M. Mueller and J. Niehren. Ordering constraints over feature trees expressed in second-order monadic logic. Information and Computation, 159(1/2):22--58, 2000.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... from the need to simultaneously represent 1) selector operations on trees (which require operations that manipulate the initial segments of paths in a tree) and the prefix closure property of the tree domain (which requires operations that manipulate the terminal segment of paths in a tree) see [31], 25, Section 7] Preliminaries. If A is a set, write A to denote the cardinality of A. An L structure (model) is a set together with functions and relations interpreting the language L. If S is an L structure and r L a function or relation symbol, write ar(r) to denote the arity of r. ....

M. Mueller and J. Niehren. Ordering constraints over feature trees expressed in second-order monadic logic. Information and Computation, 159(1/2):22--58, 2000.

.... can assume that no feature tree in the image of a contains features d and r and labels and unit (if a does not satisfy this condition we can always rename the features and labels in the image of a to another EF solution which does, because we have assumed infinitely many features and labels; see [21] for a detailed argument to this end) Given a feature tree t, we define t as the feature tree where The tree domain D t is the smallest prefix closed set of path containing f f 1 r f n 1 r f n r; f 1 r f n 1 r f n d j f 1 f n 2 D t g. The labeling function L t ....

M. Muller and J. Niehren. Ordering constraints over feature trees expressed in second-order monadic logic. In T. Nipkow, ed., International Conference on Rewriting Techniques and Applications, vol. 1379 of Lecture Notes in Computer Science, pp. 196--210. Springer-Verlag, Berlin, 1998. Full version to appear in special issue of Information and Computation on RTA'98.

....(relational calculi) over collections of unranked trees. Unranked trees differ from ranked trees in that there is no restriction on the arity of nodes. Although unranked trees have been considered in the 60s and 70s [28, 34] and are related to feature trees over an infinite set of features [22, 23], it was the advent of XML that initiated their systematic study [8] XML is a popular data format which is becoming the lingua franca for information exchange on the world wide web [37] and XML data is naturally modeled as unranked trees [25, 37] This connection made recent advances in unranked ....

....that conform to d. In the first part of the paper, we consider definability over automatic structures of unranked trees. We construct a structure T which turns out to be the universal one. Like the corresponding structure for ranked trees, T is based on the extension relation among trees [5, 22, 23]; however, in the unranked case we split it into two relations: extend a tree by adding siblings) and ( extend a tree by adding descendants) We also consider a weaker structure, p , that still defines all regular languages, but a smaller class of relations. We then look at restricted ....

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M. Muller, J. Niehren. Ordering constraints over feature trees expressed in second-order monadic logic. In RTA'98, pages 196--210.

....in term algebra. Let join a (T 1 ; T 2 ) be the binary tree whose root is labeled a, and whose left and right subtrees are T 1 and T 2 respectively. The structure hTrees n ( join a ) a2 ; a ) a2 i corresponds to the first order theory of FT constraints over feature trees studied in [23, 25]. Since that theory is known to be undecidable [25] we obtain: Proposition 2 join a is not definable in T. 2 One can also give a direct proof, showing that with join a one can define predicates not recognizable by tree automata. Thus, first order logic over T p and T is incomparable with ....

M. Muller, J. Niehren. Ordering constraints over feature trees expressed in second-order monadic logic. In RTA'98, pages 196--210.

....with existential quantification and gives the lower bound complexity results. Section 4 defines secondorder monadic logic and gives our reduction of entailment in FT to validity in SwS resp. WSwS. Section 5 contains the correctness proof of our reduction. Section 6 summarizes. The full paper [15] extends the conference version with two appendices that contain all omitted proofs. 2 Syntax and Semantics of FT The constraint system FT is defined by a set of constraints together with an interpretation over feature trees. We assume an infinite set of variables ranged over by x;y;z, a ....

M. Muller and J. Niehren. Ordering Constraints over Feature Trees Expressed in Second-order Monadic Logic. Full version available at http://www.ps.unisb. de/niehren/papers/EntailFTSub.ps.gz. 1997.

