14,962 research outputs found
Diagrammatic Inference
Diagrammatic logics were introduced in 2002, with emphasis on the notions of
specifications and models. In this paper we improve the description of the
inference process, which is seen as a Yoneda functor on a bicategory of
fractions. A diagrammatic logic is defined from a morphism of limit sketches
(called a propagator) which gives rise to an adjunction, which in turn
determines a bicategory of fractions. The propagator, the adjunction and the
bicategory provide respectively the syntax, the models and the inference
process for the logic. Then diagrammatic logics and their morphisms are applied
to the semantics of side effects in computer languages.Comment: 16 page
Automation of Diagrammatic Proofs in Mathematics
Theorems in automated theorem proving are usually proved by logical formal proofs. However, there is a subset of problems which can also be proved in a more informal way by the use of geometric operations on diagrams, so called diagrammatic proofs. Insight is more clearly perceived in these than in the corresponding logical proofs: they capture an intuitive notion of truthfulness that humans find easy to see and understand. The proposed research project is to identify and ultimately automate this diagrammatic reasoning on mathematical theorems. The system that we are in the process of implementing will be given a theorem and will (initially) interactively prove it by the use of geometric manipulations on the diagram that the user chooses to be the appropriate ones. These operations will be the inference steps of the proof. The constructive !-rule will be used as a tool to capture the generality of diagrammatic proofs. In this way, we hope to verify and to show that the diagra..
Active Inference in String Diagrams: A Categorical Account of Predictive Processing and Free Energy
We present a categorical formulation of the cognitive frameworks of
Predictive Processing and Active Inference, expressed in terms of string
diagrams interpreted in a monoidal category with copying and discarding. This
includes diagrammatic accounts of generative models, Bayesian updating,
perception, planning, active inference, and free energy. In particular we
present a diagrammatic derivation of the formula for active inference via free
energy minimisation, and establish a compositionality property for free energy,
allowing free energy to be applied at all levels of an agent's generative
model. Aside from aiming to provide a helpful graphical language for those
familiar with active inference, we conversely hope that this article may
provide a concise formulation and introduction to the framework
Accessible reasoning with diagrams: From cognition to automation
High-tech systems are ubiquitous and often safety and se- curity critical: reasoning about their correctness is paramount. Thus, precise modelling and formal reasoning are necessary in order to convey knowledge unambiguously and accurately. Whilst mathematical mod- elling adds great rigour, it is opaque to many stakeholders which leads to errors in data handling, delays in product release, for example. This is a major motivation for the development of diagrammatic approaches to formalisation and reasoning about models of knowledge. In this paper, we present an interactive theorem prover, called iCon, for a highly expressive diagrammatic logic that is capable of modelling OWL 2 ontologies and, thus, has practical relevance. Significantly, this work is the first to design diagrammatic inference rules using insights into what humans find accessible. Specifically, we conducted an experiment about relative cognitive benefits of primitive (small step) and derived (big step) inferences, and use the results to guide the implementation of inference rules in iCon
How to combine diagrammatic logics
This paper is a submission to the contest: How to combine logics? at the
World Congress and School on Universal Logic III, 2010. We claim that combining
"things", whatever these things are, is made easier if these things can be seen
as the objects of a category. We define the category of diagrammatic logics, so
that categorical constructions can be used for combining diagrammatic logics.
As an example, a combination of logics using an opfibration is presented, in
order to study computational side-effects due to the evolution of the state
during the execution of an imperative program
Adjunctions for exceptions
An algebraic method is used to study the semantics of exceptions in computer
languages. The exceptions form a computational effect, in the sense that there
is an apparent mismatch between the syntax of exceptions and their intended
semantics. We solve this apparent contradiction by efining a logic for
exceptions with a proof system which is close to their syntax and where their
intended semantics can be seen as a model. This requires a robust framework for
logics and their morphisms, which is provided by categorical tools relying on
adjunctions, fractions and limit sketches.Comment: In this Version 2, minor improvements are made to Version
Picturing classical and quantum Bayesian inference
We introduce a graphical framework for Bayesian inference that is
sufficiently general to accommodate not just the standard case but also recent
proposals for a theory of quantum Bayesian inference wherein one considers
density operators rather than probability distributions as representative of
degrees of belief. The diagrammatic framework is stated in the graphical
language of symmetric monoidal categories and of compact structures and
Frobenius structures therein, in which Bayesian inversion boils down to
transposition with respect to an appropriate compact structure. We characterize
classical Bayesian inference in terms of a graphical property and demonstrate
that our approach eliminates some purely conventional elements that appear in
common representations thereof, such as whether degrees of belief are
represented by probabilities or entropic quantities. We also introduce a
quantum-like calculus wherein the Frobenius structure is noncommutative and
show that it can accommodate Leifer's calculus of `conditional density
operators'. The notion of conditional independence is also generalized to our
graphical setting and we make some preliminary connections to the theory of
Bayesian networks. Finally, we demonstrate how to construct a graphical
Bayesian calculus within any dagger compact category.Comment: 38 pages, lots of picture
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