About Systems of Equations, X-Separability, and Left-Invertibility in the lambda-Calculus

Abstract A system of equations in the λ-calculus is a pair (Γ, X), where Γ is a set of formulas of Λ (the equations) and X is a finite set of variables of Λ (the unknowns.) A system L = (Γ, X) is said to be solvable in the theory T (T-solvable) iff there exists a suitable simultaneous substitution for the unknowns that makes the equations of L theorems in the theory T. For any finite system and within any semisensible (sms) theory T (e.g., β, βη, H ∗) a necessary condition for T-solvability is proved. A class of systems for which this condition also becomes sufficient is shown and the sufficiency is proved constructively. This class properly contains the systems L = (Γ, {x1,…,xn}) that satisfy 0,1 or 0,2 of the following hypotheses: Hp.0. (0) If Q is a proper subterm of a LHS term of an equation and the head of Q is an unknown then the degree of Q is not too large. (1) The initial part of a LHS term never collapses with another LHS term. Hp.1. The equations of S have the shape xM1 … Mn = yx1 … xwM1 … Mn, where x ϵ {x1, …, xu,} and y does not occur in the LHS terms of the equations of L . Hp.2. The equations of L have the shape xM1 … Mn = N, where x ϵ {x1, …, xu,} and N is a βη-normal form whose free variables do not occur in the LHS terms of the equations of L . With some caution we can also mix equations having the shape in Hp.1 with equations having the shape in Hp.2. A typical result is the constructive characterization of the T-solvability (T sms) of systems having the shape L = ({xx = N0, xM1 = N1, …, xMn = Nn}, {x}), where M1, …, Mn are closed λ-terms and N0, …, Nn are βη-normal forms which do not contain the unknown x. When the equations of a system L = (Γ, X) have the shape M = y, with the RHS variables fresh and pairwise distinct, we have te X-separability problem for the LHS terms. For a class of λ-free sets (see Hp.0) the X-separability is constructively characterized within any sms theory. A single equation can be solved via a system of equations. Using this idea we characterize the βη-left-invertibility for a class of λ-terms.