The complexity of type inference for higher-order lambda calculi

We analyse the computational complexity of type inference for untyped X,-terms in the secondorder polymorphic typed X-calculus (F2) invented by Girard and Reynolds, as well as higherorder extensions F3,F4, ...,/^ proposed by Girard. We prove that recognising the i^-typable terms requires exponential time, and for Fa the problem is non-elementary. We show as well a sequence of lower bounds on recognising the i^-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.