The combinatorics of abstract container data types

Abstract

The study of abstract machines such as Turing machines, push down automata and finite state machines has played an important role in the advancement of computer science. It has led to developments in the theory of general purpose computers, compilers and string manipulation as well as many other areas. The language associated with an abstract machine characterises an important aspect of the behaviour of that machine. It is therefore the principal object of interest when studying such a machine. In this thesis we consider abstract container data types to be abstract machines. We define the concept of a language associated with an abstract container data type and investigate this in the same spirit as for other abstract machines. We also consider a model which allows us to describe various abstract container data types. This model is studied in a similar manner. There is a rich selection of problems to investigate. For instance, the data items which the abstract container data types operate on can take many forms. The input stream could consist of distinct data items, say 1, 2,..., n, or it could be a word over the binary alphabet. Alternatively it could be a sequence formed from the data items in some arbitrary multiset. Another consideration is whether or not an abstract data type has a finite storage capacity. It is shown how to construct a regular grammar which generates (an encoded form of) the set of permutations which can be realised by moving tokens through a network. A one to one correspondence is given between ordered forests of bounded height and members of the language associated with a bounded capacity priority queue operating on binary data. A number of related results are also proved; in particular for networks operating on binary data, and priority queues of capacity 2.