In the literature, there exist several interesting hybrid models of finite automata which have both quantum and classical states. We call them semi-quantum automata. In this paper, we compare the descriptional power of these models with that of DFA. Specifically, we present a uniform method that gives a lower bound on the size of the three existing main models of semi-quantum automata, and this bound shows that semi-quantum automata can be at most exponentially more concise than DFA. Compared with a recent work (Bianchi, Mereghetti, Palano, Theoret. Comput. Sci., 551(2014), 102-115), our method shows the following two advantages: (i) our method is much more concise; and (ii) our method is universal, since it is applicable to the three existing main models of semi-quantum automata, instead of only a specific model.
Some of the most interesting and important results concerning quantum finite automata are those showing that they can recognize certain languages with (much) less resources than corresponding classical finite automata \cite{Amb98,Amb09,AmYa11,Ber05,Fre09,Mer00,Mer01,Mer02,Yak10,ZhgQiu112,Zhg12}. This paper shows three results of such a type that are stronger in some sense than other ones because (a) they deal with models of quantum automata with very little quantumness (so-called semi-quantum one- and two-way automata with one qubit memory only); (b) differences, even comparing with probabilis…
Eilenberg's holonomy decomposition is useful to ascertain the structural properties of automata. Using this method, Egri-Nagy and Nehaniv characterized the absence of certain types of cycles in automata. In the direction of studying the structure of automata with cycles, this work focuses on a special class of semi-flower automata and establish the holonomy decompositions of certain circular semi-flower automata.
We investigate finite deterministic automata in sets with non-homogeneous atoms: integers with successor. As there are uncount- ably many deterministic finite automata in this setting, we restrict our attention to automata with semilinear transition function. The main re- sults is a minimization procedure for semilinear automata. The proof is subtle and refers to decidability of existential Presburger arithmetic with divisibility predicates. Interestingly, the minimization is not obtained by the standard partition refinement procedure, and we demonstrate that this procedure does not necessaril…
We present a new series of examples of binary slowly synchronizing automata with sink state. The reset threshold of the $n$-state automaton in this series is $\frac{n^2}{4}+2n-9$. This improves on the previously known lower bound for the maximum reset threshold of binary synchronizing $n$-state automata with sink state.
We make a connection between the subgroup membership and identity problems for matrix groups and extended finite automata. We provide an alternative proof for the decidability of the subgroup membership problem for $ 2 \times 2 $ integer matrices. We show that the emptiness problem for extended finite automata over $ 4 \times 4 $ integer matrix semigroups is undecidable. We prove that the decidability of the universe problem for extended finite automata is a sufficient condition for the decidability of the subgroup membership and identity problems.
Pin proved that every circular automaton with a prime number of states containing a non-permutation is synchronizing. In this paper, we investigate the synchronization of circular semi-flower automata. We first prove that every semi-flower automaton is a one-cluster automaton. Subsequently, we prove that every semi-flower automaton containing a 1-cycle is synchronizing. Further, we prove that every circular semi-flower automaton with an odd number of states containing a 2-cycle is synchronizing.
We investigate the syntactic complexity of certain types of finitely generated submonoids of a free monoid. In fact, we consider those submonoids which are accepted by circular semi-flower automata (CSFA). Here, we show that the syntactic complexity of CSFA with at most one `branch point going in' (bpi) is linear. Further, we prove that the syntactic complexity of $n$-state CSFA with two bpis over a binary alphabet is $2n(n+1)$.
We explore a natural class of semigroups that have word problem decidable by finite state automata. Among the main results are invariance of this property under change of generators, invariance under basic algebraic constructions and algebraic properties of these semigroups.
