Single-tape nondeterministic Turing machines that are allowed to replace the symbol in each tape cell only when it is scanned for the first time are also known as 1-limited automata. These devices characterize, exactly as finite automata, the class of regular languages. However, they can be extremely more succinct. Indeed, in the worst case the size gap from 1-limited automata to one-way deterministic finite automata is double exponential. Here we introduce two restricted versions of 1-limited automata, once-marking 1-limited automata and always-marking 1-limited automata, and study their descriptional complexity. We prove that once-marking 1-limited automata still exhibit a double exponential size gap to one-way deter-ministic finite automata. However, their deterministic restriction is polynomially related in size to two-way deterministic finite automata, in contrast to deterministic 1-limited automata, whose equivalent two-way deterministic finite automata in the worst case are exponentially larger. For always -marking 1-limited automata, we prove that the size gap to one-way deterministic finite automata is only a single exponential. The gap remains exponential even in the case the given machine is deterministic. We obtain other size relationships between different variants of these machines and finite automata and we present some problems that deserve investigation.
We associate lattices to the sets of unions and intersections of left and right quotients of a regular language. For both unions and intersections, we show that the lattices we produce using left and right quotients are dual to each other. We also give necessary and sufficient conditions for these lattices to have maximal possible complexity.
Graph-controlled insertion-deletion (GCID) systems are regulated extensions of insertion-deletion systems. Such a system has several components and each component contains some insertion-deletion rules. The components are the vertices of a directed control graph. A rule is applied to a string in a component and the resultant string is moved to the target component specified in the rule. The language of the system is the set of all terminal strings collected in the final component. We impose the restriction in the structure of the underlying graph to be a star structure where there is a central, control component which acts like a master and transmits a string (after applying one of its rules) to one of the components specified in the (applied) rule. A component which receives the string can process the obtained string with any applicable rule available in it and sends back the resultant string only to the center component. With this restriction, we obtain computational completeness for some descriptional complexity measures.
We investigate the descriptional complexity of basic regular operations on languages represented by Boolean and alternating finite automata. In particular, we consider the operations of difference, symmetric difference, star, reversal, left quotient, and right quotient, and get tight upper bounds m+n, m+n, 2^n, 2^n, m, and 2^m , respectively, for Boolean automata, and m+n+1, m+n, 2^n, 2^n, m+1 , and 2^m+1 , respectively, for alternating finite automata. To describe witnesses for symmetric difference, we use a ternary alphabet. All the remaining witnesses are defined over binary or unary alphabets that are shown to be optimal.
Pumping lemmata are the main tool to prove that a certain language does not belong to a class of languages like the recognizable languages or the context-free languages. Essentially two pumping lemmata exist for the recognizable weighted languages: the classical one for the BOOLEAN semi -ring (i.e., the unweighted case), which can be generalized to zero-sum free semirings, and the one for fields. A joint generalization of these two pumping lemmata is provided that applies to all AR- TINIAN semirings, over which all finitely generated semimodules have a finite bound on the length of chains of strictly increasing subsemimodules. Since ARTINIAN rings are exactly those that satisfy the Descending Chain Condition, the ARTINIAN semirings include all fields and naturally also all finite semirings (like the BOOLEAN semiring). The new pumping lemma thus covers most previously known pumping lemmata for recognizable weighted languages.
We provide a method, based on automata theory, to mechanically prove the correctness of many numeration systems based on Fibonacci numbers. With it, long case-based and induction-based proofs of correctness can be replaced by simply constructing a regular expression (or finite automaton) specifying the rules for valid representations, followed by a short computation. Examples of the systems that can be handled using our technique include Brown's lazy representation (1965), the far -difference representation developed by Alpert (2009), and three representations proposed by Hajnal (2023). We also provide three additional systems and prove their validity.
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.
In this paper we deal with three models of weighted automata that take weights in the field of real numbers. The first of these models are classical weighted finite automata, the second one are crisp -deterministic weighted automata, and the third one are weighted automata over a vector space. We explore the interrelationships between weighted automata over a vector space and other two models.
We study 1-tag systems with states obeying the freezing property that only allows constant bounded number of rewrites of symbols. We look at examples of languages accepted by such systems, the accepting power of the model, as well as certain closure properties and decision problems. Finally we discuss a restriction of the system where the working alphabet must match the input alphabet.
Reversible computing is a new paradigm that has emerged recently and extends the traditional forwards-only computing mode with the ability to execute in backwards, so that computation can run in reverse as easily as in forward. Two approaches to developing transition system (automaton -like) semantics for event structure models are distinguished in the literature. In the first case, states are considered as configurations (sets of already executed events), and transitions between states are built by starting from the initial configuration and repeatedly adding executable events. In the second approach, states are understood as residuals (model fragments that have not yet been executed), and transitions are constructed by starting from the given event structure as the initial state and deleting already executed (and conflicting) parts thereof during execution. The present paper focuses on an investigation of how the two approaches are interrelated for the model of prime event structures extended with cause-respecting reversibility. The bisimilarity of the resulting transition systems is proved, taking into account step semantics of the model under consideration.
We show that a well-known family of deterministic finite automata can be used to distinguish distinct binary strings of the same length from every start state. Further, we establish almost matching lower and upper bounds on the number of states of such automata necessary to achieve this type of separation. Our result improves the currently best known linear upper bound for arbitrary DFA.
