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International Conference on Communications, Circuits and Systems

Regular quantum grammars and regular grammars

Wei Cheng, Jue Wang

Published Jan 1, 2004• 1 citation
Publisher

Abstract

To study quantum computation, Moore and Crutchfield (Theoret. Comput. Sci. vol. 237, pp. 275-306, 2000) proposed quantum versions of finite-state and push-down automata, as well as regular and context-free grammars. In this note, we show that the languages generated by regular quantum grammars are equivalent to the languages generated by regular grammars. Therefore, from the perspective of generating languages, regular quantum grammars are not more powerful than classical regular grammars. This implies that it may be necessary to look for more well-formed definitions of quantum grammars and maybe those of quantum automata.

DOI
10.1109/icccas.2004.1345943

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To study quantum computation, Moore and Crutchfield (Theoret. Comput. Sci. vol. 237, pp. 275-306, 2000) proposed quantum versions of finite-state and push-down automata, as well as regular and context-free grammars. In this note, we show that the languages generated by regular quantum grammars are equivalent to the languages generated by regular grammars. Therefore, from the perspective of generating languages, regular quantum grammars are not more powerful than classical regular grammars. This implies that it may be necessary to look for more well-formed definitions of quantum grammars and maybe those of quantum automata.
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).
The relationships among various quantum automata are clarified, and in particular, variously equivalent characterizations of quantum automata are established. The G-quantum automata, g-quantum automata, (generalized) quantum automata, G-quantum grammars and g-quantum grammars are presented, and their connections to some of the other existing quantum automata are expounded. Under a certain condition, the equivalence between the G(g)-quantum automata and G(g)-quantum grammars is discussed. Therefore the problem of quantum grammars generating quantum regular languages is solved. The relation between quantum and regular languages is dealt with, and particularly, two open problems proposed by Gudder are answered. Finally, the method of decreasing the dimensions of spaces of states on quantum automata is given.
Quantum finite automata, as well as quantum pushdown automata (QPA) were first introduced by C. Moore and J. P. Crutchfield. In this paper we introduce the notion of QPA in a non-equivalent way, including unitarity criteria, by using the definition of quantum finite automata of Kondacs and Watrous. It is established that the unitarity criteria of QPA are not equivalent to the corresponding unitarity criteria of quantum Turing machines. We show that QPA can recognize every regular language. Finally we present some simple languages recognized by QPA, not recognizable by deterministic pushdown automata.
Quantum computing is a relatively new field of computing, which utilises the fundamental concepts of quantum mechanics to process data. The seminal paper of Moore et al. [2000] introduced quantum grammars wherein a set of amplitudes was attached to each production. However they did not study the final probability of the derived word. Aruja et al. [2025] considered conditions for the well-formedness of quantum context-free grammars (QCFGs), in order to ensure that the probabilty of the derived word does not exceed one. In this paper we propose certain necessary and sufficient conditions (also known as unitary conditions) for well-formedness of QCFGs
In general, quantum computation models are expected to be more powerful than classical counterparts. However, sometimes this is not the case.It is known that there exists some regular language which one-way quantum finite automata and one-way quantum counter automata cannot recognize. This is due to the restriction of reversibility which quantum models must satisfy. Thus, it is an interesting question: what kinds of quantum models suffer from/overcome the restriction? To tackle this problem, we focus on (empty-stack acceptance) one-way quantum pushdown automata, and show that there exists a regular language which one-way quantum pushdown automata cannot recognize. This implies adding a stack to one-way finite automata cannot overcome the restriction of reversibility if we adopt empty-stack acceptance as the acceptance mode.