On Intuitionistic Linear Logic - Microsoft Research

Chapter 1
Introduction
1 Background
An important problem in theoretical computer science is discovering logical foundations of programming
languages. Such foundations provide programmers with techniques for reasoning logically
rather than informally about their programs and not only tells implementors precisely what they are
trying to implement but also enables them to reason formally about possible optimizations. One of
the most fruitful methods used to explore such logical foundations has been to utilize a fascinating
relationship between various typed λ-calculi, constructive logics and structures, or models, from
category theory. Despite their apparent independence, various work has shown how these areas are
related.
Logic with typed λ-calculi. Curry [23, Section 9E] and Howard [41] noticed that an axiomatic
formulation of Intuitionistic Logic (IL) corresponds to the type scheme for (S,K) combinatory
logic. It was also noted that the natural deduction formulation for the (⊃, ∧)-fragment of
minimal logic corresponds to the typing rules for the simply typed λ-calculus with pairs.
More importantly, the notion of normalization for minimal logic corresponds to the notion of
reduction of the λ-terms. This relationship is known as the ‘propositions-as-types analogy’ or
the Curry-Howard correspondence.
Category theory with logic. Lambek [48, 49, 50] first showed how formal deductions for propositional
logics could be given in a categorical framework. In particular he considered the
relationship between intuitionistic (propositional) logic and cartesian closed categories. Lawvere
[54] showed how these techniques could be extended to handle predicate logics by considering
more powerful categorical structures.
Category theory with typed λ-calculi. Given the relationship between logic and typed λcalculi,
Lambek was able to show how category theory could give a semantics for typed λcalculi;
this is demonstrated for various calculi in his book with Scott [52]. Curien, along with
co-workers, has shown how this semantics can be seen to suggest an abstract machine: the
categorical abstract machine [21]. This idea has been used to provide a complete compiler for
a dialect of ML [72].
These relationships can be pictured as
Typed λ-calculus Constructive Logic
❅ ❅❅❅❅
Categorical Model.
The exciting aspect of these relationships is not just that the three areas are related but that certain
concepts within them are related also, as shown below.
Logic Typed λ-calculus Categorical Model
Proposition Type Object
Proof Term Morphism
Normalization Reduction Equality of morphisms
1