Deterministic top-down tree automata with Boolean deterministic look-ahead
Introduction
Several variants of deterministic top-down tree automata models have been introduced, Martens et al. [1] presented an overview of top-down determinism in ranked and unranked tree automata, and explored several connections between them, and outlined some new research directions in Section “Conclusions and Discussion” as well. For more research on deterministic top-down unranked tree automata, see [2].
Deterministic top-down tree automata recognize the class of deterministic top-down tree languages (DTR for short), which is a proper subclass of the regular tree languages (REG for short) [3]. Deterministic top-down tree automata are less powerful than deterministic bottom-up tree automata, because in a deterministic top-down tree automaton no two rules have the same left-hand side, and the choice of the state assigned to each child is determined at the parent node without previously examining the subtree at the child. That is, a deterministic top-down tree automaton assigns states to the children of the current node depending solely on the current node's label and the state assigned to the current node.
In order to examine the subtrees at the children before assigning states to them, Fülöp and Vágvölgyi [4] took over the concept of look-ahead from [5] and introduced the notion of a deterministic top-down tree automaton with deterministic top-down look-ahead. In this model, deterministic top-down tree automata inspect the subtrees at the children, and then the automaton can select the rule to be applied at their parent in the processing of the input tree. Fülöp and Vágvölgyi [4] equipped the rules with look-ahead deterministic top-down tree languages. A rule can be applied at a node of a tree if and only if the direct subtrees of that node are in the tree languages given in the rule. For any two different rules with the same left-hand side there is a subtree for which the corresponding look-ahead tree languages are disjoint, hence at each node the automaton can apply at most one rule. This ensures determinism. The class of tree languages recognized by deterministic top-down tree automata with deterministic top-down look-ahead is called the class of deterministic top-down tree languages with deterministic top-down look-ahead, and is denoted by . Fülöp and Vágvölgyi [4] proved that .
A sub-regular language class is a proper subclass of the regular languages described without employing the full power of automata. For a language L in a given sub-regular language class, the method that distinguishes the strings that are in L from those that are not is simpler than an automaton. Rogers and his colleagues [6], [7], [8], and Truthe [9] presented numerous hierarchies of sub-regular language classes.
We call proper subclasses of the regular tree languages a sub-regular tree language class. Some researchers presented the following three hierarchies of sub-regular tree languages. (a) The Boolean closure of DTR is called the class of Boolean deterministic tree languages, and is denoted by BD. It is well known that , and Thomas [10] showed that BD is a proper subclass of the chain definable tree languages, denoted by CD, which form a proper subclass of REG. Thus, . (b) Jurvanen [11] also proved the proper inclusion of , but directly using only the pigeon hole principle. With respect to the inclusion relation, Jurvanen [11] compared BD, REG, the class of definite tree languages, the class of reverse definite tree languages, the class of generalized definite tree languages, the class of local tree languages, the class of trivial tree languages, and the Boolean closure of the class of finite tree languages. She presented an inclusion diagram for the above eight tree language classes. (c) Fülöp and Vágvölgyi [12] iterated the look-ahead tree languages as follows. Let and let, for each , be the class of tree languages recognizable by deterministic top-down tree automata with look-ahead. (So, .) They [12] showed that Hence, by iterating the look-ahead tree languages for deterministic top-down tree automaton, we obtain more and more powerful recognizing devices, and the resulting hierarchy does not exhaust REG.
Similarly to the definition of a deterministic top-down tree automata with deterministic top-down look-ahead, we introduce the concept of a deterministic top-down tree automaton with Boolean deterministic look-ahead. The class of tree languages recognized by deterministic top-down tree automata with Boolean deterministic look-ahead is called the class of deterministic top-down tree languages with Boolean deterministic look-ahead, and is denoted by . We continue the research mentioned above, and show the following statements:
- 1.
BD and are incomparable under the inclusion relation.
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.
- 3.
.
- 4.
Fig. 1 shows an inclusion diagram for the tree language classes DTR, BD, , , , and REG. We establish a four level hierarchy of sub-regular tree language classes, which is a contribution to the theory of tree languages.
- 5.
Both and are closed under binary intersection.
In Section 2 we recall some well-known concepts concerning deterministic top-down tree automata. In Section 3 we describe the Boolean closure BC of finitely many sets . We show that every can be expressed in the form where for all , , or . Moreover, the operands , , of the set union above are pairwise disjoint. In Section 4 we show that . In Section 5 we show that BD and are incomparable under the inclusion relation. In Section 6 we sum up our results presenting an inclusion diagram for the tree language classes DTR, BD, , , , and REG. In Section 7 we show that both and are closed under binary intersection. In Section 8 we overview our results and raise some open problems.
Section snippets
Preliminaries
In this section we review the notions, notations, and preliminary results used in the paper.
Sets and relations. stands for the set of all nonnegative integers. Let A be a set. Then denotes the powerset of A. The complement of A, denoted by , is the set of elements not in A within a larger set that is defined beforehand. The difference of set B from A, denoted by , is the set of all the elements of A that are not in set B. The cardinality of A is denoted by . denotes the free
Boolean sets
For the sake of completeness, we now cite a well-known result. Proposition 3.1 [15] Let be a family of subsets of a given set U. A subset T of U is in the Boolean closure of if and only if there exist , such that T can be expressed in the form where for all , , or .
Boolean deterministic look-ahead
In this section we show that . To this end, we introduce some concepts and show four preparatory results.
In a nutshell, we reason as follows. For a dta and a tree , we map a bit to each direct subtree of t. Here 0 shows that gets stuck in the subtree, and 1 shows that A successfully reads the subtree. If , then at least one bit is 0. To every tree t, we assign the resulting bit array. Generalizing this calculation, to any tree t in the intersection of the complements of n
BD and are incomparable
In this section we show that BD and are incomparable under the inclusion relation. We use the results of Jurvanen [11] when giving a tree language in . To this end, we now adopt some notions and notations from [11]. We consider the ranked alphabet , where , . Let be the set of all trees over Σ with root symbol σ. A balanced tree is a left £$-tree, if
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,
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, , and
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does not appear in t to the right
Deterministic top-down tree languages with Boolean deterministic look-ahead
In this section we show that the union of BD and does not exhaust , i.e., , and that . Then, we sum up our results presenting in Fig. 1, the inclusion diagram for the tree language classes DTR, BD, , , , and REG. Theorem 6.1 . Proof By Consequence 3.9 and Theorem 4.6, . We now show that this inclusion is proper. Let , , and . We defined the set J right before Lemma 5.2, and we defined the sets K and M in the
Closure under binary intersection
We show that both and are closed under binary intersection. Theorem 7.1 For every ranked alphabet Σ, is closed under binary intersection. Proof Let and be s. Consider the , where We now show that is deterministic,
Conclusion and open problems
We introduced the concept of a deterministic top-down tree automaton with Boolean deterministic look-ahead. In Fig. 1 we presented an inclusion diagram for the tree language classes DTR, BD, , , , and REG. We established a four level hierarchy of sub-regular tree language classes. We showed that and are closed under binary intersection.
We raise the following questions: Is it decidable for a given tree language (, , respectively) whether
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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