....this means that every word of features in the tree domain of t 1 belongs to the tree domain of t 2 , and that the (partial) labeling function of t 1 is contained in the labeling function of t 2 . In this case we write t 1 t 2 . We consider the system FT of ordering constraints over feature trees [15, 17, 18]. Its constraints j are given by the following abstract syntax: j : xx j x[ f ]x j a(x) j jj The constraints of FT are interpreted in the structure of feature trees with the weak subsumption ordering. We distinguish two cases, the structure of finite feature trees and the structure of ....

....constraint system FT , which provides for equality constraints x=y rather than more general ordering constraints xy. The full first order theory of FT is decidable [3] and has non elementary complexity [34] The decidability question for the first order theory of FT has been raised in [17]. There, two indications in favour of decidability have been formulated: its analogy to FT and its relationship to second order monadic logic (we will discuss this below) In contrast, we show in this paper that the the first order theory of FT is undecidable. Our result holds in the structure of ....

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M. Muller, J. Niehren. Ordering constraints over feature trees expressed in second-order monadic logic. In T. Nipkow (ed.), Internat. Conf. on Rewriting Techniques and Applications, no. 1379 in LNCS, 196--210, Tsukuba, Japan, 1998.

....subsumes a tree t 2 , written t 1 t 2 , if every word of features in the tree domain of t 1 belongs to the tree domain of t 2 and the (partial) labeling function of t 1 is contained in the labeling function of t 2 . We consider the system FT of ordering constraints over feature trees [18, 19, 17]. Its constraints j are given by the following abstract syntax j : x x 0 j x[ f ]x 0 j a(x) j j j 0 where f denotes a feature symbol and a a label symbol. The constraints of FT are interpreted in the structure of feature trees with the weak subsumption ordering. We distinguish two cases, ....

....be seen as a sub system of FT since x = y can be expressed as x y y x thanks to anti symmetry of the weak subsumption order. The full first order theory of FT is decidable [4] and has non elementary complexity [37] The decidability question for the first order theory of FT has been raised in [17]. There, two indications in favour of decidability have been formulated: its analogy to FT and its relationship to second order monadic logic. However, we show in this paper that the the first order theory of FT is undecidable. Our result holds in the structure of possibly infinite feature trees ....

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M. Mller and J. Niehren. Ordering constraints over feature trees expressed in secondorder monadic logic. Information and Computation, 159, May 5, 2000. Special Issue on RTA'98.

....[6] A feature tree t 1 is smaller than another feature tree t 2 if t 1 has fewer edges and node labels than t 2 . In this case we wine red color wine red 1998 color year write t 1 t 2 . An example is given in the picture. We consider the system FT of ordering constraints over feature trees [16, 14, 15]. Its constraints j are given by the following abstract syntax: j : xx 0 j x[ f ]x 0 j a(x) j jj 0 The constraints of FT are interpreted in the structure of feature trees with the weak subsumption ordering. We distinguish two cases, the structure of finite feature trees and the structure ....

....decidable in cubic time. The next step towards larger fragments of the theory of FT was to consider the judgments with existential quantification j j=9x 1 : 9x n j 0 which are equivalent to unsatisfiability judgments j :9x 1 : 9x n j 0 with quantification below negation. As shown in [15], this problem is decidable, coNP hard in case of finite trees, and PSPACE hard in case of arbitrary trees. Decidability is proved by reduction to the entailment problem with existential quantifiers in the related structure of so called sufficiently labeled feature trees. Since the full ....

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M. Muller, J. Niehren. Ordering constraints over feature trees expressed in second-order monadic logic. In T. Nipkow (ed.), Internat. Conf. on Rewriting Techniques and Applications, Tsukuba, Japan, 1998. http://www.ps.uni-sb.de/Papers/ abstracts/SWS97.html.

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M. Mueller and J. Niehren. Ordering constraints over feature trees expressed in second-order monadic logic. Information and Computation, 159(1/2):22--58, 2000.

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