## The Descriptive Complexity of Generalized Local Sets (0)

Venue: | In Uwe Moennich and Hans-Peter Kolb, editors, The Mathematics of Syntactic Structure. Mouton /deGruyter |

Citations: | 6 - 1 self |

### BibTeX

@INPROCEEDINGS{Rogers_thedescriptive,

author = {James Rogers},

title = {The Descriptive Complexity of Generalized Local Sets},

booktitle = {In Uwe Moennich and Hans-Peter Kolb, editors, The Mathematics of Syntactic Structure. Mouton /deGruyter},

year = {},

pages = {21--40}

}

### Years of Citing Articles

### OpenURL

### Abstract

Context-free grammars and tree automata, because they are required to be finite, are limited to defining sets of trees in which the branching is bounded by a finite constant. As a result they cannot capture accounts of syntactic phenomena in which no such a priori bound exists---in flat accounts of coordination, for instance. This mismatch led Langendoen in 1976 and Gazdar, et al., in 1985 (GPSG) to propose varieties of two level grammars, in the one case infinite grammars that are themselves generated by other grammars, in the other grammars that permit regular expressions on the right-hand side of rewrite rules. In earlier work, we have characterized the local sets (the sets of trees generated by CFGs) and the recognizable sets (those accepted by tree automata) by definability in the logical language L 2 K;P . In defining such sets of trees in L 2 K;P , however, one must explicitly bound the branching. In this paper we explore the consequences of relaxing these bounds. We show, f...

### Citations

471 |
Generalized Phrase Structure Grammar
- Gazdar, Klein, et al.
- 1984
(Show Context)
Citation Context ... in which there is no finite bound on the branching are characteristic of some linguistic analyses, in particular flat analyses of coordination. Perhaps the best known presentation of this is in GPSG =-=[GKPS85]-=-. The relevant component consists of three (finite) rule schemas and a linear precedence rule: X \Gamma! H[CONJ a 0 ]; H[CONJ a i ] + ha 0 ; a 1 i 2 fhand ; NILi ; hNIL; andi ; hneither; nori ; hor; N... |

295 |
Decidability of second-order theories and automata on infinite trees
- Rabin
- 1969
(Show Context)
Citation Context ... which is, in essence, a characterization of the recognizable sets by definability in L 2 K;P . The result was originally established for SnS, the monadic second-order theory of n successor functions =-=[Rab69]-=-. As we will show shortly, L 2 K;P and SnS are equivalent in their descriptive power; we work in L 2 K;P because it has a relatively natural signature for reasoning about trees in the manner typical o... |

229 |
Weak second-order arithmetic and finite automata. Zeitschrift für mathematische Logik und Grundlagen der Mathematik
- Büchi
- 1960
(Show Context)
Citation Context ...n , the theory in which second-order quantification is restricted to finite sets. Finiteness is a definable property of subsets in SnS, therefore wSnS is a definable fragment of SnS. Theorem 2 (Buchi =-=[Buc60]-=-) S1S is decidable. Theorem 3 (Rabin [Rab69]) SnS is decidable for all ns!. Buchi's result is a special case of Rabin's. We will sketch Rabin's proof. Rabin first establishes this result for the n = 2... |

178 |
Tree acceptors and some of their applications
- Doner
- 1970
(Show Context)
Citation Context ...rongly context-free sets: Theorem 1 A set of finite trees with bounded branching is definable in L 2 K;P iff it is a projection of a local set. This is a variation of a result originally due to Doner =-=[Don70]-=- and, independently, Thatcher and Wright [TW68]. It characterizes the sets of derivation trees generated by context-free grammars (the local sets) in terms of their definability as sets of models for ... |

171 |
Generalized finite automata theory with an application to a decision problem of second-order logic
- Thatcher, Wright
- 1968
(Show Context)
Citation Context ...ite trees with bounded branching is definable in L 2 K;P iff it is a projection of a local set. This is a variation of a result originally due to Doner [Don70] and, independently, Thatcher and Wright =-=[TW68]-=-. It characterizes the sets of derivation trees generated by context-free grammars (the local sets) in terms of their definability as sets of models for a restricted logical language, in this case L 2... |

