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arXiv:1507.01346

Time-space tradeoffs for two-way finite automata

Shenggen Zheng, Daowen Qiu, Jozef Gruska

Published Jul 6, 2015• 1 citation
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Abstract

We explore bounds of {\em time-space tradeoffs} in language recognition on {\em two-way finite automata} for some special languages. We prove: (1) a time-space tradeoff upper bound for recognition of the languages $L_{EQ}(n)$ on {\em two-way probabilistic finite automata} (2PFA): $TS={\bf O}(n\log n)$, whereas a time-space tradeoff lower bound on {\em two-way deterministic finite automata} is ${\bf Ω}(n^2)$, (2) a time-space tradeoff upper bound for recognition of the languages $L_{INT}(n)$ on {\em two-way finite automata with quantum and classical states} (2QCFA): $TS={\bf O}(n^{3/2}\log n)$, whereas a lower bound on 2PFA is $TS={\bf Ω}(n^2)$, (3) a time-space tradeoff upper bound for recognition of the languages $L_{NE}(n)$ on exact 2QCFA: $TS={\bf O}(n^{1.87} \log n)$, whereas a lower bound on 2PFA is $TS={\bf Ω}(n^2)$. It has been proved (Klauck, STOC'00) that the exact one-way quantum finite automata have no advantage comparing to classical finite automata in recognizing languages. However, the result (3) shows that the exact 2QCFA do have an advantage in comparison with their classical counterparts, which has been the first example showing that the exact quantum computing have advantage in time-space tradeoff comparing to classical computing. Usually, two communicating parties, Alice and Bob, are supposed to have an access to arbitrary computational power in {\em communication complexity} model that is used. Instead of that we will consider communication complexity in such a setting that two parties are using only finite automata and we prove in this setting that quantum automata are better than classical automata and also probabilistic automata are better than deterministic automata for some well known tasks.

