1,173 research outputs found
Tarski's influence on computer science
The influence of Alfred Tarski on computer science was indirect but
significant in a number of directions and was in certain respects fundamental.
Here surveyed is the work of Tarski on the decision procedure for algebra and
geometry, the method of elimination of quantifiers, the semantics of formal
languages, modeltheoretic preservation theorems, and algebraic logic; various
connections of each with computer science are taken up
On the Polytope Escape Problem for Continuous Linear Dynamical Systems
The Polyhedral Escape Problem for continuous linear dynamical systems
consists of deciding, given an affine function and a convex polyhedron ,
whether, for some initial point in , the
trajectory of the unique solution to the differential equation
,
, is entirely contained in .
We show that this problem is decidable, by reducing it in polynomial time to
the decision version of linear programming with real algebraic coefficients,
thus placing it in , which lies between NP and PSPACE. Our
algorithm makes use of spectral techniques and relies among others on tools
from Diophantine approximation.Comment: Accepted to HSCC 201
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
Better Answers to Real Questions
We consider existential problems over the reals. Extended quantifier
elimination generalizes the concept of regular quantifier elimination by
providing in addition answers, which are descriptions of possible assignments
for the quantified variables. Implementations of extended quantifier
elimination via virtual substitution have been successfully applied to various
problems in science and engineering. So far, the answers produced by these
implementations included infinitesimal and infinite numbers, which are hard to
interpret in practice. We introduce here a post-processing procedure to
convert, for fixed parameters, all answers into standard real numbers. The
relevance of our procedure is demonstrated by application of our implementation
to various examples from the literature, where it significantly improves the
quality of the results
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