Narrow Results

In this paper we give the cost, in terms of states, of some basic operations (union, intersection, concatenation, and Kleene star) on regular languages in the unary case (where the alphabet contains only one symbol). These costs are given by explicitly determining the number of states in the noncyclic and cyclic parts of the resulting automata. Furthermore, we prove that our bounds are optimal. We also present an interesting connection to Jacobsthal's function from number theory.

Regular expressions are used in many practical applications. Practical regular expressions are commonly called "regex". It is known that regex are different from regular expressions. In this paper, we give regex a formal treatment. We make a distinction between regex and extended regex; while regex represent regular languages, extended regex represent a family of languages larger than regular languages. We prove a pumping lemma for the languages expressed by extended regex. We show that the languages represented by extended regex are incomparable with context-free languages and a proper subset of context-sensitive languages. Other properties of the languages represented by extended regex are also studied.

We study binary word operations of the insertion, deletion and substitution type. Many of these operations can be generalized into a unified framework by introducing so-called trajectory condition. This generalization has been previously made for insertion and deletion operations. In this paper we naturally extend this approach also to substitution operations. We study closure properties and decision problems of substitutions on trajectories. The obtained results are then applied to model complex noisy channels and a cryptanalysis problem. Another application concerns the design of sets of DNA strands without undesired bonds.

Random context picture grammars (rcpgs) are a method of syntactic picture generation. The productions of such a grammar are context-free, but their application is regulated—permitted or forbidden—by context randomly distributed in the developing picture. In previous work we studied three natural subclasses of rcpgs, namely context-free picture grammars (cfpgs), random permitting context picture grammars (rPcpgs) and random forbidding context picture grammars (rFcpgs). We now introduce the notion of table-driven context-free picture grammars (Tcfpgs), and compare them to the above subclasses. We show that Tcfpgs are more powerful than cfpgs, and that they can generate a picture set (more commonly known as a gallery) that rPcpgs cannot. We present two conditions necessary for a gallery to be generated by a Tcfpg, and use these results to find two galleries that cannot be made by any Tcfpg. The second of these galleries can be generated by an rFcpg. Since it is easily shown that any gallery generated by a Tcfpg can be generated by an rFcpg, we can conclude that Tcfpgs are strictly weaker than rFcpgs.

A pumping condition for ultralinear languages is proved. Some examples are given using that condition to state lower bounds on the rank of some ultralinear languages and to prove that a certain language is not ultralinear. The restriction to the metalinear case is also investigated.

We present a formal model inspired by the epigenetic process of gene annotation via histone modification. In particular, we study the generative capacity of a system in which annotations on a set of strings control which substrings are ultimately produced by the system and in which only the annotations, and not the strings themselves, may be rewritten. On a biological level this represents a first attempt to better understand the computational limits of this form of epigenetic regulation. We introduce two different derivation modes for our formal system and show that these systems are actually quite weak. The weaker of the derivation modes is directly capable only of generating a subset of the regular languages while the more powerful derivation mode is also only capable of generating all regular languages modulo a begin- and an end-marker.

The simulation of deterministic pushdown automata defined over a one-letter alphabet by finite state automata is investigated from a descriptional complexity point of view. We show that each unary deterministic pushdown automaton of size s can be simulated by a deterministic finite automaton with a number of states that is exponential in s. We prove that this simulation is tight. Furthermore, its cost cannot be reduced even if it is performed by a two-way nondeterministic automaton. We also prove that there are unary languages for which deterministic pushdown automata cannot be exponentially more succinct than finite automata. In order to state this result, we investigate the conversion of deterministic pushdown automata into context-free grammars. We prove that in the unary case the number of variables in the resulting grammar is strictly smaller than the number of variables needed in the case of nonunary alphabets.

Iterative arrays (IAs) are a parallel computational model with a sequential processing of the input. They are one-dimensional arrays of interacting identical deterministic finite automata. In this paper, realtime-IAs with sublinear space bounds are used to recognize formal languages. The existence of an infinite proper hierarchy of space complexity classes between logarithmic and linear space bounds is proved. Some decidability questions on logarithmically space bounded realtime-IAs are investigated, and an optimal logarithmic space lower bound for non-regular language recognition on realtime-IAs is shown. Finally, some non-recursive trade-offs between space bounded realtime-IAs are emphasized.

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.

Parallel communicating finite automata (PCFAs) are systems of several finite state automata which process a common input string in a parallel way and are able to communicate by sending their states upon request. We consider deterministic and nondeterministic variants and distinguish four working modes. It is known that these systems in the most general mode are as powerful as one-way multi-head finite automata. It is additionally known that the number of heads corresponds to the number of automata in PCFAs in a constructive way. Thus, undecidability results as well as results on the hierarchies induced by the number of heads carry over from multi-head finite automata to PCFAs in the most general mode. Here, we complement these undecidability and hierarchy results also for the remaining working modes. In particular, we show that classical decidability questions are not semi-decidable for any type of PCFAs under consideration. Moreover, it is proven that the number of automata in the system induces infinite hierarchies for deterministic and nondeterministic PCFAs in three working modes.

