Recognizable Tree Languages Research Articles

Finite-image property of weighted tree automata over past-finite monotonic strong bimonoids

We consider weighted tree automata over strong bimonoids (for short: wta). A wta A has the finite-image property if its recognized weighted tree language 〚A〛 has finite image; moreover, A has the preimage property if the preimage under 〚A〛 of each element of the underlying strong bimonoid is a recognizable tree language. For each wta A over a past-finite monotonic strong bimonoid we prove the following results. In terms of A's structural properties, we characterize whether it has the finite-image property. We characterize those past-finite monotonic strong bimonoids such that for each wta A it is decidable whether A has the finite-image property. In particular, the finite-image property is decidable for wta over past-finite monotonic semirings. Moreover, we prove that A has the preimage property. All our results also hold for weighted string automata.

Some decidability results on one-pass reductions

We show second-order decidability results on recognizable tree languages and one-pass reduction sequences with special term rewriting systems. For some classes of term rewriting systems the second-order IO one-pass inclusion problem, the second-order IO one-pass reachability problem, and the IO and OI one-pass common ancestor problems are decidable.

TREE AUTOMATA BASED ON COMPLETE RESIDUATED LATTICE-VALUED LOGIC: REDUCTION ALGORITHM AND DECISION PROBLEMS

In this paper, at first we define the concepts of response function and accessible states of a complete residuated lattice-valued (for simplicity we write $mathcal{L}$-valued) tree automaton with a threshold $c.$ Then, related to these concepts, we prove some lemmas and theorems that are applied in considering some decision problems such as finiteness-value and emptiness-value of recognizable tree languages. Moreover, we propose a reduction algorithm for $mathcal{L}$-valued tree automata with a threshold $c.$ The goal of reducing an $ mathcal{L}$-valued tree automaton is to obtain an $mathcal{L}$-valued tree automaton with reduced number of states%that all of its states are accessible all of which are accessible, in addition it recognizes the same language as the first one given. We compare our algorithm with some other algorithms in the literature. Finally, utilizing the obtained results, we consider some fundamental decision problems for $mathcal{L}$-valued tree automata including the membership-value, the emptiness-value, the finiteness-value, the intersection-value and the equivalence-value problems.

Descendants of a recognizable tree language for prefix constrained linear monadic term rewriting with position cutting strategy

For a tree language L, a finite set Z of regular Σ-path languages, and a set S of Z-prefix constrained linear monadic term rewriting rules over Σ, the position cutting descendant of L for S is the set S↑⁎(L) of trees reachable from a tree in L by rewriting in S by position cutting strategy. If L is recognizable, then S↑⁎(L) is recognizable as well. Moreover, if S is finite, then we can construct a tree automaton recognizing S↑⁎(L). For a recognizable tree language L and a finite set Z of regular Σ-path languages, we study the set DZ,↑(L) of position cutting descendants of L for all sets of Z-prefix constrained linear monadic term rewriting rules over Σ. We show that DZ,↑(L) is finite, and that if L is given by a tree automaton A and each element of Z is given by an automaton, then we can construct a set {R1,…,Rk} of Z-prefix constrained linear monadic term rewriting systems over Σ such that DZ,↑(L)={R1⁎↑(L),…,Rk⁎↑(L)}.

One-Pass Reductions

We study OI and IO one-pass reduction sequences with term rewritesystems. We present second order decidability and undecidabilityresults on recognizable tree languages and one-pass reductions. Forleft-linear TRSs, the second order OI inclusion problem and the secondorder OI reachability problem are decidable, the second order OIjoinability problem is undecidable. For right-linear TRSs, the secondorder common IO ancestor problem is undecidable.

Characterizations of recognizable weighted tree languages by logic and bimorphisms

We give a general definition of weighted tree automata (wta) and define three instances which differ in the underlying weight algebras: semirings, multi-operator monoids, and tree-valuation monoids. Also, we define a general concept of weighted expressions based on monadic second-order logics. In the same way as for wta, we define three instances corresponding to the above-mentioned weight algebras. We prove that wta over semirings are equivalent to weighted expressions over semirings, and prove the same equivalence over tree-valuation monoids. For wta over semirings and for wta over tree-valuation monoids we prove characterizations in terms of bimorphisms.

