Sándor Vágvölgyi

Bottom-up tree pushdown automata: classification and connection with rewrite systems

We define different types of bottom-up tree pushdown automata and study their connections with rewrite systems. Along this line of research we complete and generalize the results of Gallier, Book and Salomaa. We define the notion of a tail-reduction-free (trf) rewrite system. Using the decidability of ground reducibility, we prove the decidability of the trf property. Monadic rewrite systems of Book, Gallier and Salomaa become a natural particular case of trf rewrite systems. We associate a deterministic bottom-up tree pushdown automaton with any left-linear trf rewrite system. Finally, we generalize monadic rewrite systems by introducing the notion of a semi-monadic rewrite system and show that, like a monadic rewrite system, it preserves recognizability.

Linear generalized semi-monadic rewrite systems effectively preserve recognizability

Abstract We introduce the notion of the generalized semi-monadic rewrite system, which is a generalization of well-known rewrite systems: the ground rewrite system, the monadic rewrite system, and the semi-monadic rewrite system. We show that linear generalized semi-monadic rewrite systems effectively preserve recognizability. We show that a tree language L is recognizable if and only if there exists a rewrite system R such that R ∪ R −1 is a linear generalized semi-monadic rewrite system and that L is the union of finitely many ↔ R ★ -classes. We show several decidability and undecidability results on rewrite systems effectively preserving recognizability and on generalized semi-monadic rewrite systems. For example, we show that for a rewrite system R effectively preserving recognizability, it is decidable if R is locally confluent. Moreover, we show that preserving recognizability and effectively preserving recognizability are modular properties of linear collapse-free rewrite systems. Finally, as a consequence of our results on trees we get that restricted right-left overlapping string rewrite systems effectively preserve recognizability.

Bottom-Up Tree Pushdown Automata and Rewrite Systems

Studying connections between term rewrite systems and bottom-up tree pushdown automata (tpda), we complete and generalize results of Gallier, Book and K. Salomaa. We define the notion of tail reduction free rewrite systems (trf rewrite systems). Using the decidability of inductive reducibility (Plaisted), we prove the decidability of the trf property. Monadic rewrite systems of Book, Gallier and K. Salomaa become an obvious particular case of trf rewrite systems. We define also semi-monadic rewrite systems which generalize monadic systems but keep their fair properties. We discuss different notions of bottom-up tree pushdown automata, that can be seen as the algorithmic aspect of classes of problems specified by trf rewrite systems. Especially, we associate a deterministic tpda with any left-linear trf rewrite system.

Variants of top-down tree transducers with look-ahead

In this article we consider deterministic and strongly deterministic top-down tree transducers with regular look-ahead, with regular check, with deterministic top-down look-ahead, and with deterministic top-down check. We compare the transformational power of these tree transducer classes by giving a correct inclusion diagram of the tree transformation classes induced by them. Along with the comparison we decompose some of the examined classes into simpler classes and we introduce the concept of the deterministic top-down tree automata with deterministic top-down look-ahead. We show that these recognizers recognize a tree language class which is strictly between the class of regular tree languages and the class of tree languages recognizable by deterministic top-down tree automata. We also study the closure properties of the examined tree transformation classes. We show that some classes are closed under composition while others, for example the class of tree transformations induced by deterministic top-down tree transducers with deterministic top-down look-ahead, are not.

Tree transducers with external functions

Abstract In this paper we investigate the computational power of particular tree transducers, viz., macro tree transducers and attributed tree transducers. The former tree transducers formalize the idea of syntax-directed translation, with the possibility of handling context; the latter tree transducers can serve as a formal model for the reduction semantics of attribute grammars. Here we generalize these tree transducers by allowing the invocation of external functions during the usual rewriting process. The main result of this paper is the characterization of macro tree transducers with external function calls in terms of attributed tree transducers with external function calls. Furthermore, such tree transducers with external function calls induce, in an obvious way, two operators on the set of all classes of tree functions. According to this point of view, we define two classes of tree functions inductively in the same way as the class PREC of primitive recursive tree functions, except that the closure under the scheme of primitive recursion is replaced by the closure under macro tree transducers with external function calls and attributed tree transducers with external function calls, respectively. As a second result of this paper we prove that these two classes are equal to PREC.

