One-way simple multihead finite automata

The first half of this paper investigates the accepting powers of various types of simple one-way multihead finite automata. It is shown that(1)for each k?1, simple one-way (k+1)-head finite automata are more powerful than simple one-way k-head finite automata.(2)for each k?2, nondeterministic simple one-way k-head finite automata are more powerful than deterministic ones, and(3)for each k?2, sensing simple one-way k-head finite automata are more powerful than non-sensing ones.In the latter half, closure properties for various types of simple one-way multihead finite automata are investigated.Finally, we demonstrate that languages accepted by nondeterministic sensing simple one-way 2-head finite automata are related to some open problem concerning deterministic and nondeterministic tape-bounded Turing computations.     

Multihead two-way probabilistic finite automata

We present properties of multihead two-way probabilistic finite automata that parallel those of their deterministic and nondeterministic counterparts. We define multihead probabilistic finite automata withlogspace constructible transition probabilities, and we describe a technique to simulate these automata by standard logspace probabilistic Turing machines. Next, we represent logspace probabilistic complexity classes as proper hierarchies based on corresponding multihead two-way probabilistic finite automata, and we show their (deterministic logspace) reducibility to the second levels of these hierarchies. We obtain a simple formula for the maximum inherent bandwidth of the configuration transition matrices associated with thek-head probabilistic finite automata processing a length-n input string. (The inherent bandwidth of the configuration transition matrices associated with an automaton processing a length-n input string is the smallest bandwidth we can get by changing the enumeration order of the automaton’s configurations.) Partially based on this relation, we find an apparently easier logspace complete problem forPL (the class of languages recognized by logspace unbounded-error probabilistic Turing machines), and we discuss possibilities for a space-efficient deterministic simulation of probabilistic automata.     

On 3-head versus 2-head finite automata

Summary A direct proof is given that shows that (one-way) 3-head deterministic finite automata are computationally more powerful than 2-head finite automata.This research was supported by the National Science Foundation Grant GJ-35614.     

One way finite visit automata

A one-way preset Turing machine with base L is a nondeterministic on-line Turing machine with one working tape preset to a member of L. FINITEREVERSAL($\text{L}$) (FINITEVISIT ($\text{L}$)) is the class of languages accepted by one-way preset Turing machines with bases in $\text{L}$ which are limited to a finite number of reversals (visits). For any full semiAFL $\text{L}$, FINITEREVERSAL ($\text{L}$) is the closure of $\text{L}$ under homomorphic replication or, equivalently, the closure of $\text{L}$ under iteration of controls on linear context-free grammars while FINITEVISIT ($\text{L}$) is the result of applying controls from $\text{L}$ to absolutely parallel grammars or, equivalently, the closure of $\text{L}$ under deterministic two-way finite state transductions. If $\text{L}$ is a full AFL with $\text{L}$ ≠ FINITEVISIT($\text{L}$), then FINITEREVERSAL($\text{L}$) ≠ FINITEVISIT($\text{L}$). In particular, for one-way checking automata, k + 1 reversals are more powerful than k reversals, k + 1 visits are more powerful than k visits, k visits and k + 1 reversals are incomparable and there are languages definable within 3 visits but no finite number of reversals. Finite visit one-way checking automaton languages can be accepted by nondeterministic multitape Turing machines in space log₂n. Results on finite visit checking automata provide another proof that not all context-free languages can be accepted by one-way nonerasing stack automata.     

Multihead Two-Way Probabilistic Finite Automata

We present properties of multihead two-way probabilistic finite automata that parallel those of their deterministic and nondeterministic counterparts. We define multihead probabilistic finite automata withlogspace constructible transition probabilities, and we describe a technique to simulate these automata by standard logspace probabilistic Turing machines. Next, we represent logspace probabilistic complexity classes as proper hierarchies based on corresponding multihead two-way probabilistic finite automata, and we show their (deterministic logspace) reducibility to the second levels of these hierarchies. We obtain a simple formula for the maximum inherent bandwidth of the configuration transition matrices associated with thek-head probabilistic finite automata processing a length-n input string. (The inherent bandwidth of the configuration transition matrices associated with an automaton processing a length-n input string is the smallest bandwidth we can get by changing the enumeration order of the automaton’s configurations.) Partially based on this relation, we find an apparently easier logspace complete problem forPL (the class of languages recognized by logspace unbounded-error probabilistic Turing machines), and we discuss possibilities for a space-efficient deterministic simulation of probabilistic automata. This research was supported by the National Science Foundation under Grant No. CDA 8822724 while the author was at the University of Rochester. An extended abstract of this paper appeared in Proceedings, Second Latin American Symposium, LATIN ’95: Theoretical Informatics, Valparaiso, Chile, April 1995.     

Head and state hierarchies for unary multi-head finite automata

Unary deterministic one-way multi-head finite automata characterize the unary regular languages. Here they are studied with respect to the existence of head and state hierarchies. It turns out that for any fixed number of states, there is an infinite proper head hierarchy. In particular, the head hierarchy for stateless deterministic one-way multi-head finite automata is obtained using unary languages. On the other hand, it is shown that for a fixed number of heads, \(m+1\) states are more powerful than \(m\) states. Finally, the open question of whether emptiness is undecidable for stateless one-way two-head finite automata is addressed and, as a partial answer, undecidability can be shown if at least four states are provided.     

