15 research outputs found
The isomorphism problem for tree-automatic ordinals with addition
This paper studies tree-automatic ordinals (or equivalently, well-founded
linearly ordered sets) together with the ordinal addition operation +.
Informally, these are ordinals such that their elements are coded by finite
trees for which the linear order relation of the ordinal and the ordinal
addition operation can be determined by tree automata. We describe an algorithm
that, given two tree-automatic ordinals with the ordinal addition operation,
decides if the ordinals are isomorphic
Isomorphisms of scattered automatic linear orders
We prove that the isomorphism of scattered tree automatic linear orders as
well as the existence of automorphisms of scattered word automatic linear
orders are undecidable. For the existence of automatic automorphisms of word
automatic linear orders, we determine the exact level of undecidability in the
arithmetical hierarchy
The strength of Ramsey Theorem for coloring relatively large sets
We characterize the computational content and the proof-theoretic strength of
a Ramsey-type theorem for bi-colorings of so-called {\em exactly large} sets.
An {\it exactly large} set is a set X\subset\Nat such that
\card(X)=\min(X)+1. The theorem we analyze is as follows. For every infinite
subset of \Nat, for every coloring of the exactly large subsets of
in two colors, there exists and infinite subset of such that is
constant on all exactly large subsets of . This theorem is essentially due
to Pudl\`ak and R\"odl and independently to Farmaki. We prove that --- over
Computable Mathematics --- this theorem is equivalent to closure under the
Turing jump (i.e., under arithmetical truth). Natural combinatorial
theorems at this level of complexity are rare. Our results give a complete
characterization of the theorem from the point of view of Computable
Mathematics and of the Proof Theory of Arithmetic. This nicely extends the
current knowledge about the strength of Ramsey Theorem. We also show that
analogous results hold for a related principle based on the Regressive Ramsey
Theorem. In addition we give a further characterization in terms of truth
predicates over Peano Arithmetic. We conjecture that analogous results hold for
larger ordinals
Complexity of equivalence relations and preorders from computability theory
We study the relative complexity of equivalence relations and preorders from
computability theory and complexity theory. Given binary relations , a
componentwise reducibility is defined by R\le S \iff \ex f \, \forall x, y \,
[xRy \lra f(x) Sf(y)]. Here is taken from a suitable class of effective
functions. For us the relations will be on natural numbers, and must be
computable. We show that there is a -complete equivalence relation, but
no -complete for .
We show that preorders arising naturally in the above-mentioned
areas are -complete. This includes polynomial time -reducibility
on exponential time sets, which is , almost inclusion on r.e.\ sets,
which is , and Turing reducibility on r.e.\ sets, which is .Comment: To appear in J. Symb. Logi
Tree-Automatic Well-Founded Trees
We investigate tree-automatic well-founded trees. Using Delhomme's
decomposition technique for tree-automatic structures, we show that the
(ordinal) rank of a tree-automatic well-founded tree is strictly below
omega^omega. Moreover, we make a step towards proving that the ranks of
tree-automatic well-founded partial orders are bounded by omega^omega^omega: we
prove this bound for what we call upwards linear partial orders. As an
application of our result, we show that the isomorphism problem for
tree-automatic well-founded trees is complete for level Delta^0_{omega^omega}
of the hyperarithmetical hierarchy with respect to Turing-reductions.Comment: Will appear in Logical Methods of Computer Scienc
A Hierarchy of Tree-Automatic Structures
We consider -automatic structures which are relational structures
whose domain and relations are accepted by automata reading ordinal words of
length for some integer . We show that all these structures
are -tree-automatic structures presentable by Muller or Rabin tree
automata. We prove that the isomorphism relation for -automatic
(resp. -automatic for ) boolean algebras (respectively, partial
orders, rings, commutative rings, non commutative rings, non commutative
groups) is not determined by the axiomatic system ZFC. We infer from the proof
of the above result that the isomorphism problem for -automatic
boolean algebras, , (respectively, rings, commutative rings, non
commutative rings, non commutative groups) is neither a -set nor a
-set. We obtain that there exist infinitely many -automatic,
hence also -tree-automatic, atomless boolean algebras , ,
which are pairwise isomorphic under the continuum hypothesis CH and pairwise
non isomorphic under an alternate axiom AT, strengthening a result of [FT10].Comment: To appear in The Journal of Symbolic Logic. arXiv admin note:
substantial text overlap with arXiv:1007.082