminals are computationally complete; however, the matrix length is unbounded. Matrix SSC grammars (MSSC) put matrix grammar control on SSC rules. In this paper, we show that MSSC grammars with degrees (2,1), (2,0) and (3,0) are computationally complete, restricting several other descriptional complexity measures. With our constructions, we even bound the matrix length for MSSC grammars.

Relationships between bounded languages, counter machines, finite-index grammars, ambiguity, and commutative regularity

It is shown that for every language family that is a trio containing only semilinear languages, all bounded languages in it can be accepted by one-way deterministic reversal-bounded multicounter machines (DCM). This implies that for every semilinear trio (where these properties are effective), it is possible to decide containment, equivalence, and disjointness concerning its bounded languages. A condition is also provided for when the bounded languages in a semilinear trio coincide exactly with those accepted by DCM machines, and it is used to show that many grammar systems of finite index — such as finite-index matrix grammars (Mfin) and finite-index ET0L (ET0Lfin) — have identical bounded languages as DCM.Then connections between ambiguity, counting regularity, and commutative regularity are made, as many machines and grammars that are unambiguous can only generate/accept counting regular or commutatively regular languages. Thus, such a system that can generate/accept a non-counting regular or non-commutatively regular language implies the existence of inherently ambiguous languages over that system. In addition, it is shown that every language generated by an unambiguous Mfin has a rational characteristic series in commutative variables, and is counting regular. This result plus the connections are used to demonstrate that the grammar systems Mfin and ET0Lfin can generate inherently ambiguous languages (over their grammars), as do several machine models. It is also shown that all bounded languages generated by these two grammar systems (those in any semilinear trio) can be generated unambiguously within the systems. Finally, conditions on Mfin and ET0Lfin languages implying commutative regularity are obtained. In particular, it is shown that every finite-index ED0L language is commutatively regular.

Splicing Context-Free Matrix Grammars using Parallel Communicating Grammar Systems

Extensive research on splicing of strings in DNA computing has established important theoretical results in computational theory. Further, splicing on strings has been extended to arrays in[2]. In this context, we propose, a grammar system, using queries to splice context-free matrix grammars and show that the language generated by this grammar system is incomparable to the language given in [3] and has more generative power than in [2].

Multiset Controlled Grammars: A Simple Method in Regulated Rewriting Theory

<p>This study focus on defining a new variant of regulated grammars called multiset controlled grammars as well as investigating their computational power. We apply a constructive theoretical approach; the intent of which is to provide new theories based on computational methods where the results are appeared in the form of examples, lemmas and theorems. In the study, we have found that multiset is powerful and yet a simple method in regulated rewriting theory. We have proved that multiset controlled grammars are at least as powerful as additive valence grammars, and they are at most powerful as matrix grammars.</p>

Place-labelled Petri net controlled grammars

A place-labelled Petri net (pPN) controlled grammar is a context-free grammar equipped with a Petri net and a function which maps places of the net to the productions of the grammar. The language consists of all terminal strings that can be obtained by simultaneously applying the rules of multisets which are the images of the sets of the input places of transitions in a successful occurrence sequence of the Petri net. In this paper, we study the generative power and structural properties of pPN-controlled grammars. We show that pPN-controlled grammars have the same generative power as matrix grammars. Moreover, we prove that for each pPN-controlled grammar, we can construct an equivalent place-labelled ordinary net controlled grammar.

A class of restricted P colonies with string environment

In this paper we continue the study of variants of P colonies, called APCol systems, where the environment is given as a string and the model is an accepting system. We develop the concept by introducing the notion of the generating working mode. We then compare the power of a special subclass of APCol systems working in the generating mode to the power of register machines and context-free matrix grammars without appearance checking.

Simple Matrix Grammars and Their Leftmost Variants

In essence, simple matrix grammars can be seen as sequences of context-free grammars, referred to as their components, which work in parallel. The present paper demonstrates that two-component simple matrix grammars are as powerful as ordinary matrix grammars. Then, it places three leftmost derivation restrictions upon these grammars and demonstrates that under two of these restrictions, simple matrix grammars are computational complete — that is, they are equivalent with Turing machines. From a historical perspective, concerning simple matrix grammars, the paper also makes several remarks that correct false statements published about them in the past.

Shuffle on array languages generated by array grammars

Motivated by the studies done by G. Siromoney et al. (1973) and Alexan- dru Mateescu et al. (1998) we examine the language theoretic results related to shuf- e on trajectories by making use of Siromoney array grammars such as ( R : R)AG, (R : CF )AG, (CF : R)AG, (CF : CF )AG, (CS : R)AG, (CS : CS)AG and (CF : CS)AG which are more powerful than the Siromoney matrix grammars (1972) and are used to make digital pictures.

The Complexity of Szilard Languages of Matrix Grammars Revisited

The regulated rewriting mechanism is one of the most efficient methods to augment the Chomsky hierarchy with a large variety of language classes. In this paper we investigate the derivation mechanism in regulated rewriting grammars such as matrix grammars, by studying their Szilard languages. We focus on the complexity of Szilard languages associated with unrestricted and leftmost-like derivations in matrix grammars, with or without appearance checking. The reason is twofold. First, to relate these classes of languages to parallel complexity classes such as NC1 and AC1, and, second, to improve some previous results. We prove that unrestricted Szilard languages and certain leftmost Szilard languages of context-free matrix grammars, without appearance checking, can be accepted by indexing alternating Turing machines in logarithmic time and space. Consequently, these classes are included in UE*-uniform NC1. Unrestricted Szilard languages of matrix grammars with appearance checking can be accepted by deterministic Turing machines in On log n time and Olog n space. Leftmost-like Szilard languages of context-free matrix grammars, with appearance checking, can be recognized by nondeterministic Turing machines by using the same time and space resources. Hence, all these classes are included in AC1.

