[Paper Review] Partitioning in the space of antimonotonic functions

Patrick De Causmaecker, Stefan De Wannemacker|arXiv (Cornell University)|Mar 15, 2011

Advanced Algebra and Logic1 references3 citations

TL;DR

This paper introduces a novel algebraic decomposition of intervals in the space of antimonotonic Boolean functions, leveraging a partial order and a new external product operator to partition the lattice into disjoint unions of intervals derived from lower-dimensional subspaces. The key contribution is a recursive framework that yields new recursion formulas for Dedekind numbers and enables output-polynomial time enumeration algorithms for monotonic and antimonotonic functions.

ABSTRACT

This paper studies partitions in the space of antimonotonic boolean functions on sets of n elements. The antimonotonic functions are the antichains of the partially ordered set of subsets. We analyse and characterise a natural partial ordering on this set. We study the inter- vals according to this ordering. We show how intervals of antimonotonic functions, and a fortiori the whole space of antimonotonic functions can be partitioned as disjoint unions of certain classes of intervals. These in- tervals are uniquely determined by antimonotonic functions on smaller sets. This leads to recursive enumeration algorithms and new recursion relations. Using various decompositions, we derive new recursion formu- lae for the number of antimonotonic functions and hence for the number of monotonic functions (i.e. the Dedekind number).

Motivation & Objective

• To develop a systematic algebraic framework for partitioning the space of antimonotonic Boolean functions.
• To characterize the natural partial order on antimonotonic functions and define intervals within this order.
• To derive new recursion relations for Dedekind numbers by decomposing intervals into lower-dimensional components.
• To design recursive enumeration algorithms for antimonotonic functions with output-polynomial time complexity.
• To generalize the decomposition to complete distributive lattices, illustrated via Young’s lattice.

Proposed method

• Introduces a partial order on antimonotonic functions based on inclusion of accepted sets, defining intervals as sets of functions between comparable bounds.
• Defines a new external product operator × that combines intervals from different subspaces, enabling decomposition of higher-dimensional intervals.
• Uses the base(α) and top(α) operators to identify canonical representatives of interval classes, ensuring disjointness in the partition.
• Applies two decomposition theorems: one based on the sets within a function, and another based on a partition of the ground set, to generate interval decompositions.
• Proves that intervals of uniform span and general intervals can be expressed as disjoint unions of subintervals, each tied to lower-dimensional antimonotonic functions.
• Derives recursion formulae for Dedekind numbers by recursively counting elements in decomposed intervals, leveraging the lattice structure of antichains.

Experimental results

Research questions

• RQ1 How can the space of antimonotonic functions be systematically partitioned into disjoint intervals based on a natural partial order?
• RQ2 What structural properties of antimonotonic functions allow for recursive decomposition into intervals defined on smaller sets?
• RQ3 Can the decomposition lead to new recursion relations for the Dedekind numbers, which count monotonic Boolean functions?
• RQ4 What is the computational complexity of enumerating all antimonotonic functions using the proposed interval decomposition?
• RQ5 How does the external product operator × enable a canonical, disjoint decomposition of intervals in complete distributive lattices?

Key findings

• The paper establishes that intervals of antimonotonic functions can be partitioned as disjoint unions of subintervals, each uniquely determined by antimonotonic functions on smaller sets.
• A new external product operator × is defined, which allows the decomposition of intervals into canonical, non-overlapping components based on top and base functions.
• The decomposition leads to new recursion formulae for the number of antimonotonic functions, and thus for the Dedekind numbers, which are new to the best of the authors' knowledge.
• The interval decomposition enables the design of recursive enumeration algorithms that run in output-polynomial time, with a working implementation provided.
• The framework generalizes to complete distributive lattices, and is illustrated on Young’s lattice, where the entire infinite lattice of nonempty Young diagrams is shown as a disjoint union of intervals.
• The proof of disjointness in the partition relies on the property that γ ≤ α×β implies γ ∧ top(α) ≤ α, ensuring unique assignment of elements to components.