## Theory of Combinatorial Algorithms

Prof. Emo Welzl and Prof. Bernd Gärtner

# Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

 Mittagsseminar Talk Information

Date and Time: Thursday, March 20, 2008, 12:15 pm

Duration: This information is not available in the database

Location: OAT S15/S16/S17

Speaker: Matthew Cook (Institute of Neuroinformatics)

## The Correspondence between Closed Sets of Functions and Closed Sets of Relations (Part II)

Given some boolean functions, we can consider the set of all functions that they generate. For example, if we start with AND and OR (but not NOT), their closure yields all monotone functions. If the closure includes all projection functions (functions that just return one of their arguments), then the closed set of functions is called a "clone" (origin of term unknown). Clones can be arranged in a lattice according to supersets/subsets, and Post gave the full lattice for boolean functions in 1920 (published in 1941). We can do the same for boolean relations, where two relations can be combined to form a new one by sharing variables and/or using existential quantification. As an intuitive example, given the relation is-a- brother-of and the relation is-a-parent-of, we can define a new relation is-an-uncle-of(A,B) as "there exists a C such that is-a- brother-of(A,C) and is-a-parent-of(C,B)". Going back to the world of boolean variables, another familiar example of combining relations can be seen with 3-SAT expressions, which can be viewed as large relations that are built out of the unequal relation on two variables and the OR relation on three variables. Given this notion of relational composition, we can consider closed sets of relations, which are called co-clones, and co-clones can be arranged in a lattice, just like clones. A surprising result from Geiger (1968) and Bodnarchuk et al. (1969) is that the lattices for clones and for co-clones have exactly the same structure---except that they are upside down from each other! I will describe a new and improved version of the proof of this deep connection between functions and relations.

Information for students and suggested topics for student talks

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