Rule-restricted Automaton-grammar transducers: Power and Linguistic Applications

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.

An infinite hierarchy of languages defined by dP systems

Here, we continue the study of the recently introduced dP automata. They are symport/antiport P systems consisting of a number of components, each one accepting a string, and working together in recognizing the concatenation of these separate strings; the overall string is distributed to the dP automaton components in a balanced way, i.e., in equal parts up to one symbol, like in the communication complexity area. The question whether or not the number of components induces an infinite hierarchy of the recognized languages was formulated as an open problem in the literature. We solve here affirmatively this question (by connecting P automata with right linear simple matrix grammars), then we also briefly discuss the relation between the balanced and the non-balanced way of splitting the input string among components; settling this latter problem remains as a research topic. Some other open problems are also formulated.

On the Power of Small Size Insertion P Systems

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

TOWARD UNDERSTANDING THE GENERATIVE CAPACITY OF ERASING RULES IN MATRIX GRAMMARS

This article presents approaches to the open problem of whether erasing rules can be eliminated in matrix grammars. The class of languages generated by non-erasing matrix grammars is characterized by the newly introduced linear Petri net grammars. Petri net grammars are known to be equivalent to arbitrary matrix grammars (without appearance checking). In linear Petri net grammars, the marking has to be linear in size with respect to the length of the sentential form. The characterization by linear Petri net grammars is then used to show that applying linear erasing to a Petri net language yields a language generated by a non-erasing matrix grammar. It is also shown that in Petri net grammars (with final markings and arbitrary labeling), erasing rules can be eliminated, which yields two reformulations of the problem of whether erasing rules in matrix grammars can be eliminated.

Slender Siromoney matrix languages

A Siromoney matrix grammar is a simple device for the generation of rectangular pictures. In this paper we discuss Siromoney matrix grammars whose picture language are slender, i.e., they contain only a bounded number of pictures for each size. In particular, it is shown that the slenderness decision problem is decidable for Siromoney matrix grammars with context-free rules. Moreover, some closure and decidability questions for slender Siromoney matrix languages are discussed.

P colonies and prescribed teams

We further investigate the computational power of P colonies working in the maximally parallel and sequential modes. It turns out that there is a trade-off between maximal parallelism and checking programs: using checking programs (i.e. priorities on the communication rules in the programs of the agents), P colonies working in the sequential mode with a height of at most five are computationally complete, whereas when working in the maximally parallel mode, P colonies (again with height five) already obtain the same computational power without using checking programs. Moreover, when allowing an arbitrary number of programs for each agent, we can prove that P colonies with only one agent (thus these P colonies are working in the sequential mode) are already computationally complete. On the other hand, P colonies with an arbitrary number of agents working in the sequential mode, as well as even P colonies with only one agent using an arbitrary number of non-checking programs, characterize the family of Parikh sets generated by matrix grammars without appearance checking. The completeness results can also be obtained for P colonies with prescribed teams where rules work in parallel at each agent. P colonies with prescribed teams working in the t 0 mode yield computational completeness even with one agent without priorities on the teams. On the other hand, the family of Parikh sets generated by matrix grammars without appearance checking is also characterized by P colonies with prescribed teams consisting of only (at most) two sets containing exactly one evolution rule or one antiport rule when working in the modes *,=1,≥1, and ≤ k for k ≥1, in one or an arbitrary number of agents.

P SYSTEMS WORKING IN THE SEQUENTIAL MODE ON ARRAYS AND STRINGS

Based on a quite general definition of P systems where the rules are applied in a sequential way (and not in the maximally parallel way as it usually happens in most models of P systems considered so far in the literature), we investigate the generative power of various models of such P systems working in the sequential mode on arrays and strings, respectively. P systems working in the sequential mode on arrays/strings without priority relations for the rules reveal the same computational power as the corresponding matrix grammars without appearance checking working on arrays/strings. For obtaining the computational power of matrix grammars with appearance checking, priority relations for the rules (as one of many other possible additional features) are needed.

Tissue P systems with channel states

We consider tissue-like P systems with states associated with the links (we call them synapses) between cells, controlling the passage of objects across the links. We investigate the computing power of such devices for the case of using—in a sequential manner—antiport rules of small weights. Systems with two cells are proved to be universal when having arbitrarily many states and minimal antiport rules, or one state and antiport rules of weight two. Also the systems with arbitrarily many cells, three states, and minimal antiport rules are universal. In contrast, the systems with one cell and any number of states and rules of any weight only compute Parikh sets of matrix languages (generated by matrix grammars without appearance checking); characterizations of Parikh images of matrix languages are obtained for such one-cell systems with antiport rules of a reduced weight.

Rewriting P systems: improved hierarchies

Generally, for proving universality results about rewriting P systems one considers matrix grammars in the strong binary normal form. Such grammars contain both matrices with rules used in the appearance checking mode and matrices without appearance checking rules. In the proofs of most of the universality theorems reported in the literature, appearance checking matrices are simulated by using only two membranes, while four membranes are used for simulating matrices without appearance checking rules. Thus, a way to improve these theorems is to diminish the number of membranes used for simulating matrices without appearance checking rules. In this paper we address this problem, and give first a general improved result about simulating matrix grammars without appearance checking: three membranes are shown to suffice. This result is then used to improve several universality results from various membrane computing papers, for instance, about P systems with replicated rewriting, with leftmost rewriting, with conditional communication, as well as for hybrid P systems with finite choice.