We present a series of improvements to the Hybrid system, a formal theory implemented in Isabelle/HOL to support specifying and reasoning about formal systems using higher-order abstract syntax (HOAS). We modify Hybrid's type of terms, which is built definitionally in terms of de Bruijn indices, to exclude at the type level terms with `dangling' indices. We strengthen the injectivity property for Hybrid's variable-binding operator, and develop rules for compositional proof of its side condition, avoiding conversion from HOAS to de Bruijn indices. We prove representational adequacy of Hybrid (with these improvements) for a lambda-calculus-like subset of Isabelle/HOL syntax, at the level of set-theoretic semantics and without unfolding Hybrid's definition in terms of de Bruijn indices. In further work, we prove an induction principle that maintains some of the benefits of HOAS even for open terms. We also present a case study of the formalization in Hybrid of a small programming language, Mini-ML with mutable references, including its operational semantics and a type-safety property. This is the largest case study in Hybrid to date, and the first to formalize a language with mutable references. We compare four variants of this formalization based on the two-level approach adopted by Felty and Momigliano in other recent work on Hybrid, with various specification logics (SLs), including substructural logics, formalized in Isabelle/HOL and used in turn to encode judgments of the object language. We also compare these with a variant that does not use an intermediate SL layer. In the course of the case study, we explore and develop new proof techniques, particularly in connection with context invariants and induction on SL statements.

We present a series of improvements to the Hybrid system, a formal theory implemented in Isabelle/HOL to support specifying and reasoning about formal systems using higher-order abstract syntax (HOAS). We modify Hybrid's type of terms, which is built definitionally in terms of de Bruijn indices, to exclude at the type level terms with `dangling' indices. We strengthen the injectivity property for Hybrid's variable-binding operator, and develop rules for compositional proof of its side condition, avoiding conversion from HOAS to de Bruijn indices. We prove representational adequacy of Hybrid (with these improvements) for a lambda-calculus-like subset of Isabelle/HOL syntax, at the level of set-theoretic semantics and without unfolding Hybrid's definition in terms of de Bruijn indices. In further work, we prove an induction principle that maintains some of the benefits of HOAS even for open terms. We also present a case study of the formalization in Hybrid of a small programming language, Mini-ML with mutable references, including its operational semantics and a type-safety property. This is the largest case study in Hybrid to date, and the first to formalize a language with mutable references. We compare four variants of this formalization based on the two-level approach adopted by Felty and Momigliano in other recent work on Hybrid, with various specification logics (SLs), including substructural logics, formalized in Isabelle/HOL and used in turn to encode judgments of the object language. We also compare these with a variant that does not use an intermediate SL layer. In the course of the case study, we explore and develop new proof techniques, particularly in connection with context invariants and induction on SL statements.

We present a series of improvements to the Hybrid system, a formal theory implemented in Isabelle/HOL to support specifying and reasoning about formal systems using higher-order abstract syntax (HOAS). We modify Hybrid's type of terms, which is built definitionally in terms of de Bruijn indices, to exclude at the type level terms with `dangling' indices. We strengthen the injectivity property for Hybrid's variable-binding operator, and develop rules for compositional proof of its side condition, avoiding conversion from HOAS to de Bruijn indices. We prove representational adequacy of Hybrid (with these improvements) for a lambda-calculus-like subset of Isabelle/HOL syntax, at the level of set-theoretic semantics and without unfolding Hybrid's definition in terms of de Bruijn indices. In further work, we prove an induction principle that maintains some of the benefits of HOAS even for open terms. We also present a case study of the formalization in Hybrid of a small programming language, Mini-ML with mutable references, including its operational semantics and a type-safety property. This is the largest case study in Hybrid to date, and the first to formalize a language with mutable references. We compare four variants of this formalization based on the two-level approach adopted by Felty and Momigliano in other recent work on Hybrid, with various specification logics (SLs), including substructural logics, formalized in Isabelle/HOL and used in turn to encode judgments of the object language. We also compare these with a variant that does not use an intermediate SL layer. In the course of the case study, we explore and develop new proof techniques, particularly in connection with context invariants and induction on SL statements.

