Is Multiplication Harder than Addition? Arithmetical Definability over Finite Structures
Troy Lee
Abstract:
Is Multiplication Harder than Addition?
Arithmetical Definability over Finite Structures
Troy Lee
Abstract:
In this thesis, we show that two definability results of Julia Robinson,
namely that multiplication and the successor relation can first-order
define addition, and that the divisibility relation and successor can
first-order define multiplication, also hold over finite structures (where
ordering is used instead of successor). The first result is obtained by
showing that the BIT predicate can be defined with TIMES and ordering,
thus FO(<,BIT)=FO(<,TIMES). Then the ``carry-look-ahead'' construction
can be used to define PLUS with BIT. We also show that there is no
first-order sentence with TIMES but without ordering which can define
PLUS. A corollary is then that TIMES cannot first-order define ordering.
Our results, together with recent results on the Crane Beach conjecture
show that there is a language definable in FO(<,TIMES) but not in
FO(<,PLUS). In other words, there is no first-order sentence with < and
PLUS which can express TIMES.