In this paper we combine the algebraic properties of Mealy machines generating self-similar groups and the combinatorial properties of the corresponding deterministic finite automata (DFA). In particular, we relate bounded automata to finitely generated synchronizing automata and characterize finite automata groups in terms of nilpotency of the corresponding DFA. Moreover, we present a decidable sufficient condition to have free semigroups in an automaton group. A series of examples and applications is widely discussed, in particular we show a way to color the De Bruijn automata into Mealy aut…
Parikh automata extend automata with counters whose values can only be tested at the end of the computation, with respect to membership into a semi-linear set. Parikh automata have found several applications, for instance in transducer theory, as they enjoy decidable emptiness problem. In this paper, we study two-way Parikh automata. We show that emptiness becomes undecidable in the non-deterministic case. However, it is PSpace-C when the number of visits to any input position is bounded and the semi-linear set is given as an existential Presburger formula. We also give tight complexity bounds…
Learning Weighted Finite Automata over the Max-Plus Semiring and its Termination
Graduate School of Informatics, Kyoto University（京都大学信息学研究科）；National Institute of Informatics（信息学研究所）；The Graduate University for Advanced Studies (SOKENDAI)（高级研究大学院大学（SOKENDAI））
Reversibility of a one-dimensional finite cellular automaton (CA) is dependent on lattice size. A finite CA can be reversible for a set of lattice sizes. On the other hand, reversibility of an infinite CA, which is decided by exploring the rule only, is different in its kind from that of finite CA. Can we, however, link the reversibility of finite CA to that of infinite CA? In order to address this issue, we introduce a new notion, named semi-reversibility. We classify the CAs into three types with respect to reversibility property -- reversible, semi-reversible and strictly irreversible. A to…
We show that if a semisimple synchronizing automaton with $n$ states has a minimal reachable non-unary subset of cardinality $r\ge 2$, then there is a reset word of length at most $(n-1)D(2,r,n)$, where $D(2,r,n)$ is the $2$-packing number for families of $r$-subsets of $[1,n]$.
Krohn-Rhodes theory encompasses the techniques for the study of finite automata and their decomposition into elementary automata. The famous result of Krohn and Rhodes roughly states that each finite automaton can be decomposed into elementary components which correspond to permutation and reset automata connected by a cascade product. However, this outcome is not easy to access for the working computer scientist. This paper provides a short introduction into Krohn-Rhodes theory based on the valuable work of Ginzburg.
We use results from communication complexity, both new and old ones, to prove lower bounds for unambiguous finite automata (UFAs). We show three results. $\textit{Complement:}$ There is a language $L$ recognised by an $n$-state UFA such that the complement language $\overline{L}$ requires NFAs with $n^{\tildeΩ(\log n)}$ states. This improves on a lower bound by Raskin. $\textit{Union:}$ There are languages $L_1$, $L_2$ recognised by $n$-state UFAs such that the union $L_1\cup L_2$ requires UFAs with $n^{\tildeΩ(\log n)}$ states. $\textit{Separation:}$ There is a language $L$ such that both $L$…
We give a geometric approach to groups defined by automata via the notion of enriched dual of an inverse transducer. Using this geometric correspondence we first provide some finiteness results, then we consider groups generated by the dual of Cayley type of machines. Lastly, we address the problem of the study of the action of these groups in the boundary. We show that examples of groups having essentially free actions without critical points lie in the class of groups defined by the transducers whose enriched dual generate a torsion-free semigroup. Finally, we provide necessary and sufficien…
This paper gives answers to questions formulated as open in the paper "On State Observability in Deterministic Finite Automata" by A. Mateescu and Gh. Paun. Specifically, it demonstrates that for all k >= 2, the families of regular languages acceptable by deterministic finite automata with no more than k semi-observable states, denoted by Tk, are anti-AFL's, and that the family T1 differs in the closure property under Kleene +.
