1,637 research outputs found
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
Mixin Composition Synthesis based on Intersection Types
We present a method for synthesizing compositions of mixins using type
inhabitation in intersection types. First, recursively defined classes and
mixins, which are functions over classes, are expressed as terms in a lambda
calculus with records. Intersection types with records and record-merge are
used to assign meaningful types to these terms without resorting to recursive
types. Second, typed terms are translated to a repository of typed combinators.
We show a relation between record types with record-merge and intersection
types with constructors. This relation is used to prove soundness and partial
completeness of the translation with respect to mixin composition synthesis.
Furthermore, we demonstrate how a translated repository and goal type can be
used as input to an existing framework for composition synthesis in bounded
combinatory logic via type inhabitation. The computed result is a class typed
by the goal type and generated by a mixin composition applied to an existing
class
Changing a semantics: opportunism or courage?
The generalized models for higher-order logics introduced by Leon Henkin, and
their multiple offspring over the years, have become a standard tool in many
areas of logic. Even so, discussion has persisted about their technical status,
and perhaps even their conceptual legitimacy. This paper gives a systematic
view of generalized model techniques, discusses what they mean in mathematical
and philosophical terms, and presents a few technical themes and results about
their role in algebraic representation, calibrating provability, lowering
complexity, understanding fixed-point logics, and achieving set-theoretic
absoluteness. We also show how thinking about Henkin's approach to semantics of
logical systems in this generality can yield new results, dispelling the
impression of adhocness. This paper is dedicated to Leon Henkin, a deep
logician who has changed the way we all work, while also being an always open,
modest, and encouraging colleague and friend.Comment: 27 pages. To appear in: The life and work of Leon Henkin: Essays on
his contributions (Studies in Universal Logic) eds: Manzano, M., Sain, I. and
Alonso, E., 201
Street smarts
A pluralistic approach to folk psychology must countenance the evaluative, regulatory, predictive, and explanatory roles played by attributions of intelligence in social practices across cultures. Building off of the work of the psychologist Robert Sternberg and the philosophers Gilbert Ryle and Daniel Dennett, I argue that a relativistic interpretivism best accounts for the many varieties of intelligence that emerge from folk discourse. To be intelligent is to be comparatively good at solving intellectual problems that an interpreter deems worth solving
The weakness of being cohesive, thin or free in reverse mathematics
Informally, a mathematical statement is robust if its strength is left
unchanged under variations of the statement. In this paper, we investigate the
lack of robustness of Ramsey's theorem and its consequence under the frameworks
of reverse mathematics and computable reducibility. To this end, we study the
degrees of unsolvability of cohesive sets for different uniformly computable
sequence of sets and identify different layers of unsolvability. This analysis
enables us to answer some questions of Wang about how typical sets help
computing cohesive sets.
We also study the impact of the number of colors in the computable
reducibility between coloring statements. In particular, we strengthen the
proof by Dzhafarov that cohesiveness does not strongly reduce to stable
Ramsey's theorem for pairs, revealing the combinatorial nature of this
non-reducibility and prove that whenever is greater than , stable
Ramsey's theorem for -tuples and colors is not computably reducible to
Ramsey's theorem for -tuples and colors. In this sense, Ramsey's
theorem is not robust with respect to his number of colors over computable
reducibility. Finally, we separate the thin set and free set theorem from
Ramsey's theorem for pairs and identify an infinite decreasing hierarchy of
thin set theorems in reverse mathematics. This shows that in reverse
mathematics, the strength of Ramsey's theorem is very sensitive to the number
of colors in the output set. In particular, it enables us to answer several
related questions asked by Cholak, Giusto, Hirst and Jockusch.Comment: 31 page
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