11 research outputs found
Permutations on the random permutation
The random permutation is the Fra\"iss\'e limit of the class of finite
structures with two linear orders. Answering a problem stated by Peter Cameron
in 2002, we use a recent Ramsey-theoretic technique to show that there exist
precisely 39 closed supergroups of the automorphism group of the random
permutation, and thereby expose all symmetries of this structure. Equivalently,
we classify all structures which have a first-order definition in the random
permutation.Comment: 18 page
The Complexity of Combinations of Qualitative Constraint Satisfaction Problems
The CSP of a first-order theory is the problem of deciding for a given
finite set of atomic formulas whether is satisfiable. Let
and be two theories with countably infinite models and disjoint
signatures. Nelson and Oppen presented conditions that imply decidability (or
polynomial-time decidability) of under the
assumption that and are decidable (or
polynomial-time decidable). We show that for a large class of
-categorical theories the Nelson-Oppen conditions are not
only sufficient, but also necessary for polynomial-time tractability of
(unless P=NP)
Canonical functions: a proof via topological dynamics
Canonical functions are a powerful concept with numerous applications in the study of groups, monoids, and clones on countable structures with Ramsey-type properties. In this short note, we present a proof of the existence of canonical functions in certain sets using topological dynamics, providing a shorter alternative to the original combinatorial argument. We moreover present equivalent algebraic characterisations of canonicity
The Complexity of Combinations of Qualitative Constraint Satisfaction Problems
The CSP of a first-order theory is the problem of deciding for a given
finite set of atomic formulas whether is satisfiable. Let
and be two theories with countably infinite models and disjoint
signatures. Nelson and Oppen presented conditions that imply decidability (or
polynomial-time decidability) of under the
assumption that and are decidable (or
polynomial-time decidable). We show that for a large class of
-categorical theories the Nelson-Oppen conditions are not
only sufficient, but also necessary for polynomial-time tractability of
(unless P=NP).Comment: Version 2: stronger main result with better presentation of the
proof; multiple improvements in other proofs; new section structure; new
example
The Universal Homogenous Binary Tree
A partial order is called semilinear if the upper bounds of each element are linearly ordered and any two elements have a common upper bound. There exists, up to isomorphism, a unique countable existentially closed semilinear order, which we denote by (S 2 ;≤). We study the reducts of (S 2 ;≤), that is, the relational structures with domain S 2, all of whose relations are first-order definable in (S 2 ;≤). Our main result is a classification of the model-complete cores of the reducts of S 2. From this, we also obtain a classification of reducts up to first-order interdefinability, which is equivalent to a classification of all subgroups of the full symmetric group on S 2 that contain the automorphism group of (S 2 ;≤) and are closed with respect to the pointwise convergence topology