5,964 research outputs found
Logical Relations for Monadic Types
Logical relations and their generalizations are a fundamental tool in proving
properties of lambda-calculi, e.g., yielding sound principles for observational
equivalence. We propose a natural notion of logical relations able to deal with
the monadic types of Moggi's computational lambda-calculus. The treatment is
categorical, and is based on notions of subsconing, mono factorization systems,
and monad morphisms. Our approach has a number of interesting applications,
including cases for lambda-calculi with non-determinism (where being in logical
relation means being bisimilar), dynamic name creation, and probabilistic
systems.Comment: 83 page
Relations Without Polyadic Properties: Albert the Great On the Nature and Ontological Status of Relations
I think it would be fair to say that, until about 1900, philosophers were generally reluctant to admit the existence of what are nowadays called polyadic properties.1 It is important to recognize, however, that this reluctance on the part of pre-twentieth-century philosophers did not prevent them from theorizing about relations. On the contrary, philosophers from the ancient through the modern period have had much to say about both the nature and the ontological status of relations. In this paper I examine the views of one such philosopher, namely, Albert the Grea
Dynamic IFC Theorems for Free!
We show that noninterference and transparency, the key soundness theorems for
dynamic IFC libraries, can be obtained "for free", as direct consequences of
the more general parametricity theorem of type abstraction. This allows us to
give very short soundness proofs for dynamic IFC libraries such as faceted
values and LIO. Our proofs stay short even when fully mechanized for Agda
implementations of the libraries in terms of type abstraction.Comment: CSF 2021 final versio
Possible Patterns
âThere are no gaps in logical space,â David Lewis writes, giving voice to sentiment shared by many philosophers. But different natural ways of trying to make this sentiment precise turn out to conflict with one another. One is a *pattern* idea: âAny pattern of instantiation is metaphysically possible.â Another is a *cut and paste* idea: âFor any objects in any worlds, there exists a world that contains any number of duplicates of all of those objects.â We use resources from model theory to show the inconsistency of certain packages of combinatorial principles and the consistency of others
On Second-Order Monadic Monoidal and Groupoidal Quantifiers
We study logics defined in terms of second-order monadic monoidal and
groupoidal quantifiers. These are generalized quantifiers defined by monoid and
groupoid word-problems, equivalently, by regular and context-free languages. We
give a computational classification of the expressive power of these logics
over strings with varying built-in predicates. In particular, we show that
ATIME(n) can be logically characterized in terms of second-order monadic
monoidal quantifiers
On Spatial Conjunction as Second-Order Logic
Spatial conjunction is a powerful construct for reasoning about dynamically
allocated data structures, as well as concurrent, distributed and mobile
computation. While researchers have identified many uses of spatial
conjunction, its precise expressive power compared to traditional logical
constructs was not previously known. In this paper we establish the expressive
power of spatial conjunction. We construct an embedding from first-order logic
with spatial conjunction into second-order logic, and more surprisingly, an
embedding from full second order logic into first-order logic with spatial
conjunction. These embeddings show that the satisfiability of formulas in
first-order logic with spatial conjunction is equivalent to the satisfiability
of formulas in second-order logic. These results explain the great expressive
power of spatial conjunction and can be used to show that adding unrestricted
spatial conjunction to a decidable logic leads to an undecidable logic. As one
example, we show that adding unrestricted spatial conjunction to two-variable
logic leads to undecidability. On the side of decidability, the embedding into
second-order logic immediately implies the decidability of first-order logic
with a form of spatial conjunction over trees. The embedding into spatial
conjunction also has useful consequences: because a restricted form of spatial
conjunction in two-variable logic preserves decidability, we obtain that a
correspondingly restricted form of second-order quantification in two-variable
logic is decidable. The resulting language generalizes the first-order theory
of boolean algebra over sets and is useful in reasoning about the contents of
data structures in object-oriented languages.Comment: 16 page
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
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