Linear lambda calculus and PTIME-completeness

We give transparent proofs of the PTIME-completeness of two decision problems for terms in the λ-calculus. The first is a reproof of the theorem that type inference for the simplytyped λ-calculus is PTIME-complete. Our proof is interesting because it uses no more than the standard combinators Church knew of some 70 years ago, in which the terms are linear affine—each bound variable occurs at most once. We then derive a modification of Church’s coding of Booleans that is linear, where each bound variable occurs exactly once. A consequence of this construction is that any interpreter for linear λ-calculus requires polynomial time. The logical interpretation of this consequence is that the problem of normalizing proofnets for multiplicative linear logic (MLL) is also PTIME-complete. 1 Type inference for simply typed λ-calculus The Circuit Value Problem (CVP) is to determine the output of a circuit, given an input to that circuit. CVP is complete for PTIME, because polynomial-time computations can be described by polynomial-sized circuits (Ladner, 1975). The Cook-Levin NP-completeness theorem, it should be noticed, merely augments these circuits with extra inputs which correspond to nondeterministic choices during a polynomial-time computation. We show how to code CVP into simply-typed λterms, where both type inference and term evaluation are synonymous with circuit evaluation. The programs we write to evaluate circuits are not perverse: they are completely natural, and are built out of the standard Church coding of Boolean logic; see, e.g., (Hindley & Seldin, 1986). We use ML as a presentation device, without exploiting its let-polymorphism. That is, we use the convenience of naming to identify λterms of constant size, used to build circuits. Had we expanded the definitions, the term representing the circuit would grow by only a constant factor, and become harder to read. Here, then, are the standard, classical combinators, coded in ML: