95 research outputs found
Existential Definability over the Subword Ordering
We study first-order logic (FO) over the structure consisting of finite words over some alphabet A, together with the (non-contiguous) subword ordering. In terms of decidability of quantifier alternation fragments, this logic is well-understood: If every word is available as a constant, then even the ?? (i.e., existential) fragment is undecidable, already for binary alphabets A.
However, up to now, little is known about the expressiveness of the quantifier alternation fragments: For example, the undecidability proof for the existential fragment relies on Diophantine equations and only shows that recursively enumerable languages over a singleton alphabet (and some auxiliary predicates) are definable.
We show that if |A| ? 3, then a relation is definable in the existential fragment over A with constants if and only if it is recursively enumerable. This implies characterizations for all fragments ?_i: If |A| ? 3, then a relation is definable in ?_i if and only if it belongs to the i-th level of the arithmetical hierarchy. In addition, our result yields an analogous complete description of the ?_i-fragments for i ? 2 of the pure logic, where the words of A^* are not available as constants
Existential Definability over the Subword Ordering
We study first-order logic (FO) over the structure consisting of finite words
over some alphabet , together with the (non-contiguous) subword ordering. In
terms of decidability of quantifier alternation fragments, this logic is
well-understood: If every word is available as a constant, then even the
(i.e., existential) fragment is undecidable, already for binary
alphabets . However, up to now, little is known about the expressiveness of
the quantifier alternation fragments: For example, the undecidability proof for
the existential fragment relies on Diophantine equations and only shows that
recursively enumerable languages over a singleton alphabet (and some auxiliary
predicates) are definable. We show that if , then a relation is
definable in the existential fragment over with constants if and only if it
is recursively enumerable. This implies characterizations for all fragments
: If , then a relation is definable in if and
only if it belongs to the -th level of the arithmetical hierarchy. In
addition, our result yields an analogous complete description of the
-fragments for of the pure logic, where the words of
are not available as constants
Advances and applications of automata on words and trees : abstracts collection
From 12.12.2010 to 17.12.2010, the Dagstuhl Seminar 10501 "Advances and Applications of Automata on Words and Trees" was held in Schloss Dagstuhl - Leibniz Center for Informatics. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available
Existential Definability over the Subword Ordering
We study first-order logic (FO) over the structure consisting of finite words
over some alphabet , together with the (non-contiguous) subword ordering. In
terms of decidability of quantifier alternation fragments, this logic is
well-understood: If every word is available as a constant, then even the
(i.e., existential) fragment is undecidable, already for binary
alphabets . However, up to now, little is known about the expressiveness of
the quantifier alternation fragments: For example, the undecidability proof for
the existential fragment relies on Diophantine equations and only shows that
recursively enumerable languages over a singleton alphabet (and some auxiliary
predicates) are definable. We show that if , then a relation is
definable in the existential fragment over with constants if and only if it
is recursively enumerable. This implies characterizations for all fragments
: If , then a relation is definable in if and
only if it belongs to the -th level of the arithmetical hierarchy. In
addition, our result yields an analogous complete description of the
-fragments for of the pure logic, where the words of
are not available as constants
On Relaxing Metric Information in Linear Temporal Logic
Metric LTL formulas rely on the next operator to encode time distances,
whereas qualitative LTL formulas use only the until operator. This paper shows
how to transform any metric LTL formula M into a qualitative formula Q, such
that Q is satisfiable if and only if M is satisfiable over words with
variability bounded with respect to the largest distances used in M (i.e.,
occurrences of next), but the size of Q is independent of such distances.
Besides the theoretical interest, this result can help simplify the
verification of systems with time-granularity heterogeneity, where large
distances are required to express the coarse-grain dynamics in terms of
fine-grain time units.Comment: Minor change
On shuffle products, acyclic automata and piecewise-testable languages
We show that the shuffle L \unicode{x29E2} F of a piecewise-testable
language and a finite language is piecewise-testable. The proof relies
on a classic but little-used automata-theoretic characterization of
piecewise-testable languages. We also discuss some mild generalizations of the
main result, and provide bounds on the piecewise complexity of L
\unicode{x29E2} F
Languages ordered by the subword order
We consider a language together with the subword relation, the cover
relation, and regular predicates. For such structures, we consider the
extension of first-order logic by threshold- and modulo-counting quantifiers.
Depending on the language, the used predicates, and the fragment of the logic,
we determine four new combinations that yield decidable theories. These results
extend earlier ones where only the language of all words without the cover
relation and fragments of first-order logic were considered
10501 Abstracts Collection -- Advances and Applications of Automata on Words and Trees
From 12.12.2010 to 17.12.2010, the Dagstuhl Seminar 10501
``Advances and Applications of Automata on Words and Trees\u27\u27 was held
in Schloss Dagstuhl~--~Leibniz Center for Informatics.
During the seminar, several participants presented their current
research, and ongoing work and open problems were discussed. Abstracts of
the presentations given during the seminar as well as abstracts of
seminar results and ideas are put together in this paper. The first section
describes the seminar topics and goals in general.
Links to extended abstracts or full papers are provided, if available
On the decidability and complexity of Metric Temporal Logic over finite words
Metric Temporal Logic (MTL) is a prominent specification formalism for
real-time systems. In this paper, we show that the satisfiability problem for
MTL over finite timed words is decidable, with non-primitive recursive
complexity. We also consider the model-checking problem for MTL: whether all
words accepted by a given Alur-Dill timed automaton satisfy a given MTL
formula. We show that this problem is decidable over finite words. Over
infinite words, we show that model checking the safety fragment of MTL--which
includes invariance and time-bounded response properties--is also decidable.
These results are quite surprising in that they contradict various claims to
the contrary that have appeared in the literature
- …