Finite automata on infinite words ($ω$-automata) proved to be a powerful weapon for modeling and reasoning infinite behaviors of reactive systems. Complementation of $ω$-automata is crucial in many of these applications. But the problem is non-trivial; even after extensive study during the past four decades, we still have an important type of $ω$-automata, namely Streett automata, for which the gap between the current best lower bound $2^{Ω(n \lg nk)}$ and upper bound $2^{Ω(nk \lg nk)}$ is substantial, for the Streett index size $k$ can be exponential in the number of states $n$. In arXiv:1102…
We study the reachability problem for continuous one-counter automata, COCA for short. In such automata, transitions are guarded by upper and lower bound tests against the counter value. Additionally, the counter updates associated with taking transitions can be (non-deterministically) scaled down by a nonzero factor between zero and one. Our three main results are as follows: (1) We prove that the reachability problem for COCA with global upper and lower bound tests is in NC2; (2) that, in general, the problem is decidable in polynomial time; and (3) that it is decidable in the polynomial hie…
We present an infinite series of $n$-state Eulerian automata whose reset words have length at least $(n^2-3)/2$. This improves the current lower bound on the length of shortest reset words in Eulerian automata. We conjecture that $(n^2-3)/2$ also forms an upper bound for this class and we experimentally verify it for small automata by an exhaustive computation.
We consider the complexity of equivalence and learning for multiplicity tree automata, i.e., weighted tree automata over a field. We first show that the equivalence problem is logspace equivalent to polynomial identity testing, the complexity of which is a longstanding open problem. Secondly, we derive lower bounds on the number of queries needed to learn multiplicity tree automata in Angluin's exact learning model, over both arbitrary and fixed fields. Habrard and Oncina (2006) give an exact learning algorithm for multiplicity tree automata, in which the number of queries is proportional to t…
Unambiguous non-deterministic finite automata have intermediate expressive power and succinctness between deterministic and non-deterministic automata. It has been conjectured that every unambiguous non-deterministic one-way finite automaton (1UFA) recognizing some language L can be converted into a 1UFA recognizing the complement of the original language L with polynomial increase in the number of states. We disprove this conjecture by presenting a family of 1UFAs on a single-letter alphabet such that recognizing the complements of the corresponding languages requires superpolynomial increase…
Geometric semigroup theory is the systematic investigation of finitely-generated semigroups using the topology and geometry of their associated automata. In this article we show how a number of easily-defined expansions on finite semigroups and automata lead to simplifications of the graphs on which the corresponding finite semigroups act. We show in particular that every finite semigroup can be finitely expanded so that the expansion acts on a labeled directed graph which resembles the right Cayley graph of a free Burnside semigroup in many respects.
We investigate commutative images of languages recognised by register automata and grammars. Semi-linear and rational sets can be naturally extended to this setting by allowing for orbit-finite unions instead of only finite ones. We prove that commutative images of languages of one-register automata are not always semi-linear, but they are always rational. We also lift the latter result to grammars: commutative images of one-register context-free languages are rational, and in consequence commutatively equivalent to register automata. We conjecture analogous results for automata and grammars w…
The minimum circuit size problem (MCSP) is a string compression problem with a parameter $s$ in which, given the truth table of a Boolean function over inputs of length $n$, one must answer whether it can be computed by a Boolean circuit of size at most $s(n) \ge n$. Recently, McKay, Murray, and Williams (STOC, 2019) proved a hardness magnification result for MCSP involving (one-pass) streaming algorithms: For any reasonable $s$, if there is no $\mathsf{poly}(s(n))$-space streaming algorithm with $\mathsf{poly}(s(n))$ update time for $\mathsf{MCSP}[s]$, then $\mathsf{P} \neq \mathsf{NP}$. We p…
The Parikh finite word automaton model (PA) was introduced and studied by Klaedtke and Ruess in 2003. Here, by means of related models, it is shown that the bounded languages recognized by PA are the same as those recognized by deterministic PA. Moreover, this class of languages is the class of bounded languages whose set of iterations is semilinear.
