About systems of equations, X-separability, and left-invertibility in the λ-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 $\text{L}$ = (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 $\text{L}$ theorems in the theory T. For any finite system and within any semisensible (sms) theory T (e.g., β, βη, $\text{H}$^∗) 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 $\text{L}$ = (Γ, {x₁,…,x_n}) 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 … M_n = yx₁ … x_wM₁ … M_n, where x ϵ {x₁, …, x_u,} and y does not occur in the LHS terms of the equations of $\text{L}$.

• Hp.2. The equations of $\text{L}$ have the shape xM₁ … M_n = N, where x ϵ {x₁, …, x_u,} and N is a βη-normal form whose free variables do not occur in the LHS terms of the equations of $\text{L}$.

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 $\text{L}$ = ({xx = N₀, xM₁ = N₁, …, xM_n = N_n}, {x}), where M₁, …, M_n are closed λ-terms and N₀, …, N_n are βη-normal forms which do not contain the unknown x. When the equations of a system $\text{L}$ = (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.