arXiv subjects

Benedek Nagy

Publications and source records attributed to Benedek Nagy.

16 recordsLinked to original sources

arXiv2026

Reverse Post Correspondence Problem and Undecidability of $5' \rightarrow 3'$ String Assembly Systems

The Post Correspondence Problem is as follows: having a set of dominoes, is there any (maybe repeating) sequence of them such that the words formed by the upper parts and the lower parts by the sequence of dominoes are identical. It is one of the most known problems that is algorithmically undecidable. In this paper, the reverse Post Correspondence Problem is defined, that is, where the two assembled words of the dominos are reversals of each other. Undecidability about this new, modified problem is proven. Further, based on this result, it is also proven that the emptiness problem for 5'->3' String Assembly Systems is undecidable too. 5'->3' String Assembly Systems belong to String Assembly Systems type formal language generating models. The 5'->3' denotes that, in these variants, the derivations of the generated words start from the two extremes. The notation and the new model are bio-motivated as any double stranded DNA has two opposite oriented 5'->3' strands.

cs.FL↗

arXiv2026

2-Head 2D Returning Finite Automata

We introduce and study a family of two-head finite automata called two head returning finite automata (2-HRFA) operating on rectangular arrays of picture languages, in which both heads move in opposite directions. We show that the class of picture languages accepted by 2-HRFA is incomparable with the class of languages generated by context-free matrix grammars (CFMG), while it forms a proper subset of the class of languages accepted by returning pushdown automata (RPDA). In addition, we define a constrained variant, both head stepping two head returning finite automata (B2-HRFA), in which both heads are required to move in a synchronized, stepwise fashion. We prove that the class of languages accepted by returning finite automata (RFA) is a proper subset of the class of languages accepted by B2-HRFA, which in turn is a proper subset of the class of languages accepted by 2-HRFA. Closure properties for both the families of languages accepted by 2-HRFA and B2-HRFA are also investigated.

cs.FL↗

arXiv2025

Hilbert basis in the face-centered cubic grid -- mathematical proofs

The Hilbert basis is fundamental in describing the structure of the integer points of a polyhedral cone. The face-centered cubic grid is one of the densest packing of the 3-dimensional space. The cycles of a grid satisfy the constraint set of a pointed, polyhedral cone which contains only non-negative integer vectors. The Hilbert basis of a grid gives the structure of the basic cycles in the grid. It is shown in this paper that the basic cycles of the FCC grid belong to 11 types. It is also discussed that how many elements are contained in the individual types. The proofs of the paper use geometric, combinatorial, algebraic, and operations research methods.

math.CO↗

arXiv2025

A Myhill-Nerode Type Characterization of 2detLIN Languages

Linear automata are automata with two reading heads starting from the two extremes of the input, are equivalent to 5' -> 3' Watson-Crick (WK) finite automata. The heads read the input in opposite directions and the computation finishes when the heads meet. These automata accept the class LIN of linear languages. The deterministic counterpart of these models, on the one hand, is less expressive, as only a proper subset of LIN, the class 2detLIN is accepted; and on the other hand, they are also equivalent in the sense of the class of the accepted languages. Now, based on these automata models, we characterize the class of 2detLIN languages with a Myhill-Nerode type of equivalence classes. However, as these automata may do the computation of both the prefix and the suffix of the input, we use prefix-suffix pairs in our classes. Additionally, it is proven that finitely many classes in the characterization match with the 2detLIN languages, but we have some constraints on the used prefix-suffix pairs, i.e., the characterization should have the property to be complete and it must not have any crossing pairs.

cs.FL↗

arXiv2025

On some Classes of Reversible 2-head Automata

Deterministic 2-head finite automata which are machines that process an input word from both ends are analyzed for their ability to perform reversible computations. This implies that the automata are backward deterministic, enabling unique forward and backward computation. We explore the computational power of such automata, discovering that, while some regular languages cannot be accepted by these machines, they are capable of accepting some characteristic linear languages, e.g., the language of palindromes. Additionally, we prove that restricted variants, i.e., both 1-limited reversible 2-head finite automata and complete reversible 2-head finite automata are less powerful and they form a proper hierarchy. In the former, in each computation step exactly one input letter is being processed, i.e., only one of the heads can read a letter. These automata are also characterized by putting their states to classes based on the head(s) used to reach and to leave the state. In the complete reversible 2-head finite automata, it is required that any input can be fully read by the automaton. The accepted families are also compared to the classes generated by left deterministic linear grammars.

