# Mittagsseminar (in cooperation with J. Lengler, A. Steger, and D. Steurer)

__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.

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