5,757 research outputs found
Abstract Interpretation of Indexed Grammars.
Indexed grammars are a generalization of context-free grammars and recognize a proper subset of context-sensitive languages. The class of languages recognized by indexed grammars are called indexed languages and they correspond to the languages recognized by nested stack automata. For example indexed grammars can recognize the language {a^n b^n c^n | n > = 1} which is not context-free, but they cannot recognize {(ab^n)^n) | n >= 1} which is context-sensitive. Indexed grammars identify a set of languages that are more expressive than context-free languages, while having decidability results that lie in between the ones of context-free and context-sensitive languages. In this work we study indexed grammars in order to formalize the relation between indexed languages and the other classes of languages in the Chomsky hierarchy. To this end, we provide a fixpoint characterization of the languages recognized by an indexed grammar and we study possible ways to abstract, in the abstract interpretation sense, these languages and their grammars into context-free and regular languages
Partial (In)Completeness in Abstract Interpretation
In the abstract interpretation framework, completeness represents an optimal simulation by the abstract operators over the behavior of the concrete operators. This corresponds to an ideal (often rare) feature where there is no loss of information accumulated in abstract computations with respect to the properties encoded by the underlying abstract domains. In this thesis, we deal with the opposite notion of completeness in abstract interpretation, that is, incompleteness, applied to two different contexts: static program analysis and formal languages over the Chomsky's hierarchy. In static program analysis, completeness is a very rare condition to be satisfied in practice and only the straightforward abstractions are complete for all programs, thus, we usually deal with incompleteness. For this reason, we introduce the notion of partial completeness. Partial completeness is a weaker notion of completeness which requires the imprecision of the analysis to be limited. A partially complete abstract interpretation allows some false alarms to be reported, but their number is bounded by a constant. We collect in partial completeness classes all the programs whose abstract interpretations share the same upper bound of imprecision. We then focus on the investigation of the computational limits of the class of partially complete programs with respect to a given abstract domain. Moreover, we show that the class of all partially complete programs is non-recursively enumerable, and its complement is productive whenever we allow an unlimited imprecision in the abstract domain. Finally, we formalize the local partial completeness class within which we require partial completeness only on some specific inputs. We prove that this last class of programs is a recursively enumerable set under a structural hypothesis on the underlying abstract domain, by showing an algorithm capable of proving the local partial completeness of a program with respect to a given abstract domain and an upper bound of imprecision. In formal language theory, we want to study a possible reformulation, by abstract interpretation, of classes of languages in the Chomsky's hierarchy, and, by exploiting the incompleteness of languages abstractions, we want to define separation results between classes of languages. To this end, we do a first step into this direction by studying the relation between indexed languages (recognized by indexed grammars) and context-free languages. Indexed grammars are a generalization of context-free grammars which recognize a proper subset of context-sensitive languages, the so called indexed languages. %The class of languages recognized by indexed grammars is called indexed languages and they correspond to the languages recognized by nested stack automata. For example, indexed grammars can recognize the language which is not context-free, but they cannot recognize which is context-sensitive. Indexed grammars identify a set of languages that are more expressive than context-free languages, while having decidability results that lie in between the ones of context-free and context-sensitive languages. We provide a fixpoint characterization of the languages recognized by an indexed grammar and we study possible ways to abstract, in the abstract interpretation sense, these languages and their grammars into context-free and regular languages. We formalize the separation class between indexed and context-free languages, i.e., all the languages that cannot be generated by a context-free grammar, as an instance of incompleteness of stack elimination abstraction over indexed grammars
Graph-Based Shape Analysis Beyond Context-Freeness
We develop a shape analysis for reasoning about relational properties of data
structures. Both the concrete and the abstract domain are represented by
hypergraphs. The analysis is parameterized by user-supplied indexed graph
grammars to guide concretization and abstraction. This novel extension of
context-free graph grammars is powerful enough to model complex data structures
such as balanced binary trees with parent pointers, while preserving most
desirable properties of context-free graph grammars. One strength of our
analysis is that no artifacts apart from grammars are required from the user;