A fast algorithm for constructing a tree automaton recognizing a congruential tree language

Abstract It is well known that congruential tree languages are the same as recognizable tree languages. In this paper we construct, in O( n log n ) time, for a given ground term equation system E and given ground trees p 1 ,…, p k , a deterministic tree automaton A recognizing the congruential tree language [p1] ↔ ∗ E ∪…∪[pk] ↔ ∗ E , where n is the number of occurrences of symbols in E and p 1 ,…, p k .

Minimal equational representations of recognizable tree languages

Abstract. A tree language is congruential if it is the union of finitely many classes of a finitely generated congruence on the term algebra. It is well known that congruential tree languages are the same as recognizable tree languages. An equational representation is an ordered pair (E, P) , where E is either a ground term equation system or a ground term rewriting system, and P is a finite set of ground terms. We say that (E, P) represents the congruential tree language L which is the union of those ?*E-classes containing an element of P, i.e., for which L=⋃{[p]?*E∣p∈P}. We define two sorts of minimality for equational representations. We introduce the cardinality vector (∣E∣, ∣P∣) of an equational representation (E, P). Let ?l and ?a denote the lexicographic and antilexicographic orders on the set of ordered pairs of nonnegative integers, respectively. Let L be a congruential tree language. An equational representation (E, P) of L with ?l-minimal (?a-minimal) cardinality vector is called ?l-minimal (?a-minimal). We compute, for an L given by a deterministic bottom-up tree automaton, both a ?l-minimal and a ?a-minimal equational representation of L.

Top-down tree transducers with deterministic top-down look-ahead

We introduce a new type of tree transducers called strongly deterministic top-down tree transducer with deterministic top-down look-ahead

Congruential complements of ground term rewrite systems

Abstract Let A and B be ground term rewrite systems over some ranked alphabet Σ with ↔ A ∗ ⊆↔ B ∗ . We say that a ground term rewrite system C over Σ is a congruential complement of A for B , if ↔ A∪C ∗ =↔ B ∗ and ↔ A ∗ ∩↔ C ∗ is the identity relation over T Σ . We show that, given ground term rewrite systems A , B , C over some ranked alphabet Σ with ↔ A ∗ ⊆↔ B ∗ , one can effectively decide if C is a congruential complement of A for B .

On One-Pass Term Rewriting

Reducing a term with a term rewriting system (TRS) is a highly nondeterministic process and usually no bound for the lengths of the possible reduction sequences can be given in advance. Here we consider two very restrictive strategies of term rewriting, one-pass root-started rewriting and one-pass leaf-started rewriting. If the former strategy is followed, rewriting starts at the root of the given term t and proceeds continuously towards the leaves without ever rewriting any part of the current term which has been produced in a previous rewrite step. When no more rewriting is possible, a one-pass root-started normal form of the term t has been reached. The leaf-started version is similar, but the rewriting is initiated at the leaves and proceeds towards the root. The requirement that rewriting should always concern positions immediately adjacent to parts of the term rewritten in previous steps distinguishes our rewriting strategies from the IO and OI rewriting schemes considered in [5] or [2]. It also implies that the top-down and bottom-up cases are different even for a linear TRS. Let ~ = (E, R) be a TRS over a ranked alphabet E. For any E-tree language T, we denote the sets of one-pass root-started sententiM forms, one-pass root-started normal forms, one-pass leaf-started sentential forms and one-pass leaf-started normal forms of trees in T by lrSn(T), lrNT~(T), I~Sn(T) and I~Nn(T), respectively. We show that the following inclusion problems, where T~ = (E, R) is a left-linear TRS and T1 and T2 are two regular E-tree languages, are decidable. The one-pass root-started sentential form inclusion problem: lrSn(T1) c T2? The one-pass root-started normal form inclusion problem: lrNn(T1) c T2? The one-pass leaf-started sentential form inclusion problem: 1~ Sn(T1) c_ T2? The one-pass leaf-started normal form inclusion problem: 1iNn(T1) c T2? In [9] the inclusion problem for ordinary sentential forms is called the secondorder reachability problem and the problem is shown to be decidable for a TRS T~ which preserves recognizability, i.e. if the set of sentential forms of the trees of