13,731 research outputs found
Labelled transition systems as a Stone space
A fully abstract and universal domain model for modal transition systems and
refinement is shown to be a maximal-points space model for the bisimulation
quotient of labelled transition systems over a finite set of events. In this
domain model we prove that this quotient is a Stone space whose compact,
zero-dimensional, and ultra-metrizable Hausdorff topology measures the degree
of bisimilarity such that image-finite labelled transition systems are dense.
Using this compactness we show that the set of labelled transition systems that
refine a modal transition system, its ''set of implementations'', is compact
and derive a compactness theorem for Hennessy-Milner logic on such
implementation sets. These results extend to systems that also have partially
specified state propositions, unify existing denotational, operational, and
metric semantics on partial processes, render robust consistency measures for
modal transition systems, and yield an abstract interpretation of compact sets
of labelled transition systems as Scott-closed sets of modal transition
systems.Comment: Changes since v2: Metadata updat
The Expectation Monad in Quantum Foundations
The expectation monad is introduced abstractly via two composable
adjunctions, but concretely captures measures. It turns out to sit in between
known monads: on the one hand the distribution and ultrafilter monad, and on
the other hand the continuation monad. This expectation monad is used in two
probabilistic analogues of fundamental results of Manes and Gelfand for the
ultrafilter monad: algebras of the expectation monad are convex compact
Hausdorff spaces, and are dually equivalent to so-called Banach effect
algebras. These structures capture states and effects in quantum foundations,
and also the duality between them. Moreover, the approach leads to a new
re-formulation of Gleason's theorem, expressing that effects on a Hilbert space
are free effect modules on projections, obtained via tensoring with the unit
interval.Comment: In Proceedings QPL 2011, arXiv:1210.029
Computable decision making on the reals and other spaces via partiality and nondeterminism
Though many safety-critical software systems use floating point to represent
real-world input and output, programmers usually have idealized versions in
mind that compute with real numbers. Significant deviations from the ideal can
cause errors and jeopardize safety. Some programming systems implement exact
real arithmetic, which resolves this matter but complicates others, such as
decision making. In these systems, it is impossible to compute (total and
deterministic) discrete decisions based on connected spaces such as
. We present programming-language semantics based on constructive
topology with variants allowing nondeterminism and/or partiality. Either
nondeterminism or partiality suffices to allow computable decision making on
connected spaces such as . We then introduce pattern matching on
spaces, a language construct for creating programs on spaces, generalizing
pattern matching in functional programming, where patterns need not represent
decidable predicates and also may overlap or be inexhaustive, giving rise to
nondeterminism or partiality, respectively. Nondeterminism and/or partiality
also yield formal logics for constructing approximate decision procedures. We
implemented these constructs in the Marshall language for exact real
arithmetic.Comment: This is an extended version of a paper due to appear in the
proceedings of the ACM/IEEE Symposium on Logic in Computer Science (LICS) in
July 201
Abstract Hidden Markov Models: a monadic account of quantitative information flow
Hidden Markov Models, HMM's, are mathematical models of Markov processes with
state that is hidden, but from which information can leak. They are typically
represented as 3-way joint-probability distributions.
We use HMM's as denotations of probabilistic hidden-state sequential
programs: for that, we recast them as `abstract' HMM's, computations in the
Giry monad , and we equip them with a partial order of increasing
security. However to encode the monadic type with hiding over some state
we use rather
than the conventional that suffices for
Markov models whose state is not hidden. We illustrate the
construction with a small
Haskell prototype.
We then present uncertainty measures as a generalisation of the extant
diversity of probabilistic entropies, with characteristic analytic properties
for them, and show how the new entropies interact with the order of increasing
security. Furthermore, we give a `backwards' uncertainty-transformer semantics
for HMM's that is dual to the `forwards' abstract HMM's - it is an analogue of
the duality between forwards, relational semantics and backwards,
predicate-transformer semantics for imperative programs with demonic choice.
Finally, we argue that, from this new denotational-semantic viewpoint, one
can see that the Dalenius desideratum for statistical databases is actually an
issue in compositionality. We propose a means for taking it into account
Loops and Knots as Topoi of Substance. Spinoza Revisited
The relationship between modern philosophy and physics is discussed. It is
shown that the latter develops some need for a modernized metaphysics which
shows up as an ultima philosophia of considerable heuristic value, rather than
as the prima philosophia in the Aristotelian sense as it had been intended, in
the first place. It is shown then, that it is the philosophy of Spinoza in
fact, that can still serve as a paradigm for such an approach. In particular,
Spinoza's concept of infinite substance is compared with the philosophical
implications of the foundational aspects of modern physical theory. Various
connotations of sub-stance are discussed within pre-geometric theories,
especially with a view to the role of spin networks within quantum gravity. It
is found to be useful to intro-duce a separation into physics then, so as to
differ between foundational and empirical theories, respectively. This leads to
a straightforward connection bet-ween foundational theories and speculative
philosophy on the one hand, and between empirical theories and sceptical
philosophy on the other. This might help in the end, to clarify some recent
problems, such as the absence of time and causality at a fundamental level. It
is implied that recent results relating to topos theory might open the way
towards eventually deriving logic from physics, and also towards a possible
transition from logic to hermeneutic.Comment: 42 page
Density Matrices with Metric for Derivational Ambiguity
Recent work on vector-based compositional natural language semantics has
proposed the use of density matrices to model lexical ambiguity and (graded)
entailment (e.g. Piedeleu et al 2015, Bankova et al 2019, Sadrzadeh et al
2018). Ambiguous word meanings, in this work, are represented as mixed states,
and the compositional interpretation of phrases out of their constituent parts
takes the form of a strongly monoidal functor sending the derivational
morphisms of a pregroup syntax to linear maps in FdHilb. Our aims in this paper
are threefold. Firstly, we replace the pregroup front end by a Lambek
categorial grammar with directional implications expressing a word's
selectional requirements. By the Curry-Howard correspondence, the derivations
of the grammar's type logic are associated with terms of the (ordered) linear
lambda calculus; these terms can be read as programs for compositional meaning
assembly with density matrices as the target semantic spaces. Secondly, we
extend on the existing literature and introduce a symmetric, nondegenerate
bilinear form called a "metric" that defines a canonical isomorphism between a
vector space and its dual, allowing us to keep a distinction between left and
right implication. Thirdly, we use this metric to define density matrix spaces
in a directional form, modeling the ubiquitous derivational ambiguity of
natural language syntax, and show how this alows an integrated treatment of
lexical and derivational forms of ambiguity controlled at the level of the
interpretation.Comment: 24 pages, 10 figures. SemSpace 2019, to appear in J. of Applied
Logic
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