Finiteness theorems in stochastic integer programming

We study Graver test sets for families of linear multi-stage stochastic integer programs with varying number of scenarios. We show that these test sets can be decomposed into finitely many ``building blocks'', independent of the number of scenarios, and we give an effective procedure to compute these building blocks. The paper includes an introduction to Nash-Williams' theory of better-quasi-orderings, which is used to show termination of our algorithm. We also apply this theory to finiteness results for Hilbert functions.Comment: 36 p

Hyperset Approach to Semi-structured Databases and the Experimental Implementation of the Query Language Delta

This thesis presents practical suggestions towards the implementation of the hyperset approach to semi-structured databases and the associated query language Delta. This work can be characterised as part of a top-down approach to semi-structured databases, from theory to practice. The main original part of this work consisted in implementation of the hyperset Delta query language to semi-structured databases, including worked example queries. In fact, the goal was to demonstrate the practical details of this approach and language. The required development of an extended, practical version of the language based on the existing theoretical version, and the corresponding operational semantics. Here we present detailed description of the most essential steps of the implementation. Another crucial problem for this approach was to demonstrate how to deal in reality with the concept of the equality relation between (hyper)sets, which is computationally realised by the bisimulation relation. In fact, this expensive procedure, especially in the case of distributed semi-structured data, required some additional theoretical considerations and practical suggestions for efficient implementation. To this end the 'local/global' strategy for computing the bisimulation relation over distributed semi-structured data was developed and its efficiency was experimentally confirmed.Comment: Technical Report (PhD thesis), University of Liverpool, Englan

Hyperset approach to semi-structured databases and the experimental implementation of the query language Delta

This thesis presents practical suggestions towards the implementation of the hyperset approach to semi-structured databases and the associated query language Delta. This work can be characterised as part of a top-down approach to semi-structured databases, from theory to practice. The main original part of this work consisted in implementation of the hyperset Delta query language to semi-structured databases, including worked example queries. In fact, the goal was to demonstrate the practical details of this approach and language. The required development of an extended, practical version of the language based on the existing theoretical version, and the corresponding operational semantics. Here we present detailed description of the most essential steps of the implementation. Another crucial problem for this approach was to demonstrate how to deal in reality with the concept of the equality relation between (hyper)sets, which is computationally realised by the bisimulation relation. In fact, this expensive procedure, especially in the case of distributed semi-structured data, required some additional theoretical considerations and practical suggestions for efficient implementation. To this end the 'local/global' strategy for computing the bisimulation relation over distributed semi-structured data was developed and its efficiency was experimentally confirmed. Finally, the XML-WDB format for representing any distributed WDB as system of set equations was developed so that arbitrary XML elements can participate and, hence, queried by the -language. The query system with the syntax of the language and several example queries from this thesis is available online at http://www.csc.liv.ac.uk/˜molyneux/t

Term rewriting systems from Church-Rosser to Knuth-Bendix and beyond

Term rewriting systems are important for computability theory of abstract data types, for automatic theorem proving, and for the foundations of functional programming. In this short survey we present, starting from first principles, several of the basic notions and facts in the area of term rewriting. Our treatment, which often will be informal, covers abstract rewriting, Combinatory Logic, orthogonal systems, strategies, critical pair completion, and some extended rewriting formats

Computability and Complexity Properties of Automatic Structures and their Applications

Finite state automata are Turing machines with fixed finite bounds on resource use. Automata lend themselves well to real-time computations and efficient algorithms. Continuing a tradition of studying computability in mathematics, we examine automatic structures, mathematical objects which can be represented by automata, and apply resulting observations to computer science. We measure the complexity of automatic structures via well-established concepts from model theory, topology, and set theory. We prove the following results. The ordinal height of any automatic well-founded partial order is bounded by \omega^\omega. The ordinal heights of automatic well-founded relations are unbounded below the first uncomputable ordinal. For any computable ordinal, there is an automatic structure of Scott rank at least that ordinal. Moreover, there are automatic structures of Scott rank the first uncomputable ordinal and the successor of the first uncomputable ordinal. For any computable ordinal, there is an automatic successor tree of Cantor-Bendixson rank that ordinal. Next, we study infinite graphs produced from a natural unfolding operation applied to finite graphs. Graphs produced via such operations have finite degree and can be described by finite automata over a one-letter alphabet. We investigate algorithmic properties of such graphs in terms of their finite presentations. In particular, we ask how hard it is to check whether a given node belongs to an infinite component, whether two given nodes in the graph are reachable from one another, and whether the graph is connected. We give polynomial-time algorithms answering each of these questions. For a fixed input graph, the algorithm for infinite component membership works in constant time and reachability is decided uniformly by a single automaton. Hence, we improve on previous work, in which nonelementary or nonuniform algorithms were found. We turn our attention to automata techniques for deciding first-order logical theories. These techniques are useful in Integer Linear Programming and Mixed Integer Linear Programming, which in turn have applications in diverse areas of computer science and engineering. We extend known work to address the enumeration problem for linear programming solutions. Then, we apply a similar paradigm to give an automata theoretic decision procedure for the p-adic valued ring under addition and for formal Laurent series over a finite field with valuation and addition