Notions of Computation and Monads

The i.-calculus is considered a useful mathematical tool in the study of programming languages, since programs can be identified with I-terms. However, if one goes further and uses bn-conversion to prove equivalence of programs, then a gross simplification is introduced (programs are identified with total functions from calues to values) that may jeopardise the applicability of theoretical results, In this paper we introduce calculi. based on a categorical semantics for computations, that provide a correct basis for proving equivalence of programs for a wide range of notions of computation

Robustness in Metric Spaces over Continuous Quantales and the Hausdorff-Smyth Monad

Generalized metric spaces are obtained by weakening the requirements (e.g., symmetry) on the distance function and by allowing it to take values in structures (e.g., quantales) that are more general than the set of non-negative real numbers. Quantale-valued metric spaces have gained prominence due to their use in quantitative reasoning on programs/systems, and for defining various notions of behavioral metrics. We investigate imprecision and robustness in the framework of quantale-valued metric spaces, when the quantale is continuous. In particular, we study the relation between the robust topology, which captures robustness of analyses, and the Hausdorff-Smyth hemi-metric. To this end, we define a preorder-enriched monad $\mathsf{P}_S$, called the Hausdorff-Smyth monad, and when $Q$ is a continuous quantale and $X$ is a $Q$-metric space, we relate the topology induced by the metric on $\mathsf{P}_S(X)$ with the robust topology on the powerset $\mathsf{P}(X)$ defined in terms of the metric on $X$.Comment: 19 pages, 1 figur

A Semantic Account of Rigorous Simulation

Hybrid systems are a powerful formalism for modeling cyber-physical systems. Reachability analysis is a general method for checking safety properties, especially in the presence of uncertainty and non-determinism. Rigorous simulation is a convenient tool for reachability analysis of hybrid systems. However, to serve as proof tool, a rigorous simulator must be correct wrt a clearly defined notion of reachability,which captures what is intuitively eachable in finite time. As a step towards addressing this challenge, this paper presents a rigorous simulator in the form of an operational semantics and a specification in the form of a denotational semantics. We show that, under certain conditions about the representation of enclosures, the rigorous simulator is correct. We also show that finding a representation satisfying these assumptions is non-trivial

Sound over-approximation of probabilities

Safety analysis of high confidence systems requires guaranteed bounds on the probabilities of events of interest. Establishing the correctness of algorithms that aim to compute such bounds is challenging. We address this problem in three steps. First, we use monadic transition systems (MTS) in the category of sets as a framework for modeling discrete time systems. MTS can capture different types of system behaviors, but we focus on a combination of non-deterministic and probabilistic behaviors that often arises when modeling complex systems. Second, we use the category of posets and monotonic maps as a setting to define and compare approximations. In particular, for the MTS of interest, we consider approximations of their configurations based on complete lattices. Third, by restricting to finite lattices, we obtain algorithms that compute over-approximations, i.e., bounds from above within some partial order of approximants, of the system configuration after n steps. Interestingly, finite lattices of “interval probabilities” may fail to accurately approximate configurations that are both non-deterministic and probabilistic, even for deterministic (and continuous) system dynamics. However, better choices of finite lattices are available

Safe & robust reachability analysis of hybrid systems

Hybrid systems—more precisely, their mathematical models—can exhibit behaviors, like Zeno behaviors, that are absent in purely discrete or purely continuous systems. First, we observe that, in this context, the usual definition of reachability—namely, the reflexive and transitive closure of a transition relation—can be unsafe, i.e., it may compute a proper subset of the set of states reachable in finite time from a set of initial states. Therefore, we propose safe reachability, which always computes a superset of the set of reachable states. Second, in safety analysis of hybrid and continuous systems, it is important to ensure that a reachability analysis is also robust w.r.t. small perturbations to the set of initial states and to the system itself, since discrepancies between a system and its mathematical models are unavoidable. We show that, under certain conditions, the best Scott continuous approximation of an analysis A is also its best robust approximation. Finally, we exemplify the gap between the set of reachable states and the supersets computed by safe reachability and its best robust approximation

