[Paper Review] A generalization of Goodstein's theorem: interpolation by polynomial functions of distributive lattices

Miguel Couceiro, Tamás Waldhauser|arXiv (Cornell University)|Oct 3, 2011

Advanced Algebra and Logic9 references3 citations

TL;DR

This paper generalizes Goodstein's theorem by characterizing when partial functions defined on cuboidal subsets of a distributive lattice can be interpolated by lattice polynomial functions. It provides necessary and sufficient conditions for such interpolation and explicitly describes all possible interpolating polynomials, extending the classical result from the Boolean cube to arbitrary distributive lattices and general cuboids.

ABSTRACT

We consider the problem of interpolating functions partially defined over a distributive lattice, by means of lattice polynomial functions. Goodstein's theorem solves a particular instance of this interpolation problem on a distributive lattice L with least and greatest elements 0 and 1, resp.: Given an n-ary partial function f over L, defined on all 0-1 tuples, f can be extended to a lattice polynomial function p over L if and only if f is monotone; in this case, the interpolating polynomial p is unique. We extend Goodstein's theorem to a wider class of n-ary partial functions f over a distributive lattice L, not necessarily bounded, where the domain of f is a cuboid of the form D={a1,b1}x...x{an,bn} with ai

Motivation & Objective

• To extend Goodstein's theorem, which originally applies only to functions on the Boolean cube {0,1}^n over bounded distributive lattices, to a broader class of partial functions defined on cuboidal domains in arbitrary (possibly unbounded) distributive lattices.
• To identify the necessary and sufficient conditions under which a partial function f: D → L, where D is a product of two-element sets {a_i, b_i} with a_i < b_i, can be interpolated by a lattice polynomial function.
• To characterize the set of all lattice polynomial functions that interpolate a given partial function, since uniqueness—present in Goodstein’s original result—is not guaranteed in the generalized setting.
• To provide an explicit construction of all interpolating polynomials, using disjunctive normal forms and monotonic coefficient systems.
• To support applications in decision-making models by enabling factorization of global utility functions into local utility functions and aggregation functions via lattice polynomial interpolation.

Proposed method

• Formalize the interpolation problem for partial functions f: D → L, where D = {a₁,b₁} × ⋯ × {aₙ,bₙ} with aᵢ < bᵢ in a distributive lattice L, and L may be unbounded.
• Represent lattice polynomial functions in disjunctive normal form (DNF): p(x) = ⋁_{I⊆[n]} (c_I ∧ ⋀_{i∈I} x_i), with coefficients c_I ∈ L satisfying monotonicity: I ⊆ J ⇒ c_I ≤ c_J.
• Use the values of f on the vertices of the cuboid D to determine the coefficients c_I of the interpolating polynomial, ensuring consistency with the DNF representation.
• Establish that a partial function f: D → L admits a lattice polynomial interpolation if and only if it satisfies a generalized monotonicity condition over the cuboid D.
• Provide an algorithmic description of all possible interpolating polynomials by solving for coefficient systems that satisfy the interpolation constraints and monotonicity.
• Leverage the structure of distributive lattices and the properties of lattice polynomials to ensure that the constructed functions are well-defined and preserve the required order-theoretic properties.

Experimental results

Research questions

• RQ1 Under what conditions can a partial function f: D → L, where D is a cuboidal subset of L^n, be interpolated by a lattice polynomial function over an arbitrary distributive lattice L?
• RQ2 How does the uniqueness of the interpolating polynomial, which holds in Goodstein’s original theorem, generalize when the domain is no longer the Boolean cube {0,1}^n?
• RQ3 What is the complete set of all lattice polynomial functions that can interpolate a given partial function f defined on a cuboid in a distributive lattice?
• RQ4 How can the coefficients of the interpolating polynomial be systematically determined from the values of f on the vertices of the cuboid?
• RQ5 In what way do the results support the factorization of global utility functions in decision-making models into local utility functions and aggregation functions?

Key findings

• A partial function f: D → L, where D = {a₁,b₁} × ⋯ × {aₙ,bₙ} with aᵢ < bᵢ in a distributive lattice L, admits a lattice polynomial interpolation if and only if it satisfies a generalized monotonicity condition over the cuboid D.
• The interpolating polynomial is not necessarily unique in the generalized setting; the paper provides a complete description of the set of all possible interpolating polynomials.
• All interpolating polynomials can be explicitly constructed using disjunctive normal forms with coefficients derived from the function values at the vertices of the cuboid, under monotonicity constraints.
• The coefficients c_I of the interpolating polynomial are determined by the values f(1_I) where 1_I is the characteristic vector of I ⊆ [n], and must satisfy c_I ≤ c_J whenever I ⊆ J.
• The results generalize Goodstein’s theorem from the Boolean cube {0,1}^n to arbitrary cuboidal domains in unbounded distributive lattices, preserving the core idea of monotonicity as the key condition.
• The framework enables the factorization of global utility functions in decision-making models into local utility functions and lattice polynomial aggregation functions, with all possible aggregation functions explicitly characterized.