80 |
Characterizing derivation trees of context-free grammars through a generalization of finite automata theory
- Thatcher
- 1967
(Show Context)
Citation Context ...hT ; i is a tree and the range ofsis a set of ntuples. The i th -projection of T , for 0 ! isn, is hT ;si ffi i---T with nodes labeled with the i th component of their labels in T . Lemma 1 (Thatcher =-=[Tha67]-=-) Every recognizable set is the projection of a local set. Proof: The proof simply carries out the idea of the example. Suppose T = A(QS ) for some tree automaton A over \Sigma and Q and set of initia... |

55 | A Descriptive Approach to Language-Theoretic Complexity
- Rogers
- 1998
(Show Context)
Citation Context ... sets of finite trees, in S!S in which there is no finite bound on the branching factor; in fact, unless a formula explicitly includes such a bound, if it has models with n-ary 4 For full details see =-=[Rog96a]-=- 11 branching it will have models with branching factors greater than n as well. Such sets are not recognizable. Theorem 5 A set of finite trees with bounded branching is definable in L 2 K;P iff it i... |

29 |
Explicit definitions and linguistic dominoes
- Gorn
- 1967
(Show Context)
Citation Context ...a member of the set assigned to the variable X . We interpret these formulae over trees in the form of model-theoretic structuressT = hT ; P T ; D T ; L T ; I K ; I P i. The domain T is a tree domain =-=[Gor67]--- a-=- set of strings of natural numbers which are interpreted as node addresses: the root of the tree occurs at address " and the children of a node at addresssw are at addresses w0; w1; : : : ; wn in... |

17 |
Connectedness and Binary Branching. Foris
- Kayne
- 1984
(Show Context)
Citation Context ... are restricted to expanding non-terminals into finite strings of terminals and non-terminals such a bound must always exist. Furthermore, it has be argued (on learnability grounds, for instance, see =-=[Kay84]-=-) that such a bound should exist for human languages. Nonetheless, tree sets in which no finite bound on the branching exists are linguistically interesting figuring, for instance, in all flat account... |

13 | A model-theoretic framework for theories of syntax
- Rogers
- 1996
(Show Context)
Citation Context ...ual formal consequences. Distinctions that have been previously overlooked, on the other hand, in particular differences in the nature of linguistic universals, appear to be formally more significant =-=[Rog96b]-=-. As this model-theoretic approach is extended to more complex classes of sets of structures and is applied to a broader range of theories it promises to offer considerable insight into the regulariti... |

12 | Grammarless" phrase structure grammar
- Rogers
- 1997
(Show Context)
Citation Context ...nt notions in the realm of tree automata as generalized tree automata and generalized recognizable sets. We are interested, in particular, 5 For details of this and other aspects of the treatment see =-=[Rog97a]-=- 14 in the variants in which the generated grammar (or automaton) is a regular set. This gives a slight generalization of the GPSG style grammars, in that, rather than restricting application of the K... |

3 |
On the weak generative capacity of infinite grammars
- Langendoen
- 1976
(Show Context)
Citation Context ...infinite set of rewrite rules, but that set itself is generated by a finite grammar, in this case a regular grammar. Such a notion of grammar is not new. Langendoen refers to them as hypergrammars in =-=[Lan76]-=-, where he motivates them with a flat account of coordination quite similar to the GPSG account. A class of hypergrammars is determined by both the class of the generated grammar and the class of the ... |

3 | Strict LT2 : regular :: local : recognizable
- Rogers
- 1997
(Show Context)
Citation Context ...tic significance. 2 The effect of the projection is to permit the refinement of categories on the basis of local context that is typical of GPSG and HPSG (as in slashed categories). Exploring this in =-=[Rog97b]-=- we establish some formal consequences of the question of whether to require all categories to be empirically justified or whether to admit their arbitrary refinement in this way. In this paper we tur... |