Let G be an abelian group acting on a set X, and suppose that no element of G has any finite orbit of size greater than one. We show that every partial order on X invariant under G extends to a linear order on X also invariant under G. We then discuss extensions to linear preorders when the orbit condition is not met, and show that for any abelian group acting on a set X, there is a G-invariant...

We extend RBPP, the state-of-the-art, translation-based planner for conformant probabilistic planning (CPP) with deterministic actions, to handle a wide set of CPPs with stochastic actions. Our planner uses relevance analysis to divide a probabilistic ”failure-allowance” between the initial state and the stochastic actions. Using its ”initial-state allowance,” it uses relevance analysis to sele...

In standard measure theory the measure on the total space Ω is normalised to one, which encodes the statement that “Ω happens”. The standard rules of measure theory tell us that the measure on Ω is strictly positive if and only if it cannot be covered by a collection of sets of zero measure. On the other hand, in quantum measure theory, simple examples suffice to demonstrate that a cover of set...

framework for powerset semantics. We now cast the powerset semantics in our general relation-based framework, which enables us to establish when naı̈ve evaluation works for it. For a set D of database objects and a set C of complete objects, we have a pairR “ pRval,Rsemq of relations withRval Ď Dˆ2 C andRsem Ď 2 ˆC. The first relation corresponds to applying multiple valuations (e.g., relating D...

It is shown how Lawvere’s one-to-one translation between Birkhoff’s description of varieties and the categorical one (see [6]) turns Hu’s theorem on varieties generated by a primal algebra (see [4], [5]) into a simple reformulation of the classical representation theorem of finite Boolean algebras as powerset algebras.

Papert Strauss (Proc. London Math. Soc. 18(3), 217–230, 1968) used Pontryagin duality to prove that a compact Hausdorff topological Boolean algebra is a powerset algebra. We give a more elementary proof of this result that relies on a version of Bogolyubov’s lemma.

Lax monoidal powerset-enriched monads yield a monoidal structure on the category of monoids in the Kleisli category of a monad. Exponentiable objects in this category are identified as those Kleisli monoids with algebraic structure. This result generalizes the classical identification of exponentiable topological spaces as those whose lattice of open subsets forms a continuous lattice.

Query generalization is one option to implement flexible query answering. In this paper, we introduce a generalization operator (called powerset-AI) that extends conventional Anti-Instantiation (AI). We analyze structural modifications imposed by the generalization to obtain syntactic similarity measures (based on the star feature) that rank generalized queries with regard to their closeness to...

Multirelations provide a semantic domain for computing systems that involve two dual kinds of nondeterminism. This paper presents relational formalisations of Kleisli, Parikh and Peleg compositions and liftings of multirelations. These liftings are similar to those that arise in the Kleisli category of the powerset monad. We show that Kleisli composition of multirelations is associative, but ne...

In this paper we give an explicit construction of n × n matrices over finite fields which are somewhat rigid, in that if we change at most k entries in each row, its rank remains at least Cn(logq k)/k, where q is the size of the field and C is an absolute constant. Our matrices satify a somewhat stronger property, we which explain and call “strong rigidity.” We introduce and briefly discuss str...