We now define the notion, already discussed, of an effectively calculable function of positive integers by identifying it with the notion of a recursive function of positive integers (or of a lambdadefinable function of positive integers). The phrase in parentheses refers to the apparatus which Church had developed to investigate this and other problems in the foundations of mathematics: the calculus of lambda conversion. Both the Thesis and the lambda calculus have been of seminal influence on the development of Computing Science. The main subject of this article is the lambda calculus but I will begin with a brief sketch of the emergence of the Thesis. The epistemological status of Church’s Thesis is not immediately clear from the above quotation and remains a matter of debate, as is explored in other papers of this volume. My own view, which I will state but not elaborate here, is that the thesis is empirical because it relies for its significance on a claim about what can be calculated by mechanisms. This becomes clearer in

. In lazy functional languages, ? is typically an element of every type. While this provides great flexibility, it also comes at a cost. In this paper we explore the consequences of allowing unpointed types in a lazy functional language like Haskell. We use the type (and class) system to keep track of pointedness, and show the consequences for parametricity and for controlling evaluation order and unboxing. 1 Introduction Ever since Scott and others showed how to use pointed CPOs (i.e. with bottoms) to give meaning to general recursion, both over values (including functions), and over types themselves, functional languages seem to have been wedded to the concept. Languages like Haskell [5] model types by appropriate CPOs and, because non-terminating computations can happen at any type, all the CPOs are pointed. This gives significant flexibility. In particular, values of any type may be defined using recursion. 1.1 Parametricity There are associated costs, however. When reason...

Pattern matching is an important operation used in many applications such as functional programming, rewriting and rule-based expert systems. By preprocessing the patterns into a DFA-like automaton, we can rapidly select the matching pattern(s) in a single scan of the relevant portions of the input term. This automaton is typically based on left-to-right traversal of the patterns. By adapting the traversal order to suit the set of input patterns, it is possible to considerably reduce the space and matching time requirements of the automaton.

Many language theoreticians have taken great efforts in designing higher-level programming languages that are more elegant and more expressive than conventional languages. However, few of these new languages have been implemented very efficiently. The result is that most software engineers still prefer to use conventional languages, even though the new higherlevel languages offer a better and simpler programming model. This dissertation concentrates on improving the performance of programs written in Standard ML (SML)---a statically typed functional language---on today's RISC machines. SML poses tough challenges to efficient implementations: very frequent function calls, polymorphic types, recursive data structures, higher-order functions, and first-class continuations. This dissertation presents the design and evaluation of several new compilation techniques that meet these challenges by taking advantage of some of the higher-level language features in SML. Type-directed compilation ...

Our objective is to describe a formal proof of correctness for the following Haskell [13] program in a type theory-based proof verification system, such as the Coq system [10, 1]. sieve (p:rest) = p:sieve [r | r <- rest, r ‘rem ‘ p / = 0] primes = sieve [2..] This program is a functional implementation of Eratosthenes ’ sieve that consists in removing all multiples of previously found primes from the sequence of natural numbers. We want to prove that the expression primes is the stream containing all the prime numbers in increasing order. This work relies on co-inductive types [5, 11, 12] because the program manipulates infinite lists, also known as streams. It first uses the infinite list of natural numbers larger than 2, then the infinite list of numbers larger than 3 and containing no multiples of 2, then the infinite list of numbers larger than 4 and containing no multiples of prime numbers smaller than 4, and so on. This example was initially proposed as a challenge by G. Kahn and used as an illustration of a program and its proof of correctness in a

. Modern compilers for ML-like polymorphic languages have used explicit run-time type passing to support advanced optimizations such as intensional type analysis, representation analysis and tagless garbage collection. Unfortunately, maintaining type information at run time can incur a large overhead to the time and space usage of a program. In this paper, we present an optimal type-lifting algorithm that lifts all type applications in a program to the top level. Our algorithm eliminates all run-time type constructions within any core-language functions. In fact, it guarantees that the number of types built at run time is strictly a static constant. We present our algorithm as a type-preserving source-to-source transformation and show how to extend it to handle the entire SML'97 with higher-order modules. 1 Introduction Modern compilers for ML-like polymorphic languages [16, 17] usually use variants of the Girard-Reynolds polymorphic -calculus [5, 26] as their intermediate language (I...

A simple formal semantics for the standard hardware description language vhdl is set out in functional style. The presentation comprises an executable specification for a synchronously clocked vhdl simulator. 1 Introduction This paper attempts to lay the basis for simpler formal semantics for vhdl than available in the literature. Vhdl is a standardized hardware description language developed in the 1980s with the support of the US DoD, and it has gained wide acceptance. No formal semantics underpins the standard [4], and there has been great interest in developing a formal semantics for the simulation model it defines. Timing aspects are formalized in [7], but [9] appears to be the most complete and satisfactory formalization to date. It sets out an asynchronous operational semantics for vhdl simulators. In contrast, we set out a synchronous semantics. Vhdl is seen as a side-effect on a clocked sequence of global states; one for every unit interval of time. A good formal semantics...

m networks and the recent works of Thierry Coquand in type theory have been the most important sources of motivation for the ideas presented here. I wish to specially thank Roberto Amadio, who read the manuscript in a very short delay, providing many helpful comments and remarks. Many thanks also to Luc Boug'e, who accepted to be my oficial supervisor, and to the chair of the jury, Michel Cosnard, who opened to me the doors of the LIP. During these last three years in Lyon I met many wonderful people, who then become wonderful friends. Miguel, Nuria, Veronique, Patricia, Philippe, Pia, Rodrigo, Salvador, Sophie : : : with you I have shared the happiness and sadness of everyday life, those little things which make us to remember someone forever. I also would like to thank the people from "Tango de Soie", for all those funny nights at the Caf'e Moulin Joly. Thanks too to the Uruguayan research community in Computer Science (specially to Cristina Cornes and Alberto Pardo) w

: We derive a compiler for ML-style pattern matches. It is conceptually simple and produces reasonably good compiled matches. The derivation is inspired by the instrumentation and partial evaluation of naive string matchers. Following that paradigm, we first present a general and naive ML pattern matcher, instrument it to collect and exploit extra information, and show that partial evaluation of the instrumented general matcher with respect to a given match produces an efficient specialized matcher. We then discard the partial evaluator and show that a match compiler can be obtained just by slightly modifying the instrumented general matcher. The resulting match compiler is interesting in its own right, and naturally detects inexhaustive matches and redundant match rules. 1 Introduction Consel and Danvy [3] studied a general string matcher, taking two inputs: a pattern string to look for, and an object string to look in. The matcher finds the first occurrence (if any) of the pattern ...

Tupling is a transformation tactic to obtain new functions, without redundant calls and/or multiple traversals of common inputs. It achieves this feat by allowing each set (tuple) of function calls to be computed recursively from its previous set. In previous works by Chin and Khoo [8, 9], a safe (terminating) fold/unfold transformation algorithm was developed for some classes of functions which are guaranteed to be successfully tupled. However, these classes of functions currently use static-sized tables for eliminating the redundant calls. As shown by Richard Bird in [3], there are also other classes of programs whose redundant calls could only be eliminated by using dynamic-sized tabulation. This paper proposes a new solution to dynamic-sized tabulation by an extension to the tupling tactic. Our extension uses lambda abstractions which can be viewed as either dynamic-sized tables or applications of the higher-order generalisation technique to facilitate tupling. Significant speedups could be obtained after the transformed programs were vectorised, as confirmed by experiment.