Second-Order Simple Grammars

Stirling, Colin

doi:10.1007/11817949_34

Cited by 4 publications

(3 citation statements)

References 26 publications

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“…The decidability question for higher-order DPDA equivalence remains an open problem; some progress in this direction was made by Stirling in [27]. Bisimilarity of second-order RT-PDA is undecidable [5].…”

Section: Related Work and Some Open Questionsmentioning

confidence: 99%

Equivalences of Pushdown Systems Are Hard

Jancar

2014

Lecture Notes in Computer Science

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Abstract. Language equivalence of deterministic pushdown automata (DPDA) was shown to be decidable by Sénizergues (1997Sénizergues ( , 2001Stirling (2002) then showed that the problem is primitive recursive.Sénizergues (1998, 2005) also generalized his proof to show decidability of bisimulation equivalence of (nondeterministic) PDA where ε-rules can be only deterministic and popping; this problem was shown to be nonelementary by Benedikt, Göller, Kiefer, and Murawski (2013), even for PDA with no ε-rules.Here we refine Stirling's analysis and show that DPDA equivalence is in TOWER, i.e., in the "least" nonelementary complexity class. The basic proof ideas remain the same but the presentation and the analysis are simplified, in particular by using a first-order term framework.The framework of (nondeterministic) first-order grammars, with term root-rewriting rules, is equivalent to the model of PDA with restricted ε-rules to which Sénizergues's decidability proof applies. We show that bisimulation equivalence is here Ackermann-hard, and thus not primitive recursive.

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Section: Related Work and Some Open Questionsmentioning

confidence: 99%

Equivalences of Pushdown Systems Are Hard

Jancar

2014

Lecture Notes in Computer Science

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“…For example it seems that we cannot avoid using several control states in the above undecidability proofs (though we can surely decrease their number by extending the stack alphabet). Hence (normed) second-order simple grammars, studied in [6], are a possible target for exploring.…”

Section: Additional Commentsmentioning

confidence: 99%

Note on Undecidability of Bisimilarity for Second-Order Pushdown Processes

Jančar,

Srba

2013

Preprint

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Broadbent and Göller (FSTTCS 2012) proved the undecidability of bisimulation equivalence for processes generated by ε-free second-order pushdown automata. We add a few remarks concerning the used proof technique, called Defender's forcing, and the related undecidability proof for first-order pushdown automata with ε-transitions (Jančar and Srba, JACM 2008).Language equivalence of pushdown automata (PDA) is a well-known problem in computer science community. There are standard textbook proofs showing the undecidability even for ε-free PDA, i.e. for PDA that have no ε-transitions; such PDA are sometimes called real-time PDA. The decidability question of language equivalence for deterministic PDA (DPDA) was a famous long-standing open problem. It was positively answered by Oyamaguchi [3] for ε-free DPDA and later by Sénizergues [4] for the whole class of DPDA.Besides their role of language acceptors, PDA can be also viewed as generators of (infinite) labelled transition systems; in this context it is natural to study another fundamental equivalence, namely bisimulation equivalence, also called bisimilarity. This equivalence is finer than language equivalence, but the two equivalences in principle coincide on deterministic systems.Sénizergues [5] showed an involved proof of the decidability of bisimilarity for (nondeterministic) ε-free PDA but also for PDA in which ε-transitions are deterministic, popping and do not collide with ordinary input a-transitions. It turned out that a small relaxation, allowing for nondeterministic popping ε-transitions, already leads to undecidability [2]. In [2], the authors of this note also explicitly describe a general proof technique called Defender's forcing. It is a simple, yet powerful, idea related to the bisimulation game played between Attacker and Defender; it was used, sometimes implicitly, also in context of other hardness results for bisimilarity on various classes of infinite state systems.The classical PDA, to which we have been referring so far, are the first-order PDA in the hierarchy of higher-order PDA that were introduced in connection with higher-order recursion schemes already in 1970s. The decidability question for equivalence of deterministic nth-order PDA, where n ≥ 2, seems to be open so far. A step towards a solution was made by Stirling [6] who showed the decidability for a subclass of ε-free second-order DPDA.Recently Broadbent and Göller [1] noted that the results in [2], or anywhere else in the literature, do not answer the decidability question for bisimilarity of ε-free second-order PDA.

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“…For second-order PDA the undecidability is established without ε-steps [21]. We can refer to the survey papers [22,23] for the work on higher-order PDA, and in particular mention that the decidability of equivalence of deterministic higher-order PDA remains open; some progress in this direction was made by Stirling in [24].…”

Section: Introductionmentioning

confidence: 99%

Equivalence of pushdown automata via first-order grammars

Jancar

2018

Preprint

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A decidability proof for bisimulation equivalence of first-order grammars is given. It is an alternative proof for a result by Sénizergues (1998Sénizergues ( , 2005 that subsumes his affirmative solution of the famous decidability question for deterministic pushdown automata. The presented proof is conceptually simpler, and a particular novelty is that it is not given as two semidecision procedures but it provides an explicit algorithm that might be amenable to a complexity analysis.

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Second-Order Simple Grammars

Cited by 4 publications

References 26 publications

Equivalences of Pushdown Systems Are Hard

Equivalences of Pushdown Systems Are Hard

Note on Undecidability of Bisimilarity for Second-Order Pushdown Processes

Equivalence of pushdown automata via first-order grammars

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