拟（h,k）存贮有限自动机的可逆性

主要研究拟（h,k）阶存贮有限自动机的延迟k步与k＋1步弱可逆性,以及它的弱逆,得到了拟（h,k）阶存贮有限自动机的延迟k步与k＋1步弱可逆的充分必要条件,并且通过所得结果可以比较简便地构造出延迟k步与k＋1步弱可逆拟（h,k）阶存贮有限自动机的延迟k步与k＋1步弱逆。     

Cellular automata with limited inter-cell bandwidth

A d-dimensional cellular automaton is a d-dimensional grid of interconnected interacting finite automata. There are models with parallel and sequential input modes. In the latter case, the distinguished automaton at the origin, the communication cell, is connected to the outside world and fetches the input sequentially. Often in the literature this model is referred to as an iterative array. In this paper, d-dimensional iterative arrays and one-dimensional cellular automata are investigated which operate in real and linear time and whose inter-cell communication bandwidth is restricted to some constant number of different messages independent of the number of states. It is known that even one-dimensional two-message iterative arrays accept rather complicated languages such as {a^p∣p prime} or {a²ⁿ∣n∈N} (H. Umeo, N. Kamikawa, Real-time generation of primes by a 1-bit-communication cellular automaton, Fund. Inform. 58 (2003) 421-435). Here, the computational capacity of d-dimensional iterative arrays with restricted communication is investigated and an infinite two-dimensional hierarchy with respect to dimensions and messages is shown. Furthermore, the computational capacity of the one-dimensional devices in question is compared with the power of two-way and one-way cellular automata with restricted communication. It turns out that the relations between iterative arrays and cellular automata are quite different from the relations in the unrestricted case. Additionally, an infinite strict message hierarchy for real-time two-way cellular automata is obtained as well as a very dense time hierarchy for k-message two-way cellular automata. Finally, the closure properties of one-dimensional iterative arrays with restricted communication are investigated and differences to the unrestricted case are shown as well.     

Stateless multicounter 5′?→?3′ Watson–Crick automata: the deterministic case

We consider stateless counter machines which mix the features of one-head counter machines and special two-head Watson?CCrick automata (WK-automata). These biologically motivated machines have heads that read the input starting from the two extremes. The reading process is finished when the heads meet. The machine is realtime or non-realtime depending on whether the heads are required to advance at each move. A counter machine is k -reversal if each counter makes at most k alternations between increasing mode and decreasing mode on any computation, and reversal bounded if it is k-reversal for some k. In this paper we concentrate on the properties of deterministic stateless realtime WK-automata with counters that are reversal bounded. We give examples and establish hierarchies with respect to counters and reversals.     

Der programmierbare endliche automat

Summary The paper deals with finite automata with two tapes, the second tape of which is interpreted as programme tape. For fixed automata those classes: of languages are considered, which can be accepted by variation of the programmes. The representation of all regular sets over some alphabet X by a fixed one-way automaton proves to be impossible. This problem has not yet been solved for automata with rewind instructions and two-way automata, which are strongly more powerful than one-way automata.

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Diese Arbeit ist ein Auszug aus der Diplomarbeit, die von Prof. Dr. G. Hotz vergeben und betreut wurde. Ihm möchte der Verfasser seinen Dank aussprechen für das sehr interessante Thema.     

A hierarchy theorem for multihead stack-counter automata

Summary Let L _b = {w ₁ ^*...* w _2b ¦w _i is in {0,1}^* and w _i = w _2b+1–i for 1i2b for b1. We show that the language L _b is not recognizable by any nondeterministic one-way k-head stack-counter automata if \left( {\begin{array}{*{20}c} k \\ 2 \\ \end{array} } \right)$$ " align="middle" border="0"> . As a corollary, we show that k+1 heads are better than k for one-way multihead stack-counter automata.     

On real-time cellular automata and trellis automata

Summary It is shown that f(n)-time one-way cellular automata are equivalent to f(n)-time trellis automata, the real-time one-way cellular automata languages are closed under reversal, the 2n-time one-way cellular automata are equivalent to real-time cellular automata and the latter are strictly more powerful than the real-time one-way cellular automata.This work has been done during the second author's visit at the University of Paris and during both authors' visit at the Institute für Informationsverarbeitung Graz, Austria     

Unbounded-error quantum computation with small space bounds

We prove the following facts about the language recognition power of quantum Turing machines (QTMs) in the unbounded error setting: QTMs are strictly more powerful than probabilistic Turing machines for any common space bound s satisfying s(n)=o(loglogn). For “one-way” Turing machines, where the input tape head is not allowed to move left, the above result holds for s(n)=o(logn). We also give a characterization for the class of languages recognized with unbounded error by real-time quantum finite automata (QFAs) with restricted measurements. It turns out that these automata are equal in power to their probabilistic counterparts, and this fact does not change when the QFA model is augmented to allow general measurements and mixed states. Unlike the case with classical finite automata, when the QFA tape head is allowed to remain stationary in some steps, more languages become recognizable. We define and use a QTM model that generalizes the other variants introduced earlier in the study of quantum space complexity.     

