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We consider logics on
and which are weaker than
Presburger arithmetic and
we settle the following decision
problem: given a k-ary
relation on and
which are first order definable in
Presburger arithmetic, are they definable in these
weaker logics? These logics, intuitively,
are obtained by considering modulo and threshold counting predicates for differences of two variables.
We consider the defect theorem in the context of labelled
polyominoes, i.e., two-dimensional figures. The classical version of
this property states that if a set of n words is not a code then
the words can be expressed as a product of at most n - 1 words, the
smaller set being a code. We survey several two-dimensional
extensions exhibiting the boundaries where the theorem fails. In
particular, we establish the defect property in the case of three
dominoes (n × 1 or 1 × n rectangles).
We introduce the notion of nested distance desert automata as a joint generalization of distance automata and desert automata. We show that limitedness of nested distance desert automata is PSPACE-complete. As an application, we show that it is decidable in space whether the language accepted by an -state non-deterministic automaton is of a star height less than a given integer (concerning rational expressions with union, concatenation and iteration), which is the first ever complexity bound...
We introduce the notion of nested distance desert automata as a joint generalization of distance automata and desert automata. We show that limitedness of nested distance desert automata is PSPACE-complete. As an application, we show that it is decidable in 22O(n) space whether the language accepted by an n-state non-deterministic automaton is of a star height less than a given integer h (concerning rational expressions with union, concatenation and iteration), which is the first ever complexity...
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