cs.FL↗

arXiv2024

5' -> 3' Watson-Crick Automata accepting Necklaces

Watson-Crick (WK) finite automata work on a Watson-Crick tape representing a DNA molecule. They have two reading heads. In 5'->3' WK automata, the heads move and read the input in opposite physical directions. In this paper, we consider such inputs which are necklaces, i.e., they represent circular DNA molecules. In sensing 5'->3' WK automata, the computation on the input is finished when the heads meet. As the original model is capable of accepting the linear context-free languages, the necklace languages we are investigating here have strong relations to that class. Here, we use these automata in two different acceptance modes. On the one hand, in weak acceptance mode the heads are starting nondeterministically at any point of the input, like the necklace is cut at a nondeterministically chosen point), and if the input is accepted, it is in the accepted necklace language. These languages can be seen as the languages obtained from the linear context-free languages by taking their closure under cyclic shift operation. On the other hand, in strong acceptance mode, it is required that the input is accepted starting the heads in the computation from every point of the cycle. These languages can be seen as the maximal cyclic shift closed languages included in a linear language. On the other hand, as it will be shown, they have a kind of locally testable property. We present some hierarchy results based on restricted variants of the WK automata, such as stateless or all-final variants.

cs.FL↗

arXiv2023

On Languages Generated by Signed Grammars

We consider languages defined by signed grammars which are similar to context-free grammars except productions with signs associated to them are allowed. As a consequence, the words generated also have signs. We use the structure of the formal series of yields of all derivation trees over such a grammar as a method of specifying a formal language and study properties of the resulting family of languages.

cs.FL↗

arXiv2023

Proceedings of the 13th International Workshop on Non-Classical Models of Automata and Applications

The Thirteenth International Workshop on Non-Classical Models of Automata and Applications (NCMA 2023) was held in Famagusta, North Cyprus, on September 18 and 19, 2023, organized by the Eastern Mediterranean University. The NCMA workshop series was established in 2009 as an annual event for researchers working on non-classical and classical models of automata, grammars or related devices. Such models are investigated both as theoretical models and as formal models for applications from various points of view.

cs.FL↗

arXiv2023

State-deterministic Finite Automata with Translucent Letters and Finite Automata with Nondeterministically Translucent Letters

Deterministic and nondeterministic finite automata with translucent letters were introduced by Nagy and Otto more than a decade ago as Cooperative Distributed systems of a kind of stateless restarting automata with window size one. These finite state machines have a surprisingly large expressive power: all commutative semi-linear languages and all rational trace languages can be accepted by them including various not context-free languages. While the nondeterministic variant defines a language class with nice closure properties, the deterministic variant is weaker, however it contains all regular languages, some non-regular context-free languages, as the Dyck language, and also some languages that are not even context-free. In all those models for each state, the letters of the alphabet could be in one of the following categories: the automaton cannot see the letter (it is translucent), there is a transition defined on the letter (maybe more than one transitions in nondeterministic case) or none of the above categories (the automaton gets stuck by seeing this letter at the given state and this computation is not accepting). State-deterministic automata are recent models, where the next state of the computation determined by the structure of the automata and it is independent of the processed letters. In this paper our aim is twofold, on the one hand, we investigate state-deterministic finite automata with translucent letters. These automata are specially restricted deterministic finite automata with translucent letters. In the other novel model we present, it is allowed that for a state the set of translucent letters and the set of letters for which transition is defined are not disjoint. One can interpret this fact that the automaton has a nondeterministic choice for each occurrence of such letters to see them (and then erase and make the transition) or not to see that occurrence at that time. Based on these semi-translucent letters, the expressive power of the automata increases, i.e., in this way a proper generalization of the previous models is obtained.

cs.FL↗

arXiv2022

Quasi-deterministic 5' -> 3' Watson-Crick Automata

Watson-Crick (WK) finite automata are working on a Watson-Crick tape, that is, on a DNA molecule. A double stranded DNA molecule contains two strands, each having a 5' and a 3' end, and these two strands together form the molecule with the following properties. The strands have the same length, their 5' to 3' directions are opposite, and in each position, the two strands have nucleotides that are complement of each other (by the Watson-Crick complementary relation). Consequently, WK automata have two reading heads, one for each strand. In traditional WK automata both heads read the whole input in the same physical direction, but in 5'->3' WK automata the heads start from the two extremes and read the input in opposite direction. In sensing 5'->3' WK automata, the process on the input is finished when the heads meet, and the model is capable to accept the class of linear context-free languages. Deterministic variants are weaker, the class named 2detLIN, a proper subclass of linear languages is accepted by them. Recently, another specific variants, the state-deterministic sensing 5'->3' WK automata are investigated in which the graph of the automaton has the special property that for each node of the graph, all out edges (if any) go to a sole node, i.e., for each state there is (at most) one state that can be reached by a direct transition. It was shown that this concept is somewhat orthogonal to the usual concept of determinism in case of sensing 5'->3' WK automata. In this paper a new concept, the quasi-determinism is investigated, that is in each configuration of a computation (if it is not finished yet), the next state is uniquely determined although the next configuration may not be, in case various transitions are enabled at the same time. We show that this new concept is a common generalisation of the usual determinism and the state-determinism, i.e., the class of quasi-deterministic sensing 5'->3' WK automata is a superclass of both of the mentioned other classes. There are various usual restrictions on WK automata, e.g., stateless or 1-limited variants. We also prove some hierarchy results among language classes accepted by various subclasses of quasi-deterministic sensing 5'->3' WK automata and also some other already known language classes.