We present a series of improvements to the Hybrid system, a formal theory implemented in Isabelle/HOL to support specifying and reasoning about formal systems using higher-order abstract syntax (HOAS). We modify Hybrid's type of terms, which is built definitionally in terms of de Bruijn indices, to exclude at the type level terms with `dangling' indices. We strengthen the injectivity property for Hybrid's variable-binding operator, and develop rules for compositional proof of its side condition, avoiding conversion from HOAS to de Bruijn indices. We prove representational adequacy of Hybrid (with these improvements) for a lambda-calculus-like subset of Isabelle/HOL syntax, at the level of set-theoretic semantics and without unfolding Hybrid's definition in terms of de Bruijn indices. In further work, we prove an induction principle that maintains some of the benefits of HOAS even for open terms. We also present a case study of the formalization in Hybrid of a small programming language, Mini-ML with mutable references, including its operational semantics and a type-safety property. This is the largest case study in Hybrid to date, and the first to formalize a language with mutable references. We compare four variants of this formalization based on the two-level approach adopted by Felty and Momigliano in other recent work on Hybrid, with various specification logics (SLs), including substructural logics, formalized in Isabelle/HOL and used in turn to encode judgments of the object language. We also compare these with a variant that does not use an intermediate SL layer. In the course of the case study, we explore and develop new proof techniques, particularly in connection with context invariants and induction on SL statements.

This thesis consists of two parts. In the first part, we study a class of special functions Aik (y), k = 2, 4, 6, ··· generalising the classical Airy function Ai(y) to higher orders and in the second part, we apply expressions and properties of Ai4(y) to spectral problem of a specific operator. The first part is however motivated by latter part. We establish regularity properties of Aik (y) and particularly show that Aik (y) is smooth, bounded, and extends to the complex plane as an entire function, and obtain pointwise bounds on Aik (y) for all k. Some analytic properties of Aik (y) are also derived allowing one to express Aik (y) as a finite sum of certain generalised hypergeometric functions. We further obtain full asymptotic expansions of Aik (y) and their first derivative Ai'(y) both for y > 0 and for y < 0. Using these expansions, we derive expressions for the negative real zeroes of Aik (y) and Ai'(y). Using expressions and properties of Ai4(y), we extensively study spectral properties of a non-local operator H whose physical interpretation is the massless relativistic quartic anharmonic oscillator in one dimension. Various spectral results for H are derived including estimates of eigenvalues, spectral gaps and trace formula, and a Weyl-type asymptotic relation. We study asymptotic behaviour, analyticity, and uniform boundedness properties of the eigenfunctions Ψn(x) of H. The Fourier transforms of these eigenfunctions are expressed in two terms, one involving Ai4(y) and another term derived from Ai4(y) denoted by Āi4(y). By investigating the small effect generated by Āi4(y) this work shows that eigenvalues λn of H are exponentially close, with increasing n Ε N, to the negative real zeroes of Ai4(y) and those of its first derivative Ai'4(y) arranged in alternating and increasing order of magnitude. The eigenfunctions Ψ(x) are also shown to be exponentially well-approximated by the inverse Fourier transform of Ai4(|y| - λn) in its normalised form.

Subject searching—seeking information with a subject or topic in mind—is often involved in carrying out undergraduate assignments such as term papers and research reports. It is also an important component of information literacy—the abilities and experiences of effectively finding and evaluating, and appropriately using, needed information—which universities hope to cultivate in undergraduates by the time they complete their degree programs. By exploring the subject searching of a small group of upper-level, academically successful undergraduates over a school year I sought to acquire a deeper understanding of the contexts and characteristics of their subject searching, and of the extent to which it was similar in quality to that of search and domain experts. Primary data sources for this study comprised subject searching diaries maintained by participants, and three online subject searches they demonstrated at the beginning, middle, and end of the study during which they talked aloud while I observed, followed by focused interviews. To explore the quality of study participants’ subject searching I looked for indications of advanced thinking in thoughts they spoke aloud during demonstration sessions relating to using strategy, evaluating, and creating personal understanding, which represent three of the most challenging and complex aspects of information literacy. Applying a layered interpretive process, I identified themes within several hundred instances of participants’ advanced thinking relating to these three information literacy elements, with evaluative themes occurring most often. I also noted three factors influencing the extent of similarity between the quality of participants’ advanced thinking and that of search and domain experts which reflected matters that tended to be i) pragmatic or principled, , ii) technical or conceptual, and iii) externally or internally focused. Filtered through these factors, participants’ instances of advanced thinking brought to mind three levels of subject searching abilities: the competent student, the search expert, and the domain expert. Although relatively few in number, I identified at least some advanced thinking evincing domain expert qualities in voiced thoughts of all but one participant, suggesting the gap between higher order thinking abilities of upper-level undergraduates and information literate individuals is not always dauntingly large.