An upper bound on the complexity of recognizable tree languages

The third author noticed in his 1992 Ph.D. thesis [P. Simonnet, Automates et théorie descriptive (1992).] that every regular tree language of infinite trees is in a class for some natural number n ≥ 1, where is the game quantifier. We first give a detailed exposition of this result. Next, using an embedding of the Wadge hierarchy of non self-dual Borel subsets of the Cantor space 2ω into the class ∆21, and the notions of Wadge degree and Veblen function, we argue that this upper bound on the topological complexity of regular tree languages is much better than the usual ∆21.

Forward and backward application of symbolic tree transducers

We consider symbolic tree automata (sta) and symbolic regular tree grammars and their corresponding classes of tree languages: s-recognizable tree languages and s-regular tree languages. We prove that the following three classes are equal: the class of s-recognizable tree languages, the class of s-regular tree languages, and the class of images of classical recognizable tree languages under tree relabelings. Moreover, the sta and the recently introduced variable tree automata are incomparable with respect to their recognition power. Also, we consider symbolic tree transducers (stt) and prove the following theorems. The syntactic composition of two stt computes the composition of the tree transformations computed by each stt, provided that (1) the first one is deterministic or the second one is linear and (2) the first one is total or the second one is nondeleting. Backward application of an stt to any s-recognizable tree language yields an s-recognizable tree language. There is a linear stt of which the range is not an s-recognizable tree language. Forward application of simple and linear stt preserves s-recognizability. A restricted version of both the type checking problem of simple and linear stt, and the inverse type checking problem of arbitrary stt is decidable. Since we deal with trees over infinite alphabets, we require appropriate conditions on sta and stt such that all the proofs are constructive.

Tree shuffle

We introduce the mix operation for hedges and the shuffle operation for unranked trees. We show that the shuffle of recognizable tree languages is also a recognizable tree language.

Construction of tree automata from regular expressions

Since recognizable tree languages are closed under the rational operations, every regular tree expression denotes a recognizable tree language. We provide an alternative proof to this fact that results in smaller tree automata. To this aim, we transfer Antimirov's partial derivatives from regular word expressions to regular tree expressions. For an analysis of the size of the resulting automaton as well as for algorithmic improvements, we also transfer the methods of Champarnaud and Ziadi from words to trees.

Classes of Tree Languages and DR Tree Languages Given by Classes of Semigroups

In the first section of the paper we give general conditions under which a class of recognizable tree languages with a given property can be defined by a class of monoids or semigroups defining the class of string languages having the same property. In the second part similar questions are studied for classes of (DR) tree languages recognized by deterministic root-to-frontier tree recognizers.

DECOMPOSITION OF WEIGHTED MULTIOPERATOR TREE AUTOMATA

Weighted multioperator tree automata (for short: wmta) are finite-state bottom-up tree automata in which the transitions are weighted with an operation taken from some multioperator monoid. A wmta recognizes a tree series which is a mapping from the set of trees to some commutative monoid. We prove that every wmta recognizable tree series can be decomposed into a relabeling tree transformation, a recognizable tree language, and a tree series computed by a homomorphism wmta; vice versa, the composition of an arbitrary relabeling tree transformation, a recognizable tree language, and a tree series computed by a homomorphism wmta yields a wmta recognizable tree series. We use this characterization result for specific multioperator monoids and prove (1) a new decomposition of polynomial bottom-up tree series transducers over semirings and (2) a new characterization of tree series which are recognizable by weighted tree automata over semirings, in terms of projections of local tree languages.

On Recognizable Tree Languages Beyond the Borel Hierarchy

We investigate the topological complexity of non Borel recognizable tree languages with regard to the difference hierarchy of analytic sets. We show that, for each integer n g 1, there is a D$_{ω^n}$(Σ$^1_1$)-complete tree language L$_n$ accepted by a (non deterministic) Muller tree automaton. On the other hand, we prove that a tree language accepted by an unambiguous Buchi tree automaton must be Borel. Then we consider the game tree languages W$_{(i,κ)}$, for Mostowski-Rabin indices (iκ). We prove that the D$_{ω^n}$(Σ$^1_1$)-complete tree languages L$_n$ are Wadge reducible to the game tree language W$_{(i,κ)}$ for κ−ig 2. In particular these languages W$_{(i,κ)}$ are not in any class D$_{α}$(Σ$^1_1$) for α<ω$^{ω}$.

