The programming language Standard ML is an amalgam of two, largely orthogonal, languages. The Core language expresses details of algorithms and data structures. The Modules language expresses the modular architecture of a software system. Both languages are statically typed, with their static and dynamic semantics specified by a formal definition.

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Folds are appreciated by functional programmers. Their dual, unfolds, are not new, but they are not nearly as well appreciated. We believe they deserve better. To illustrate, we present (indeed, we calculate) a number of algorithms for computing the breadth-first traversal of a tree. We specify breadth-first traversal in terms of level-order traversal, which we characterize first as a fold. The presentation as a fold is simple, but it is inefficient, and removing the inefficiency makes it no longer a fold. We calculate a characterization as an unfold from the characterization as a fold; this unfold is equally clear, but more efficient. We also calculate a characterization of breadth-first traversal directly as an unfold; this turns out to be the `standard' queue-based algorithm.

A logic for specification and verification is derived from the axioms of Zermelo-Fraenkel set theory. The proofs are performed using the proof assistant Isabelle. Isabelle is generic, supporting several different logics. Isabelle has the flexibility to adapt to variants of set theory. Its higher-order syntax supports the definition of new binding operators. Unknowns in subgoals can be instantiated incrementally. The paper describes the derivation of rules for descriptions, relations and functions, and discusses interactive proofs of Cantor’s Theorem, the Composition of Homomorphisms challenge [9], and Ramsey’s Theorem [5]. A generic proof assistant can stand up against provers dedicated to particular logics. Key words. Isabelle, set theory, generic theorem proving, Ramsey’s Theorem,

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Scientists and engineers must ensure that the equations and formulae which they use are dimensionally consistent, but existing programming languages treat all numeric values as dimensionless. This thesis investigates the extension of programming languages to support the notion of physical dimension. A type system is presented similar to that of the programming language ML but extended with polymorphic dimension types. An algorithm which infers most general dimension types automatically is then described and proved correct. The semantics of the language is given by a translation into an explicitlytyped language in which dimensions are passed as arguments to functions. The operational semantics of this language is specified in the usual way by an evaluation relation defined by a set of rules. This is used to show that if a program is well-typed then no dimension errors can occur during its evaluation. More abstract properties of the language are investigated using a denotational semantics: these include a notion of invariance under changes in the units of measure used, analogous to parametricity in the polymorphic lambda calculus. Finally the dissertation is summarised and many possible directions for future research in dimension types and related type systems are described. i ii

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Abstract. A long standing problem in logic programming is how to impose directionality on programs in a safe fashion. The benefits of directionality include freedom from explicit sequential control, the ability to reason about algorithmic properties of programs (such as termination, complexity and deadlock-freedom) and controlling concurrency. By using Girard’s linear logic, we are able to devise a type system that combines types and modes into a unified framework, and enables one to express directionality declaratively. The rich power of the type system allows outputs to be embedded in inputs and vice versa. Type checking guarantees that values have unique producers, but multiple consumers are still possible. From a theoretical point of view, this work provides a “logic programming interpretation ” of (the proofs of) linear logic, adding to the concurrency and functional programming interpretations that are already known. It also brings logic programming into the broader world of typed languages and types-as-propositions paradigm, enriching it with static scoping and higher-order features.

A type of ‘non-traditional constituents’ motivates an extended class of Combinatory Categorial Grammars, CCGs with Generalized Type-Raised Categories (CCG-GTRC) involving variables. Although the class of standard CCGs is known to be polynomially parsable, use of variables suggests more complexity for processing GTRCs. This paper argues that polynomial parsing is still possible for CCG-GTRC from practical and theoretical points of view. First, we show that an experimental parser runs polynomially in practice on a realistic fragment of Japanese by eliminating spurious ambiguity and excluding genuine ambiguities. Then, we present a worst-case polynomial recognition algorithm for CCG-GTRC by extending the polynomial algorithm for the standard CCGs.

We present a constructive formalization of the Myhill-Nerode theorem on the minimization of finite automata that follows the account in Hopcroft and Ullman's book Formal Languages and Their Relation to Automata. We chose to formalize this theorem because it illustrates many points critical to formalization of computational mathematics, especially the extraction of an important algorithm from a proof as a method of knowing that the algorithm is correct. It also gave us an opportunity to experiment with a constructive implementation of quotient sets. We carried out the formalization in Nuprl, an interactive theorem prover based on constructive type theory. Nuprl borrows an implementation of the ML language from the LCF system of Milner, Gordon, and Wadsworth, and makes heavy use of the notion of tactic pioneered by Milner in LCF. We are interested in the pedagogical value of electronic formal mathematical texts and have put our formalization on the World Wide Web. Readers are invited to ...