Recursion does not always help

We show that adding recursion does not increase the total functions definable in the typed $\lambda\beta\eta$-calculus or the partial functions definable in the $\lambda\Omega$-calculus. As a consequence, adding recursion does not increase the class of partial or total definable functions on free algebras and so, in particular, on the natural numbers.Comment: Improved presentation a littl

A Model of Cooperative Threads

We develop a model of concurrent imperative programming with threads. We focus on a small imperative language with cooperative threads which execute without interruption until they terminate or explicitly yield control. We define and study a trace-based denotational semantics for this language; this semantics is fully abstract but mathematically elementary. We also give an equational theory for the computational effects that underlie the language, including thread spawning. We then analyze threads in terms of the free algebra monad for this theory.Comment: 39 pages, 5 figure

Smart Choices and the Selection Monad

Describing systems in terms of choices and their resulting costs and rewards offers the promise of freeing algorithm designers and programmers from specifying how those choices should be made; in implementations, the choices can be realized by optimization techniques and, increasingly, by machine learning methods. We study this approach from a programming-language perspective. We define two small languages that support decision-making abstractions: one with choices and rewards, and the other additionally with probabilities. We give both operational and denotational semantics. In the case of the second language we consider three denotational semantics, with varying degrees of correlation between possible program values and expected rewards. The operational semantics combine the usual semantics of standard constructs with optimization over spaces of possible execution strategies. The denotational semantics, which are compositional and can also be viewed as an implementation by translation to a simpler language, rely on the selection monad, to handle choice, combined with an auxiliary monad, to handle other effects such as rewards or probability. We establish adequacy theorems that the two semantics coincide in all cases. We also prove full abstraction at ground types, with varying notions of observation in the probabilistic case corresponding to the various degrees of correlation. We present axioms for choice combined with rewards and probability, establishing completeness at ground types for the case of rewards without probability

Three Inadequate Models

The connection between operational and denotational semantics is of longstanding interest in the study of programming languages. One naturally seeks positive results. For example in [FP94, Sim99] adequacy results are given for models in a variety of categories. Again, the failure of full abstraction in the standard models constructed using complete partial orders and continuous functions [Plo77, Mil77] prompted the exploration of other categories (see, e.g., [BCL85, FJM96, AM98, AC98]) with varying degrees of success. In this paper we interest ourselves in counterexamples in order to make a case that these natural avenues of research had a degree of necessity. To this end, we construct inadequate models and investigate whether one can do better than the standard model, but still stay in the category of complete partial orders. (In contrast, an inadequate standard model of PCF is given in [Sim99]—but in a specially constructed category.

Automatic Methods of Inductive Inference

This thesis is concerned with algorithms for generating generalisations-from experience. These algorithms are viewed as examples of the general concept of a hypothesis discovery system which, in its turn, is placed in a framework in which it is seen as one component in a multi-stage process which includes stages of hypothesis criticism or justification, data gathering and analysis and prediction. Formal and informal criteria, which should be satisfied by the discovered hypotheses are given. In particular, they should explain experience and be simple. The formal work uses the first-order predicate calculus. These criteria are applied to the case of hypotheses which are generalisations from experience. A formal definition of generalisation from experience, relative to a body of knowledge is developed and several syntactical simplicity measures are defined. This work uses many concepts taken from resolution theory (Robinson, 1965). We develop a set of formal criteria that must be satisfied by any hypothesis generated by an algorithm for producing generalisation from experience. The mathematics of generalisation is developed. In particular, in the case when there is no body of knowledge, it is shown that there is always a least general generalisation of any two clauses, in the generalisation ordering. (In resolution theory, a clause is an abbreviation for a disjunction of literals.) This least general generalisation is effectively obtainable. Some lattices induced by the generalisation ordering, in the case where there is no body of knowledge, are investigated. The formal set of criteria is investigated. It is shown that for a certain simplicity measure, and under the assumption that there is no body of knowledge, there always exist hypotheses which satisfy them. Generally, however, there is no algorithm which, given the sentences describing experience, will produce as output a hypothesis satisfying the formal criteria. These results persist for a wide range of other simplicity measures. However several useful cases for which algorithms are available are described, as are some general properties of the set of hypotheses which satisfy the criteria. Some connections with philosophy are discussed. It is shown that, with sufficiently large experience, in some cases, any hypothesis which satisfies the formal criteria is acceptable in the sense of Hintikka and Hilpinen (1966). The role of simplicity is further discussed. Some practical difficulties which arise because of Goodman's (1965) "grue" paradox of confirmation theory are presented. A variant of the formal criteria suggested by the work of Meltzer (1970) is discussed. This allows an effective method to be developed when this was not possible before. However, the possibility is countenanced that inconsistent hypotheses might be proposed by the discovery algorithm. The positive results on the existence of hypotheses satisfying the formal criteria are extended to include some simple types of knowledge. It is shown that they cannot be extended much further without changing the underlying simplicity ordering. A program which implements one of the decidable cases is described. It is used to find definitions in the game of noughts and crosses and in family relationships. An abstract study is made of the progression of hypothesis discovery methods through time. Some possible and some impossible behaviours of such methods are demonstrated. This work is an extension of that of Gold (1967) and Feldman (1970). The results are applied to the case of machines that discover generalisations. They are found to be markedly sensitive to the underlying simplicity ordering employed