If all proper factors of a word u are β-power-free while u itself is not, then u is a minimal β-power. We consider the following general problem: for which numbers k,β, and p there exists a k-ary minimal β-power of period p? For the case β ≥ 2 we completely solve this problem. If the number β < 2 is relatively "big" w.r.t. k, we show that any number p can be the period of a minimal β-power. Finally, for "small" β we describe some sets of forbidden periods and provide a numerical evidence that for k ≥ 9 these sets are almost exhaustive.

With this contribution we would like to commemorate Chandra M. R. Kintala, who passed away in November 2009. We will give short overviews of his CV and his contributions to the field of theoretical and applied computer science and, given the opportunity, will attempt to present his influence on areas like limited nondeterminism and resources as well as software reliability in an exemplary fashion. Finally, we will briefly touch on some research topics which hopefully will be addressed in the not-so-distant future.

Systems of parallel finite automata communicating by states are investigated. We consider deterministic and nondeterministic devices and distinguish four working modes. It is known that systems in the most general mode are as powerful as one-way multi-head finite automata. Here we solve some open problems on the computational capacity of systems working in the remaining modes. In particular, it is shown that deterministic returning and non-returning devices are equivalent, and that there are languages which are accepted by deterministic returning and centralized systems but cannot be accepted by deterministic non-returning centralized systems. Furthermore, we show that nondeterministic systems are strictly more powerful than their deterministic variants in all the four working modes. Finally, incomparability with the classes of (deterministic) (linear) context-free languages as well as the Church-Rosser languages is derived.

A recent language definition device named consensual is based on agreement between similar words. Considering a language over a bipartite alphabet made by pairs of unmarked/marked letters, the match relation specifies when such words agree. Thus a set (the “base”) over the bipartite alphabet consensually specifies another language that includes any terminal word such that a set of corresponding matching words is in the base. We show that all and only the regular languages are consensually generated by a strictly locally testable base; the result is based on a generalization of Medvedev's homomorphic characterization of regular languages. Consensually context-free languages strictly include the base family. The consensual and the base families collapse together if the base is context-sensitive.

This paper discusses research frontiers of membrane computing by presenting current open problems and research topics, together with the relevant background and motivation.

Limited automata are one-tape Turing machines that are allowed to rewrite the content of any tape cell only in the first d visits, for a fixed constant d. In the case d = 1, namely, when a rewriting is possible only during the first visit to a cell, these models have the same power of finite state automata. We prove state upper and lower bounds for the conversion of 1-limited automata into finite state automata. In particular, we prove a double exponential state gap between nondeterministic 1-limited automata and one-way deterministic finite automata. The gap reduces to a single exponential in the case of deterministic 1-limited automata. This also implies an exponential state gap between nondeterministic and deterministic 1-limited automata. Another consequence is that 1-limited automata can have less states than equivalent two-way nondeterministic finite automata. We show that this is true even if we restrict to the case of the one-letter input alphabet. For each d ≥ 2, d-limited automata are known to characterize the class of context-free languages. Using the Chomsky-Schützenberger representation for contextfree languages, we present a new conversion from context-free languages into 2-limited automata.

We extend the concept of factorization on finite words to ω-rational languages and show how to compute them. We define a normal form for Büchi automata and introduce a universal automaton for Büchi automata in normal form. We prove that, for every ω-rational language, this Büchi automaton, based on factorization, is canonical and that it is the smallest automaton that contains the morphic image of every equivalent Büchi automaton in normal form.

Idempotency languages are generated from a single word by iterated application of rules of the form $u^m\to u^n$ for natural numbers m and n. We investigate these languages over alphabets of only one or two letters. The conditions under which the underlying rewrite relations are confluent are fully characterized. Then for many combinations of the parameters m and n we answer the question, whether the corresponding idempotency languages are regular or not. What remains open are only the cases where $2\le m<n$.

Let $\mathrm{\Sigma }$ be an alphabet having at least three symbols. Let $CFL$ denote the family of all context-free languages over $\mathrm{\Sigma }$, $PEG$ denote the family of all parsing expression grammars over $\mathrm{\Sigma }$, and $PEL$ denote the family of all parsing expression languages over $\mathrm{\Sigma }$. Let $K$ satisfy $K\subseteq CFL$ and “$K\cap PEL\ne \varnothing$”, where we use the double-quoted “$K\cap PEL\ne \varnothing$” in the sense that we can effectively find some $G\in PEG$ such that $L(G)\in K\cap PEL$. Then the following problem is shown to be undecidable: Given $G\in PEG$, decide if $L(G)\in K$. In particular, the following basic problem is shown to be undecidable: Given $G\in PEG$, decide if $L(G)\in CFL$.

M-rate 0L systems are interactionless Lindenmayer systems together with a function assigning to every string a set of multisets of productions that may be applied simultaneously to the string. Some questions that have been left open in the forerunner papers are examined, and the computational power of deterministic M-rate 0L systems is investigated, where also tabled and extended variants are taken into consideration.