DOI
10.48550/arxiv.1507.01346

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We explore bounds of {\em time-space tradeoffs} in language recognition on {\em two-way finite automata} for some special languages. We prove: (1) a time-space tradeoff upper bound for recognition of the languages LEQ(n)L_{EQ}(n)LEQ​(n) on {\em two-way probabilistic finite automata} (2PFA): TS=O(nlog⁡n)TS={\bf O}(n\log n)TS=O(nlogn), whereas a time-space tradeoff lower bound on {\em two-way deterministic finite automata} is Ω(n2){\bf Ω}(n^2)Ω(n2), (2) a time-space tradeoff upper bound for recognition of the languages LINT(n)L_{INT}(n)LINT​(n) on {\em two-way finite automata with quantum and classical states} (2QCFA): TS=O(n3/2log⁡n)TS={\bf O}(n^{3/2}\log n)TS=O(n3/2logn), whereas a lower bound on 2PFA is TS=Ω(n2)TS={\bf Ω}(n^2)TS=Ω(n2), (3) a time-space tradeoff upper bound for recognition of the languages LNE(n)L_{NE}(n)LNE​(n) on exact 2QCFA: TS=O(n1.87log⁡n)TS={\bf O}(n^{1.87} \log n)TS=O(n1.87logn), whereas a lower bound on 2PFA is TS=Ω(n2)TS={\bf Ω}(n^2)TS=Ω(n2). It has been proved (Klauck, STOC'00) that the exact one-way quantum finite automata have no advantage comparing to classical finite automata in recognizing languages. However, the result (3) shows that the exact 2QCFA do have an advantage in comparison with their classical counterparts, which has been the first example showing that the exact quantum computing have advantage in time-space tradeoff comparing to classical computing. Usually, two communicating parties, Alice and Bob, are supposed to have an access to arbitrary computational power in {\em communication complexity} model that is used. Instead of that we will consider communication complexity in such a setting that two parties are using only finite automata and we prove in this setting that quantum automata are better than classical automata and also probabilistic automata are better than deterministic automata for some well known tasks.
Two-way finite automata with quantum and classical states (2QCFA) were introduced by Ambainis and Watrous, and it was shown that 2QCFA have superiority over two-way probabilistic finite automata (2PFA) for recognizing some non-regular languages such as the language Leq = {anbn ∣ n ∈ N} and the palindrome language Lpal = {w ∈ {a,b}* ∣ w=wR}, where xR is x in the reverse order. It is interesting to find more languages like these that witness the superiority of 2QCFA over 2PFA. In this paper, we consider the language Lm = {xcy ∣ ∑ = {a,b,c},x,y ∈ {a,b}*, ∣x∣ = ∣y∣} that is similar to the middle language Lmiddle = {xay ∣ x,y ∈ {a,b}*, ∣x∣ = ∣y∣}. We prove that the language Lm show that Lm can be recognized by 2PFA with bounded error, but only in exponential expected time. Thus Lm is another witness of the fact that 2QCFA are more powerful than their classical counterparts.
The two-way finite automaton with quantum and classical states (2QCFA), defined by Ambainis and Watrous, is a model of quantum computation whose quantum part is extremely limited; however, as they showed, 2QCFA are surprisingly powerful: a 2QCFA with only a single-qubit can recognize the language Lpal={w∈{a,b}∗:w is a palindrome}L_{pal}=\{w \in \{a,b\}^*:w \text{ is a palindrome}\}Lpal​={w∈{a,b}∗:w is a palindrome} with bounded error in expected time 2O(n)2^{O(n)}2O(n), on inputs of length nnn. We prove that their result essentially cannot be improved upon: a 2QCFA (of any size) cannot recognize LpalL_{pal}Lpal​ with bounded error in expected time 2o(n)2^{o(n)}2o(n). To our knowledge, this is the first example of a language that can be recognized with bounded error by a 2QCFA in exponential time but not in subexponential time. Moreover, we prove that a quantum Turing machine (QTM) running in space o(log⁡n)o(\log n)o(logn) and expected time 2n1−Ω(1)2^{n^{1-Ω(1)}}2n1−Ω(1) cannot recognize LpalL_{pal}Lpal​ with bounded error; again, this is the first lower bound of its kind. Far more generally, we establish a lower bound on the running time of any 2QCFA or o(log⁡n)o(\log n)o(logn)-space QTM that recognizes any language LLL in terms of a natural "hardness measure" of LLL. This allows us to exhibit a large family of languages for which we have asymptotically matching lower and upper bounds on the running time of any such 2QCFA or QTM recognizer.
One-way quantum finite automata together with classical states (1QFAC) proposed in [Journal of Computer and System Sciences 81(2) (2015) 359--375] is a new one-way quantum finite automata (1QFA) model that integrates quantum finite automata (QFA) and deterministic finite automata (DFA). This model uses classical states to control the evolution and measurement of quantum states. As a quantum-classical hybrid model, 1QFAC recognize all regular languages. It was shown that the state complexity of 1QFAC for some languages is essentially superior to that of DFA and other 1QFA. In this paper, our goal is to clarify state complexity problems for 1QFAC. We obtain the following results: (1) We optimize the bound given by Qiu et al. that characterizes the relationship between quantum basis state number and classical state number of 1QFAC as well as the state number of its