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Types for Modules
, 1998
"... 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 dy ..."
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Cited by 68 (9 self)
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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.
The UnderAppreciated Unfold
 In Proceedings of the Third ACM SIGPLAN International Conference on Functional Programming
, 1998
"... 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 breadthfirst traversal of a tree. We specify brea ..."
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Cited by 48 (11 self)
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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 breadthfirst traversal of a tree. We specify breadthfirst traversal in terms of levelorder 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 breadthfirst traversal directly as an unfold; this turns out to be the `standard' queuebased algorithm.
Set theory for verification: I. From foundations to functions
 J. Auto. Reas
, 1993
"... A logic for specification and verification is derived from the axioms of ZermeloFraenkel 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 higherord ..."
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Cited by 45 (17 self)
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A logic for specification and verification is derived from the axioms of ZermeloFraenkel 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 higherorder 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,
Programming Languages and Dimensions
, 1996
"... 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. ..."
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Cited by 35 (3 self)
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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 welltyped 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
Safe Functional Reactive Programming through Dependent Types
"... Functional Reactive Programming (FRP) is an approach to reactive programming where systems are structured as networks of functions operating on signals. FRP is based on the synchronous dataflow paradigm and supports both continuoustime and discretetime signals (hybrid systems). What sets FRP apart ..."
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Cited by 12 (0 self)
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Functional Reactive Programming (FRP) is an approach to reactive programming where systems are structured as networks of functions operating on signals. FRP is based on the synchronous dataflow paradigm and supports both continuoustime and discretetime signals (hybrid systems). What sets FRP apart from most other languages for similar applications is its support for systems with dynamic structure and for higherorder reactive constructs. Statically guaranteeing correctness properties of programs is an attractive proposition. This is true in particular for typical application domains for reactive programming such as embedded systems. To that end, many existing reactive languages have type systems or other static checks that guarantee domainspecific properties, such as feedback loops always being wellformed. However, they are limited in their capabilities to support dynamism and higherorder dataflow compared with FRP. Thus, the onus of ensuring such properties of FRP programs has so far been on the programmer as established static techniques do not suffice. In this paper, we show how dependent types allow this concern to be addressed. We present an implementation of FRP embedded in the dependentlytyped language Agda, leveraging the type system of the host language to craft a domainspecific (dependent) type system for FRP. The implementation constitutes a discrete, operational semantics of FRP, and as it passes the Agda type, coverage, and termination checks, we know the operational semantics is total, which means our type system is safe. Categories and Subject Descriptors D.3.2 [Programming Languages]: Language Classifications—applicative (functional) languages, dataflow languages, specialized application languages General Terms Languages Keywords dependent types, domainspecific languages, DSELs, FRP, functional programming, reactive programming, synchronous dataflow
Dependent types at work
 LERNET 2008. LNCS
, 2009
"... In these lecture notes we give an introduction to functional programming with dependent types. We use the dependently typed programming language Agda which is an extension of MartinLöf type theory. First we show how to do simply typed functional programming in the style of Haskell and ML. Some dif ..."
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Cited by 12 (1 self)
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In these lecture notes we give an introduction to functional programming with dependent types. We use the dependently typed programming language Agda which is an extension of MartinLöf type theory. First we show how to do simply typed functional programming in the style of Haskell and ML. Some differences between Agda’s type system and the HindleyMilner type system of Haskell and ML are also discussed. Then we show how to use dependent types for programming and we explain the basic ideas behind typechecking dependent types. We go on to explain the CurryHoward identification of propositions and types. This is what makes Agda a programming logic and not only a programming language. According to CurryHoward, we identify programs and proofs, something which is possible only by requiring that all program terminate. However, at the end of these notes we present a method for encoding partial and general recursive functions as total functions using dependent types.
A typed foundation for directional logic programming
 In Proc. Workshop on Extensions to Logic Programming
, 1992
"... 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 d ..."
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Cited by 12 (1 self)
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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 deadlockfreedom) 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 typesaspropositions paradigm, enriching it with static scoping and higherorder features.