Murg term rewrite systems

The union of a monadic and a right-ground term rewrite system is called a murg term rewrite system. We show that for murg TRSs the ground common ancestor problem is undecidable. We show that for a murg term rewrite system it is undecidable whether the set of descendants of a ground tree is a recognizable tree language. We show that it is undecidable whether a murg term rewrite system over Σ preserves Σ-recognizability.

A note on cut-worthiness of recognizable tree series

The topic of this short note is tree series over semirings with partially ordered carrier set. Thus, tree series become poset-valued fuzzy sets. Starting with a collection of recognizable tree languages we construct a tree series recognizable over any semiring, such that the carrier set is a poset generated by the collection. In the framework of fuzzy structures, the starting collection becomes a subset of the collection of the corresponding cut sets. Under some stricter conditions, it is even equal to this collection. We also partially solve an open problem which was posed in Borchardt et al. [Cut sets as recognizable tree languages, Fuzzy Sets and Systems 157 (2006) 1560–1571]. Namely, we show that if ϕ is a given tree series over a partially ordered, locally finite semiring A , then ϕ is recognizable if and only if there are finitely many cut sets of ϕ and every cut set is a recognizable tree language.

CLASSES OF TREE LANGUAGES DETERMINED BY CLASSES OF MONOIDS

In this paper finite state recognizers are considered as unary tree recognizers with unary operational symbols. We introduce translation recognizers of a tree recognizer, which are finite state recognizers whose operations are the elementary translations of the underlying algebra of the considered tree recognizer. In terms of translation recognizers we give general conditions under which a class of recognizable tree languages with a given property can be determined by a class of monoids determining the class of string languages having the same property.

Fuzzy tree automata

In this paper, we map an arbitrary algebra to a completely distributive lattice and define algebras of fuzzy sets. When the algebra is a term algebra, we obtain fuzzy tree languages. After defining fuzzy equational sets, fuzzy rational sets and fuzzy recognizable tree languages, we derive a “Kleene theorem” for fuzzy tree languages.

Descendants of a recognizable tree language for sets of linear monadic term rewrite rules

For a tree language L and a set S of term rewrite rules over Σ, the descendant of L for S is the set S ∗ ( L ) of trees reachable from a tree in L by rewriting in S. For a recognizable tree language L, we study the set D ( L ) of descendants of L for all sets of linear monadic term rewrite rules over Σ. We show that D ( L ) is finite. For each tree automaton A over Σ, we can effectively construct a set { R 1 , … , R k } of linear monadic term rewrite systems over Σ such that D ( L ( A ) ) = { R 1 ∗ ( L ( A ) ) , … , R k ∗ ( L ( A ) ) } and for any 1 ⩽ i < j ⩽ k , R i ∗ ( L ( A ) ) ≠ R j ∗ ( L ( A ) ) .

Cut sets as recognizable tree languages

A tree series over a semiring with partially ordered carrier set can be considered as a fuzzy set. We investigate conditions under which it can also be understood as a fuzzified recognizable tree language. In this sense, sufficient conditions are presented which, when imposed, ensure that every cut set, i.e., the pre-image of a prime filter of the carrier set, is a recognizable tree language. Moreover, such conditions are also presented for cut sets of recognizable tree series.

Algebraic recognizability of regular tree languages

We propose a new algebraic framework to discuss and classify recognizable tree languages, and to characterize interesting classes of such languages. Our algebraic tool, called preclones, encompasses the classical notion of syntactic Σ -algebra or minimal tree automaton, but adds new expressivity to it. The main result in this paper is a variety theorem à la Eilenberg, but we also discuss important examples of logically defined classes of recognizable tree languages, whose characterization and decidability was established in recent papers (by Benedikt and Ségoufin, and by Bojańczyk and Walukiewicz) and can be naturally formulated in terms of pseudovarieties of preclones. Finally, this paper constitutes the foundation for another paper by the same authors, where first-order definable tree languages receive an algebraic characterization.