FUNCTIONAL PEARL LinearlambdacalculusandPTIME-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 (e.g. see 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 λ-terms 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: