Two-dimensional Languages Research Articles

Rely more or less, for better or for worse: Intertwining reliability and preferences

Abstract Dealing with preferences is a ubiquitous part of our daily lives, and thus, preference forms an integral part of our daily decision-making processes. If we consider preferences among groups of agents, deliberation often leads to changes in preferences of an agent, influenced by the opinions of others, depending on how reliable these agents are according to the agent under consideration. Sometimes, it also leads to changes in the opposite direction, i.e. reliability over agents gets updated depending on their preferences. There are various formal studies of preference change based on reliability and/or trust, but not the other way around—this work contributes to the formal study of the latter aspect, i.e. on reliability change based on agent preferences. In process, some policies of preference change based on agent reliabilities are also discussed. We introduce a novel formal study of the relevant changes in agent reliabilities in decision-making scenarios, which is an integral part of artificial intelligence studies in the current day. A two-dimensional hybrid language is proposed to describe such processes, and axiomatizations and decidability are discussed.

A Bio-Inspired Model of Picture Array Generating P System with Restricted Insertion Rules

In the bio-inspired area of membrane computing, a novel computing model with a generic name of P system was introduced around the year 2000. Among its several variants, string or array language generating P systems involving rewriting rules have been considered. A new picture array model of array generating P system with a restricted type of picture insertion rules and picture array objects in its regions, is introduced here. The generative power of such a system is investigated by comparing with the generative power of certain related picture array grammar models introduced and studied in two-dimensional picture language theory. It is shown that this new model of array P system can generate picture array languages which cannot be generated by many other array grammar models. The theoretical model developed is for handling the application problem of generation of patterns encoded as picture arrays over a finite set of symbols. As an application, certain floor-design patterns are generated using such an array P system.

Membership Problem for Two-Dimensional General Row Jumping Finite Automata

Two-dimensional general row jumping finite automata were recently introduced as an interesting computational model for accepting two-dimensional languages. These automata are nondeterministic. They guess an order in which rows of the input array are read and they jump to the next row only after reading all symbols in the previous row. In each row, they choose, also nondeterministically, an order in which segments of the row are read. In this paper, we study the membership problem for these automata. We show that each general row jumping finite automaton can be simulated by a nondeterministic Turing machine with space bounded by the logarithm. This means that the fixed membership problems for such automata are in NL, and so in P. On the other hand, we show that the uniform membership problem is NP-complete.

A Common Framework to Recognize Two-dimensional Languages*

A Common Framework to Recognize Two-dimensional Languages*

Equi-triangular array grammars with equal matrix type of rules

Generation of two-dimensional languages of picture arrays, rectangular, triangular or hexagonal in shape, has been considered in several studies motivated by problems in picture processing and pattern recognition. Picture arrays in the form of equi-triangular arrays (et-arrays) composed of equilateral triangles and their generation by context-free type of array grammars have been considered recently. Here generation of picture languages of equi-triangular arrays based on equal matrix grammar type of rules is introduced and investigated. The rules in the matrix are regular type of rules except for initial rules. It has been proved that these equi-triangular array grammars with equal matrix grammar type of rules can generate picture languages of et-arrays that cannot be generated by context-free equi-triangular array grammars. Also the family of equi-triangular array grammars with equal matrix type of rules is shown to properly include the family of regular picture languages of et-arrays.

Equi-triangular array grammars with equal matrix type of rules

Generation of two-dimensional languages of picture arrays, rectangular, triangular or hexagonal in shape, has been considered in several studies motivated by problems in picture processing and pattern recognition. Picture arrays in the form of equi-triangular arrays (et-arrays) composed of equilateral triangles and their generation by context-free type of array grammars have been considered recently. Here generation of picture languages of equi-triangular arrays based on equal matrix grammar type of rules is introduced and investigated. The rules in the matrix are regular type of rules except for initial rules. It has been proved that these equi-triangular array grammars with equal matrix grammar type of rules can generate picture languages of et-arrays that cannot be generated by context-free equi-triangular array grammars. Also the family of equi-triangular array grammars with equal matrix type of rules is shown to properly include the family of regular picture languages of et-arrays.

Actuality, Tableaux, and Two-Dimensional Modal Logics

In this paper we present tableau methods for two-dimensional modal logics. Although models for such logics are well known, proof systems remain rather unexplored as most of their developments have been purely axiomatic. The logics herein considered contain first-order quantifiers with identity, and all the formulas in the language are doubly-indexed in the proof systems, with the upper indices intuitively representing the actual or reference worlds, and the lower indices representing worlds of evaluation—first and second dimensions, respectively. The tableaux modulate over different notions of validity such as local, general, and diagonal, besides being general enough for several two-dimensional logics proposed in the literature. We also motivate the introduction of a new operator into two-dimensional languages and explore some of the philosophical questions raised by it concerning the relations there are between actuality, necessity, and the a priori, that seem to undermine traditional intuitive interpretations of two-dimensional operators.

Visualisation of Collage Grammar to CellWorks: ET0L Mode and Part Sensitive Mode

Images are an important aspect of human life as one remembers pictures better than words. Informally, a twodimensional string is called a picture. A two-dimensional language (or picture language) is a set of pictures. Picture generation and analysis has become a widely investigated field in Theoretical Computer Science and in Mathematics. Collage grammars are studied as devices that generate pictures by rewriting based on hyperedge replacement. A cell-work is a finite set of cells where each cell (being a three dimensional entity) is surrounded by one or more faces. This paper focuses on how cell work languages can be captured by collage grammar in ET0L and Part Sensitive modes.

Two-dimensional double jumping finite automata

In this paper, we extend the concept of double jumping automata for string languages recently introduced in the literature to two-dimensional languages, also called as picture languages. The paper is a modification and study of jumping finite automata for two-dimensional languages already introduced in the literature, where they always perform two simultaneous row-jumps over a row of an input array according to the same rule which is defined. The direction of either of the two simultaneous row-jumps is: 1) left; 2) right or 3) in either direction. Based on the directions of the two simultaneous jumps, we give the classification of the jumping finite automata for two-dimensional languages. The mutual relationship between the families of languages accepted by the two-dimensional jumping finite automata performing the jumps in the ways as mentioned above are investigated. We also give the comparison of these families with that of local and recognisable families of languages and families of Siromoney matrix languages.

Expressiveness and complexity of regular pure two-dimensional context-free languages

Picture languages generalize classical string languages to two-dimensional arrays. Several approaches have been proposed during the years; consequently, a general classification and a detailed comparison of the proposed classes turn to be necessary. In this paper, we study some closure properties of (regularly controlled) pure two-dimensional context-free grammars, for which we also prove that the parsing is NP-hard. Moreover, we draw some comparisons with other interesting picture grammars like (regional) tile grammars, Pruša grammars and local languages, clarifying, in some cases, their mutual relationship with respect to expressiveness.

A stronger recognizability condition for two-dimensional languages

Automata, Logic and Semantics The paper presents a condition necessarily satisfied by (tiling system) recognizable two-dimensional languages. The new recognizability condition is compared with all the other ones known in the literature (namely three conditions), once they are put in a uniform setting: they are stated as bounds on the growth of some complexity functions defined for two-dimensional languages. The gaps between such functions are analyzed and examples are shown that asymptotically separate them. Finally the new recognizability condition results to be the strongest one, while the remaining ones are its particular cases. The problem of deciding whether a two-dimensional language is recognizable is here related to the one of estimating the minimal size of finite automata recognizing a sequence of (one-dimensional) string languages.

Mind the Gap

The spaces and tensions between races, ethnic groups, and communities in late Apartheid and post-1994 South African society, and the co-existence of different languages, religions and cultures, generated a society so fractured that cultural translation became a formidably difficult task. The concept of translation in a transforming society is examined through an analysis of two-dimensional language as a means of translating political events and experiences into visual forms, which attempt to communicate across cultural gaps. Iconic documentary photographs by Sam Nzima (1976) and Truth and Reconciliation Commission (TRC) photographers (1997), struggle posters by co-operatives and formally trained designers, and artworks by Kevin Brand, Sue Williamson and Marion Arnold are discussed. The images reveal that different forms of visual representation encode different relationships of signifying content and aesthetic form to offer alternatives to speech and writing in communicating some implications of Apartheid politics, leaving a legacy that validates art and design as tools of political activism.

A global learning approach for an online handwritten mathematical expression recognition system

Despite the recent advances in handwriting recognition, handwritten two-dimensional (2D) languages are still a challenge. Electrical schemas, chemical equations and mathematical expressions (MEs) are examples of such 2D languages. In this case, the recognition problem is particularly difficult due to the two dimensional layout of the language. This paper presents an online handwritten mathematical expression recognition system that handles mathematical expression recognition as a simultaneous optimization of expression segmentation, symbol recognition, and 2D structure recognition under the restriction of a mathematical expression grammar. The originality of the approach is a global strategy allowing learning mathematical symbols and spatial relations directly from complete expressions. A new contextual modeling is proposed for combining syntactic and structural information. Those models are used to find the most likely combination of segmentation/recognition hypotheses proposed by a 2D segmentation scheme. Thus, models are based on structural information concerning the symbol layout. The system is tested with a new public database of mathematical expressions which was used in the CHROME competition. We have also produced a large base of semi-synthetic expressions which are used to train and test the global learning approach. We obtain very promising results on both synthetic and real expressions databases, as well as in the recent CHROME competition.

