Disjunctive Probabilistic Modal Logic is Enough for Bisimilarity on Reactive Probabilistic Systems

Larsen and Skou characterized probabilistic bisimilarity over reactive probabilistic systems with a logic including true, negation, conjunction, and a diamond modality decorated with a probabilistic lower bound. Later on, Desharnais, Edalat, and Panangaden showed that negation is not necessary to characterize the same equivalence. In this paper, we prove that the logical characterization holds also when conjunction is replaced by disjunction, with negation still being not necessary. To this end, we introduce reactive probabilistic trees, a fully abstract model for reactive probabilistic systems that allows us to demonstrate expressiveness of the disjunctive probabilistic modal logic, as well as of the previously mentioned logics, by means of a compactness argument.Comment: Aligned content with version accepted at ICTCS 2016: fixed minor typos, added reference, improved definitions in Section 3. Still 10 pages in sigplanconf forma

The Structure of Differential Invariants and Differential Cut Elimination

The biggest challenge in hybrid systems verification is the handling of differential equations. Because computable closed-form solutions only exist for very simple differential equations, proof certificates have been proposed for more scalable verification. Search procedures for these proof certificates are still rather ad-hoc, though, because the problem structure is only understood poorly. We investigate differential invariants, which define an induction principle for differential equations and which can be checked for invariance along a differential equation just by using their differential structure, without having to solve them. We study the structural properties of differential invariants. To analyze trade-offs for proof search complexity, we identify more than a dozen relations between several classes of differential invariants and compare their deductive power. As our main results, we analyze the deductive power of differential cuts and the deductive power of differential invariants with auxiliary differential variables. We refute the differential cut elimination hypothesis and show that, unlike standard cuts, differential cuts are fundamental proof principles that strictly increase the deductive power. We also prove that the deductive power increases further when adding auxiliary differential variables to the dynamics

An Axiomatic Approach to Liveness for Differential Equations

This paper presents an approach for deductive liveness verification for ordinary differential equations (ODEs) with differential dynamic logic. Numerous subtleties complicate the generalization of well-known discrete liveness verification techniques, such as loop variants, to the continuous setting. For example, ODE solutions may blow up in finite time or their progress towards the goal may converge to zero. Our approach handles these subtleties by successively refining ODE liveness properties using ODE invariance properties which have a well-understood deductive proof theory. This approach is widely applicable: we survey several liveness arguments in the literature and derive them all as special instances of our axiomatic refinement approach. We also correct several soundness errors in the surveyed arguments, which further highlights the subtlety of ODE liveness reasoning and the utility of our deductive approach. The library of common refinement steps identified through our approach enables both the sound development and justification of new ODE liveness proof rules from our axioms.Comment: FM 2019: 23rd International Symposium on Formal Methods, Porto, Portugal, October 9-11, 201

Learning Tuple Probabilities

Learning the parameters of complex probabilistic-relational models from labeled training data is a standard technique in machine learning, which has been intensively studied in the subfield of Statistical Relational Learning (SRL), but---so far---this is still an under-investigated topic in the context of Probabilistic Databases (PDBs). In this paper, we focus on learning the probability values of base tuples in a PDB from labeled lineage formulas. The resulting learning problem can be viewed as the inverse problem to confidence computations in PDBs: given a set of labeled query answers, learn the probability values of the base tuples, such that the marginal probabilities of the query answers again yield in the assigned probability labels. We analyze the learning problem from a theoretical perspective, cast it into an optimization problem, and provide an algorithm based on stochastic gradient descent. Finally, we conclude by an experimental evaluation on three real-world and one synthetic dataset, thus comparing our approach to various techniques from SRL, reasoning in information extraction, and optimization

Regularity of the Einstein Equations at Future Null Infinity

When Einstein's equations for an asymptotically flat, vacuum spacetime are reexpressed in terms of an appropriate conformal metric that is regular at (future) null infinity, they develop apparently singular terms in the associated conformal factor and thus appear to be ill-behaved at this (exterior) boundary. In this article however we show, through an enforcement of the Hamiltonian and momentum constraints to the needed order in a Taylor expansion, that these apparently singular terms are not only regular at the boundary but can in fact be explicitly evaluated there in terms of conformally regular geometric data. Though we employ a rather rigidly constrained and gauge fixed formulation of the field equations, we discuss the extent to which we expect our results to have a more 'universal' significance and, in particular, to be applicable, after minor modifications, to alternative formulations.Comment: 43 pages, no figures, AMS-TeX. Minor revisions, updated to agree with published versio