We show by elementary methods that given any finite partition of the set ℕ of positive integers, there is one cell that is both additively and multiplicatively rich. In particular, this cell must contain a sequence and all of its finite sums, and another sequence and all of its finite products, a fact that was previously known only by utilizing the algebraic structure of the Stone–Čech compactification βℕ of ℕ.

Notions of deletion and contraction for the class of set functions from finite sets into the integers are defined. An operation on a subclass of such set functions is a function from the subclass into itself that preserves ground sets and respects isomorphism. The operations on set functions that interchange deletion and contraction are characterised, as are those with the further property of being involutary. Similar results are given for polymatroids. There is a unique involutary operation on the class of k-polymatroids that interchanges deletion and contraction. The results generalise those of Kung [3].

Feature unification in parsing has previously used either inefficient Prolog programs, or LISP programs implementing early pre-WAM Prolog models of unification involving searches of binding lists, and the copying of rules to generate edges: features within rules and edges have traditionally been expressed as lists or functions, with clarity being preferred to speed of processing. As a result, parsing takes about 0·5 seconds for a 7-word sentence. Our earlier work produced an optimised chart parser for a non-unification context-free-grammar that achieved 5 ms parses, with high-ambiguity sentences involving hundreds of edges, using the grammar and sentences from Tomita's work on shift-reduce parsing with multiple stack branches. A parallel logic card design resulted that would speed this by a further factor of at least 17. The current paper extends this parser to treat a much more complex unification grammar with structures, using extensive indexing of rules and edges and the optimisations of top-down filtering and look-ahead, to demonstrate where unification occurs during parsing. Unification in parsing is distinguished from that in Prolog, and four alternative schemes for storing features and performing unification are considered, including the traditional binding-list method and three other methods optimised for speed for which overall unification times are calculated. Parallelisation of unification using cheap logic hardware is considered, and estimates show that unification will negligibly increase the parse time of our parallel parser card. Preliminary results are reported from a prototype serial parser that uses the fourth most efficient unification method, and achieves 7 ms for 7-word sentences, and under 1 s for a 36-word 360-way ambiguous sentence with 10,000 edges, on a conventional workstation.

A polynomial-time randomised algorithm for uniformly generating forests in a dense graph is presented. Using this, a fully polynomial randomised approximation scheme (fpras) for counting the number of forests in a dense graph is created.

The partition number of a product hypergraph is introduced as the minimal size of a partition of its vertex set into sets that are edges. This number is shown to be multiplicative if all factors are graphs with all loops included.

Jackson [10] gave a polynomial sufficient condition for a bipartite tournament to contain a cycle of a given length. The question arises as to whether deciding on the maximum length of a cycle in a bipartite tournament is polynomial. The problem was considered by Manoussakis [12] in the slightly more general setting of 2-edge coloured complete graphs: is it polynomial to find a longest alternating cycle in such coloured graphs? In this paper, strong evidence is given that such an algorithm exists. In fact, using a reduction to the well known exact matching problem, we prove that the problem is random polynomial.

Artificial languages for person-machine communication seldom display the most characteristic properties of natural languages, such as the use of anaphoric or other referring expressions, or ellipsis. This paper argues that useful use could be made of such devices in artificial languages, and proposes a mechanism for the resolution of ellipsis and anaphora in them using finite state transduction techniques. This yields an interpretation system displaying many desirable properties: easily implementable, efficient, incremental and reversible.

Linguists in general, and computational linguists in particular, do well to employ finite state devices wherever possible. They are theoretically appealing because they are computationally weak and best understood from a mathematical point of view. They are computationally appealing because they make for simple, elegant, and highly efficient implementations. In this paper, I hope I have shown how they can be applied to a problem… which seems initially to require heavier machinery.

Text-to-speech systems are currently designed to work on complete sentences and paragraphs, thereby allowing front end processors access to large amounts of linguistic context. Problems with this design arise when applications require text to be synthesized in near real time, as it is being typed. How does the system decide which incoming words should be collected and synthesized as a group when prior and subsequent word groups are unknown? We describe a rule-based parser that uses a three cell buffer and phrasing rules to identify break points for incoming text. Words up to the break point are synthesized as new text is moved into the buffer; no hierarchical structure is built beyond the lexical level. The parser was developed for use in a system that synthesizes written telecommunications by Deaf and hard of hearing people. These are texts written entirely in upper case, with little or no punctuation, and using a nonstandard variety of English (e.g. WHEN DO I WILL CALL BACK YOU). The parser performed well in a three month field trial utilizing tens of thousands of texts. Laboratory tests indicate that the parser exhibited a low error rate when compared with a human reader.

Dowling lattices are a class of geometric lattices, based on groups, which have been shown to share many properties with projective geometries. In this paper we show that the automorphisms of Dowling lattices are analogs of the automorphisms of projective geometries. We also treat similar results for several related geometric lattices.

A harmonious colouring of a simple graph G is a proper vertex colouring such that each pair of colours appears together on at most one edge. The harmonious chromatic number h(G) is the least number of colours in such a colouring.

For any positive integer m, let Q(m) be the least positive integer k such that ≥ m. We show that for almost all unlabelled, unrooted trees T, h(T) = Q(m), where m is the number of edges of T.

We consider a random digraph Din, out(n) on vertices 1, …, n, where, for each vertex v, we choose at random one of the n possible arcs with head v and one of the n possible arcs with tail v. We show that the expected size of the largest component of Din, out is .

We give some sufficient conditions for an (S, U)-outline T-factorization of Kn to be an (S, U)-amalgamated T-factorization of Kn. We then apply these to give various necessary and sufficient conditions for edge coloured graphs G to have recoverable embeddings in T-factorized Kn's.

The study of asymptotics of random permutations was initiated by Erdős and Turáan, in a series of papers from 1965 to 1968, and has been much studied since. Recent developments in permutation group theory make it reasonable to ask questions with a more group-theoretic flavour. Two examples considered here are membership in a proper transitive subgroup, and the intersection of a subgroup with a random conjugate. These both arise from other topics (quasigroups, bases for permutation groups, and design constructions).

This paper deals with infinite binary sequences. Each sequence is treated as generated by a nondeterministic shift register. A measure-theoretic criterion helpful in finding a deterministic generator of the set of sequences is proposed.