We take Abramsky's term assignment for Intuitionistic Linear Logic (the linear term calculus) as the basis of a functional programming language. This is a language where the programmer must embed explicitly the resource and control information of an algorithm. We give a type reconstruction algorithm for our language in the style of Milner's W algorithm, together with a description of the implementation and examples of use.

We analyse the computational complexity of type inference for untyped λ-terms in the second-order polymorphic typed λ-calculus (F2) invented by Girard and Reynolds, as well as higher-order extensions F3, F4, …, Fω proposed by Girard. We prove that recognising the F2-typable terms requires exponential time, and for Fω the problem is non-elementary. We show as well a sequence of lower bounds on recognising the Fk-typable terms, where the bound for Fk+1 is exponentially larger than that for Fk.

The lower bounds are based on generic simulation of Turing Machines, where computation is simulated at the expression and type level simultaneously. Non-accepting computations are mapped to non-normalising reduction sequences, and hence non-typable terms. The accepting computations are mapped to typable terms, where higher-order types encode reduction sequences, and first-order types encode the entire computation as a circuit, based on a unification simulation of Boolean logic. A primary technical tool in this reduction is the composition of polymorphic functions having different domains and ranges.

These results are the first nontrivial lower bounds on type inference for the Girard/Reynolds system as well as its higher-order extensions. We hope that the analysis provides important combinatorial insights which will prove useful in the ultimate resolution of the complexity of the type inference problem.

Unfolding is a common technique in program transformations. Here, we present a computation model where unfolding is a simple generalisation of the usual concept of evaluation. The model is a variant of the well-known full substitution evaluation rule for recursive programs. The evaluation mechanism involved is symbolic substitution of function definitions followed by simplification. Simplification is expressed as a confluent rewrite strategy which uses three kinds of reductions: β-reduction, non-erasing reduction, and erasing reduction. Non-erasing reductions include simplification of constant subexpressions. Erasing reductions formalize the behaviour of nonstrict operations. In this computation model, we prove a termination theorem of symbolic unfolding relative to more instantiated calls. A possible application of the model is a technique called total unfolding, where a partially instantiated function call is unfolded until no more calls exist. Under certain conditions the result will be a first order term: such terms correspond to basic blocks in imperative programs and can be efficiently implemented by scheduling techniques. Possible applications are hardware synthesis, and code generation for parallel machines.

Self-applicable specializers have been used successfully to automate the generation of compilers. Specializers are often rather sophisticated, for which reason one would like to adapt and transform them with the aid of the computer. But how to automate this process? The answer to this question is given by three specializer projections. While the Futamura projections define the generation of compilers from interpreters, the specializer projections define the generation of specializers from interpreters. We discuss the potential applications of the specializer projections, and argue that their realization is a real touchstone for the effectiveness of the specialization principle. In particular, we discuss generic specializers, bootstrapping of subject languages and the generation of optimizing specializers from interpretive specifications. The Futamura projections are regarded as a special case of the specializer projections. Recent results confirm that the specializer projections can be performed in practice using partial evaluators.

Large functional programs are often constructed by decomposing each big task into smaller tasks which can be performed by simpler functions. This hierarchical style of developing programs has been found to improve programmers' productivity because smaller functions are easier to construct and reuse. However, programs written in this way tend to be less efficient. Unnecessary intermediate data structures may be created. More function invocations may be required.

To reduce such performance penalties, Phil Wadler proposed a transformation algorithm, called deforestation, which could automatically fuse certain composed expressions together to eliminate intermediate tree-like data structures. However, his technique is currently safe (terminates with no loss of efficiency) for only a subset of first-order expressions.

This paper will generalise the deforestation technique to make it safe for all first-order and higher-order functional programs. Our generalisation is explained using a model for safe fusion which views each function as a producer and its parameters as consumers. Through this model, syntactic program properties are proposed to classify producers and consumers as either safe or unsafe. This classification is used to identify sub-terms that can be safely fused/eliminated. We present the generalised transformation algorithm, illustrate it with examples and provide a termination proof for the transformation algorithm of first-order programs. This paper also contains a suite of additional techniques to further improve the basic safe fusion method. These improvements could be viewed as enhancements to compensate for some inadequacies of the syntactic analyses used.