More on quantum,stochastic, and pseudo stochastic languages with few states

Stochastic languages are the languages recognized by probabilistic finite automata (PFAs) with cutpoint over the field of real numbers. More general computational models over the same field such as generalized finite automata (GFAs) and quantum finite automata (QFAs) define the same class. In 1963, Rabin proved the set of stochastic languages to be uncountable presenting a single 2-state PFA over the binary alphabet recognizing uncountably many languages depending on the cutpoint. In this paper, we show the same result for unary stochastic languages. Namely, we exhibit a 2-state unary GFA, a 2-state unary QFA, and a family of 3-state unary PFAs recognizing uncountably many languages; all these numbers of states are optimal. After this, we completely characterize the class of languages recognized by 1-state GFAs, which is the only nontrivial class of languages recognized by 1-state automata. Finally, we consider the variations of PFAs, QFAs, and GFAs based on the notion of inclusive/exclusive cutpoint, and present some results on their expressive power.     

On a class of languages recognizable by probabilistic reversible decide-and-halt automata

We analyze the properties of probabilistic reversible decide-and-halt automata (DH-PRA) and show that there is a strong relationship between DH-PRA and 1-way quantum automata. We show that a general class of regular languages is not recognizable by DH-PRA by proving that two “forbidden” constructions in minimal deterministic automata correspond to languages not recognizable by DH-PRA. The shown class is identical to a class known to be not recognizable by 1-way quantum automata. We also prove that the class of languages recognizable by DH-PRA is not closed under union and other non-trivial Boolean operations.     

Algebraic Results on Quantum Automata

We use tools from the algebraic theory of automata to investigate the class of languages recognized by two models of Quantum Finite Automata (QFA): Brodsky and Pippenger's end-decisive model (which we call BPQFA) and a new QFA model (which we call LQFA) whose definition is motivated by implementations of quantum computers using nucleo-magnetic resonance (NMR). In particular, we are interested in LQFA since NMR was used to construct the most powerful physical quantum machine to date. We give a complete characterization of the languages recognized by LQFA and by Boolean combinations of BPQFA. It is a surprising consequence of our results that LQFA and Boolean combinations of BPQFA are equivalent in language recognition power.     

Spinal-Formed Context-Free Tree Grammars

In this paper we introduce a restricted model of context-free tree grammars called spine grammars, and study their formal properties including considerably simple normal forms. Recent research on natural languages has suggested that formalisms for natural languages need to generate a slightly larger class of languages than context-free grammars, and for that reason tree adjoining grammars have been widely studied relating them to natural languages. It is shown that the class of string languages generated by spine grammars coincides with that of tree adjoining grammars. We also introduce acceptors called linear pushdown tree automata, and show that linear pushdown tree automata accept exactly the class of tree languages generated by spine grammars. Linear pushdown tree automata are obtained from pushdown tree automata with a restriction on duplicability for the pushdown stacks. Received May 29, 1998, and in revised form April 27, 1999, and in final form May 10, 1999.     

On the minimization of XML Schemas and tree automata for unranked trees

Automata for unranked trees form a foundation for XML Schemas, querying and pattern languages. We study the problem of efficiently minimizing such automata. First, we study unranked tree automata that are standard in database theory, assuming bottom-up determinism and that horizontal recursion is represented by deterministic finite automata. We show that minimal automata in that class are not unique and that minimization is np-complete. Second, we study more recent automata classes that do allow for polynomial time minimization. Among those, we show that bottom-up deterministic stepwise tree automata yield the most succinct representations. Third, we investigate abstractions of XML schema languages. In particular, we show that the class of one-pass preorder typeable schemas allows for polynomial time minimization and unique minimal models.     

量子自动机的刻画

澄清了各类量子自动机之间的相互关系,并给出了量子自动机的各种等价刻画定理.引入G-量子自动机、g-量子自动机、(广义)量子自动机及G-量子文法和g-量子文法,并阐明了它们与其他量子自动机之间的等价关系.在一定条件下讨论了G(g)-量子自动机与G(g)-量子文法的等价性,从而解决了关于量子文法产生量子正规语言的问题.讨论了量子语言与正规语言的关系,特别是回答了Gudder提出的两个公开问题.最后,给出了一种减少状态空间维数的方法.     

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].     

Probabilistic asynchronous automata

Asynchronous automata were introduced by W. Zielonka as an algebraic model of distributed systems, showing that the class of trace languages recognizable by automata over free partially commutative monoids coincides with the class of trace languages recognizable by deterministic asynchronous automata. In this paper we extend the notion of asynchronous automata to the probabilistic case. Our main result is a nontrivial generalization to Zielonka's theorem: we prove that the sets of behaviors of probabilistic automata and of probabilistic asynchronous automata coincide in the case of concurrent alphabets with acyclic dependency graphs. This research has been supported by European Projects EBRA Nos. 3148 (DEMON), 3166 (ASMICS), and 6317 (ASMICS2), by MURST 40%, and by the CNR Project “Modelli di Computazione Parallela.”     

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.     

Asynchronous Mappings and Asynchronous Cellular Automata

The aim of this paper is the study of asynchronous automata, a special kind of automata which encode the independency relation between actions and which enable their concurrent execution. These automata, introduced by Zielonka (RAIRO Inform. Theor. Appl.21, 99-135 (1987)), constitute a natural extension of finite automata to the case of asynchronous parallelism. Their behaviour is described by trace languages, subsets of partially commutative monoids. The main result concerning this class of automata states that they accept exactly all recognizable trace languages. In this paper we give new improved constructions of asynchronous automata. In the final part of the paper we present a distributed system of messages with bounded time-stamps based on asynchronous automata.     