Parity word automata and their determinisation play an important role in automata and game theory. We discuss a determinisation procedure for nondeterministic parity automata through deterministic Rabin to deterministic parity automata. We prove that the intermediate determinisation to Rabin automata is optimal. We show that the resulting determinisation to parity automata is optimal up to a small constant. Moreover, the lower bound refers to the more liberal Streett acceptance. We thus show that determinisation to Streett would not lead to better bounds than determinisation to parity. As a si…
We present a strongly exponential lower bound that applies both to the subset synchronization threshold for binary deterministic automata and to the careful synchronization threshold for binary partial automata. In the later form, the result finishes the research initiated by Martyugin (2013). Moreover, we show that both the thresholds remain strongly exponential even if restricted to strongly connected binary automata. In addition, we apply our methods to computational complexity. Existence of a subset reset word is known to be PSPACE-complete; we show that this holds even under the restricti…
We consider finite two-way automata and measure the use of two-way motion by counting the number of left moves in accepting computations. Restriction of the automata according to this measure allows us to study in detail the use of two-way motion for the acceptance of regular languages in terms of state complexity. The two-way spectrum of a given regular language is introduced. This quantity reflects the change of size of minimal accepting devices if the use of two-way motion is increased incrementally. We give examples for spectra, prove uniform upper and lower bounds and study their sharpnes…
Several distinct techniques have been proposed to design quasi-polynomial algorithms for solving parity games since the breakthrough result of Calude, Jain, Khoussainov, Li, and Stephan (2017): play summaries, progress measures and register games. We argue that all those techniques can be viewed as instances of the separation approach to solving parity games, a key technical component of which is constructing (explicitly or implicitly) an automaton that separates languages of words encoding plays that are (decisively) won by either of the two players. Our main technical result is a quasi-polyn…
We introduce the category of dependency automata. A dependency automaton consists of two nondeterministic finite automata, with a relation between their states satisfying conditions. This category is equivalent to deterministic finite automata interpreted in join-semilattices i.e. JSL-dfas. The canonical dependency automaton accepting $L$ amounts to the state-minimal dfas for $L$ and $rev(L)$ connected by the `dependency relation'. We describe many canonical JSL-dfas as dependency automata and also explain/extend Brzozowski's algorithm. Call an nfa `subatomic' if its individual states accept a…
We consider the following problem: given that a finite automaton $M$ of $N$ states accepts at least one $k$-power-free (resp., overlap-free) word, what is the length of the shortest such word accepted? We give upper and lower bounds which, unfortunately, are widely separated.
We show that some results from the theory of group automata and monoid automata still hold for more general classes of monoids and models. Extending previous work for finite automata over commutative groups, we demonstrate a context-free language that can not be recognized by any rational monoid automaton over a finitely generated permutable monoid. We show that the class of languages recognized by rational monoid automata over finitely generated completely simple or completely 0-simple permutable monoids is a semi-linear full trio. Furthermore, we investigate valence pushdown automata, and pr…
The equivalence of finite automata and regular expressions dates back to the seminal paper of Kleene on events in nerve nets and finite automata from 1956. In the present paper we tour a fragment of the literature and summarize results on upper and lower bounds on the conversion of finite automata to regular expressions and vice versa. We also briefly recall the known bounds for the removal of spontaneous transitions (epsilon-transitions) on non-epsilon-free nondeterministic devices. Moreover, we report on recent results on the average case descriptional complexity bounds for the conversion of…
In this dissertation, we study two of the global properties of 1-dimensional cellular automata (CAs) under periodic boundary condition, namely, reversibility and randomness. To address reversibility of finite CAs, we develop a mathematical tool, named reachability tree, which can efficiently characterize those CAs. A decision algorithm is proposed using minimized reachability tree which takes a CA rule and size n as input and verifies whether the CA is reversible for that n. To decide reversibility of a finite CA, we need to know both the rule and the CA size. However, for infinite CAs, revers…
Systems of deterministic finite automata communicating by sending their states upon request are investigated, when the amount of communication is restricted. The computational power and decidability properties are studied for the case of returning centralized systems, when the number of necessary communications during the computations of the system is bounded by a function depending on the length of the input. It is proved that an infinite hierarchy of language families exists, depending on the number of messages sent during their most economical recognitions. Moreover, several properties are …
Antonenko and Russyev independently have shown that any Mealy automaton with no cycles with exit--that is, where every cycle in the underlying directed graph is a sink component--generates a fi- nite (semi)group, regardless of the choice of the production functions. Antonenko has proved that this constitutes a characterization in the non-invertible case and asked for the invertible case, which is proved in this paper.
