193 research outputs found
Trees over Infinite Structures and Path Logics with Synchronization
We provide decidability and undecidability results on the model-checking
problem for infinite tree structures. These tree structures are built from
sequences of elements of infinite relational structures. More precisely, we
deal with the tree iteration of a relational structure M in the sense of
Shelah-Stupp. In contrast to classical results where model-checking is shown
decidable for MSO-logic, we show decidability of the tree model-checking
problem for logics that allow only path quantifiers and chain quantifiers
(where chains are subsets of paths), as they appear in branching time logics;
however, at the same time the tree is enriched by the equal-level relation
(which holds between vertices u, v if they are on the same tree level). We
separate cleanly the tree logic from the logic used for expressing properties
of the underlying structure M. We illustrate the scope of the decidability
results by showing that two slight extensions of the framework lead to
undecidability. In particular, this applies to the (stronger) tree iteration in
the sense of Muchnik-Walukiewicz.Comment: In Proceedings INFINITY 2011, arXiv:1111.267
The Isomorphism Relation Between Tree-Automatic Structures
An -tree-automatic structure is a relational structure whose domain
and relations are accepted by Muller or Rabin tree automata. We investigate in
this paper the isomorphism problem for -tree-automatic structures. We
prove first that the isomorphism relation for -tree-automatic boolean
algebras (respectively, partial orders, rings, commutative rings, non
commutative rings, non commutative groups, nilpotent groups of class n >1) is
not determined by the axiomatic system ZFC. Then we prove that the isomorphism
problem for -tree-automatic boolean algebras (respectively, partial
orders, rings, commutative rings, non commutative rings, non commutative
groups, nilpotent groups of class n >1) is neither a -set nor a
-set
Cardinality and counting quantifiers on omega-automatic structures
We investigate structures that can be represented by omega-automata, so
called omega-automatic structures, and prove that relations defined over such
structures in first-order logic expanded by the first-order quantifiers `there
exist at most many', 'there exist finitely many' and 'there exist
modulo many' are omega-regular. The proof identifies certain algebraic
properties of omega-semigroups. As a consequence an omega-regular equivalence
relation of countable index has an omega-regular set of representatives. This
implies Blumensath's conjecture that a countable structure with an
-automatic presentation can be represented using automata on finite
words. This also complements a very recent result of Hj\"orth, Khoussainov,
Montalban and Nies showing that there is an omega-automatic structure which has
no injective presentation
Tree games with regular objectives
We study tree games developed recently by Matteo Mio as a game interpretation
of the probabilistic -calculus. With expressive power comes complexity.
Mio showed that tree games are able to encode Blackwell games and,
consequently, are not determined under deterministic strategies.
We show that non-stochastic tree games with objectives recognisable by
so-called game automata are determined under deterministic, finite memory
strategies. Moreover, we give an elementary algorithmic procedure which, for an
arbitrary regular language L and a finite non-stochastic tree game with a
winning objective L decides if the game is determined under deterministic
strategies.Comment: In Proceedings GandALF 2014, arXiv:1408.556
On the logical definability of certain graph and poset languages
We show that it is equivalent, for certain sets of finite graphs, to be
definable in CMS (counting monadic second-order logic, a natural extension of
monadic second-order logic), and to be recognizable in an algebraic framework
induced by the notion of modular decomposition of a finite graph. More
precisely, we consider the set of composition operations on graphs
which occur in the modular decomposition of finite graphs. If is a subset
of , we say that a graph is an \calF-graph if it can be
decomposed using only operations in . A set of -graphs is recognizable if
it is a union of classes in a finite-index equivalence relation which is
preserved by the operations in . We show that if is finite and its
elements enjoy only a limited amount of commutativity -- a property which we
call weak rigidity, then recognizability is equivalent to CMS-definability.
This requirement is weak enough to be satisfied whenever all -graphs are
posets, that is, transitive dags. In particular, our result generalizes Kuske's
recent result on series-parallel poset languages
Cardinality and counting quantifiers on omega-automatic structures
We investigate structures that can be represented by
omega-automata, so called omega-automatic structures, and prove
that relations defined over such structures in first-order logic
expanded by the first-order quantifiers `there exist at most
many\u27, \u27there exist finitely many\u27 and \u27there exist
modulo many\u27 are omega-regular. The proof identifies certain
algebraic properties of omega-semigroups.
As a consequence an omega-regular equivalence relation of countable
index has an omega-regular set of representatives. This implies
Blumensath\u27s conjecture that a countable structure with an
-automatic presentation can be represented using automata
on finite words. This also complements a very recent result of
Hj"orth, Khoussainov, Montalban and Nies showing that there is an
omega-automatic structure which has no injective presentation
On factorisation forests
The theorem of factorisation forests shows the existence of nested
factorisations -- a la Ramsey -- for finite words. This theorem has important
applications in semigroup theory, and beyond. The purpose of this paper is to
illustrate the importance of this approach in the context of automata over
infinite words and trees. We extend the theorem of factorisation forest in two
directions: we show that it is still valid for any word indexed by a linear
ordering; and we show that it admits a deterministic variant for words indexed
by well-orderings. A byproduct of this work is also an improvement on the known
bounds for the original result. We apply the first variant for giving a
simplified proof of the closure under complementation of rational sets of words
indexed by countable scattered linear orderings. We apply the second variant in
the analysis of monadic second-order logic over trees, yielding new results on
monadic interpretations over trees. Consequences of it are new caracterisations
of prefix-recognizable structures and of the Caucal hierarchy.Comment: 27 page
Ambiguity Hierarchy of Regular Infinite Tree Languages
An automaton is unambiguous if for every input it has at most one accepting
computation. An automaton is k-ambiguous (for k>0) if for every input it has at
most k accepting computations. An automaton is boundedly ambiguous if there is
k, such that for every input it has at most k accepting computations. An
automaton is finitely (respectively, countably) ambiguous if for every input it
has at most finitely (respectively, countably) many accepting computations.
The degree of ambiguity of a regular language is defined in a natural way. A
language is k-ambiguous (respectively, boundedly, finitely, countably
ambiguous) if it is accepted by a k-ambiguous (respectively, boundedly,
finitely, countably ambiguous) automaton. Over finite words, every regular
language is accepted by a deterministic automaton. Over finite trees, every
regular language is accepted by an unambiguous automaton. Over -words
every regular language is accepted by an unambiguous B\"uchi automaton and by a
deterministic parity automaton. Over infinite trees, Carayol et al. showed that
there are ambiguous languages.
We show that over infinite trees there is a hierarchy of degrees of
ambiguity: For every k>1 there are k-ambiguous languages which are not k-1
ambiguous; and there are finitely (respectively countably, uncountably)
ambiguous languages which are not boundedly (respectively finitely, countably)
ambiguous.Comment: Revised according to the reviewers comment
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