4,533 research outputs found
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
Completeness of Flat Coalgebraic Fixpoint Logics
Modal fixpoint logics traditionally play a central role in computer science,
in particular in artificial intelligence and concurrency. The mu-calculus and
its relatives are among the most expressive logics of this type. However,
popular fixpoint logics tend to trade expressivity for simplicity and
readability, and in fact often live within the single variable fragment of the
mu-calculus. The family of such flat fixpoint logics includes, e.g., LTL, CTL,
and the logic of common knowledge. Extending this notion to the generic
semantic framework of coalgebraic logic enables covering a wide range of logics
beyond the standard mu-calculus including, e.g., flat fragments of the graded
mu-calculus and the alternating-time mu-calculus (such as alternating-time
temporal logic ATL), as well as probabilistic and monotone fixpoint logics. We
give a generic proof of completeness of the Kozen-Park axiomatization for such
flat coalgebraic fixpoint logics.Comment: Short version appeared in Proc. 21st International Conference on
Concurrency Theory, CONCUR 2010, Vol. 6269 of Lecture Notes in Computer
Science, Springer, 2010, pp. 524-53
The Complexity of Synthesizing Uniform Strategies
We investigate uniformity properties of strategies. These properties involve
sets of plays in order to express useful constraints on strategies that are not
\mu-calculus definable. Typically, we can state that a strategy is
observation-based. We propose a formal language to specify uniformity
properties, interpreted over two-player turn-based arenas equipped with a
binary relation between plays. This way, we capture e.g. games with winning
conditions expressible in epistemic temporal logic, whose underlying
equivalence relation between plays reflects the observational capabilities of
agents (for example, synchronous perfect recall). Our framework naturally
generalizes many other situations from the literature. We establish that the
problem of synthesizing strategies under uniformity constraints based on
regular binary relations between plays is non-elementary complete.Comment: In Proceedings SR 2013, arXiv:1303.007
A History of Until
Until is a notoriously difficult temporal operator as it is both existential
and universal at the same time: A until B holds at the current time instant w
iff either B holds at w or there exists a time instant w' in the future at
which B holds and such that A holds in all the time instants between the
current one and w'. This "ambivalent" nature poses a significant challenge when
attempting to give deduction rules for until. In this paper, in contrast, we
make explicit this duality of until to provide well-behaved natural deduction
rules for linear-time logics by introducing a new temporal operator that allows
us to formalize the "history" of until, i.e., the "internal" universal
quantification over the time instants between the current one and w'. This
approach provides the basis for formalizing deduction systems for temporal
logics endowed with the until operator. For concreteness, we give here a
labeled natural deduction system for a linear-time logic endowed with the new
operator and show that, via a proper translation, such a system is also sound
and complete with respect to the linear temporal logic LTL with until.Comment: 24 pages, full version of paper at Methods for Modalities 2009
(M4M-6
About Norms and Causes
Knowing the norms of a domain is crucial, but there exist no repository of
norms. We propose a method to extract them from texts: texts generally do not
describe a norm, but rather how a state-of-affairs differs from it. Answers
concerning the cause of the state-of-affairs described often reveal the
implicit norm. We apply this idea to the domain of driving, and validate it by
designing algorithms that identify, in a text, the "basic" norms to which it
refers implicitly
Dyck algebras, interval temporal logic and posets of intervals
We investigate a natural Heyting algebra structure on the set of Dyck paths
of the same length. We provide a geometrical description of the operations of
pseudocomplement and relative pseudocomplement, as well as of regular elements.
We also find a logic-theoretic interpretation of such Heyting algebras, which
we call Dyck algebras, by showing that they are the algebraic counterpart of a
certain fragment of a classical interval temporal logic (also known as
Halpern-Shoham logic). Finally, we propose a generalization of our approach,
suggesting a similar study of the Heyting algebra arising from the poset of
intervals of a finite poset using Birkh\"off duality. In order to illustrate
this, we show how several combinatorial parameters of Dyck paths can be
expressed in terms of the Heyting algebra structure of Dyck algebras together
with a certain total order on the set of atoms of each Dyck algebra.Comment: 17 pages, 3 figure
Diamonds are Forever
We defend the thesis that every necessarily true proposition is always true. Since not every proposition that is always true is necessarily true, our thesis is at odds with theories of modality and time, such as those of Kit Fine and David Kaplan, which posit a fundamental symmetry between modal and tense operators. According to such theories, just as it is a contingent matter what is true at a given time, it is likewise a temporary matter what is true at a given possible world; so a proposition that is now true at all worlds, and thus necessarily true, may yet at some past or future time be false in the actual world, and thus not always true. We reconstruct and criticize several lines of argument in favor of this picture, and then argue against the picture on the grounds that it is inconsistent with certain sorts of contingency in the structure of time
A decidable weakening of Compass Logic based on cone-shaped cardinal directions
We introduce a modal logic, called Cone Logic, whose formulas describe
properties of points in the plane and spatial relationships between them.
Points are labelled by proposition letters and spatial relations are induced by
the four cone-shaped cardinal directions. Cone Logic can be seen as a weakening
of Venema's Compass Logic. We prove that, unlike Compass Logic and other
projection-based spatial logics, its satisfiability problem is decidable
(precisely, PSPACE-complete). We also show that it is expressive enough to
capture meaningful interval temporal logics - in particular, the interval
temporal logic of Allen's relations "Begins", "During", and "Later", and their
transposes
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