corresponding minimal DFA for recognizing any given regular language. (2) We give an upper bound showing that how many classical states are needed if the quantum basis states of 1QFAC are reduced without changing its recognition ability. (3) We give a lower bound of the classical state number of 1QFAC for recognizing any given regular language, and the lower bound is exact if the given language is finite. (4) We show that 1QFAC are exponentially more succinct than DFA and probabilistic finite automata (PFA) for recognizing some regular languages that can not be recognized by measure-once 1QFA (MO-1QFA), measure-many 1QFA (MM-1QFA) or multi-letter 1QFA. (5) We reveal essential relationships between 1QFAC, MO-1QFA and multi-letter 1QFA, and induce a result regarding a quantitative relationship between the state number of multi-letter 1QFA and DFA.
{\it Two-way quantum automata with quantum and classical states} (2QCFA) were introduced by Ambainis and Watrous in 2002. In this paper we study state succinctness of 2QCFA. For any m∈Z+m\in {\mathbb{Z}}^+m∈Z+ and any ε<1/2ε<1/2ε<1/2, we show that: {enumerate} there is a promise problem Aeq(m)A^{eq}(m)Aeq(m) which can be solved by a 2QCFA with one-sided error εεε in a polynomial expected running time with a constant number (that depends neither on mmm nor on ε\varepsilonε) of quantum states and O(log⁡1ε)\mathbf{O}(\log{\frac{1}ε)}O(logε1​) classical states, whereas the sizes of the corresponding {\it deterministic finite automata} (DFA), {\it two-way nondeterministic finite automata} (2NFA) and polynomial expected running time {\it two-way probabilistic finite automata} (2PFA) are at least 2m+22m+22m+2, log⁡m\sqrt{\log{m}}logm​, and (log⁡m)/b3\sqrt[3]{(\log m)/b}3(logm)/b​, respectively; there exists a language Ltwin(m)={wcw∣w∈{a,b}∗}L^{twin}(m)=\{wcw| w\in\{a,b\}^*\}Ltwin(m)={wcw∣w∈{a,b}∗} over the alphabet Σ={a,b,c}Σ=\{a,b,c\}Σ={a,b,c} which can be recognized by a 2QCFA with one-sided error εεε in an exponential expected running time with a constant number of quantum states and O(log⁡1ε)\mathbf{O}(\log{\frac{1}ε)}O(logε1​) classical states, whereas the sizes of the corresponding DFA, 2NFA and polynomial expected running time 2PFA are at least 2m2^m2m, m\sqrt{m}m​, and m/b3\sqrt[3]{m/b}3m/b​, respectively; {enumerate} where bbb is a constant.
Two-way finite automata with quantum and classical states (2QCFA’s), introduced by Ambainis and Watrous, extend deterministic two-way finite automata with a constant-size quantum register and provide a natural hybrid model of computation. In this paper, we introduce two-way nondeterministic finite automata with quantum and classical states (2QCNFA’s), extending this framework to the nondeterministic setting. We show that 2QCNFA’s are closed under union, intersection, and marked concatenation, and that they can recognize certain comparison languages whose status for 2QCFA’s remains open. We further initiate the study of decision problems for this model by proving that 2QCNFA’s can recognize the set of invalid computations of a deterministic Turing machine, which implies that the universality problem is undecidable. In addition, we investigate descriptional complexity, showing that there is no approximation algorithm for minimization between 2QCNFA’s and classical models such as DFA’s and context-free grammars, and that the corresponding trade-offs are non-recursive.
The two-way finite automaton with quantum and classical states (2QCFA), defined by Ambainis and Watrous, is a model of quantum computation whose quantum part is extremely limited; however, as they showed, 2QCFA are surprisingly powerful: a 2QCFA, with a single qubit, can recognize, with bounded error, the language Leq={ambm:m∈N}L_{eq}=\{a^m b^m :m \in \mathbb{N}\}Leq​={ambm:m∈N} in expected polynomial time and the language Lpal={w∈{a,b}∗:w is a palindrome}L_{pal}=\{w \in \{a,b\}^*:w \text{ is a palindrome}\}Lpal​={w∈{a,b}∗:w is a palindrome} in expected exponential time. We further demonstrate the power of 2QCFA by showing that they can recognize the word problems of many groups. In particular 2QCFA, with a single qubit and algebraic number transition amplitudes, can recognize, with bounded error, the word problem of any finitely generated virtually abelian group in expected polynomial time, as well as the word problems of a large class of linear groups in expected exponential time. This latter class (properly) includes all groups with context-free word problem. We also exhibit results for 2QCFA with any constant number of qubits. As a corollary, we obtain a direct improvement on the original Ambainis and Watrous result by showing that LeqL_{eq}Leq​ can be recognized by a 2QCFA with better parameters. As a further corollary, we show that 2QCFA can recognize certain non-context-free languages in expected polynomial time. In a companion paper, we prove matching lower bounds, thereby showing that the class of languages recognizable with bounded error by a 2QCFA in expected subexponential\mathit{subexponential}subexponential time is properly contained in the class of languages recognizable with bounded error by a 2QCFA in expected exponential\mathit{exponential}exponential time.