Indexing the Event Calculus with Kd-trees to Monitor Diabetes

Personal Health Systems (PHS) are mobile solutions tailored to monitoring patients affected by chronic non communicable diseases. A patient affected by a chronic disease can generate large amounts of events. Type 1 Diabetic patients generate several glucose events per day, ranging from at least 6 events per day (under normal monitoring) to 288 per day when wearing a continuous glucose monitor (CGM) that samples the blood every 5 minutes for several days. This is a large number of events to monitor for medical doctors, in particular when considering that they may have to take decisions concerning adjusting the treatment, which may impact the life of the patients for a long time. Given the need to analyse such a large stream of data, doctors need a simple approach towards physiological time series that allows them to promptly transfer their knowledge into queries to identify interesting patterns in the data. Achieving this with current technology is not an easy task, as on one hand it cannot be expected that medical doctors have the technical knowledge to query databases and on the other hand these time series include thousands of events, which requires to re-think the way data is indexed. In order to tackle the knowledge representation and efficiency problem, this contribution presents the kd-tree cached event calculus (\ceckd) an event calculus extension for knowledge engineering of temporal rules capable to handle many thousands events produced by a diabetic patient. \ceckd\ is built as a support to a graphical interface to represent monitoring rules for diabetes type 1. In addition, the paper evaluates the \ceckd\ with respect to the cached event calculus (CEC) to show how indexing events using kd-trees improves scalability with respect to the current state of the art.Comment: 24 pages, preliminary results calculated on an implementation of CECKD, precursor to Journal paper being submitted in 2017, with further indexing and results possibilities, put here for reference and chronological purposes to remember how the idea evolve

Model-Checking Process Equivalences

Process equivalences are formal methods that relate programs and system which, informally, behave in the same way. Since there is no unique notion of what it means for two dynamic systems to display the same behaviour there are a multitude of formal process equivalences, ranging from bisimulation to trace equivalence, categorised in the linear-time branching-time spectrum. We present a logical framework based on an expressive modal fixpoint logic which is capable of defining many process equivalence relations: for each such equivalence there is a fixed formula which is satisfied by a pair of processes if and only if they are equivalent with respect to this relation. We explain how to do model checking, even symbolically, for a significant fragment of this logic that captures many process equivalences. This allows model checking technology to be used for process equivalence checking. We show how partial evaluation can be used to obtain decision procedures for process equivalences from the generic model checking scheme.Comment: In Proceedings GandALF 2012, arXiv:1210.202

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

Model-Checking the Higher-Dimensional Modal mu-Calculus

The higher-dimensional modal mu-calculus is an extension of the mu-calculus in which formulas are interpreted in tuples of states of a labeled transition system. Every property that can be expressed in this logic can be checked in polynomial time, and conversely every polynomial-time decidable problem that has a bisimulation-invariant encoding into labeled transition systems can also be defined in the higher-dimensional modal mu-calculus. We exemplify the latter connection by giving several examples of decision problems which reduce to model checking of the higher-dimensional modal mu-calculus for some fixed formulas. This way generic model checking algorithms for the logic can then be used via partial evaluation in order to obtain algorithms for theses problems which may benefit from improvements that are well-established in the field of program verification, namely on-the-fly and symbolic techniques. The aim of this work is to extend such techniques to other fields as well, here exemplarily done for process equivalences, automata theory, parsing, string problems, and games.Comment: In Proceedings FICS 2012, arXiv:1202.317

Lewis meets Brouwer: constructive strict implication

C. I. Lewis invented modern modal logic as a theory of "strict implication". Over the classical propositional calculus one can as well work with the unary box connective. Intuitionistically, however, the strict implication has greater expressive power than the box and allows to make distinctions invisible in the ordinary syntax. In particular, the logic determined by the most popular semantics of intuitionistic K becomes a proper extension of the minimal normal logic of the binary connective. Even an extension of this minimal logic with the "strength" axiom, classically near-trivial, preserves the distinction between the binary and the unary setting. In fact, this distinction and the strong constructive strict implication itself has been also discovered by the functional programming community in their study of "arrows" as contrasted with "idioms". Our particular focus is on arithmetical interpretations of the intuitionistic strict implication in terms of preservativity in extensions of Heyting's Arithmetic.Comment: Our invited contribution to the collection "L.E.J. Brouwer, 50 years later