TWO-DIMENSIONAL LANGUAGES AND CELLULAR AUTOMATA

Space-time diagrams of a one-dimensional cellular automaton can be visualized as half-plane arrays of symbols. The set of rectangular blocks extracted from such arrays forms a two-dimensional (picture) language. We initiate a study of cellular automata through the associated two dimensional languages by investigating cellular automata whose two-dimensional languages are factorial-local. We show that these cellular automata have the same characterization as one-sided cellular automata with SFT traces.

EXPLORING INSIDE TILING RECOGNIZABLE PICTURE LANGUAGES TO FIND DETERMINISTIC SUBCLASSES

Tiling recognizable two-dimensional languages, also known as REC, generalize recognizable string languages to two dimensions and share with them several theoretical properties. Nevertheless family REC is not closed under complementation and this implies that it is intrinsically non-deterministic. We consider different notions of unambiguity and determinism and the corresponding REC subclasses: they define a hierarchy inside REC. We show that some definitions of unambiguity are equivalent to particular notions of determinism and therefore the corresponding classes have linear parsing algorithms and are closed under complementation.

Classes of two-dimensional languages and recognizability conditions

The paper deals with some classes of two-dimensional recognizable languages of “high complexity”, in a sense specified in the paper and motivated by some necessary conditions holding for recognizable and unambiguous languages. For such classes we can solve some open questions related to unambiguity, finite ambiguity and complementation. Then we reformulate a necessary condition for recognizability stated by Matz, introducing a new complexity function. We solve an open question proposed by Matz, showing that all the known necessary conditions for recognizability of a language and its complement are not sufficient. The proof relies on a family of languages defined by functions.

Deterministic and Unambiguous Families within Recognizable Two-dimensional Languages

Recognizable two-dimensional languages (REC) are defined by tiling systems that generalize to two dimensions non-deterministic finite automata for strings. We introduce the notion of deterministic tiling system and the corresponding family of languages (DREC) and study its structural and closure properties. Furthermore we show that, in contrast with the one-dimensional case, there exist other classes between deterministic and non-deterministic families that we separate by means of examples and decidability properties.

TRANSDUCER GENERATED ARRAYS OF ROBOTIC NANO-ARMS.

We consider sets of two-dimensional arrays, called here transducer generated languages, obtained by iterative applications of transducers (finite state automata with output). Each transducer generates a set of blocks of symbols such that the bottom row of a block is an input string accepted by the transducer and, by iterative application of the transducer, each row of the block is an output of the transducer on the preceding row. We show how these arrays can be implemented through molecular assembly of triple crossover DNA molecules. Such assembly could serve as a scaffold for arranging molecular robotic arms capable for simultaneous movements. We observe that transducer generated languages define a class of languages which is a proper subclass of recognizable picture languages, but it containing the class of all factorial local two-dimensional languages. By taking the average growth rate of the number of blocks in the language as a measure of its complexity, we further observe that arrays with high complexity patterns can be generated in this way.

Weighted Picture Automata and Weighted Logics

We investigate formal power series on pictures. These are functions that map pictures to elements of a semiring and provide an extension of two-dimensional languages to a quantitative setting. We establish a notion of a weighted MSO logics over pictures. The semantics of a weighted formula will be a picture series. We introduce weighted 2-dimensional on-line tessellation automata (W2OTA) and prove that for commutative semirings, the class of picture series defined by sentences of the weighted logics coincides with the family of picture series that are computable by W2OTA. Moreover, we show that the family of behaviors of W2OTA coincide precisely with the class of picture series characterized by weighted (quadrapolic) picture automata and consequently, the notion of weighted recognizability presented here is robust. However, the weighted structures can not be used to get better decidability properties than in the language case. For every commutative semiring, it is undecidable whether a given MSO formula has restricted structure or whether the semantics of a formula has empty support.

On the Tiling System Recognizability of Various Classes of Convex Polyominoes

We consider some problems concerning two relevant classes of two-dimensional languages, i.e., the tiling recognizable languages, and the local languages, recently introduced by Giammarresi and Restivo and already extensively studied. We show that various classes of convex and column-convex polyominoes can be naturally represented as two-dimensional words of tiling recognizable languages. Moreover, we investigate the nature of the generating function of a tiling recognizable language, providing evidence that such a generating function need not be D-finite.