Parikh automata extend finite automata by counters that can be tested for membership in a semilinear set, but only at the end of a run, thereby preserving many of the desirable algorithmic properties of finite automata. Here, we study the extension of the classical framework onto infinite inputs: We introduce reachability, safety, Büchi, and co-Büchi Parikh automata on infinite words and study expressiveness, closure properties, and the complexity of verification problems. We show that almost all classes of automata have pairwise incomparable expressiveness, both in the deterministic and the n…
We propose several heuristics for mitigating one of the main causes of combinatorial explosion in rank-based complementation of Büchi automata (BAs): unnecessarily high bounds on the ranks of states. First, we identify elevator automata, which is a large class of BAs (generalizing semi-deterministic BAs), occurring often in practice, where ranks of states are bounded according to the structure of strongly connected components. The bounds for elevator automata also carry over to general BAs that contain elevator automata as a sub-structure. Second, we introduce two techniques for refining bound…
Multi-head finite automata were introduced in (Rabin, 1964) and (Rosenberg, 1966). Since that time, a vast literature on computational and descriptional complexity issues on multi-head finite automata documenting the importance of these devices has been developed. Although multi-head finite automata are a simple concept, their computational behavior can be already very complex and leads to undecidable or even non-semi-decidable problems on these devices such as, for example, emptiness, finiteness, universality, equivalence, etc. These strong negative results trigger the study of subclasses and…
Nondeterministic automata may be viewed as succinct programs implementing deterministic automata, i.e. complete specifications. Converting a given deterministic automaton into a small nondeterministic one is known to be computationally very hard; in fact, the ensuing decision problem is PSPACE-complete. This paper stands in stark contrast to the status quo. We restrict attention to subatomic nondeterministic automata, whose individual states accept unions of syntactic congruence classes. They are general enough to cover almost all structural results concerning nondeterministic state-minimality…
An automata network is a finite graph where each node holds a state from some finite alphabet and is equipped with an update function that changes its state according to the configuration of neighboring states. More concisely, it is given by a finite map $f:Q^n\rightarrow Q^n$. In this paper we study how some (sets of) automata networks can be simulated by some other (set of) automata networks with prescribed update mode or interaction graph. Our contributions are the following. For non-Boolean alphabets and for any network size, there are intrinsically non-sequential transformations (i.e. tha…
This paper studies the complexity of languages of finite words using automata theory. To go beyond the class of regular languages, we consider infinite automata and the notion of state complexity defined by Karp. Motivated by the seminal paper of Rabin from 1963 introducing probabilistic automata, we study the (deterministic) state complexity of probabilistic languages and prove that probabilistic languages can have arbitrarily high deterministic state complexity. We then look at alternating automata as introduced by Chandra, Kozen and Stockmeyer: such machines run independent computations on …
We show that a cellular automaton on a mixing subshift of finite type is a Von Neumann regular element in the semigroup of cellular automata if and only if it is split epic onto its image in the category of sofic shifts and block maps. It follows from [S.-Törmä, 2015] that Von Neumann regularity is decidable condition, and we decide it for all elementary CA.
In the present work, we tackle the regular language indexing problem by first studying the hierarchy of $p$-sortable languages: regular languages accepted by automata of width $p$. We show that the hierarchy is strict and does not collapse, and provide (exponential in $p$) upper and lower bounds relating the minimum widths of equivalent NFAs and DFAs. Our bounds indicate the importance of being able to index NFAs, as they enable indexing regular languages with much faster and smaller indexes. Our second contribution solves precisely this problem, optimally: we devise a polynomial-time algorith…
