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A Polynomial-Time Algorithm for Checking the Equivalence for Real-Time Deterministic Restricted One-Counter Transducers Which Accept by Final State
Abstract
This paper is concerned with a subclass of deterministic pushdown transducers, called deterministic restricted one-counter transducers (droct's), and studies the inclusion problem for droct's which accept by final state. In the previous study, we presented a polynomialtime algorithm for checking the equivalence of these droct's. By extending this technique, we present a polynomial-time algorithm for checking the inclusion of these droct's. Copyright © 2015 by The International Society for Computers and Their Applications (ISCA).
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A deterministic pushdown automaton (dpda) having just one stack symbol is called a deterministic restricted one-counter automaton (droca). When it accepts by empty stack, it is called strict. A deterministic one-counter automaton (doca) is a dpda having only one stack symbol, with the exception of a bottom-of-stack marker. The class of languages accepted by strict droca's is a subclass of the class of languages accepted by doca's. Valiant has proved the decidability of the equivalence problem for doca's and the undecidability of the inclusion problem for doca's. Hence the decidability of the equivalence problem for strict droca's is obvious. In this paper, we present a new direct branching algorithm for checking the inclusion for a pair of languages accepted by strict droca's. Then we show that the worst-case time complexity of our algorithm is polynomial with respect to these automata.
We propose a new class of non-real-time deterministic pushdown automata (dpda's), named dpda's having the weak segmental property (WSP), and show that the equivalence problem is solvable for two dpda's, one of which is in this class. The equivalence checking algorithm to prove this problem is a further extended direct branching algorithm of Tomita; and with the new skipping step combined with the type B replacement of Oyamaguchi, Inagaki and Honda. The algorithm is still relatively simple. The class of dpda's given above is one of the widest known subclasses of proper dpda's (introduced by Ukkonen); with the decidable extended equivalence problem.
The class of deterministic one-counter automata is a natural extension of the class of finite-state machines. We have shown that in contrast with the inclusion and nullity of intersection problems, which become undecidable under this generalisation the equivalence problem remains decidable.
We have established that these automata have certain periodic structural properties, and have derived an upper bound for the associated period that is asymptotically achievable. The resulting bound on the time complexity of the derived decision procedure is exponential in about the square root of the number of states of the tested machines. Whether a polynomial time test exists, remains open.
In his paper (Theoretical Computer Science, Vol 251 (2001), pages 1—166), the author gives a positive solution to the decidability of the equivalence problem for deterministic pushdown automata: Given two languages L
1 and L
2 accepted by deterministic pushdown automata decide whether L
1=L
2. The problem was posed by S. Ginsburg and S. Greibach in 1966 and various subcases were shown to be decidable over years. However, the full question remained elusive until it was finally settled by the awarded. He showed the problem to be decidable. The paper not only settles the equivalence problem for deterministic context-free languages, but also develops an entire machinery of new techniques which are likely to be useful in other contexts. They have already found useful in semantics of programming languages.
We show that simple deterministic languages are polynomial time learnable via membership queries if the learner knows a special finite set of positive examples. This finite set is called a representative sample and has been introduced by Angluin Inform. Control 51 (1981) to show that regular languages are polynomial time learnable via membership queries. If simple deterministic languages are learnable in polynomial time via membership and equivalence queries, we can obtain a representative sample of a target language in polynomial time from a correct hypothesis. Thus, our result implies that the polynomial time learning problem of simple deterministic languages via membership and equivalence queries is solvable if and only if we can find a representative sample in polynomial time via these queries. We show the learnability of simple deterministic languages by giving a learning algorithm. The algorithm, at the first stage, makes all possible candidate rules to generate the target language and a set of simple deterministic grammars which are little different each other. Then, comparing them, the algorithm eliminates inappropriate rules.
A deterministic pushdown automaton (dpda) having just one stack symbol is called a deterministic restricted one-counter automaton (droca). When it accepts an input by empty stack, it is called strict. This paper is concerned with a subclass of real-time strict droca's, called Szilard strict droca's, and studies the problem of identifying the subclass in the limit from positive data. The class of languages accepted by Szilard strict droca's coincides with the class of Szilard languages (or, associated languages) of strict droca's and is incomparable to each of the class of regular languages and that of simple languages. After providing some properties of languages accepted by Szilard strict droca's, we show that the class of Szilard strict droca's is polynomial time identifiable in the limit from positive data in the sense of Yokomori. This identifiability is proved by giving an exact characteristic sample of polynomial size for a language accepted by a Szilard strict droca. The class of very simple languages, which is a proper subclass of simple languages, is also proved to be polynomial time identifiable in the limit from positive data by Yokomori, but it is yet unknown whether there exists a characteristic sample of polynomial size for any very simple language.
A new direct algorithm is presented for checking equivalence of some classes of deterministic pushdown automata (dpda's), after Korenjak and Hopcroft's branching algorithm. It is not only powerful enough to be applicable to two dpda's accepting by empty stack, one of which is real-time, but also simple even for dpda's in lower subclasses. This is the first time the branching algorithm has been used to give such a general decision procedure without ever “mixing” the two languages in question. In other words, it deals with only the equivalence equation whose left-hand side consists of a pure reachable configuration of one dpda and whose right-hand side that of the other.
A deterministic pushdown automaton (dpda) having just one stack symbol is called a deterministic restricted one-counter automaton (droca). A deterministic one-counter automaton (doca) is a dpda having only one stack symbol, with the exception of a bottom-of-stack marker. The class of languages accepted by droca's which accept by final state is a proper subclass of the class of languages accepted by doca's. Valiant has proved the decidability of the equivalence problem for doca's and the undecidability of the inclusion problem for doca's. Thus the decidability of the equivalence problem for droca's is obvious. In this paper, we evaluate the upper bound of the length of the shortest input string (shortest witness) that disproves the inclusion for a pair of real-time droca's which accept by accept mode, and present a direct branching algorithm for checking the inclusion for a pair of languages accepted by these droca's. Then we show that the worst-case time complexity of our algorithm is polynomial in the size of these droca's.
A new direct branching algorithm is presented for checking the equivalence of deterministic pushdown transducers, one of whose associated deterministic pushdown automata (DPDAs) is real-time strict. It is a straightforward extension of the corresponding algorithm for DPDAs (Tomita, 1982), and then proves the general applicability of our approach.