it thus offers a high degree of automation. We implemented our analysis and
successfully applied it to various programs manipulating AVL trees,
(doubly-linked) lists, and combinations of both
An Alternative Conception of Tree-Adjoining Derivation
The precise formulation of derivation for tree-adjoining grammars has
important ramifications for a wide variety of uses of the formalism, from
syntactic analysis to semantic interpretation and statistical language
modeling. We argue that the definition of tree-adjoining derivation must be
reformulated in order to manifest the proper linguistic dependencies in
derivations. The particular proposal is both precisely characterizable through
a definition of TAG derivations as equivalence classes of ordered derivation
trees, and computationally operational, by virtue of a compilation to linear
indexed grammars together with an efficient algorithm for recognition and
parsing according to the compiled grammar.Comment: 33 page
Calibrating Generative Models: The Probabilistic Chomsky-SchĂĽtzenberger Hierarchy
A probabilistic Chomsky–Schützenberger hierarchy of grammars is introduced and studied, with the aim of understanding the expressive power of generative models. We offer characterizations of the distributions definable at each level of the hierarchy, including probabilistic regular, context-free, (linear) indexed, context-sensitive, and unrestricted grammars, each corresponding to familiar probabilistic machine classes. Special attention is given to distributions on (unary notations for) positive integers. Unlike in the classical case where the "semi-linear" languages all collapse into the regular languages, using analytic tools adapted from the classical setting we show there is no collapse in the probabilistic hierarchy: more distributions become definable at each level. We also address related issues such as closure under probabilistic conditioning
The Computational Complexity of Symbolic Dynamics at the Onset of Chaos
In a variety of studies of dynamical systems, the edge of order and chaos has
been singled out as a region of complexity. It was suggested by Wolfram, on the
basis of qualitative behaviour of cellular automata, that the computational
basis for modelling this region is the Universal Turing Machine. In this paper,
following a suggestion of Crutchfield, we try to show that the Turing machine
model may often be too powerful as a computational model to describe the
boundary of order and chaos. In particular we study the region of the first
accumulation of period doubling in unimodal and bimodal maps of the interval,
from the point of view of language theory. We show that in relation to the
``extended'' Chomsky hierarchy, the relevant computational model in the
unimodal case is the nested stack automaton or the related indexed languages,
while the bimodal case is modeled by the linear bounded automaton or the
related context-sensitive languages.Comment: 1 reference corrected, 1 reference added, minor changes in body of
manuscrip
CHR Grammars
A grammar formalism based upon CHR is proposed analogously to the way
Definite Clause Grammars are defined and implemented on top of Prolog. These
grammars execute as robust bottom-up parsers with an inherent treatment of
ambiguity and a high flexibility to model various linguistic phenomena. The
formalism extends previous logic programming based grammars with a form of
context-sensitive rules and the possibility to include extra-grammatical
hypotheses in both head and body of grammar rules. Among the applications are
straightforward implementations of Assumption Grammars and abduction under
integrity constraints for language analysis. CHR grammars appear as a powerful
tool for specification and implementation of language processors and may be
proposed as a new standard for bottom-up grammars in logic programming.
To appear in Theory and Practice of Logic Programming (TPLP), 2005Comment: 36 pp. To appear in TPLP, 200
On Descriptive Complexity, Language Complexity, and GB
We introduce , a monadic second-order language for reasoning about
trees which characterizes the strongly Context-Free Languages in the sense that
a set of finite trees is definable in iff it is (modulo a
projection) a Local Set---the set of derivation trees generated by a CFG. This
provides a flexible approach to establishing language-theoretic complexity
results for formalisms that are based on systems of well-formedness constraints
on trees. We demonstrate this technique by sketching two such results for
Government and Binding Theory. First, we show that {\em free-indexation\/}, the
mechanism assumed to mediate a variety of agreement and binding relationships
in GB, is not definable in and therefore not enforcible by CFGs.
Second, we show how, in spite of this limitation, a reasonably complete GB
account of English can be defined in . Consequently, the language
licensed by that account is strongly context-free. We illustrate some of the
issues involved in establishing this result by looking at the definition, in
, of chains. The limitations of this definition provide some insight
into the types of natural linguistic principles that correspond to higher
levels of language complexity. We close with some speculation on the possible
significance of these results for generative linguistics.Comment: To appear in Specifying Syntactic Structures, papers from the Logic,
Structures, and Syntax workshop, Amsterdam, Sept. 1994. LaTeX source with
nine included postscript figure
- …