[Paper Review] On the reconstructibility of totally symmetric functions and of other functions with a unique identification minor

Erkko Lehtonen|arXiv (Cornell University)|Aug 15, 2012

Mathematical Analysis and Transform Methods19 references3 citations

TL;DR

This paper investigates the reconstructibility of Boolean and multi-valued functions from their identification minors—functions formed by collapsing two arguments. It proves that totally symmetric functions of sufficiently large arity are reconstructible, and that functions weakly determined by the order of first occurrence (especially over binary domains) are also reconstructible, establishing foundational results in the minor-based reconstruction problem for discrete functions.

ABSTRACT

We investigate the problem whether a function of several arguments can be reconstructed from its identification minors. We focus on functions with a unique identification minor, and we establish some positive and negative results on the reconstruction problem. In particular, we show that totally symmetric functions (of sufficiently large arity) are reconstructible and the class of functions weakly determined by the order of first occurrence (of sufficiently large arity) is weakly reconstructible.

Motivation & Objective

• To determine whether functions can be reconstructed from their identification minors, a generalization of graph reconstruction to discrete functions.
• To study functions with a unique identification minor, focusing on structural properties that enable or prevent reconstructibility.
• To establish sufficient conditions under which functions—particularly symmetric and order-dependent functions—are reconstructible.
• To explore the role of permutation groups and minor relations in function reconstruction, extending classical reconstruction ideas to algebraic function theory.

Proposed method

• Uses the minor relation to define identification minors: functions obtained by identifying two input variables via a surjective map.
• Applies group-theoretic techniques, particularly permutation group actions and parity arguments (even/odd permutations), to compare function values under variable relabeling.
• Employs the concept of 'order of first occurrence' (ofo) to characterize functions whose output depends on the sequence in which input values first appear.
• Introduces the notion of weak reconstructibility, where equivalent functions (up to variable permutation) can be reconstructed from minors.
• Analyzes specific function classes such as those determined by support (supp) or odd support (oddsupp), showing their reconstructibility for large arity.
• Uses counterexample constructions and case analysis (e.g., via Table 3) to demonstrate non-reconstructibility in certain symmetric settings, especially when arity is small.

Experimental results

Research questions

• RQ1 Can a totally symmetric function of sufficiently large arity be reconstructed from its identification minors?
• RQ2 Are functions weakly determined by the order of first occurrence reconstructible, particularly over binary domains?
• RQ3 Under what conditions is a function with a unique identification minor reconstructible?
• RQ4 Can functions determined by supp or oddsupp be reconstructed from their minors?
• RQ5 Is every function determined by the order of first occurrence reconstructible when the input set has size k ≥ 3 and arity is sufficiently large?

Key findings

• Totally symmetric functions of arity at least n ≥ k+2 (for k ≥ 2) are reconstructible, provided the domain size is k.
• Functions weakly determined by the order of first occurrence are weakly reconstructible for sufficiently large arity, and this holds in particular for binary domains.
• Functions determined by supp or oddsupp (of sufficiently large arity) are reconstructible, forming a subclass of order-of-occurrence functions.
• For k = 2, functions weakly determined by the order of first occurrence are reconstructible, as shown by Corollary 6.3.
• The paper constructs explicit counterexamples using permutation parity to show that certain functions (e.g., φ⁺ₖ and ψ⁺ₖ) are not equivalent despite sharing minors, proving non-reconstructibility in specific cases.
• An open question remains: whether all functions determined by the order of first occurrence are reconstructible when k ≡ 1,2 mod 4 and n ≥ k+2, or k ≡ 0,3 mod 4 and n ≥ k+3.