Three write heads are as good ask

Nondeterministic one-wayk-head writing finite automata are investigated. It is shown that 3-head automata with one read-write head followed by two read-only heads are as powerful as any multihead automaton. Furthermore, a new way to detect a coincidence of heads is introduced, which has as a consequence that it makes no difference, whether the heads make stationary moves ore-moves.This research was done while the author visited the Department of Mathematics, University of California at Santa Barbara.The work was supported in part by the National Science Foundation under Grant MCS 77-11360 and by the German Academic Exchange Service under NATO Research Grant 430/402/777/9.     

Tree-stack automata

We introduce a new model of stack automata, the “tree-stack automata,” extending the linear stack to a tree-stack. A main subject of our investigations is to explore the relationship between tree-stack automata and stack automata. The main result of this paper is that tree-stack have the same recognition power as stack-pushdown automata, another (well-known) extension of stack automata. Therefore we obtain that the class of languages accepted by the one-way (linear) stack automata is a proper subset of the class of languages accepted by the one-way tree-stack automata and that two-way tree-stack automata have the same recognition power as two-way (linear) stack automata. As a special case of tree-stack automata we consider tree-pushdown automata. As opposed to stack automata the tree-pushdown storage does not extend the recognition power of one-way (resp. two-way) pushdown automata.     

Lower-bound-constrained runs in weighted timed automata

We investigate a number of problems related to infinite runs of weighted timed automata (with a single weight variable), subject to lower-bound constraints on the accumulated weight. Closing an open problem from Bouyer et al. (2008), we show that the existence of an infinite lower-bound-constrained run is—for us somewhat unexpectedly—undecidable for weighted timed automata with four or more clocks.This undecidability result assumes a fixed and known initial credit. We show that the related problem of existence of an initial credit for which there exists a feasible run is decidable in PSPACE. We also investigate the variant of these problems where only bounded-duration runs are considered, showing that this restriction makes our original problem decidable in  NEXPTIME. We prove that the universal versions of all those problems (i.e, checking that all the considered runs satisfy the lower-bound constraint) are decidable in PSPACE.Finally, we extend this study to multi-weighted timed automata: the existence of a feasible run becomes undecidable even for bounded duration, but the existence of initial credits remains decidable (in PSPACE).     

Two-way unary automata versus logarithmic space

We show that if L=NL (the classical logarithmic space classes), then each unary 2nfa (a two-way nondeterministic finite automaton) can be converted into an equivalent 2dfa (a deterministic two-way automaton), keeping the number of states polynomial. (Unlike other results of this kind, here the deterministic simulation is valid for inputs of all lengths, not only polynomially long ones.) This shows a connection between the standard logarithmic space complexity and the state complexity of two-way unary automata: it indicates that L could be separated from NL by proving a superpolynomial gap, in the number of states, for the conversion from unary 2nfas to 2dfa. Moreover, without any unproven assumptions, we show that each n-state unary 2nfa can be simulated by an equivalent 2ufa (an unambiguous 2nfa) with a polynomial number of states.     

Hybrid automata, reachability, and Systems Biology

Hybrid automata are a powerful formalism for the representation of systems evolving according to both discrete and continuous laws. Unfortunately, undecidability soon emerges when one tries to automatically verify hybrid automata properties. An important verification problem is the reachability one that demands to decide whether a set of points is reachable from a starting region.If we focus on semi-algebraic hybrid automata the reachability problem is semi-decidable. However, high computational costs have to be afforded to solve it. We analyse this problem by exploiting some existing tools and we show that even simple examples cannot be efficiently solved. It is necessary to introduce approximations to reduce the number of variables, since this is the main source of runtime requirements. We propose some standard approximation methods based on Taylor polynomials and ad hoc strategies. We implement our methods within the software SAHA-Tool and we show their effectiveness on two biological examples: the Repressilator and the Delta-Notch protein signaling.     

Constant-space quantum interactive proofs against multiple provers

We present upper and lower bounds of the computational complexity of the two-way communication model of multiple-prover quantum interactive proof systems whose verifiers are limited to measure-many two-way quantum finite automata. We prove that (i) the languages recognized by those multiple-prover systems running in expected polynomial time are exactly the ones in NEXP, the nondeterministic exponential-time complexity class, (ii) if we further require verifiers to be one-way quantum finite automata, then their associated proof systems recognize context-free languages but not beyond languages in NE, the nondeterministic linear exponential-time complexity class, and moreover, (iii) when no time bound is imposed, the proof systems become as powerful as Turing machines. The first two results answer affirmatively an open question, posed by Nishimura and Yamakami [J. Comput. System Sci. 75 (2009) 255–269], of whether multiple-prover quantum interactive proof systems are more powerful than single-prover ones. Our proofs are simple and intuitive, although they heavily rely on an earlier result on multiple-prover classical interactive proof systems of Feige and Shamir [J. Comput. System Sci. 44 (1992) 259–271].