The HOM-problem, which asks whether the image of a regular tree language under a tree homomorphism is again regular, is known to be decidable by [Godoy, Gim\'enez, Ramos, \`Alvarez: The HOM problem is decidable. STOC (2010)]. Research on the weighted version of this problem, however, is still in its infancy since it requires customized investigations. In this paper we address the weighted HOM-problem and strive to keep the underlying semiring as general as possible. In return, we restrict the input: We require the tree homomorphism h to be tetris-free, a condition weaker than injectivity, and for the given weighted tree automaton, we propose an ambiguity notion with respect to h. These assumptions suffice to ensure decidability of the thus restricted HOM-problem for all zero-sum free semirings by allowing us to reduce it to the (decidable) unweighted case.
We investigate the equational theory of Kleene algebra terms with variable complements -- (language) complement where it applies only to variables -- w.r.t. languages. While the equational theory w.r.t. languages coincides with the language equivalence (under the standard language valuation) for Kleene algebra terms, this coincidence is broken if we extend the terms with complements. In this paper, we prove the decidability of some fragments of the equational theory: the universality problem is coNP-complete, and the inequational theory t <= s is coNP-complete when t does not contain Kleene-star. To this end, we introduce words-to-letters valuations; they are sufficient valuations for the equational theory and ease us in investigating the equational theory w.r.t. languages. Additionally, we prove that for words with variable complements, the equational theory coincides with the word equivalence.
Tree-controlled grammars are context-free grammars where the derivation process is controlled in such a way that every word on a level of the derivation tree must belong to a certain control language. We investigate the generative capacity of such tree-controlled grammars where the control languages are special regular sets, especially strictly locally testable languages or languages restricted by resources of the generation (number of non-terminal symbols or production rules) or acceptance (number of states). Furthermore, the set theoretic inclusion relations of these subregular language families themselves are studied.
This work is a survey of the main results reported for the degree of extension of two models defining non-regular languages, namely the context-free grammar and the extended automaton over groups. More precisely, we recall the main results regarding the degree on non-regularity of a context-free grammar as well as the degree of extension of finite automata over groups. Finally, we consider a similar measure for the finite automata with translucent letters and present some preliminary results. This measure could be considered for many mechanisms that extend a less expressive one.
Deterministic synchronous systems consisting of two finite automata running in opposite directions on a shared read-only input are studied with respect to their ability to perform reversible computations, which means that the automata are also backward deterministic and, thus, are able to uniquely step the computation back and forth. We study the computational capacity of such devices and obtain on the one hand that there are regular languages that cannot be accepted by such systems. On the other hand, such systems can accept even non-semilinear languages. Since the systems communicate by sending messages, we consider also systems where the number of messages sent during a computation is restricted. We obtain a finite hierarchy with respect to the allowed amount of communication inside the reversible classes and separations to general, not necessarily reversible, classes. Finally, we study closure properties and decidability questions and obtain that the questions of emptiness, finiteness, inclusion, and equivalence are not semidecidable if a superlogarithmic amount of communication is allowed.
The study of the operational complexity of minimal pumping constants started in [J. DASSOW and I. JECKER. Operational complexity and pumping lemmas. Acta Inform., 59:337-355, 2022], where an almost complete picture of the operational complexity of minimal pumping constants for two different variants of pumping lemmata from the literature was given. We continue this research by considering a pumping lemma for regular languages that allows pumping of sub-words at any position of the considered word, if the sub-word is long enough [S. J. SAVITCH. Abstract Machines and Grammars. 1982]. First we improve on the simultaneous regulation of minimal pumping constants induced by different pumping lemmata including Savitch's pumping lemma. In this way we are able to simultaneously regulate four different minimal pumping constants. This is a novel result in the field of descriptional complexity. Moreover, for Savitch's pumping lemma we are able to completely classify the range of the minimal pumping constant for the operations Kleene star, reversal, complement, prefix- and suffix-closure, union, set-subtraction, concatenation, intersection, and symmetric difference. In this way, we also solve some of the open problems from the paper that initiated the study of the operational complexity of minimal pumping constants mentioned above.
In this paper we introduce a new type of approximate state reductions where the behaviors of the reduced and the original automaton do not have to be identical, but they must match on all words of length less than or equal to some given natural number. We provide four methods for performing such reductions.
Quantum fingerprinting is a technique that maps classical input word to a quantum state. The obtained quantum state is much shorter than the original word, and its processing uses less resources, making it useful in quantum algorithms, communication, and cryptography. One of the examples of quantum fingerprinting is quantum automata algorithms for MOD_p languages, where p is a prime number. However, implementing such an automaton on the current quantum hardware is not efficient. Quantum fingerprinting maps a word of length N to a state of O(log N) qubits, and uses O(N) unitary operations. Computing quantum fingerprint using all available qubits of the current quantum computers is infeasible due to a large number of quantum operations. To make quantum fingerprinting practical, we should optimize the circuit for depth instead of width in contrast to the previous works. We propose explicit methods of quantum fingerprinting based on tools from additive combinatorics, such as generalized arithmetic progressions (GAPs), and prove that these methods provide circuit depth comparable to a probabilistic method. We also compare our method to prior work on explicit quantum fingerprinting methods.
We propose a method for the construction of sets of variable dimension strong non-overlapping matrices basing on any strong non-overlapping set of strings.
