On the emptiness problem of tree automata and completeness of modal logics of programs

Different systolic tree automata (STA) with base (T(b)−STA) are compared. This is a subclass of STA with interesting properties of modularity. We give a necessary and sufficient condition for the inclusion between classes of languages accepted by T(b)− STA, (L(T(b)−STA)), as b varies. We focus on T(b)−STA obtained by varying the base b in a natural way. We prove that for every base b within this framework there exists an a such that L(T(a)−STA) is not contained in L(T(b)−STA). We characterize the family of languages accepted by T(b)−STA when the input conditions are relaxed. Moreover we show that the emptiness problem is decidable for T(b)−STA.

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The Complexity of Tree Automata and Logics of Programs

SIAM Journal on Computing ◽

10.1137/s0097539793304741 ◽

1999 ◽

Vol 29 (1) ◽

pp. 132-158 ◽

Cited By ~ 61

Author(s):

E. Allen Emerson ◽

Charanjit S. Jutla

Keyword(s):

Tree Automata ◽

Logics Of Programs

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The emptiness problem for tree automata with at least one global disequality constraint is NP-hard

Information Processing Letters ◽

10.1016/j.ipl.2016.09.007 ◽

2017 ◽

Vol 118 ◽

pp. 6-9 ◽

Cited By ~ 1

Author(s):

P.-C. Héam ◽

V. Hugot ◽

O. Kouchnarenko

Keyword(s):

Np Hard ◽

Tree Automata ◽

Emptiness Problem

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Projection for Büchi Tree Automata with Constraints between Siblings

International Journal of Foundations of Computer Science ◽

10.1142/s012905412041004x ◽

2020 ◽

Vol 31 (06) ◽

pp. 749-775

Author(s):

Patrick Landwehr ◽

Christof Löding

Keyword(s):

Decision Procedure ◽

Order Logic ◽

Tree Automata ◽

Boolean Operations ◽

Regular Tree ◽

Infinite Trees ◽

Acceptance Condition ◽

Emptiness Problem ◽

Second Order Logic ◽

Monadic Second Order Logic

We consider an extension of tree automata on infinite trees that can use equality and disequality constraints between direct subtrees of a node. Recently, it has been shown that the emptiness problem for these kind of automata with a parity acceptance condition is decidable and that the corresponding class of languages is closed under Boolean operations. In this paper, we show that the class of languages recognizable by such tree automata with a Büchi acceptance condition is closed under projection. This construction yields a new algorithm for the emptiness problem, implies that a regular tree is accepted if the language is non-empty (for the Büchi condition), and can be used to obtain a decision procedure for an extension of monadic second-order logic with predicates for subtree comparisons.

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RELATING TREE SERIES TRANSDUCERS AND WEIGHTED TREE AUTOMATA

International Journal of Foundations of Computer Science ◽

10.1142/s012905410500325x ◽

2005 ◽

Vol 16 (04) ◽

pp. 723-741 ◽

Cited By ~ 11

Author(s):

ANDREAS MALETTI

Keyword(s):

Tree Automata ◽

Formal Representation ◽

Weighted Tree ◽

Bottom Up ◽

Operation Symbol ◽

Tree Series ◽

Pumping Lemma ◽

Emptiness Problem ◽

Weighted Tree Automata ◽

The Given

Bottom-up tree series transducers (tst) over the semiring [Formula: see text] are implemented with the help of bottom-up weighted tree automata (wta) over an extension of [Formula: see text]. Therefore bottom-up [Formula: see text]-weighted tree automata ([Formula: see text]-wta) with [Formula: see text] a distributive Ω-algebra are introduced. A [Formula: see text]-wta is essentially a wta but uses as transition weight an operation symbol of the Ω-algebra [Formula: see text] instead of a semiring element. The given tst is implemented with the help of a [Formula: see text]-wta, essentially showing that [Formula: see text]-wta are a joint generalization of tst (using IO-substitution) and wta. Then a semiring and a wta are constructed such that the wta computes a formal representation of the semantics of the [Formula: see text]-wta. The applicability of the obtained presentation result is demonstrated by deriving a pumping lemma for deterministic finite [Formula: see text]-wta from a known pumping lemma for deterministic finite wta. Finally, it is observed that the known decidability results for emptiness cannot be applied to obtain decidability of emptiness for finite [Formula: see text]-wta. Thus with help of a weaker version of the derived pumping lemma, decidability of the emptiness problem for finite [Formula: see text]-wta is shown under mild conditions on [Formula: see text].

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