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Journal of Shaanxi Normal University

Algebraic properties of quantum infinite regular languages

Zhaowei Han

Published Jan 1, 2012• 0 citations
OpenAlex

Abstract

The concepts of quantum Muller automaton and quantum infinite regular language are introduced.In virtue of the fact that the image set of a quantum infinite regular language recognized by an arbitrary quantum Muller automaton is always finite and by means of semantic analysis and quantum state construction, the algebraic characterization of quantum Muller automaton is studied.It is shown that an arbitrary quantum Muller automaton is equivalent to the one which has crisp initial states and transition relation but with quantum final states. Based on this,the algebraic description and the level characterization of quantum infinite regular languages are obtained, and it is proved that an arbitrary quantum infinite language A is recognizable if and only if the image set of A is finite and A can be represented as a finite union of special quantum infinite languages.As applications,it is proved that even if the distributivity law does not hold in quantum logic itself, quantum infinite regular languages are still closed under regular operations.

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•Jan 1, 2002•0 citations
The concepts of quantum Muller automaton and quantum infinite regular language are introduced.In virtue of the fact that the image set of a quantum infinite regular language recognized by an arbitrary quantum Muller automaton is always finite and by means of semantic analysis and quantum state construction, the algebraic characterization of quantum Muller automaton is studied.It is shown that an arbitrary quantum Muller automaton is equivalent to the one which has crisp initial states and transition relation but with quantum final states. Based on this,the algebraic description and the level characterization of quantum infinite regular languages are obtained, and it is proved that an arbitrary quantum infinite language A is recognizable if and only if the image set of A is finite and A can be represented as a finite union of special quantum infinite languages.As applications,it is proved that even if the distributivity law does not hold in quantum logic itself, quantum infinite regular languages are still closed under regular operations.
The notion of quantum Bchi automaton(LVBA for short)is introduced,by means of quantum state construction, the equivalence of an LVBA and an LVSBA with crisp transition function is proved,based on this,the algebraic and level characterizations and also the Bchi characterization of quantum infinite regular languages are investigated,and also the closed properties of those quantum infinite regular languages under some regular operations are dealt with.By providing the concept of monadic secondorder quantum logic(LVMSO in short),the monadic second-order logic characterizations of infinite regular languages recognized by quantum Biichi automata are presented,which deepen and generalize the fundamental Bchi theorem to quantum setting.
We study a model of one-way quantum automaton where only measurement operations are allowed ($\mon$). We give an algebraic characterization of $\lmo(Σ)$, showing that the syntactic monoids of the languages in $\lmo(Σ)$ are exactly the JJJ-trivial literally idempotent syntactic monoids, where JJJ is the Green's relation determined by two-sided ideals. We also prove that $\lmo(Σ)$ coincides with the literal variety of literally idempotent piecewise testable regular languages. This allows us to prove the existence of a polynomial time algorithm for deciding whether a regular language belongs to $\lmo(Σ)$ and to discuss definability issues in terms of the existential first-order logic $Σ_1[
In this paper, we introduce 1-way and 2-way quantum finite state automata (1qfa's and 2qfa's), which are the quantum analogues of deterministic, nondeterministic and probabilistic 1-way and 2-way finite state automata. We prove the following facts regarding 2qfa's. 1. For any /spl epsiv/>0, there is a 2qfa M which recognizes the non-regular language L={a/sup m/b/sup m/|m/spl ges/1} with (one-sided) error bounded by E, and which halts in linear time. Specifically, M accepts any string in L with probability 1 and rejects any string not in L with probability at least 1-/spl epsiv/. 2. For every regular language L, there is a reversible (and hence quantum) 2-way finite state automaton which recognizes L and which runs in linear time. In fact, it is possible to define 2qfar's which recognize the non-context-free language {a/sup m/b/sup m/c/sup m/|m/spl ges/1}, based on the same technique used for 1. Consequently, the class of languages recognized by linear time, bounded error 2qfa's properly includes the regular languages. Since it is known that 2-way deterministic, nondeterministic and polynomial expected time, bounded error probabilistic finite automata can recognize only regular languages, it follows that 2qfa's are strictly more powerful than these "classical" models. In the case of 1-way automata, the situation is reversed. We prove that the class of languages recognizable by bounded error 1qfa's is properly contained in the class of regular languages.
Quantum finite automata were introduced by C.Moore and J.P.Crutchfield in [MC 97] and by A.Kondacs and J.Watrous in [KW 97]. This notion is not a generalization of the deterministic finite automata, but rather a generalization of deterministic reversible (permutation) automata. In [AF 98] A.Ambainis and R.Freivalds raised the question what kind of probabilistic automata can be viewed as a special case of quantum finite automata. To answer that question and study relationship between quantum finite automata and probabilistic finite automata, we introduce a notion of probabilistic reversible automata (PRA, or doubly stochastic automata). We give the necessary condition for a language to be recognized by PRA. We find that there is a strong relationship between different possible models of PRA and corresponding models of quantum finite automata. In these thesis we regard quantum automata, probabilistic reversible and deterministic reversible automata as reversible automata. At least two non-equivalent definitions of language recognition are used for one-way reversible finite automata in various papers. Both of these definitions have their own advantages and disadvantages, summarized in our classification of one-way reversible automata. Further, we introduce a notion of quantum finite multitape automata and prove that there is a language recognized by a quantum finite multitape automaton but not by deterministic finite multitape automata. Additionally we discover unexpected probabilistic automata recognizing complicated languages. Finally, we revise the concept of quantum pushdown automata (QPA), first introduced by C.Moore and J.P.Cruthchfield in [MC 97]. We give the definition of QPA in a non-equivalent way, including unitarity criteria, by using the definition of quantum finite automata of [KW 97]. It is established that the unitary criteria of QPA are not equivalent to the corresponding unitary criteria of quantum Turing machines [BV 97]. We show that QPA can recognize every regular language. We also present some simple non-context-free languages recognized by QPA (hence not recognizable by deterministic and nondeterministic pushdown automata).