A bag (literally and mathematically!) of marbles is deemed to be mingled if all the colours of the marbles are different. Given a positive integer $k$ and a collection of $m$ marbles, our objective is to try and extract as many mingled bags as possible from the collection, where each bag contains exactly $k$ marbles.

Pure Type Systems (PTS) come in two flavours: domain-free systems with untyped $\lambda$-abstractions (i.e. of the form $\lambda{x}.{M}$); and domain-free systems with typed $\lambda$-abstractions (i.e. of the form $\lambda{x}{A}{M}$). Both flavours of systems are related by an erasure function $\er{.}$ that removes types from $\lambda$-abstractions. Preservation of Equational Theory, which states the equational theories of both systems coincide through the erasure function, is a property of functional and normalizing PTSs. In this paper we establish that Preservation of Equational Theory fails for some non-normalizing PTSs, including the PTS with $\ast:\ast$. The gist of our argument is to exhibit a typable expression $Y_H$ whose erasure $\er{Y}$ is a fixpoint combinator, but which is not a fixpoint combinator itself.

This paper introduces reFLect, a functional programming language with reflection features intended for applications in hardware design and verification. The reFLect language is strongly typed and similar to ML, but has quotation and antiquotation constructs. These may be used to construct and decompose expressions in the reFLect language itself. The paper motivates and presents the syntax and type system of this language, which brings together a new combination of pattern-matching and reflection features targeted specifically at our application domain. It also gives an operational semantics based on a novel use of contexts as expression constructors, and it presents a scheme for compiling reFLect programs using the same context mechanism.

We introduce 2-3 finger trees, a functional representation of persistent sequences supporting access to the ends in amortized constant time, and concatenation and splitting in time logarithmic in the size of the smaller piece. Representations achieving these bounds have appeared previously, but 2-3 finger trees are much simpler, as are the operations on them. Further, by defining the split operation in a general form, we obtain a general purpose data structure that can serve as a sequence, priority queue, search tree, priority search queue and more.

In a typing system, there are two approaches that may be taken to the notion of equality. One can use some external relation of convertibility defined on the terms of the grammar, such as $\beta$-convertibility or $\beta \eta$-convertibility; or one can introduce a judgement form for equality into the rules of the typing system itself. For quite some time, it has been an open problem whether the two systems produced by these two choices are equivalent. This problem is essentially the problem of proving that the Subject Reduction property holds in the system with judgemental equality. In this paper, we shall prove that the equivalence holds for all functional Pure Type Systems (PTSs). The proof essentially consists of proving the Church-Rosser Theorem for a typed version of parallel one-step reduction. This method should generalise easily to many typing systems which satisfy the Uniqueness of Types property.