Higher-order transformations are ubiquitous within data management. In relational databases, higher-order queries appear in numerous aspects including query rewriting and query specification. In XML databases, higher-order functions are natural due to the close connection of XML query languages with functional programming. The thesis investigates higher-order query languages that combine higher- order transformations with ordinary database query languages. We de- fine higher-order query languages based on Relational Algebra, Monad Algebra, and XQuery. The thesis also studies basic problems for these query languages including evaluation, containment, and type inference. We show that even though evaluating these higher-order query languages is non-elementary, there are subclasses that are polynomially reducible to evaluation for ordinary query languages. Our theoretical analysis is complemented by an implementation of the languages, our Higher-Order Mapping Evaluation System (HOMES). The system integrates querying and query transformation in a single higher- order query language. It allows users to write queries that integrate and combine query transformations. The system is implemented on top of traditional database management systems. The evaluation algorithm is optimized by a combination of subquery caching techniques from relational and XML databases and sharing detection schemes from functional programming.

According to orthodoxy the study of the Soritical and semantical paradoxes belongs to the domain of the philosophy of language. To solve these paradoxes we need to investigate the nature of words like `heap' and `true.' In this thesis I criticise linguistic explanations of the state of ignorance we find ourselves in when confronted with indeterminate cases and develop a classical non-linguistic theory of indeterminacy in its stead. The view places the study of vagueness and indeterminacy squarely in epistemological terms, situating it within a theory of rational propositional attitudes. The resulting view is applied to a number of problems in the philosophy of vagueness and the semantic paradoxes.

This thesis examines the use of game semantics for the automatic equivalence checking of higher-order programs. Game semantics has proved to be a powerful method for constructing fully abstract models of logics and programming languages. Furthermore, the concrete nature of the semantics lends itself to algorithmic analysis. The game-semantic model can be used to identify fragments of languages which have a decidable observational equivalence problem. We investigate decidability results for different languages as well as the efficiency of these algorithms in practice. First we consider the call-by-value higher-order language with state, RML. This can be viewed as a canonical restriction of Standard ML to ground-type references. The O-strict fragment of RML is the largest set of type sequents for which, in the game-semantic denotation, justification pointers from O-moves are always uniquely reconstructible from the underlying move sequence. The O-strict fragment is surprisingly expressive, including higher-order types and difficult examples from the literature. By representing strategies as Visibly Pushdown Automata (VPA) we show that observational equivalence of O-strict terms is decidable (and in fact is ExpTime-complete). We then consider extensions of the O-strict fragment. Adding general recursion or using most non-O-strict types leads to undecidability. However, a limited form of recursion can be added while still preserving decidability (although the full power of DPDA is required). Next we examine languages with non-local control. This involves adding call/cc to our language and is known to correspond to dropping the game-semantic bracketing condition. In the call-by-name game-semantic model of Idealized Algol (IA), in which answers cannot justify questions, the visibility condition still implies a form of weak bracketing. By making bracketing violations explicit we show that we can still model the entire third-order fragment using VPA. We have also implemented tools based on these algorithms. Our model checkers Homer and Hector perform equivalence checking for third-order IA and O-strict RML respectively. Homer uses a naive explicit state method whereas Hector takes advantage of on-the-fly model checking. Our tools perform well on small yet challenging examples. On negative instances, the on-the-fly approach allows Hector to outperform Homer. To improve their performance, we also consider using ideas from symbolic execution. We propose a representation for finite automata using transitions labelled with formulas and guards which aims to take advantage of the symmetries of the game-semantic model so that strategies can be represented compactly. We refer to this representation as Symbolically Executed Automata (SEA). Using SEA allows much larger data types to be handled but is not as effective on larger examples with small data types.

Student Number : 9903549T - MSc dissertation - School of Process and Materials Engineering - Faculty of Engineering and the Built Environment / Preliminary evaluations during flowsheet synthesis require simple effective tools for comparison and elimination of process alternatives. This work investigates three areas of interest in distillation. Column profile map theory has simplified complex column investigations. The predictions of the difference point equation at finite reflux were experimentally verified for the acetone, methanol and ethanol system in a continuous column apparatus. Residue curve analysis is usually limited to systems with four components for distillation system analysis. An alternative representation, based on combinatorial topology and temperature sequencing, is introduced for use in high level synthesis decisions for higher component order systems. Attainable region (AR) theory is applied to an ideal binary distillation system for a geometrically based method of cost analysis. A constrained attainable region is constructed from a series of equilibrium step compositions with varying reflux and corresponding cost associations. The AR is shown to be useful for costing and optimization.