Z-style notation for Probabilities

A notation for probabilities is proposed that differs from the traditional, conventional notation by making explicit the domains and bound variables involved. The notation borrows from the Z notation, and lends itself well to calculational manipulations, with a smooth transition back and forth to set and predicate notation

A correctness proof of sorting by means of formal procedures

We consider a recursive sorting algorithm in which, in each invocation, a new variable and a new procedure (using the variable globally) are defined and the procedure is passed to recursive calls. This algorithm is proved correct with Hoare-style pre- and postassertions. We also discuss the same algorithm expressed as a functional program

Ignorance in the Relational Model

We hypothesize that an extension-with-conditioning of Dempster-Shafer theory is suitable for encoding uncertainty and ignorance in the Relational Model. We present a formal and well-motivated definition of conditioning, and show the spirit of the required change in the Relational Model and some results that then follow. It remains to be investigated whether these results are satisfactory

Datatype Laws Without Signatures

Using the well-known categorical notion of `functor' one may define the concept of datatype (algebra) without being forced to introduce a signature, that is, names and typings for the individual sorts (types) and operations involved. This has proved to be advantageous for those theory developments where one is not interested in the syntactic appearance of an algebra. The categorical notion of `transformer' developed in this paper allows the same approach to laws: without using signatures one can define the concept of law for datatypes (lawful algebras), and investigate the equational specification of datatypes in a syntax-free way. A transformer is a special kind of functor and also a natural transformation on the level of dialgebras. Transformers are quite expressive, satisfy several closure properties, and are related to naturality and Wadler's Theorems For Free. In fact, any colimit is an initial lawful algebra

Inference Optimization using Relational Algebra

Exact inference procedures in Bayesian networks can be expressed using relational algebra; this provides a common ground for optimizations from the AI and database communities. Specifically, the ability to accomodate sparse representations of probability distributions opens up the way to optimize for their cardinality instead of the dimensionality; we apply this in a sensor data model.\u

Subtyping can have a simple semantics

Consider a first order typed language, with semantics $S$ for expressions and types. Adding subtyping means that a partial order $<$; on types is defined and that the typing rules are extended to the effect that expression $e$ has type $t$ whenever $e$ has type $s$ and $s<t$ We show how to adapt the semantics $S$ in a simple set theoretic way, obtaining a semantics $S'$ that satisfies, in addition to some obvious requirements, also the property that: $S'~s$ is included in $S'~t$, whenever $s < t$

Program Calculation Properties of Continuous Algebras

Defining data types as initial algebras, or dually as final co-algebras, is beneficial, if not indispensible, for an algebraic calculus for program construction, in view of the nice equational properties that then become available. It is not hard to render finite lists as an initial algebra and, dually, infinite lists as a final co-algebra. However, this would mean that there are two distinct data types for lists, and then a program that is applicable to both finite and infinite lists is not possible, and arbitrary recursive definitions are not allowed. We prove the existence of algebras that are both initial in one category of algebras and final in the closely related category of co-algebras, and for which arbitrary (continuous) fixed point definitions ("recursion") do have a solution. Thus there is a single data type that comprises both the finite and the infinite lists. The price to be paid, however, is that partiality (of functions and values) is unavoidable. \ud We derive, for any such data type, various laws that are useful for an algebraic calculus of programs