125,024 research outputs found
Binary simple homogeneous structures are supersimple with finite rank
Suppose that M is an infinite structure with finite relational vocabulary
such that every relation symbol has arity at most 2. If M is simple and
homogeneous then its complete theory is supersimple with finite SU-rank which
cannot exceed the number of complete 2-types over the empty set
On sets with rank one in simple homogeneous structures
We study definable sets of SU-rank 1 in , where is a
countable homogeneous and simple structure in a language with finite relational
vocabulary. Each such can be seen as a `canonically embedded structure',
which inherits all relations on which are definable in , and has no
other definable relations. Our results imply that if no relation symbol of the
language of has arity higher than 2, then there is a close relationship
between triviality of dependence and being a reduct of a binary random
structure. Somewhat more preciely: (a) if for every , every -type
which is realized in is determined by its sub-2-types
, then the algebraic closure restricted to is
trivial; (b) if has trivial dependence, then is a reduct of a binary
random structure
The random graph
Erd\H{o}s and R\'{e}nyi showed the paradoxical result that there is a unique
(and highly symmetric) countably infinite random graph. This graph, and its
automorphism group, form the subject of the present survey.Comment: Revised chapter for new edition of book "The Mathematics of Paul
Erd\H{o}s
Program logics for homogeneous meta-programming.
A meta-program is a program that generates or manipulates another program; in homogeneous meta-programming, a program may generate new parts of, or manipulate, itself. Meta-programming has been used extensively since macros
were introduced to Lisp, yet we have little idea how formally to reason about metaprograms. This paper provides the first program logics for homogeneous metaprogramming
– using a variant of MiniMLe by Davies and Pfenning as underlying meta-programming language.We show the applicability of our approach by reasoning about example meta-programs from the literature. We also demonstrate that our logics are relatively complete in the sense of Cook, enable the inductive derivation of characteristic formulae, and exactly capture the observational properties induced by the operational semantics
Algebra and logic. Some problems
The paper has a form of a talk on the given topic. It consists of three
parts.
The first part of the paper contains main notions, the second one is devoted
to logical geometry, the third part describes types and isotypeness. The
problems are distributed in the corresponding parts. The whole material
oriented towards universal algebraic geometry (UAG), i.e., geometry in an
arbitrary variety of algebras . We consider logical geometry (LG) as a
part of UAG. This theory is strongly influenced by model theory and ideas of
A.Tarski and A.I.Malcev
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