4 research outputs found
Presburger Arithmetic with algebraic scalar multiplications
We consider Presburger arithmetic (PA) extended by scalar multiplication by
an algebraic irrational number , and call this extension
-Presburger arithmetic (-PA). We show that the complexity of
deciding sentences in -PA is substantially harder than in PA. Indeed,
when is quadratic and , deciding -PA sentences with
alternating quantifier blocks and at most variables and inequalities
requires space at least (tower of
height ), where the constants only depend on , and
is the length of the given -PA sentence . Furthermore
deciding -PA sentences with at
most inequalities is PSPACE-hard, where is another constant depending
only on~. When is non-quadratic, already four alternating
quantifier blocks suffice for undecidability of -PA sentences
Presburger Arithmetic with algebraic scalar multiplications
We consider Presburger arithmetic (PA) extended by scalar multiplication by
an algebraic irrational number , and call this extension
-Presburger arithmetic (-PA). We show that the complexity of
deciding sentences in -PA is substantially harder than in PA. Indeed,
when is quadratic and , deciding -PA sentences with
alternating quantifier blocks and at most variables and inequalities
requires space at least (tower of
height ), where the constants only depend on , and
is the length of the given -PA sentence . Furthermore
deciding -PA sentences with at
most inequalities is PSPACE-hard, where is another constant depending
only on~. When is non-quadratic, already four alternating
quantifier blocks suffice for undecidability of -PA sentences
A strong version of Cobham's theorem
Let be two multiplicatively independent integers. Cobham's
famous theorem states that a set is both
-recognizable and -recognizable if and only if it is definable in
Presburger arithmetic. Here we show the following strengthening: let
be -recognizable, let be
-recognizable such that both and are not definable in Presburger
arithmetic. Then the first-order logical theory of is
undecidable. This is in contrast to a well-known theorem of B\"uchi that the
first-order logical theory of is decidable