cs.FL↗

arXiv2018

Counting of Shortest Paths in Cubic Grid

The enumeration of shortest paths in cubic grid is presented herein, which could have importance in image processing and also in the network sciences. The cubic grid considers three neighborhoods - namely, 6-, 18- and 26-neighborhood related to face connectivity, edge connectivity and vertex connectivity, respectively. The formulation for distance metrics is given. L1, D18, and L_$\infty$ are the three metrics for 6-neighborhood, 18-neighborhood and 26-neighborhood. The task is to count the number of minimal paths, based on given neighborhood relations, from any given point to any other, in the three-dimensional cubic grid. Based on the coordinate triplets describing the grid, the formulations for the three neighborhoods are presented in this work. The problem both of theoretical importance and has several practical aspects.

cs.DM↗

arXiv2017

A New Sensing 5'-->3' Watson-Crick Automata Concept

Watson-Crick (WK) finite automata are working on a Watson-Crick tape, that is, on a DNA molecule. Therefore, it has two reading heads. While in traditional WK automata both heads read the whole input in the same physical direction, in 5'->3' WK automata the heads start from the two extremes and read the input in opposite direction. In sensing 5'->3' WK automata the process on the input is finished when the heads meet. Since the heads of a WK automaton may read longer strings in a transition, in previous models a so-called sensing parameter took care for the proper meeting of the heads (not allowing to read the same positions of the input in the last step). In this paper, a new model is investigated, which works without the sensing parameter (it is done by an appropriate change of the concept of configuration). Consequently, the accepted language classes of the variants are also changed. Various hierarchy results are proven in the paper.

cs.FL↗

arXiv2014

Representations of Circular Words

In this article we give two different ways of representations of circular words. Representations with tuples are intended as a compact notation, while representations with trees give a way to easily process all conjugates of a word. The latter form can also be used as a graphical representation of periodic properties of finite (in some cases, infinite) words. We also define iterative representations which can be seen as an encoding utilizing the flexible properties of circular words. Every word over the two letter alphabet can be constructed starting from ab by applying the fractional power and the cyclic shift operators one after the other, iteratively.

cs.FL↗

arXiv2014

Computing discrete logarithm by interval-valued paradigm

Interval-valued computing is a relatively new computing paradigm. It uses finitely many interval segments over the unit interval in a computation as data structure. The satisfiability of Quantified Boolean formulae and other hard problems, like integer factorization, can be solved in an effective way by its massive parallelism. The discrete logarithm problem plays an important role in practice, there are cryptographical methods based on its computational hardness. In this paper we show that the discrete logarithm problem is computable by an interval-valued computing in a polynomial number of steps (within this paradigm).

cs.DS↗

arXiv2010

Pumping lemmas for linear and nonlinear context-free languages

Pumping lemmas are created to prove that given languages are not belong to certain language classes. There are several known pumping lemmas for the whole class and some special classes of the context-free languages. In this paper we prove new, interesting pumping lemmas for special linear and context-free language classes. Some of them can be used to pump regular languages in two place simultaneously. Other lemma can be used to pump context-free languages in arbitrary many places.

cs.FL↗

arXiv2010

Approximating the Euclidean circle in the square grid using neighbourhood sequences

Distance measuring is a very important task in digital geometry and digital image processing. Due to our natural approach to geometry we think of the set of points that are equally far from a given point as a Euclidean circle. Using the classical neighbourhood relations on digital grids, we get circles that greatly differ from the Euclidean circle. In this paper we examine different methods of approximating the Euclidean circle in the square grid, considering the possible motivations as well. We compare the perimeter-, area-, curve- and noncompactness-based approximations and examine their realization using neighbourhood sequences. We also provide a table which summarizes our results, and can be used when developing applications that support neighbourhood sequences.

math.MG↗