Completeness of algebraic CPS simulations

The algebraic lambda calculus and the linear algebraic lambda calculus are two extensions of the classical lambda calculus with linear combinations of terms. They arise independently in distinct contexts: the former is a fragment of the differential lambda calculus, the latter is a candidate lambda calculus for quantum computation. They differ in the handling of application arguments and algebraic rules. The two languages can simulate each other using an algebraic extension of the well-known call-by-value and call-by-name CPS translations. These simulations are sound, in that they preserve reductions. In this paper, we prove that the simulations are actually complete, strengthening the connection between the two languages.Comment: In Proceedings DCM 2011, arXiv:1207.682

Semantics of a Typed Algebraic Lambda-Calculus

Algebraic lambda-calculi have been studied in various ways, but their semantics remain mostly untouched. In this paper we propose a semantic analysis of a general simply-typed lambda-calculus endowed with a structure of vector space. We sketch the relation with two established vectorial lambda-calculi. Then we study the problems arising from the addition of a fixed point combinator and how to modify the equational theory to solve them. We sketch an algebraic vectorial PCF and its possible denotational interpretations

Left atrial dysfunction detected by speckle tracking in patients with systemic sclerosis

BACKGROUND: Cardiac involvement is a relevant clinical finding in systemic sclerosis (SSc) and is associated with poor prognosis. Left atrial (LA) remodeling and/or dysfunction can be an early sign of diastolic dysfunction. Two-dimensional speckle tracking echocardiography (STE) is a novel and promising tool for detecting very early changes in LA myocardial performance. AIM: To assess whether STE strain parameters may detect early alterations in LA function in SSc patients. METHODS: Forty-two SSc patients (Group 1, age 50 +/- 14 years, 95% females) without clinical evidence for cardiac involvement and 42 age- and gender-matched control subjects (Group 2, age 49 +/- 13 years, 95% females) were evaluated with comprehensive 2D and Doppler echocardiography, including tissue Doppler imaging analysis. Positive peak left atrial longitudinal strain ( pos peak), second positive left atrial longitudinal strain (sec pos peak), and negative left atrial longitudinal strain ( neg peak) were measured using a 12-segment model for the LA, by commercially available semi-automated 2D speckle-tracking software (EchoPac PC version 108.1.4, GE Healthcare, Horten, Norway). RESULTS: All SSc patients had a normal left ventricular ejection fraction (63.1 +/- 4%). SSc patients did not differ from controls in E/A (Group 1 = 1.1 +/- 0.4 vs Group 2 = 1.3 +/- 0.4, p = .14) or pulmonary arterial systolic pressure (Group 1 = 24.1 +/- 8 mmHg vs Group 2 = 21 +/- 7 mmHg, p = .17). SSc patients did not show significantly different indexed LA volumes (Group 1 = 24.9 +/- 5.3 ml/m2 vs Group 2 = 24.7 +/- 4.4 ml/m2, p = .8), whereas E/e' ratio was significantly higher in SSc (Group 1 = 7.6 +/- 2.4 vs Group 2 = 6.5 +/- 1.7, p<0.05), although still within normal values. LA strain values were significantly different between the two groups ( pos peak Group 1 = 31.3 +/- 4.2% vs Group 2 = 35.0 +/- 7.6%, p < .01, sec pos peak Group 1 = 18.4 +/- 4 vs Group 2 = 21.4 +/- 7.6, p < 0.05). CONCLUSION: 2D speckle-tracking echocardiography is a sensitive tool to assess impairment of LA mechanics, which is detectable in absence of changes in LA size and volume, and may represent an early sign of cardiac involvement in patients with SSc

Acumen : an open-source testbed for cyber-physical systems research

Developing Cyber-Physical Systems requires methods and tools to support simulation and verification of hybrid (both continuous and discrete) models. The Acumen modeling and simulation language is an open source testbed for exploring the design space of what rigorousbut- practical next-generation tools can deliver to developers of Cyber- Physical Systems. Like verification tools, a design goal for Acumen is to provide rigorous results. Like simulation tools, it aims to be intuitive, practical, and scalable. However, it is far from evident whether these two goals can be achieved simultaneously. This paper explains the primary design goals for Acumen, the core challenges that must be addressed in order to achieve these goals, the “agile research method” taken by the project, the steps taken to realize these goals, the key lessons learned, and the emerging language design