5,391 research outputs found
Graph Interpolation Grammars: a Rule-based Approach to the Incremental Parsing of Natural Languages
Graph Interpolation Grammars are a declarative formalism with an operational
semantics. Their goal is to emulate salient features of the human parser, and
notably incrementality. The parsing process defined by GIGs incrementally
builds a syntactic representation of a sentence as each successive lexeme is
read. A GIG rule specifies a set of parse configurations that trigger its
application and an operation to perform on a matching configuration. Rules are
partly context-sensitive; furthermore, they are reversible, meaning that their
operations can be undone, which allows the parsing process to be
nondeterministic. These two factors confer enough expressive power to the
formalism for parsing natural languages.Comment: 41 pages, Postscript onl
The strength of countable saturation
We determine the proof-theoretic strength of the principle of countable
saturation in the context of the systems for nonstandard arithmetic introduced
in our earlier work.Comment: Corrected typos in Lemma 3.4 and the final paragraph of the
conclusio
Workshop on Verification and Theorem Proving for Continuous Systems (NetCA Workshop 2005)
Oxford, UK, 26 August 200
A non-standard analysis of a cultural icon: The case of Paul Halmos
We examine Paul Halmos' comments on category theory, Dedekind cuts, devil
worship, logic, and Robinson's infinitesimals. Halmos' scepticism about
category theory derives from his philosophical position of naive set-theoretic
realism. In the words of an MAA biography, Halmos thought that mathematics is
"certainty" and "architecture" yet 20th century logic teaches us is that
mathematics is full of uncertainty or more precisely incompleteness. If the
term architecture meant to imply that mathematics is one great solid castle,
then modern logic tends to teach us the opposite lession, namely that the
castle is floating in midair. Halmos' realism tends to color his judgment of
purely scientific aspects of logic and the way it is practiced and applied. He
often expressed distaste for nonstandard models, and made a sustained effort to
eliminate first-order logic, the logicians' concept of interpretation, and the
syntactic vs semantic distinction. He felt that these were vague, and sought to
replace them all by his polyadic algebra. Halmos claimed that Robinson's
framework is "unnecessary" but Henson and Keisler argue that Robinson's
framework allows one to dig deeper into set-theoretic resources than is common
in Archimedean mathematics. This can potentially prove theorems not accessible
by standard methods, undermining Halmos' criticisms.
Keywords: Archimedean axiom; bridge between discrete and continuous
mathematics; hyperreals; incomparable quantities; indispensability; infinity;
mathematical realism; Robinson.Comment: 15 pages, to appear in Logica Universali
- …