A NEW AUTOMATON MODEL FOR TAGS: 2-SA

In this paper, we introduce and define a new class of automata (pushdown automata with n stacks, abbreviated as n-SA). The ultimate aim is to give a new characterization of LCRFL, the class of languages accepted by a linear context-free rewriting system (LCFRS). In particular, we introduce 2-SA as a new automaton model for tree-adjoining grammars (TAG). In the simplest cases (0-SA and 1-SA), the languages that are accepted by the automata are the regular and context-free languages respectively. A more complex case is the case of a 2-SA which accepts TALs. The n-SA creates an infinite hierarchy of languages and it seems that this hierarchy corresponds to others in the class LCFRL. The 2-SA corresponds closely to the EPDA (embedded pushdown automaton, an automaton model equivalent to TAGs). Unlike the EPDA, which allows push operations "below the top stack," an n-SA allows push and pop operations only on the top of their (multiple) stacks. So n-SA trade simpler operations against an also simpler but expanded storage structure.     

Topological Automata

We give a new, topological definition of automata that extends previous definitions of probabilistic and quantum automata. We then are able to prove in a unified framework that deterministic or non-deterministic probabilistic and quantum automata recognise only regular languages with an isolated threshold.     

基于量子逻辑的自动机和文法理论

初步建立了基于量子逻辑的自动机和文法理论的基本框架.引入了量子文法(称为l值文法),特别是证明了任意l值正规文法生成的语言(称为量子语言)等价于某种基于量子逻辑且含动作(的自动机(称为l值自动机)识别的语言,反之,任意l值自动机识别的语言等价于某l值正规文法生成的语言.建立了l值泵引理,并得到量子语言的判定性刻画.最后简要讨论了正规文法与量子文法(即l值正规文法)的关系.因此,为进一步研究更复杂的量子自动机(如量子下推自动机和Turing机)和量子文法(如量子上下文无关文法和上下文有关文法)奠定了基础.     

On the Computational Complexity of P Automata

We characterize the classes of languages over finite alphabets which may be described by P automata, i.e., accepting P systems with communication rules only. Motivated by properties of natural computing systems, and the actual behavior of P automata, we study computational complexity classes with a certain restriction on the use of the available workspace in the course of computations and relate these to the language classes described by P automata. We prove that if the rules of the P system are applied sequentially, then the accepted language class is strictly included in the class of languages accepted by one-way Turing machines with a logarithmically bounded workspace, and if the rules are applied in the maximally parallel manner, then the class of context-sensitive languages is obtained.     

Globally deterministic CD-systems of stateless R-automata with window size 1

It is known that cooperating distributed systems (CD-systems) of stateless deterministic restarting automata with window size 1 accept a class of semi-linear languages that properly includes all rational trace languages. Although the component automata of such a CD-system are all deterministic, the CD-system itself is not. Here, we study CD-systems of stateless deterministic restarting automata with window size 1 that are themselves completely deterministic. In fact, we consider two such types of CD-systems: the strictly deterministic systems and the globally deterministic systems.     

Variants of top-down tree transducers with look-ahead

In this article we consider deterministic and strongly deterministic top-down tree transducers with regular look-ahead, with regular check, with deterministic top-down look-ahead, and with deterministic top-down check. We compare the transformational power of these tree transducer classes by giving a correct inclusion diagram of the tree transformation classes induced by them. Along with the comparison we decompose some of the examined classes into simpler classes and we introduce the concept of the deterministic top-down tree automata with deterministic top-down look-ahead. We show that these recognizers recognize a tree language class which is strictly between the class of regular tree languages and the class of tree languages recognizable by deterministic top-down tree automata. We also study the closure properties of the examined tree transformation classes. We show that some classes are closed under composition while others, for example the class of tree transformations induced by deterministic top-down tree transducers with deterministic top-down look-ahead, are not.     

A note on partially ordered tree automata

A recent paper by Bouajjani, Muscholl and Touili shows that the class of languages accepted by partially ordered word automata (or equivalently accepted by Σ₂-formulae) is closed under semi-commutation and it suggested the following open question: can we extend this result to tree languages? This problem can be addressed by proving (1) that the class of tree regular languages accepted by Σ₂ formulae is strictly included in the class of languages accepted by partially ordered automata, and (2) that Bouajjani and the others results cannot be extended to tree.     

基于量子逻辑的下推自动机的代数刻画

首先,本文提出量子下推自动机（简记为L-VPDA）的概念,从代数角度出发详细研究了此类自动机的性质,同时建立此类自动机的代数刻画,即利用量子状态构造证明了任意L-VPDA与状态转移为经典函数且具有量子终状态的L-VPDA间的相互等价性;其次详细研究了量子上下文无关语言的代数刻画以及对于正则运算的封闭性。     

Linearly bounded infinite graphs

Linearly bounded Turing machines have been mainly studied as acceptors for context-sensitive languages. We define a natural class of infinite automata representing their observable computational behavior, called linearly bounded graphs. These automata naturally accept the same languages as the linearly bounded machines defining them. We present some of their structural properties as well as alternative characterizations in terms of rewriting systems and context-sensitive transductions. Finally, we compare these graphs to rational graphs, which are another class of automata accepting the context-sensitive languages, and prove that in the bounded-degree case, rational graphs are a strict sub-class of linearly bounded graphs.A preliminary version of this article appeared in MFCS 2005.     

A class of measures on formal languages

Summary In this paper we introduce a class of measures on formal languages. These measures are based on the number of different ways a string of a specified finite length can be completed to obtain strings of the language. The relation with automata and grammars is established, and the polynomial measure, a special case of the general notion, is studied in detail. We give some closure properties for well-known operations on languages, and finally, we prove that the class of polynomial measurable languages is a Pre-AFL.